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Exterior Diﬀerential Systems and Euler-Lagrange Partial Diﬀerential Equations

Robert Bryant Phillip Griﬃths July 3, 2002 Daniel Grossman

ii

Contents

Preface Introduction 1 Lagrangians and Poincar´-Cartan Forms e 1.1 Lagrangians and Contact Geometry . . . . . . . . . 1.2 The Euler-Lagrange System . . . . . . . . . . . . . . 1.2.1 Variation of a Legendre Submanifold . . . . . 1.2.2 Calculation of the Euler-Lagrange System . . 1.2.3 The Inverse Problem . . . . . . . . . . . . . . 1.3 Noether’s Theorem . . . . . . . . . . . . . . . . . . . 1.4 Hypersurfaces in Euclidean Space . . . . . . . . . . . 1.4.1 The Contact Manifold over En+1 . . . . . . . 1.4.2 Euclidean-invariant Euler-Lagrange Systems . 1.4.3 Conservation Laws for Minimal Hypersurfaces 2 The 2.1 2.2 2.3 2.4 2.5 Geometry of Poincar´-Cartan Forms e The Equivalence Problem for n = 2 . . . . . . . Neo-Classical Poincar´-Cartan Forms . . . . . . e Digression on Aﬃne Geometry of Hypersurfaces The Equivalence Problem for n ≥ 3 . . . . . . . The Prescribed Mean Curvature System . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . v vii 1 1 7 7 8 10 14 21 21 24 27 37 39 52 58 65 74 79 80 80 85 93 97 102 110 111 114

3 Conformally Invariant Systems 3.1 Background Material on Conformal Geometry . . . . 3.1.1 Flat Conformal Space . . . . . . . . . . . . . 3.1.2 The Conformal Equivalence Problem . . . . . 3.1.3 The Conformal Laplacian . . . . . . . . . . . 3.2 Conformally Invariant Poincar´-Cartan e Forms . . . . . . . . . . . . . . . . . . . . . . . . . . 3.3 The Conformal Branch of the Equivalence Problem . n+2 3.4 Conservation Laws for ∆u = Cu n−2 . . . . . . . . . 3.4.1 The Lie Algebra of Inﬁnitesimal Symmetries 3.4.2 Calculation of Conservation Laws . . . . . . . iii

iv 3.5 Conservation Laws for Wave Equations . . . . . . . 3.5.1 Energy Density . . . . . . . . . . . . . . . . 3.5.2 The Conformally Invariant Wave Equation 3.5.3 Energy in Three Space Dimensions . . . . . . . . . . . . . . . . .

CONTENTS . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 118 122 123 127 133 133 133 136 139 141 145 150 151 155 158 164 168 168 172 182 191

4 Additional Topics 4.1 The Second Variation . . . . . . . . . . . . . . . . . . . . . 4.1.1 A Formula for the Second Variation . . . . . . . . . 4.1.2 Relative Conformal Geometry . . . . . . . . . . . . . 4.1.3 Intrinsic Integration by Parts . . . . . . . . . . . . . 4.1.4 Prescribed Mean Curvature, Revisited . . . . . . . . 4.1.5 Conditions for a Local Minimum . . . . . . . . . . . 4.2 Euler-Lagrange PDE Systems . . . . . . . . . . . . . . . . . 4.2.1 Multi-contact Geometry . . . . . . . . . . . . . . . . 4.2.2 Functionals on Submanifolds of Higher Codimension 4.2.3 The Betounes and Poincar´-Cartan Forms . . . . . . e 4.2.4 Harmonic Maps of Riemannian Manifolds . . . . . . 4.3 Higher-Order Conservation Laws . . . . . . . . . . . . . . . 4.3.1 The Inﬁnite Prolongation . . . . . . . . . . . . . . . 4.3.2 Noether’s Theorem . . . . . . . . . . . . . . . . . . . 4.3.3 The K = −1 Surface System . . . . . . . . . . . . . 4.3.4 Two B¨cklund Transformations . . . . . . . . . . . . a

Preface

During the 1996-97 academic year, Phillip Griﬃths and Robert Bryant conducted a seminar at the Institute for Advanced Study in Princeton, NJ, outlining their recent work (with Lucas Hsu) on a geometric approach to the calculus of variations in several variables. The present work is an outgrowth of that project; it includes all of the material presented in the seminar, with numerous additional details and a few extra topics of interest. The material can be viewed as a chapter in the ongoing development of a theory of the geometry of diﬀerential equations. The relative importance among PDEs of second-order Euler-Lagrange equations suggests that their geometry should be particularly rich, as does the geometric character of their conservation laws, which we discuss at length. A second purpose for the present work is to give an exposition of certain aspects of the theory of exterior diﬀerential systems, which provides the language and the techniques for the entire study. Special emphasis is placed on the method of equivalence, which plays a central role in uncovering geometric properties of diﬀerential equations. The Euler-Lagrange PDEs of the calculus of variations have turned out to provide excellent illustrations of the general theory.

v

vi

PREFACE

Introduction

In the classical calculus of variations, one studies functionals of the form FL(z) = L(x, z,

Ω

z) dx,

Ω ⊂ Rn ,

(1)

usually interpreted as a kind of energy. The Euler-Lagrange equation describing functions z(x) that are stationary for such a functional is the second-order partial diﬀerential equation ∆z(x) = F (z(x)).

¯ where x = (x1, . . . , xn), dx = dx1 ∧ · · · ∧ dxn, z = z(x) ∈ C 1(Ω) (for example), and the Lagrangian L = L(x, z, p) is a smooth function of x, z, and p = (p1, . . . , pn). Examples frequently encountered in physical ﬁeld theories are Lagrangians of the form 1 L = 2 ||p||2 + F (z),

For another example, we may identify a function z(x) with its graph N ⊂ Rn+1 , and take the Lagrangian L = 1 + ||p||2, whose associated functional FL (z) equals the area of the graph, regarded as a hypersurface in Euclidean space. The Euler-Lagrange equation describing functions z(x) stationary for this functional is H = 0, where H is the mean curvature of the graph N . To study these Lagrangians and Euler-Lagrange equations geometrically, one has to choose a class of admissible coordinate changes, and there are four natural candidates. In increasing order of generality, they are:

• Classical transformations, of the form x = x (x), z = z (z); in this situation, we think of (x, z, p) as coordinates on the space J 1 (Rn, R) of 1-jets of maps Rn → R.1 • Gauge transformations, of the form x = x (x), z = z (x, z); here, we think of (x, z, p) as coordinates on the space of 1-jets of sections of a bundle Rn+1 → Rn, where x = (x1, . . . , xn) are coordinates on the base Rn and z ∈ R is a ﬁber coordinate.

1 A 1-jet is an equivalence class of functions having the same value and the same ﬁrst derivatives at some designated point of the domain.

vii

viii

INTRODUCTION • Point transformations, of the form x = x (x, z), z = z (x, z); here, we think of (x, z, p) as coordinates on the space of tangent hyperplanes {dz − pidxi}⊥ ⊂ T(xi ,z) (Rn+1) of the manifold Rn+1 with coordinates (x1 , . . . , xn, z). • Contact transformations, of the form x = x (x, z, p), z = z (x, z, p), p = p (x, z, p), satisfying the equation of diﬀerential 1-forms dz − pi dxi = f · (dz − pi dxi)

for some function f(x, z, p) = 0. We will be studying the geometry of functionals FL (z) subject to the class of contact transformations, which is strictly larger than the other three classes. The eﬀects of this choice will become clear as we proceed. Although contact transformations were recognized classically, appearing most notably in studies of surface geometry, they do not seem to have been extensively utilized in the calculus of variations. Classical calculus of variations primarily concerns the following features of a functional FL . The ﬁrst variation δFL (z) is analogous to the derivative of a function, where z = z(x) is thought of as an independent variable in an inﬁnite-dimensional space of functions. The analog of the condition that a point be critical is the condition that z(x) be stationary for all ﬁxed-boundary variations. Formally, one writes δFL(z) = 0, and as we shall explain, this gives a second-order scalar partial diﬀerential equation for the unknown function z(x) of the form ∂L − ∂z d dxi ∂L ∂pi = 0.

This is the Euler-Lagrange equation of the Lagrangian L(x, z, p), and we will study it in an invariant, geometric setting. This seems especially promising in light of the fact that, although it is not obvious, the process by which we associate an Euler-Lagrange equation to a Lagrangian is invariant under the large class of contact transformations. Also, note that the Lagrangian L determines the functional FL , but not vice versa. To see this, observe that if we add to L(x, z, p) a “divergence term” and consider L (x, z, p) = L(x, z, p) + ∂K i (x, z) ∂K i (x, z) i p + ∂xi ∂z

for functions K i (x, z), then by Green’s theorem, the functionals FL and FL diﬀer by a constant depending only on values of z on ∂Ω. For many purposes,

ix such functionals should be considered equivalent; in particular, L and L have the same Euler-Lagrange equations. Second, there is a relationship between symmetries of a Lagrangian L and conservation laws for the corresponding Euler-Lagrange equations, described by a classical theorem of Noether. A subtlety here is that the group of symmetries of an equivalence class of Lagrangians may be strictly larger than the group of symmetries of any particular representative. We will investigate how this discrepancy is reﬂected in the space of conservation laws, in a manner that involves global topological issues. Third, one considers the second variation δ 2 FL , analogous to the Hessian of a smooth function, usually with the goal of identifying local minima of the functional. There has been a great deal of analytic work done in this area for classical variational problems, reducing the problem of local minimization to understanding the behavior of certain Jacobi operators, but the geometric theory is not as well-developed as that of the ﬁrst variation and the Euler-Lagrange equations. We will consider these issues and several others in a geometric setting as suggested above, using various methods from the subject of exterior diﬀerential systems, to be explained along the way. Chapter 1 begins with an introduction to contact manifolds, which provide the geometric setting for the study of ﬁrst-order functionals (1) subject to contact transformations. We then construct an object that is central to the entire theory: the Poincar´-Cartan form, e an explicitly computable diﬀerential form that is associated to the equivalence class of any Lagrangian, where the notion of equivalence includes that alluded to above for classical Lagrangians. We then carry out a calculation using the Poincar´-Cartan form to associate to any Lagrangian on a contact manifold an e exterior diﬀerential system—the Euler-Lagrange system—whose integral manifolds are stationary for the associated functional; in the classical case, these correspond to solutions of the Euler-Lagrange equation. The Poincar´-Cartan e form also makes it quite easy to state and prove Noether’s theorem, which gives an isomorphism between a space of symmetries of a Lagrangian and a space of conservation laws for the Euler-Lagrange equation; exterior diﬀerential systems provides a particularly natural setting for studying the latter objects. We illustrate all of this theory in the case of minimal hypersurfaces in Euclidean space En , and in the case of more general linear Weingarten surfaces in E3 , providing intuitive and computationally simple proofs of known results. In Chapter 2, we consider the geometry of Poincar´-Cartan forms more e ´ closely. The main tool for this is E. Cartan’s method of equivalence, by which one develops an algorithm for associating to certain geometric structures their diﬀerential invariants under a speciﬁed class of equivalences. We explain the various steps of this method while illustrating them in several major cases. First, we apply the method to hyperbolic Monge-Ampere systems in two independent variables; these exterior diﬀerential systems include many important Euler-Lagrange systems that arise from classical problems, and among other results, we ﬁnd a characterization of those PDEs that are contact-equivalent

x

INTRODUCTION

to the homogeneous linear wave equation. We then turn to the case of n ≥ 3 independent variables, and carry out several steps of the equivalence method for Poincar´-Cartan forms, after isolating those of the algebraic type arising from e classical problems. Associated to such a neo-classical form is a ﬁeld of hypersurfaces in the ﬁbers of a vector bundle, well-deﬁned up to aﬃne transformations. This motivates a digression on the aﬃne geometry of hypersurfaces, conducted using Cartan’s method of moving frames, which we will illustrate but not discuss in any generality. After identifying a number of diﬀerential invariants for Poincar´-Cartan forms in this manner, we show that they are suﬃcient for chare acterizing those Poincar´-Cartan forms associated to the PDE for hypersurfaces e having prescribed mean curvature. A particularly interesting branch of the equivalence problem for neo-classical Poincar´-Cartan forms includes some highly symmetric Poincar´-Cartan forms e e corresponding to Poisson equations, discussed in Chapter 3. Some of these equations have good invariance properties under the group of conformal transformations of the n-sphere, and we ﬁnd that the corresponding branch of the equivalence problem reproduces a construction that is familiar in conformal geometry. We will discuss the relevant aspects of conformal geometry in some detail; these include another application of the equivalence method, in which the important conceptual step of prolongation of G-structures appears for the ﬁrst time. This point of view allows us to apply Noether’s theorem in a particularly simple way to the most symmetric of non-linear Poisson equations, the one with the critical exponent: ∆u = Cu n−2 . Having calculated the conservation laws for this equation, we also consider the case of wave equations, and in particular the very symmetric example: z = Cz n−1 . Here, conformal geometry with Lorentz signature is the appropriate background, and we present the conservation laws corresponding to the associated symmetry group, along with a few elementary applications. The ﬁnal chapter addresses certain matters which are thus far not so welldeveloped. First, we consider the second variation of a functional, with the goal of understanding which integral manifolds of an Euler-Lagrange system are local minima. We give an interesting geometric formula for the second variation, in which conformal geometry makes another appearance (unrelated to that in the preceding chapter). Speciﬁally, we ﬁnd that the critical submanifolds for certain variational problems inherit a canonical conformal structure, and the second variation can be expressed in terms of this structure and an additional scalar curvature invariant. This interpretation does not seem to appear in the classical literature. Circumstances under which one can carry out in an invariant manner the usual “integration by parts” in the second-variation formula, which is crucial for the study of local minimization, turn out to be somewhat limited. n+3 n+2

xi We discuss the reason for this, and illustrate the optimal situation by revisiting the example of prescribed mean curvature systems. We also consider the problem of ﬁnding an analog of the Poincar´-Cartan e form in the case of functionals on vector-valued functions and their EulerLagrange PDE systems. Although there is no analog of proper contact transformations in this case, we will present and describe the merits of D. Betounes’ construction of such an analog, based on some rather involved multi-linear algebra. An illuminating special case is that of harmonic maps between Riemannian manifolds, for which we ﬁnd the associated forms and conservation laws. Finally, we consider the appearance of higher-order conservation laws for ﬁrst-order variational problems. The geometric setting for these is the inﬁnite prolongation of an Euler-Lagrange system, which has come to play a major role in classifying conservation laws. We will propose a generalized version of Noether’s theorem appropriate to our setting, but we do not have a proof of our statement. In any case, there are other ways to illustrate two of the most well-known but intriguing examples: the system describing Euclidean surfaces of Gauss curvature K = −1, and that corresponding to the sine-Gordon equation, z = sin z. We will generate examples of higher-order conservation laws by relating these two systems, ﬁrst in the classical manner, and then more systematically using the notions of prolongation and integrable extension, which come from the subject of exterior diﬀerential systems. Finally, having explored these systems this far, it is convenient to exhibit and relate the B¨cklund transa formations that act on each. One particularly appealing aspect of this study is that one sees in action so many aspects of the subject of exterior diﬀerential systems. There are particularly beautiful instances of the method of equivalence, a good illustration of the method of moving frames (for aﬃne hypersurfaces), essential use of prolongation both of G-structures and of diﬀerential systems, and a use of the notion of integrable extension to clarify a confusing issue. Of course, the study of Euler-Lagrange equations by means of exterior diﬀerential forms and the method of equivalence is not new. In fact, much of the 19th century material in this area is so naturally formulated in terms of diﬀerential forms (cf. the Hilbert form in the one-variable calculus of variations) that it is diﬃcult to say exactly when this approach was initiated. ´ However, there is no doubt that Elie Cartan’s 1922 work Le¸ons sur les c invariants int´graux [Car71] serves both as an elegant summary of the known e material at the time and as a remarkably forward-looking formulation of the use of diﬀerential forms in the calculus of variations. At that time, Cartan did not bring his method of equivalence (which he had developed beginning around 1904 as a tool to study the geometry of pseudo-groups) to bear on the subject. It was not until his 1933 work Les espaces m´triques fond´s sur la notion d’aire [Car33] e e and his 1934 monograph Les espaces de Finsler [Car34] that Cartan began to explore the geometries that one could attach to a Lagrangian for surfaces or for curves. Even in these works, any explicit discussion of the full method of equivalence is supressed and Cartan contents himself with deriving the needed

xii

INTRODUCTION

geometric structures by seemingly ad hoc methods. After the modern formulation of jet spaces and their contact systems was put into place, Cartan’s approach was extended and further developed by several people. One might particularly note the 1935 work of Th. de Donder [Don35] and its development. Beginning in the early 1940s, Th. Lepage [Lep46, Lep54] undertook a study of ﬁrst order Lagrangians that made extensive use of the algebra of diﬀerential forms on a contact manifold. Beginning in the early 1950s, this point of view was developed further by P. Dedecker [Ded77], who undertook a serious study of the calculus of variations via tools of homological algebra. All of these authors are concerned in one way or another with the canonical construction of diﬀerential geometric (and other) structures associated to a Lagrangian, but the method of equivalence is not utilized in any extensive way. Consequently, they deal primarily with ﬁrst-order linear-algebraic invariants of variational problems. Only with the method of equivalence can one uncover the full set of higher-order geometric invariants. This is one of the central themes of the present work; without the equivalence method, for example, one could not give our unique characterizations of certain classical, “natural” systems (cf. §2.1, §2.5, and §3.3). In more modern times, numerous works of I. Anderson, D. Betounes, R. Hermann, N. Kamran, V. Lychagin, P. Olver, H. Rund, A. Vinogradov, and their coworkers, just to name a few, all concern themselves with geometric aspects and invariance properties of the calculus of variations. Many of the results expounded in this monograph can be found in one form or another in works by these or earlier authors. We certainly make no pretext of giving a complete historical account of the work in this area in the 20th century. Our bibliography lists those works of which we were aware that seemed most relevant to our approach, if not necessarily to the results themselves, and it identiﬁes only a small portion of the work done in these areas. The most substantially developed alternative theory in this area is that of the variational bicomplex associated to the algebra of diﬀerential forms on a ﬁber bundle. The reader can learn this material from Anderson’s works [And92] and [And], and references therein, which contain results heavily overlapping those of our Chapter 4. Some terminology and notation that we will use follows, with more introduced in the text. An exterior diﬀerential system (EDS) is a pair (M, E) consisting of a smooth manifold M and a homogeneous, diﬀerentially closed ideal E ⊆ Ω∗ (M ) in the algebra of smooth diﬀerential forms on M . Some of the EDSs that we study are diﬀerentially generated by the sections of a smooth subbundle I ⊆ T ∗ M of the cotangent bundle of M ; this subbundle, and sometimes its space of sections, is called a Pfaﬃan system on M . It will be useful to use the notation {α, β, . . .} for the (two-sided) algebraic ideal generated by forms α, β, . . . , and to use the notation {I} for the algebraic ideal generated by the sections of a Pfaﬃan system I ⊆ T ∗ M . An integral manifold of an EDS (M, E) def is a submanifold immersion ι : N → M for which ϕN = ι∗ ϕ = 0 for all ϕ ∈ E. Integral manifolds of Pfaﬃan systems are deﬁned similarly.

xiii A diﬀerential form ϕ on the total space of a ﬁber bundle π : E → B is said to be semibasic if its contraction with any vector ﬁeld tangent to the ﬁbers of π ∗ vanishes, or equivalently, if its value at each point e ∈ E is the pullback via πe of some form at π(e) ∈ B. Some authors call such a form horizontal. A stronger condition is that ϕ be basic, meaning that it is locally (in open subsets of E) the pullback via π ∗ of a form on the base B. Our computations will frequently require the following multi-index notation. If (ω1 , . . . , ωn ) is an ordered basis for a vector space V , then corresponding to a multi-index I = (i1 , . . . , ik ) is the k-vector ω I = ω i1 ∧ · · · ∧ ω ik ∈ k (V ),

and for the complete multi-index we simply deﬁne ω = ω1 ∧ · · · ∧ ωn . Letting (e1 , . . . , en) be a dual basis for V ∗ , we also deﬁne the (n − k)-vector ω(I) = eI ω = e ik (eik−1 · · · (ei1 ω) · · · ).

This ω(I) is, up to sign, just ωIc , where Ic is a multi-index complementary to I. For the most frequently occurring cases k = 1, 2 we have the formulae (with “hats” ˆ indicating omission of a factor) ω(i) ω(ij) = (−1)i−1 ω1 ∧ · · · ∧ ωi ∧ · · · ∧ ωn , ˆ

= (−1)i+j−1 ω1 ∧ · · · ∧ ωi ∧ · · · ∧ ωj ∧ · · · ∧ ωn ˆ ˆ = −ω(ji) , for i < j, ωi ∧ ω(j) i δj ω, i i δk ω(j) − δj ω(k).

and the identities = =

ωi ∧ ω(jk)

We will often, but not always, use without comment the convention of summing over repeated indices. Always, n ≥ 2.2

2 For the case n = 1, an analogous geometric approach to the calculus of variations for curves may be found in [Gri83].

xiv

INTRODUCTION

Chapter 1

Lagrangians and Poincar´-Cartan Forms e

In this chapter, we will construct and illustrate our basic objects of study. The geometric setting that one uses for studying Lagrangian functionals subject to contact transformations is a contact manifold, and we will begin with its definition and relevant cohomological properties. These properties allow us to formalize an intuitive notion of equivalence for functionals, and more importantly, to replace such an equivalence class by a more concrete diﬀerential form, the Poincar´-Cartan form, on which all of our later calculations depend. In e particular, we will ﬁrst use it to derive the Euler-Lagrange diﬀerential system, whose integral manifolds correspond to stationary points of a given functional. We then use it to give an elegant version of the solution to the inverse problem, which asks when a diﬀerential system of the appropriate algebraic type is the Euler-Lagrange system of some functional. Next, we use it to deﬁne the isomorphism between a certain Lie algebra of inﬁnitesimal symmetries of a variational problem and the space conservation laws for the Euler-Lagrange system, as described in Noether’s theorem. All of this will be illustrated at an elementary level using examples from Euclidean hypersurface geometry.

1.1

Lagrangians and Contact Geometry

We begin by introducing the geometric setting in which we will study Lagrangian functionals and their Euler-Lagrange systems. Deﬁnition 1.1 A contact manifold (M, I) is a smooth manifold M of dimension 2n + 1 (n ∈ Z+ ), with a distinguished line sub-bundle I ⊂ T ∗ M of the cotangent bundle which is non-degenerate in the sense that for any local 1-form θ generating I, θ ∧ (dθ)n = 0. 1

2

´ CHAPTER 1. LAGRANGIANS AND POINCARE-CARTAN FORMS

Note that the non-degeneracy criterion is independent of the choice of θ; this is ¯ because if θ = fθ for some function f = 0, then we ﬁnd ¯ ¯ θ ∧ (dθ)n = f n+1 θ ∧ (dθ)n . For example, on the space J 1 (Rn, R) of 1-jets of functions, we can take coordinates (xi , z, pi) corresponding to the jet at (xi ) ∈ Rn of the linear function f(¯) = z + pi (¯i − xi ). Then we deﬁne the contact form x x θ = dz − for which dθ = − dpi ∧ dxi, so the non-degeneracy condition θ ∧ (dθ)n = 0 is apparent. In fact, the Pfaﬀ theorem (cf. Ch. I, §3 of [B+ 91]) implies that every contact manifold is locally isomorphic to this example; that is, every contact manifold (M, I) has local coordinates (xi , z, pi) for which the form θ = dz − pidxi generates I. More relevant for diﬀerential geometry is the example Gn(T X n+1 ), the Grassmannian bundle parameterizing n-dimensional oriented subspaces of the tangent spaces of an (n + 1)-dimensional manifold X. It is naturally a contact manifold, and will be considered in more detail later. Let (M, I) be a contact manifold of dimension 2n + 1, and assume that I is generated by a global, non-vanishing section θ ∈ Γ(I); this assumption only simpliﬁes our notation, and would in any case hold on a double-cover of M . Sections of I generate the contact diﬀerential ideal I = {θ, dθ} ⊂ Ω∗ (M ) in the exterior algebra of diﬀerential forms on M .1 A Legendre submanifold of M is an immersion ι : N → M of an n-dimensional submanifold N such that ι∗ θ = 0 for any contact form θ ∈ Γ(I); in this case ι∗dθ = 0 as well, so a Legendre submanifold is the same thing as an integral manifold of the diﬀerential ideal I. In Pfaﬀ coordinates with θ = dz − pidxi, one such integral manifold is given by N0 = {z = pi = 0}. To see other Legendre submanifolds “near” this one, note than any submanifold C 1 -close to N0 satisﬁes the independence condition dx1 ∧ · · · ∧ dxn = 0, and can therefore be described locally as a graph N = {(xi , z(x), pi(x))}. pi dxi,

1 Recall our convention that braces {·} denote the algebraic ideal generated by an object; for instance, {θ} consists of exterior multiples of any contact form θ, and is smaller than I. We sometimes use {I} as alternate notation for {θ}.

1.1. LAGRANGIANS AND CONTACT GEOMETRY In this case, we have θ|N = 0 if and only if pi (x) = ∂z (x). ∂xi

3

Therefore, N is determined by the function z(x), and conversely, every function z(x) determines such an N ; we informally say that “the generic Legendre submanifold depends locally on one arbitrary function of n variables.” Legendre submanifolds of this form, with dx|N = 0, will often be described as transverse. Motivated by (1) in the Introduction, we are primarily interested in functionals given by triples (M, I, Λ), where (M, I) is a (2n + 1)-dimensional contact manifold, and Λ ∈ Ωn (M ) is a diﬀerential form of degree n on M ; such a Λ will be referred to as a Lagrangian on (M, I).2 We then deﬁne a functional on the set of smooth, compact Legendre submanifolds N ⊂ M , possibly with boundary ∂N , by FΛ (N ) = Λ.

N

The classical variational problems described above may be recovered from this notion by taking M = J 1(Rn , R) ∼ R2n+1 with coordinates (xi , z, pi), I gener= ated by θ = dz − pidxi, and Λ = L(xi , z, pi)dx. This formulation also admits certain functionals depending on second derivatives of z(x), because there may be dpi -terms in Λ. Later, we will restrict attention to a class of functionals which, possibly after a contact transformation, can be expressed without second derivatives. There are two standard notions of equivalence for Lagrangians Λ. First, note that if the diﬀerence Λ − Λ of two Lagrangians lies in the contact ideal I then the functionals FΛ and FΛ are equal, because they are deﬁned only for Legendre submanifolds, on which all forms in I vanish. Second, suppose that the diﬀerence of two Lagrangians is an exact n-form, Λ − Λ = dϕ for some ϕ ∈ Ωn−1(M ). Then we ﬁnd FΛ(N ) = FΛ (N ) + ϕ

∂N

for all Legendre submanifolds N . One typically studies the variation of FΛ along 1-parameter families Nt with ﬁxed boundary, and the preceding equation shows that FΛ and FΛ diﬀer only by a constant on such a family. Such Λ and Λ are sometimes said to be divergence-equivalent. These two notions of equivalence suggest that we consider the class [Λ] ∈ Ωn (M )/(I n + dΩn−1(M )), where I n = I ∩ Ωn (M ). The natural setting for this space is the quotient ¯ ¯ ¯ ¯ (Ω∗ , d) of the de Rham complex (Ω∗ (M ), d), where Ωn = Ωn (M )/I n, and d is induced by the usual exterior derivative d on this quotient. We then have

2 In the Introduction, we used the term Lagrangian for a function, rather than for a diﬀerential form, but we will not do so again.

4

´ CHAPTER 1. LAGRANGIANS AND POINCARE-CARTAN FORMS

¯ ¯ ¯ characteristic cohomology groups H n = H n (Ω∗ , d). We will show in a moment that (recalling dim(M ) = 2n + 1): for k > n, I k = Ωk (M ). (1.1)

In other words, all forms on M of degree greater than n lie in the contact ideal; one consequence is that I can have no integral manifolds of dimension greater than n. The importance of (1.1) is that it implies that dΛ ∈ I n+1, and we can therefore regard our equivalence class of functionals as a characteristic cohomology class ¯ [Λ] ∈ H n. This class is almost, but not quite, our fundamental object of study. To prove both (1.1) and several later results, we need to describe some of the pointwise linear algebra associated with the contact ideal I = {θ, dθ} ⊂ Ω∗ (M ). Consider the tangent distribution of rank 2n I⊥ ⊂ T M given by the annihilator of the contact line bundle. Then the non-degeneracy condition on I implies that the 2-form Θ = dθ restricts ﬁberwise to I ⊥ as a non-degenerate, alternating bilinear form, determined by I up to scaling. This allows one to use tools from symplectic linear algebra; the main fact is the following. Proposition 1.1 Let (V 2n, Θ) be a symplectic vector space, where Θ ∈ is a non-degenerate alternating bilinear form. Then (a) for 0 ≤ k ≤ n, the map (Θ∧)k : n−k 2 def

V∗

V∗ →

n+k

V∗

(1.2)

is an isomorphism, and (b) if we deﬁne the space of primitive forms to be P n−k (V ∗ ) = Ker ((Θ∧)k+1 : n−k V∗ →

n+k+2

V ∗ ),

then we have a decomposition of Sp(n, R)-modules n−k V ∗ = P n−k(V ∗ ) ⊕ Θ ∧

n−k−2

V ∗ .3

3 Of course, we can extend this decomposition inductively to obtain the Hodge-Lepage decomposition n−k

V ∗ ∼ P n−k (V ∗ ) ⊕ P n−k−2 (V ∗ ) ⊕ · · · ⊕ =

P 1 (V ∗ ) = V ∗ for n − k odd, P 0 (V ∗ ) = R for n − k even,

1.1. LAGRANGIANS AND CONTACT GEOMETRY

5

Proposition 1.1 implies in particular (1.1), for it says that modulo {θ} (equivalently, restricted to I ⊥ ), every form ϕ of degree greater than n is a multiple of dθ, which is exactly to say that ϕ is in the algebraic ideal generated by θ and dθ. Proof. (a) Because n−k V ∗ and n+k V ∗ have the same dimension, it suﬃces to show that the map (1.2) is injective. We proceed by induction on k, downward from k = n to k = 0. In case k = n, the (1.2) is just multiplication (Θn )· : R →

2n

V ∗,

which is obviously injective, because Θ is non-degenerate. Now suppose that the statement is proved for some k, and suppose that n−(k−1) ξ∈ satisﬁes Θk−1 ∧ ξ = 0. This implies that Θk ∧ ξ = 0, so that for every vector X ∈ V , we have 0=X (Θk ∧ ξ) = k(X Θ) ∧ Θk−1 ∧ ξ + Θk ∧ (X ξ).

Now, the ﬁrst term on the right-hand side vanishes by our assumption on ξ (our second use of this assumption), so we must have 0 = Θk ∧ (X and the induction hypothesis then gives X ξ = 0. ξ),

This is true for every X ∈ V , so we conclude that ξ = 0. (b) We will show that any ξ ∈ V ∗ has a unique decomposition as the sum of a primitive form and a multiple of Θ. For the existence of such a decomposition, we apply the surjectivity in part (a) to the element Θk+1 ∧ ξ ∈ n+k+2 V ∗ , and n−k−2 ∗ ﬁnd η ∈ V for which Θk+2 ∧ η = Θk+1 ∧ ξ. under which any element ξ ∈ n−k n−k

V ∗ can be written uniquely as Θ n−k 2

ξ = ξ0 + (Θ ∧ ξ1 ) + (Θ2 ∧ ξ2 ) + · · · +

∧ξ

n−k 2

,

with each ξi ∈ P n−k−2i (V ∗ ). What we will not prove here is that the representation of Sp(n, R) on P n−k (V ∗ ) is irreducible for each k, so this gives the complete irreducible decomposition of n−k (V ∗ ).

6

´ CHAPTER 1. LAGRANGIANS AND POINCARE-CARTAN FORMS

Then we can decompose ξ = (ξ − Θ ∧ η) + (Θ ∧ η), where the ﬁrst summand is primitive by construction. To prove uniqueness, we need to show that if Θ ∧ η is primitive for some n−k−2 η∈ (V ∗ ), then Θ ∧ η = 0. In fact, primitivity means 0 = Θk ∧ Θ ∧ η, which implies that η = 0 by the injectivity in part (a). Returning to our discussion of Lagrangian functionals, observe that there is a short exact sequence of complexes ¯ 0 → I ∗ → Ω∗ (M ) → Ω∗ → 0 giving a long exact cohomology sequence n n+1 ¯ δ · · · → HdR (M ) → H n → H n+1 (I) → HdR (M ) → · · · ,

where δ is essentially exterior diﬀerentiation. Although an equivalence class ¯ [Λ] ∈ H n generally has no canonical representative diﬀerential form, we can now show that its image δ([Λ]) ∈ H n+1 (I) does. Theorem 1.1 Any class [Π] ∈ H n+1(I) has a unique global representative closed form Π ∈ I n+1 satisfying θ ∧ Π = 0 for any contact form θ ∈ Γ(I), or equivalently, Π ≡ 0 (mod {I}). Proof. Any Π ∈ I n+1 may be written locally as Π = θ ∧ α + dθ ∧ β for some α ∈ Ωn (M ), β ∈ Ωn−1(M ). But this is the same as Π = θ ∧ (α + dβ) + d(θ ∧ β), so replacing Π with the equivalent (in H n+1(I)) form Π − d(θ ∧ β), we have the local existence of a representative as claimed. For uniqueness, suppose that Π1 − Π2 = d(θ ∧ γ) for some (n − 1)-form γ (this is exactly equivalence in H n+1(I)), and that θ ∧ Π1 = θ ∧ Π2 = 0. Then θ ∧ dθ ∧ γ = 0, so dθ ∧ γ ≡ 0 (mod {I}). By symplectic linear algebra, this implies that γ ≡ 0 (mod {I}), so Π1 − Π2 = 0. Finally, global existence follows from local existence and uniqueness. We can now deﬁne our main object of study. Deﬁnition 1.2 For a contact manifold (M, I) with Lagrangian Λ, the unique representative Π ∈ I n+1 of δ([Λ]) satisfying Π ≡ 0 (mod {I}) is called the Poincar´-Cartan form of Λ. e

1.2. THE EULER-LAGRANGE SYSTEM

7

Poincar´-Cartan forms of Lagrangians will be the main object of study in e these lectures, and there are two computationally useful ways to think of them. The ﬁrst is as above: given a representative Lagrangian Λ, express dΛ locally as θ ∧ (α + dβ) + d(θ ∧ β), and then Π = θ ∧ (α + dβ). The second, which will be important for computing the ﬁrst variation and the Euler-Lagrange system of [Λ], is as an exact form: Π = d(Λ − θ ∧ β). In fact, β is the unique (n − 1)-form modulo {I} such that d(Λ − θ ∧ β) ≡ 0 (mod {I}). This observation will be used later, in the proof of Noether’s theorem.

1.2

The Euler-Lagrange System

In the preceding section, we showed how one can associate to an equivalence class [Λ] of Lagrangians on a contact manifold (M, I) a canonical (n + 1)-form Π. In this section, we use this Poincar´-Cartan form to ﬁnd an exterior dife ferential system whose integral manifolds are precisely the stationary Legendre submanifolds for the functional FΛ . This requires us to calculate the ﬁrst variation of FΛ , which gives the derivative of FΛ (Nt ) for any 1-parameter family Nt of Legendre submanifolds of (M, I). The Poincar´-Cartan form enables us e to carry out the usual integration by parts for this calculation in an invariant manner. We also consider the relevant version of the inverse problem of the calculus of variations, which asks whether a given PDE of the appropriate type is equivalent to the Euler-Lagrange equation for some functional. We answer this by giving a necessary and suﬃcient condition for an EDS of the appropriate type to be locally equivalent to the Euler-Lagrange system of some [Λ]. We ﬁnd these conditions by reducing the problem to a search for a Poincar´-Cartan form. e

1.2.1

Variation of a Legendre Submanifold

Suppose that we have a 1-parameter family {Nt } of Legendre submanifolds of a contact manifold (M, I); more precisely, this is given by a compact manifold with boundary (N, ∂N ) and a smooth map F : N × [0, 1] → M which is a Legendre submanifold Ft for each ﬁxed t ∈ [0, 1] and is independent of t ∈ [0, 1] on ∂N × [0, 1]. Because Ft∗ θ = 0 for any contact form θ ∈ Γ(I), we must have locally F ∗θ = G dt (1.3)

8

´ CHAPTER 1. LAGRANGIANS AND POINCARE-CARTAN FORMS

for some function G on N × [0, 1]. We let g = G|N ×{0} be the restriction to the initial submanifold. It will be useful to know that given a Legendre submanifold f : N → M , every function g may be realized as in (1.3) for some ﬁxed-boundary variation and some contact form θ, locally in the interior N o . This may be seen in Pfaﬀ coordinates (xi , z, pi) on M , for which θ = dz − pi dxi generates I and such that our given N is a 1-jet graph {(xi, z(x), pi(x) = zxi (x))}. Then (xi) give coordinates on N , and a variation of N is of the form F (x, t) = (xi , z(x, t), zxi (x, t)). Now F ∗(dz − pidxi ) = zt dt; and given z(x, 0), we can always extend to z(x, t) with g(x) = zt (x, 0) prescribed arbitrarily, which is what we claimed.

1.2.2

Calculation of the Euler-Lagrange System

We can now carry out a calculation that is fundamental for the whole theory. Suppose given a Lagrangian Λ ∈ Ωn (M ) on a contact manifold (M, I), and a ﬁxed-boundary variation of Legendre submanifold F : N × [0, 1] → M ; we wish d to compute dt ( Nt Λ). To do this, ﬁrst recall the calculation of the Poincar´-Cartan form for the e ¯ equivalence class [Λ] ∈ H n . Because I n+1 = Ωn+1 (M ), we can always write dΛ = = and then Π = θ ∧ (α + dβ) = d(Λ − θ ∧ β). (1.4) θ ∧ α + dθ ∧ β θ ∧ (α + dβ) + d(θ ∧ β),

We are looking for conditions on a Legendre submanifold f : N → M to be stationary for [Λ] under all ﬁxed-boundary variations, in the sense that d ( Nt Λ) = 0 whenever F |t=0 = f. We compute (without writing the dt t=0 ∗ F s) d dt Λ =

Nt

d dt

Nt

Nt

(Λ − θ ∧ β)

= =

Nt

L ∂ (Λ − θ ∧ β)

∂t

∂ ∂t ∂ ∂t

d(Λ − θ ∧ β) + Π

d

Nt

∂ ∂t

(Λ − θ ∧ β)

=

Nt

(because ∂N is ﬁxed).

One might express this result as δ(FΛ )N (v) =

N

v

f ∗ Π,

1.2. THE EULER-LAGRANGE SYSTEM

9

where the variational vector ﬁeld v, lying in the space Γ0(f ∗ T M ) of sections of ∂ f ∗ T M vanishing along ∂N , plays the role of ∂t . The condition Π ≡ 0 (mod {I}) allows us to write Π = θ ∧ Ψ for some n-form Ψ, not uniquely determined, and we have d g f ∗ Ψ, Λ= dt t=0 Nt N

∂ where g = ( ∂t F ∗θ)|t=0 . It was shown previously that this g could locally be chosen arbitrarily in the interior N o , so the necessary and suﬃcient condition for a Legendre submanifold f : N → M to be stationary for FΛ is that f ∗ Ψ = 0.

Deﬁnition 1.3 The Euler-Lagrange system of the Lagrangian Λ is the diﬀerential ideal generated algebraically as EΛ = {θ, dθ, Ψ} ⊂ Ω∗ (M ). A stationary Legendre submanifold of Λ is an integral manifold of EΛ . The functional is said to be non-degenerate if its Poincar´-Cartan form Π = θ ∧ Ψ e has no degree-1 divisors (in the exterior algebra of T ∗ M ) other than multiples of θ. Note ﬁrst that EΛ is uniquely determined by Π, even though θ and Ψ may not be.4 Note also that the ideal in Ω∗ (M ) algebraically generated by {θ, dθ, Ψ} is already diﬀerentially closed, because dΨ ∈ Ωn+1 (M ) = I n+1. We can examine this for the classical situation where M = {(xi, z, pi)}, θ = dz − pi dxi, and Λ = L(x, z, p)dx. We ﬁnd dΛ = = so referring to (1.4), Π = θ ∧ (Lz dx − d(Lpi dx(i))) = θ ∧ Ψ. Lz θ ∧ dx + Lpi dpi ∧ dx θ ∧ Lz dx − dθ ∧ Lpi dx(i),

Now, for a transverse Legendre submanifold N = {(xi , z(x), zxi (x))}, we have Ψ|N = 0 if and only if along N ∂L − ∂z d dxi ∂L ∂pi = 0,

4 Actually, given Π we have not only a well-deﬁned E , but a well-deﬁned Ψ modulo {I} Λ ¯ which is primitive on I ⊥ . There is a canonical map E : H n (Ω∗ ) → P n (T ∗ M/I) to the space of primitive forms, taking a Lagrangian class [Λ] to the corresponding Ψ in its Euler-Lagrange system; and this map ﬁts into a full resolution of the constant sheaf E ¯ ¯ ¯ 0 → R → Ω0 → · · · → Ωn−1 → H n (Ω∗ ) → P n (T ∗ M/I) → · · · → P 0 (T ∗ M/I) → 0.

This has been developed and applied in the context of CR geometry in [Rum90].

10 where

´ CHAPTER 1. LAGRANGIANS AND POINCARE-CARTAN FORMS

d ∂ ∂ = + z xi + i i dx ∂x ∂z

zxi xj j ∂ ∂pj

is the total derivative. This is the usual Euler-Lagrange equation, a secondorder, quasi-linear PDE for z(x1, . . . , xn) having symbol Lpi pj . It is an exercise to show that this symbol matrix is invertible at (xi, z, pi) if and only if Λ is non-degenerate in the sense of Deﬁnition 1.3.

1.2.3

The Inverse Problem

There is a reasonable model for exterior diﬀerential systems of “Euler-Lagrange type”. Deﬁnition 1.4 A Monge-Ampere diﬀerential system (M, E) consists of a contact manifold (M, I) of dimension 2n + 1, together with a diﬀerential ideal E ⊂ Ω∗ (M ), generated locally by the contact ideal I and an n-form Ψ ∈ Ω n (M ). Note that in this deﬁnition, the contact line bundle I can be recovered from E as its degree-1 part. We can now pose a famous question. Inverse Problem: When is a given Monge-Ampere system E on M equal to the Euler-Lagrange system EΛ of some Lagrangian Λ ∈ Ωn (M )? Note that if a given E does equal EΛ for some Λ, then for some local generators θ, Ψ of E we must have θ ∧ Ψ = Π, the Poincar´-Cartan form of Λ. Indeed, e we can say that (M, E) is Euler-Lagrange if and only if there is an exact form Π ∈ Ωn+1 (M ), locally of the form θ ∧ Ψ for some generators θ, Ψ of E. However, we face the diﬃculty that (M, E) does not determine either Ψ ∈ Ωn(M ) or θ ∈ Γ(I) uniquely. This can be partially overcome by normalizing Ψ as follows. Given only ¯ (M, E = {θ, dθ, Ψ}), Ψ is determined as an element of Ωn = Ωn(M )/I n. We can obtain a representative Ψ that is unique modulo {I} by adding the unique multiple of dθ that yields a primitive form on I ⊥ , referring to the symplectic n decomposition of (T ∗ M/I) (see Proposition 1.1). With this choice, we have a form θ ∧ Ψ which is uniquely determined up to scaling; the various multiples fθ ∧ Ψ, where f is a locally deﬁned function on M , are the candidates to be Poincar´-Cartan form. Note that using a primitive normalization is reasonable, e because our actual Poincar´-Cartan forms Π = θ ∧ Ψ satisfy dΠ = 0, which in e particular implies that Ψ is primitive on I ⊥ . The proof of Noether’s theorem in the next section will use a more reﬁned normalization of Ψ. The condition for a Monge-Ampere system to be Euler-Lagrange is therefore that there should be a globally deﬁned exact n-form Π, locally of the form fθ∧Ψ with Ψ normalized as above. This suggests the more accessible local inverse problem, which asks whether there is a closed n-form that is locally expressible as fθ ∧ Ψ. It is for this local version that we give a criterion.

1.2. THE EULER-LAGRANGE SYSTEM

11

We start with any candidate Poincar´-Cartan form Ξ = θ ∧ Ψ, and consider e the following criterion on Ξ: dΞ = ϕ ∧ Ξ for some ϕ with dϕ ≡ 0 (mod I). (1.5)

We ﬁrst note that if this holds for some choice of Ξ = θ ∧ Ψ, then it holds for all other choices fΞ; this is easily veriﬁed. Second, we claim that if (1.5) holds, then we can ﬁnd ϕ also satisfying ˜ dΞ = ϕ ∧ Ξ, and in addition, dϕ = 0. To see this, write ˜ ˜ dϕ = θ ∧ α + β dθ (here α is a 1-form and β is a function), and diﬀerentiate using d2 = 0, modulo the algebraic ideal {I}, to obtain 0 ≡ dθ ∧ (α + dβ) (mod {I}).

But with the standing assumption n ≥ 2, symplectic linear algebra implies that the 1-form α + dβ must vanish modulo {I}. As a result, d(ϕ − β θ) = θ ∧ (α + dβ) = 0, so we can take ϕ = ϕ − β θ, verifying the claim. ˜ Third, once we know that dΞ = ϕ∧Ξ with dϕ = 0, then on a possibly smaller neighborhood, we use the Poincar´ lemma to write ϕ = du for a function u, and e then d(e−uΞ) = e−u (ϕ ∧ Ξ − du ∧ Ξ) = 0. This proves the following. Theorem 1.2 A Monge-Ampere system (M, E = {θ, dθ, Ψ}) on a (2n + 1)dimensional contact manifold M with n ≥ 2, where Ψ is assumed to be primitive modulo {I}, is locally equal to an Euler-Lagrange system E Λ if and only if it satisﬁes (1.5). Example 1. Consider a scalar PDE of the form ∆z = f(x, z,

2

z),

(1.6)

∂ where ∆ = ; we ask which functions f : R2n+1 → R are such that (1.6) ∂xi2 is contact-equivalent to an Euler-Lagrange equation. To apply our framework, we let M = J 1(Rn , R), θ = dz − pidxi so dθ = − dpi ∧ dxi, and set

Ψ=

dpi ∧ dx(i) − f(x, z, p)dx.

∂z Restricted to a Legendre submanifold of the form N = {(xi, z(x), ∂xi (x)}, we ﬁnd Ψ|N = (∆z − f(x, z, z))dx.

12

´ CHAPTER 1. LAGRANGIANS AND POINCARE-CARTAN FORMS

Evidently Ψ is primitive modulo {I}, and E = {θ, dθ, Ψ} is a Monge-Ampere system whose transverse integral manifolds (i.e., those on which dx1 ∧· · ·∧dxn = 0) correspond to solutions of the equation (1.6). To apply our test, we start with the candidate Ξ = θ ∧ Ψ, for which dΞ = −θ ∧ dΨ = θ ∧ df ∧ dx. Therefore, we consider ϕ satisfying θ ∧ df ∧ dx = ϕ ∧ Ξ, or equivalently df ∧ dx ≡ −ϕ ∧ Ψ ϕ= (mod {I}), and ﬁnd that they are exactly those 1-forms of the form fpi dxi + c θ

for an arbitrary function c. The problem is reduced to describing those f(x, z, p) for which there exists some c(x, z, p) so that ϕ = fpi dxi + c θ is closed. We can determine all such forms explicitly, as follows. The condition that ϕ be closed expands to 0 = cpi dpi ∧ dz j +(fpi pj − cδi − cpj pi )dpj ∧ dxi

+(fpi z − cxi − cz pi)dz ∧ dxi .

1 + 2 (fpi xj − fpj xi − cxj pi + cxi pj )dxj ∧ dxi

These four terms must vanish separately. The vanishing of the ﬁrst term implies that c = c(xi, z) does not depend on any pi . Given this, the vanishing of the second term implies that f(xi , z, pi) is quadratic in the pi , with diagonal leading term: f(xi , z, pi) = 1 c(xi, z) p2 + ej (xi, z)pj + a(xi , z) j 2 for some functions ej (xi , z) and a(xi , z). Now the vanishing of the third term reduces to j 0 = e i j − e xi , x implying that for some function b(xj , z), ej (xi , z) = ∂b(xi , z) ; ∂xj

this b(xj , z) is uniquely determined only up to addition of a function of z. Finally, the vanishing of the fourth term reduces to (bz − c)xi = 0, so that c(xi , z) diﬀers from bz (xi, z) by a function of z alone. By adding an antiderivative of this diﬀerence to b(xi, z) and relabelling the result as b(xi, z),

1.2. THE EULER-LAGRANGE SYSTEM

13

we see that our criterion for the Monge-Ampere system to be Euler-Lagrange is that f(xi , z, pi) be of the form f(xi , z, pi) = 1 bz (x, z) 2 p2 + i bxi (x, z)pi + a(x, z)

for some functions b(x, z), a(x, z). These describe exactly those Poisson equations that are locally contact-equivalent to Euler-Lagrange equations. Example 2. An example that is not quasi-linear is given by det(

2

z) − g(x, z,

z) = 0.

The n-form Ψ = dp − g(x, z, p)dx and the standard contact system generate a Monge-Ampere system whose transverse integral manifolds correspond to solutions of this equation. A calculation similar to that in the preceding example shows that this Monge-Ampere system is Euler-Lagrange if and only if g(x, z, p) is of the form g(x, z, p) = g0 (x, z) g1(p, z − pi xi). Example 3. The linear Weingarten equation aK + bH + c = 0 for a surface in Euclidean space having Gauss curvature K and mean curvature H is EulerLagrange for all choices of constants a, b, c, as we shall see in §1.4.2. In this case, the appropriate contact manifold for the problem is M = G2(T E3 ), the Grassmannian of oriented tangent planes of Euclidean space. Example 4. Here is an example of a Monge-Ampere system which is locally, but not globally, Euler-Lagrange, suitable for those readers familiar with some complex algebraic geometry. Let X be a K3 surface; that is, X is a simply connected, compact, complex manifold of complex dimension 2 with trivial canonical bundle, necessarily of K¨hler type. Suppose also that there is a positive a holomorphic line bundle L → X with a Hermitian metric having positive ﬁrst Chern form ω ∈ Ω1,1(X). Our contact manifold M is the unit circle subbundle of L → X, a smooth manifold of real dimension 5; the contact form is θ= i 2π α,

dθ = ω,

where α is the u(1)-valued Hermitian connection form on M . Note θ∧(dθ)2 = 0, because the 4-form (dθ)2 = ω2 is actually a volume form on M (by positivity) and θ is non-vanishing on ﬁbers of M → X, unlike (dθ)2 . Now we trivialize the canonical bundle of X with a holomorphic 2-form Φ = Ψ + iΣ, and take for our Monge-Ampere system E = {θ, dθ = ω, Ψ = Re(Φ)}. We can see that E is locally Euler-Lagrange as follows. First, by reasons of type, ω ∧ Φ = 0; and ω is real, so 0 = Re(ω ∧ Φ) = ω ∧ Ψ. In particular, Ψ is primitive. With Ξ = θ ∧ Ψ, we compute dΞ = ω ∧ Ψ − θ ∧ dΨ = −θ ∧ dΨ,

14

´ CHAPTER 1. LAGRANGIANS AND POINCARE-CARTAN FORMS

but dΨ = Re(dΦ) = 0, because Φ is holomorphic and therefore closed. On the other hand, (M, E) cannot be globally Euler-Lagrange; that is, Ξ = θ ∧ Ψ cannot be exact, for if Ξ = dξ, then

M

Ξ∧Σ=

M

d(ξ ∧ Σ) = 0, ¯ Φ ∧ Φ,

but also

M

Ξ∧Σ =

M

θ∧Ψ∧Σ= c

X

for some number c = 0.

1.3

Noether’s Theorem

The classical theorem of Noether describes an isomorphism between a Lie algebra of inﬁnitesimal symmetries associated to a variational problem, and a space of conservation laws for its Euler-Lagrange equations. We will often assume without comment that our Lagrangian is non-degenerate in the sense discussed earlier. There are four reasonable Lie algebras of symmetries that we might consider in our setup. Letting V(M ) denote the Lie algebra of all vector ﬁelds on M , they are the following. • Symmetries of (M, I, Λ): gΛ = {v ∈ V(M ) : Lv I ⊆ I, Lv Λ = 0}. • Symmetries of (M, I, [Λ]): g[Λ] = {v ∈ V(M ) : Lv I ⊆ I, Lv [Λ] = 0}. • Symmetries of (M, Π): gΠ = {v ∈ V(M ) : Lv Π = 0}. (Note that Lv Π = 0 implies Lv I ⊆ I for non-degenerate Λ.) • Symmetries of (M, EΛ): gEΛ = {v ∈ V(M ) : Lv EΛ ⊆ EΛ}. (Note that Lv EΛ ⊆ EΛ implies Lv I ⊆ I.) We comment on the relationship between these spaces. Clearly, there are inclusions gΛ ⊆ g[Λ] ⊆ gΠ ⊆ gEΛ . Any of the three inclusions may be strict. For example, we locally have g [Λ] = gΠ because Π is the image of [Λ] under the coboundary δ : H n (Ω∗ /I) → H n+1(I),

1.3. NOETHER’S THEOREM

15

which is invariant under diﬀeomorphisms of (M, I) and is an isomorphism on contractible open sets. However, we shall see later that globally there is an inclusion n gΠ/g[Λ] → HdR (M ), and this discrepancy between the two symmetry algebras introduces some subtlety into Noether’s theorem. Also, there is a bound dim (gEΛ /gΠ ) ≤ 1. (1.7)

This follows from noting that if a vector ﬁeld v preserves EΛ, then it preserves Π up to multiplication by a function; that is, Lv Π = fΠ. Because Π is a closed form, we ﬁnd that df ∧ Π = 0; in the non-degenerate case, this implies df = uθ for some function u. The deﬁnition of a contact form prohibits any uθ from being closed unless u = 0, meaning that f is a constant. This constant gives a linear functional on gEΛ whose kernel is gΠ , proving (1.7). The area functional and minimal surface equation for Euclidean hypersurfaces provide an example where the two spaces are diﬀerent. In that case, the induced Monge-Ampere system is invariant not only under Euclidean motions, but under dilations of Euclidean space as well; this is not true of the Poincar´-Cartan form. e The next step in introducing Noether’s theorem is to describe the relevant spaces of conservation laws. In general, suppose that (M, J ) is an exterior differential system with integral manifolds of dimension n. A conservation law for (M, J ) is an (n − 1)-form ϕ ∈ Ωn−1(M ) such that d(f ∗ ϕ) = 0 for every integral manifold f : N n → M of J . Actually, we will only consider as conservation laws those ϕ on M such that dϕ ∈ J , which may be a strictly smaller set. This will not present any liability, as one can always “saturate” J to remove this discrepancy. The two apparent ways in which a conservation law may be trivial are when either ϕ ∈ J n−1 already or ϕ is exact on M . Factoring out these cases leads us to the following. Deﬁnition 1.5 The space of conservation laws for (M, J ) is C = H n−1(Ω∗ (M )/J ). It also makes sense to factor out those conservation laws represented by ϕ ∈ Ωn−1(M ) which are already closed on M , and not merely on integral manifolds of J . This can be understood using the long exact sequence: n−1 n · · · → HdR (M ) → C → H n(J ) → HdR (M ) → · · · . π

¯ Note that there is an inclusion C → H n(J ). In case J = EΛ is the EulerLagrange system of a non-degenerate functional Λ on a contact manifold (M, I), we have the following.

n−1 ¯ Deﬁnition 1.6 The space of proper conservation laws is C = C/π(HdR (M )).

16

´ CHAPTER 1. LAGRANGIANS AND POINCARE-CARTAN FORMS

Theorem 1.3 (Noether) Let (M, EΛ) be the Euler-Lagrange system of a nondegenerate functional Λ. There is a linear isomorphism η : gΠ → H n (EΛ ), ¯ taking the subalgebra g[Λ] ⊂ gΠ to the subspace η(g[Λ] ) = C ⊂ H n(EΛ ). Before proceeding to the proof, which will furnish an explicit formula for η, we need to make a digression on the algebra of inﬁnitesimal contact transformations gI = {v ∈ V(M ) : Lv I ⊆ I}. The key facts are that on any neighborhood where I has a non-zero generator θ, a contact symmetry v is uniquely determined by its so-called generating function g = v θ, and that given such θ, any function g is the generating function of some v ∈ gI . This can be seen on a possibly smaller neighborhood by taking Pfaﬀ coordinates with θ = dz − pidxi. Working in a basis ∂θ , ∂ i, ∂i dual to i ∗ the basis θ, dpi, dx of T M , we write v = g ∂θ + Now the condition Lv θ ≡ 0 (mod {I}) can be made explicit, and it turns out to be vi = ∂ i g = ∂ ∂ + pi ∂xi ∂z g, vi = −∂ i g = − ∂g . ∂pi v i ∂i + vi ∂ i .

This establishes our claim, because the correspondence between v and g is now given by v = g∂θ − g pi ∂ i + (gxi + pigz )∂ i . (1.8)

As we have presented it, the correspondence between inﬁnitesimal contact symmetries and their generating functions is local. But a simple patching argument shows that globally, as one moves between diﬀerent local generators θ for I, the diﬀerent generating functions g glue together to give a global section g ∈ Γ(M, I ∗ ) of the dual line bundle. In fact, the formula (1.8) describes a canonical splitting of the surjection Γ(T M ) → Γ(I ∗ ) → 0. Note that this splitting is not a bundle map, but a diﬀerential operator. Returning to Noether’s theorem, the proof that we present is slightly incomplete in that we assume given a global non-zero contact form θ ∈ Γ(I), or equivalently, that the contact line bundle is trivial. This allows us to treat generating functions of contact symmetries as functions rather than as sections of I ∗ . It is an enlightening exercise to develop the patching arguments needed

1.3. NOETHER’S THEOREM

17

to overcome this using sheaf cohomology. Alternatively, one can simply pull everything up to a double cover of M on which I has a global generator, and little will be lost. Proof of Theorem 1.3. Step 1: Deﬁnition of the map η. The map in question is given by η(v) = v Π for v ∈ gΠ ⊂ V(M ). Ψ), so that v Π),

Note that locally v Π = (v θ)Ψ − θ ∧ (v Furthermore, the condition Lv Π = 0 gives 0=v so that v dΠ + d(v

Π lies in EΛ .

Π) = d(v

Π is closed, and gives a well-deﬁned class η(v) ∈ H n(EΛ ).

Step 2: η is injective. Write η(v) = (v θ)Ψ − θ ∧ (v Ψ).

Suppose that this n-form is cohomologous to zero in H n(EΛ ); that is, (v θ)Ψ − θ ∧ (v Ψ) = = d(θ ∧ α + dθ ∧ β) −θ ∧ dα + dθ ∧ (α + dβ).

Regarding this equation modulo {I} and using the primitivity of Ψ, we conclude that v θ = 0. An inﬁnitesimal symmetry v ∈ gI of the contact system is locally determined by its generating function v θ as in (1.8), so we conclude that v = 0, proving injectivity. Step 3: η is locally surjective. We start by representing a class in H n (EΛ ) by a closed n-form Φ = gΨ + θ ∧ α. (1.9)

We can choose the unique contact vector ﬁeld v such that v θ = g, and our goals are to show that v ∈ gΠ and that η(v) = [Φ] ∈ H n(EΛ ). For this, we need a special choice of Ψ, which so far is determined only modulo {I}; this is reasonable because the presentation (1.9) is not unique. In fact, we can further normalize Ψ by the condition dθ ∧ Ψ = 0. To see why this is so, ﬁrst note that by symplectic linear algebra (Proposition 1.1), dΨ ≡ dθ ∧ Γ (mod {I}), (1.10)

18

´ CHAPTER 1. LAGRANGIANS AND POINCARE-CARTAN FORMS

for some Γ, because dΨ is of degree n + 1. Now suppose we replace Ψ by ¯ ¯ Ψ = Ψ − θ ∧ Γ, which certainly preserves the essential condition Π = θ ∧ Ψ. Then we have ¯ dθ ∧ Ψ = dθ ∧ (Ψ − θ ∧ Γ)

= (dΠ + θ ∧ dΨ) − dθ ∧ θ ∧ Γ = θ ∧ (dΨ − dθ ∧ Γ) = 0, by (1.10),

and we have obtained our reﬁned normalization. Now we combine the following three equations modulo {I}: • 0 ≡ Lv θ ≡ dg + v dθ, when multiplied by Ψ, gives dθ) ∧ Ψ ≡ 0 (mod {I}); dg ∧ Ψ + (v

• 0 = dΦ = d(gΨ + θ ∧ α), so using our normalization condition dθ ∧ Ψ = 0 (which implies dΨ ≡ 0 (mod {I})), dg ∧ Ψ + dθ ∧ α ≡ 0 (mod {I});

• Ordinary primitivity gives dθ ∧ Ψ ≡ 0 (mod {I}), and contracting with v, (v dθ) ∧ Ψ + dθ ∧ (v Ψ) ≡ 0 (mod {I}).

These three equations combine to give dθ ∧ (α + v Ψ) ≡ 0 (mod {I}),

and from symplectic linear algebra, we have α+v This allows us to conclude v Π = gΨ + θ ∧ α (1.11) Ψ ≡ 0 (mod {I}).

which would complete the proof of surjectivity, except that we have not yet shown that Lv Π = 0. However, by hypothesis gΨ + θ ∧ α is closed; with (1.11), this is enough to compute Lv Π = 0. The global isomorphism asserted in the theorem follows easily from these local conclusions, so long as we maintain the assumption that there exists a global contact form. Step 4: η maps symmetries of [Λ] to proper conservation laws. For this, ﬁrst note that there is an exact sequence n ¯ 0 → C → H n(EΛ ) → HdR (M ) → · · · i

1.3. NOETHER’S THEOREM so it suﬃces to show that for v ∈ gΠ , Lv [Λ] = 0 if and only if η(v) ∈ Ker i. Recall that Π = d(Λ − θ ∧ β) for some β, and we can therefore calculate η(v) = = ≡ v Π Lv (Λ − θ ∧ β) − d(v (Λ − θ ∧ β)) Lv Λ (mod dΩn−1(M ) + I n). i j

19

(1.12)

This proves that j ◦ i(η(v)) = Lv [Λ] in the composition n H n(EΛ ) → HdR (M ) → H n (Ω∗ (M )/I).

The conclusion (1.12) will follow if we can prove that j is injective. To see that j is injective, note that it occurs in the long exact cohomology sequence of 0 → I → Ω∗ (M ) → Ω∗ (M )/I → 0; namely, we have n · · · → H n (I) → HdR (M ) → H n(Ω∗ (M )/I) → · · · . j

So it suﬃces to show that H n(I) = 0, which we will do under the standing assumption that there is a global contact form θ. Suppose that the n-form ϕ = θ ∧ α + dθ ∧ β = θ ∧ (α + dβ) + d(θ ∧ β) is closed. Then regarding 0 = dϕ modulo {I}, we have by symplectic linear algebra that α + dβ ≡ 0 (mod {I}) so that actually This says that ϕ ∼ 0 in H n(I), and our proof is complete. It is important in practice to have a local formula for a representative in Ωn−1(M ), closed modulo EΛ , for the proper conservation law η(v). This is obtained by ﬁrst writing as usual Π = d(Λ − θ ∧ β), and also, for a given v ∈ g[Λ] , Lv Λ ≡ dγ We will show that the (n − 1)-form ϕ = −v Λ + (v θ)β + γ (1.15) (mod I). (1.14) (1.13) ϕ = d(θ ∧ β).

20

´ CHAPTER 1. LAGRANGIANS AND POINCARE-CARTAN FORMS

is satisfactory. First, compute dϕ = ≡ ≡ ≡ (−Lv Λ + v dΛ) + d((v θ)β) + dγ v (Π + d(θ ∧ β)) + d((v θ)β) (mod I) η(v) + Lv (θ ∧ β) (mod I) η(v) (mod I).

Now we have dϕ = η(v) + Ξ for some closed Ξ ∈ I n. We proved in the last part of the proof of Noether’s theorem that H n(I) = 0, which implies that Ξ = dξ for some ξ ∈ I n−1. Now we have d(ϕ − ξ) = η(v), and ϕ ∼ ϕ − ξ in C = H n−1(Ω∗ (M )/EΛ ). This justiﬁes our prescription (1.15). Note that the prescription is especially simple when v ∈ gΛ ⊆ g[Λ] , for then we can take γ = 0. Example. Let Ln+1 = {(t, y1 , . . . , yn )} ∼ Rn+1 be Minkowski space, and = let M 2n+3 = J 1 (Ln+1, R) be the standard contact manifold, with coordinates (t, yi , z, pa) (where 0 ≤ a ≤ n), θ = dz − p0 dt − pi dyi . For a Lagrangian, take Λ=

1 2 2 ||p||

+ F (z) dt ∧ dy

for some “potential” function F (z), where dy = dy 1 ∧ · · · ∧ dyn and ||p||2 = −p2 + p2 is the Lorentz-signature norm. The local symmetry group of this 0 i functional is generated by two subgroups, the translations in Ln+1 and the linear isometries SOo (1, n); as we shall see in Chapter 3, for certain F (z) the symmetry group of the associated Poincar´-Cartan form is strictly larger. For now, we e calculate the conservation law corresponding to translation in t, and begin by ﬁnding the Poincar´-Cartan form Π. Letting f(z) = F (z), we diﬀerentiate e dΛ = = (−p0 dp0 + pi dpi + f(z)θ) ∧ dt ∧ dy θ ∧ (f(z)dt ∧ dy) + dθ ∧ (p0 dy + pj dt ∧ dy(j)); dpj ∧ dt ∧ dy(j) . dpj ∧ dt ∧ dy(j);

with the usual recipe Π = θ ∧ (α + dβ) whenever dΛ = θ ∧ α + dθ ∧ β, we obtain Π = θ ∧ f(z)dt ∧ dy + dp0 ∧ dy + We see the Euler-Lagrange equation using Ψ = f(z)dt ∧ dy + dp0 ∧ dy +

an integral manifold of EΛ = {θ, dθ, Ψ} of the form {(t, y, z(t, y), zt (t, y), zyi (t, y))} must satisfy 0 = Ψ|N = ∂2z − ∂t2 ∂2z + f(z) dt ∧ dy. ∂yi2

With the independence condition dt∧dy = 0, we have the familiar wave equation z(t, y) = f(z).

1.4. HYPERSURFACES IN EUCLIDEAN SPACE Now considering the time-translation symmetry v = prescription (1.15) gives ϕ = = = −v − − Λ + (v θ)β ||p||2 + F (z) dy − p0 p0dy + 2

1 2 ∂ ∂t

21 ∈ gΛ, the Noether

pj dt ∧ dy(j) pj dy(j) ).

p2 + F (z) dy − p0dt ∧ ( a

One can verify that ϕ is closed when restricted to a solution of z = f(z). The question of how one might use this conservation law will be taken up later.

1.4

Hypersurfaces in Euclidean Space

We will apply the the theory developed so far to the study of hypersurfaces in Euclidean space N n → En+1 . We are particularly interested in the study of those functionals on such hypersurfaces which are invariant under the group E(n + 1) of orientation-preserving Euclidean motions.

1.4.1

The Contact Manifold over En+1

Points of En+1 will be denoted x = (x0 , . . . , xn), and each tangent space Tx En+1 will be canonically identiﬁed with En+1 itself via translation. A frame for En+1 is a pair f = (x, e) consisting of a point x ∈ En+1 and a positively-oriented orthonormal basis e = (e0 , . . . , en ) for Tx En+1 . The set F of all such frames is a manifold, and the right SO(n + 1, R)-action a (x, (e0, . . . , en)) · (gb ) = (x, ( a ea g0 , . . . , a ea gn))

gives the basepoint map x : F → En+1 the structure of a principal bundle.5 There is also an obvious left-action of E(n+ 1, R) on F, and a choice of reference frame gives a left-equivariant identiﬁcation F ∼ E(n + 1) of the bundle of frames with the group of Euclidean motions. = The relevant contact manifold for studying hypersurfaces in En+1 is the manifold of contact elements M 2n+1 = {(x, H) : x ∈ En+1 , H n ⊂ Tx En+1 an oriented hyperplane}.

5 Throughout

this section, we use index ranges 1 ≤ i, j ≤ n and 0 ≤ a, b ≤ n.

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´ CHAPTER 1. LAGRANGIANS AND POINCARE-CARTAN FORMS

This M will be given the structure of a contact manifold in such a way that transverse Legendre submanifolds correspond to arbitrary immersed hypersurfaces in En+1 . Note that M may be identiﬁed with the unit sphere bundle of En+1 by associating to a contact element (x, H) its oriented orthogonal complement (x, e0 ). We will use this identiﬁcation without further comment. The projection F → M taking (x, (ea )) → (x, e0) is E(n + 1, R)-equivariant (for the left-action). To describe the contact structure on M and to carry out calculations, we will actually work on F using the following structure equations. First, we deﬁne canonical 1-forms on F by diﬀerentiating the vector-valued coordinate functions x(f), ea (f) on F, and decomposing the resulting vectorvalued 1-forms at each f ∈ F with respect to the frame ea (f): dx = eb · ω b , dea = b e b · ωa .

(1.16)

Diﬀerentiating the relations ea (f), eb (f) = δab yields a b ωb + ωa = 0. a The forms ωa , ωb satisfy no other linear algebraic relations, giving a total of 1 (n + 1) + 2 n(n + 1) = dim(F) independent 1-forms. By taking the derivatives of the deﬁning relations (1.16), we obtain the structure equations

dωa +

a ωc ∧ ωc = 0,

a dωb +

a c ωc ∧ ωb = 0.

(1.17)

The forms ωa are identiﬁed with the usual tautological 1-forms on the orthonormal frame bundle of a Riemannian manifold (in this case, of En+1 ); and then the a ﬁrst equation indicates that ωb are components of the Levi-Civita connection n+1 of E , while the second indicates that it has vanishing Riemann curvature tensor. In terms of these forms, the ﬁbers of x : F → En+1 are exactly the maximal connected integral manifolds of the Pfaﬃan system {ω a }. Note that {ω a } and {dxa } are alternative bases for the space of forms on F that are semibasic over En+1 , but the former is E(n + 1)-invariant, while the latter is not. We return to an explanation of our contact manifold M , by ﬁrst distinguishing the 1-form on F θ = ω0 . def Note that its deﬁning formula θf (v) = dx(v), e0(f) , v ∈ Tf F,

shows that it is the pullback of a unique, globally deﬁned 1-form on M , which we will also call θ ∈ Ω1 (M ). To see that θ is a contact form, ﬁrst relabel the forms on F (this will be useful later, as well)

0 πi = ω i , def

1.4. HYPERSURFACES IN EUCLIDEAN SPACE and note the equation on F dθ = − πi ∧ ω i .

23

So on F we certainly have θ ∧ (dθ)n = 0, and because pullback of forms via the submersion F → M is injective, the same non-degeneracy holds on M . To understand the Legendre submanifolds of M , consider an oriented immersion ι N n → M 2n+1, y = (y1 , . . . , yn) → (x(y), e0 (y)). The Legendre condition is (ι∗ θ)y (v) = dxy (v), e0 (y) = 0, v ∈ Ty N.

In the transverse case, when the composition x ◦ ι : N n → M 2n+1 → En+1 is a hypersurface immersion (equivalently, ι∗ ( ωi ) = 0, suitably interpreted), this condition is that e0 (y) is a unit normal vector to the hypersurface x◦ι(N ). These Legendre submanifolds may therefore be thought of as the graphs of Gauss maps of oriented hypersurfaces N n → En+1 . Non-transverse Legendre submanifolds of M are sometimes of interest. To give some intuition for these, we exhibit two examples in the contact manifold over E3 . First, over an immersed curve x : I → E3 , one can deﬁne a cylinder N = S 1 × I → M ∼ E3 × S 2 by = (v, w) → (x(w), Rv (νx)), where ν is any normal vector ﬁeld along the curve x(w), and Rv is rotation through angle v ∈ S 1 about the tangent x (w). The image is just the unit normal bundle of the curve, and it is easily veriﬁed that this is a Legendre submanifold. Our second example corresponds to the pseudosphere, a singular surface of revolution in E3 having constant Gauss curvature K = −1 away from the singular locus. The map x : S 1 × R → E3 given by x : (v, w) → (sech w cos v, −sech w sin v, w − tanh w) fails to be an immersion where w = 0. However, the Gauss map of the complement of this singular locus can be extended to a smooth map e3 : S 1 × R → S 2 given by e3(v, w) = (−tanh w cos v, tanh w sin v, −sech w).

The graph of the Gauss map is the product (x, e3) : S 1 × R → M . It is a Legendre submanifold, giving a smooth surface in M whose projection to E3 is one-to-one, is an immersion almost everywhere, and has image equal to the singular pseudosphere. We will discuss in §4.3.3 the exterior diﬀerential system whose integral manifolds are graphs of Gauss maps of K = −1 surfaces in E3 . In §4.3.4, we will discuss the B¨cklund transformation for this system, which a relates this particular example to a special case of the preceding example, the unit normal bundle of a line.

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´ CHAPTER 1. LAGRANGIANS AND POINCARE-CARTAN FORMS

1.4.2

Euclidean-invariant Euler-Lagrange Systems

We can now introduce one of the most important of all variational problems, that of ﬁnding minimal-area hypersurfaces in Euclidean space. Deﬁne the nform Λ = ω1 ∧ · · · ∧ ωn ∈ Ωn (F), and observe that it is basic over M ; that is, it is the pullback of a well-deﬁned n-form on M (although its factors ω i are not basic). This deﬁnes a Lagrangian functional FΛ (N ) = Λ

N

on compact Legendre submanifolds N n → M 2n+1, which in the transverse case discussed earlier equals the area of N induced by the immersion N → En+1 . We calculate the Poincar´-Cartan form up on F using the structure equations e (1.17), as dΛ = −θ ∧ πi ∧ ω(i), so the Euler-Lagrange system EΛ is generated by I = {θ, dθ} and Ψ=− πi ∧ ω(i),

which is again well-deﬁned on M . A transverse Legendre submanifold N → M will locally have a basis of 1-forms given by pullbacks (by any section) of ω1 , . . . , ωn, so applying the Cartan lemma to 0 = dθ|N = −πi ∧ ωi shows that restricted to N there are expressions πi = j hij ωj

for some functions hij = hji. If N → M is also an integral manifold of EΛ ⊂ Ω∗ (M ), then additionally 0 = Ψ|N = − hii ω1 ∧ · · · ∧ ωn.

One can identify hij with the second fundamental form of N → En+1 in this transverse case, and we then have the usual criterion that a hypersurface is stationary for the area functional if and only if its mean curvature hii vanishes. We will return to the study of this Euler-Lagrange system shortly. Another natural E(n+1)-invariant PDE for hypersurfaces in Euclidean space is that of prescribed constant mean curvature H, not necessarily zero. We ﬁrst ask whether such an equation is even Euler-Lagrange, and to answer this we apply our inverse problem test to the Monge-Ampere system EH = {θ, dθ, ΨH }, ΨH = − πi ∧ ω(i) − Hω .

1.4. HYPERSURFACES IN EUCLIDEAN SPACE

25

Here, H is the prescribed constant and ω = ω 1 ∧ · · · ∧ ωn is the induced volume form. The transverse integral manifolds of EH correspond to the desired Euclidean hypersurfaces. To implement the test, we take the candidate Poincar´-Cartan form e ΠH = −θ ∧ πi ∧ ω(i) − Hω

and diﬀerentiate; the derivative of the ﬁrst term vanishes, as we know from the preceding case of H = 0, and we have dΠH = = = H d(θ ∧ ω1 ∧ · · · ∧ ωn ) H d(dx0 ∧ · · · ∧ dxn) 0.

So this EH is at least locally the Euler-Lagrange system for some functional ΛH , which can be taken to be an anti-derivative of ΠH . One diﬃculty in ﬁnding ΠH is that there is no such ΛH that is invariant under the Euclidean group E(n+1). The next best thing would be to ﬁnd a ΛH which is invariant under the rotation subgroup SO(n + 1, R), but not under translations. A little experimentation yields the Lagrangian ΛH = ω +

H n+1 x

Ω,

dΛH = ΠH ,

∂ where x = xa ∂xa is the radial position vector ﬁeld, ω = ω 1 ∧ · · · ∧ ωn is the hypersurface area form, and Ω = ω 1 ∧ · · · ∧ ωn+1 is the ambient volume form. The choice of an origin from which to deﬁne the position vector x reduces the symmetry group of ΛH from E(n + 1) to SO(n + 1, R). The functional N ΛH gives the area of the hypersurface N plus a scalar multiple of the signed volume of the cone on N with vertex at the origin. It is actually possible to list all of the Euclidean-invariant Poincar´-Cartan e forms on M → En+1 . Let 1 Λ−1 = − n+1 x

Ω,

Λk =

|I|=k

πI ∧ ω(I) (0 ≤ k ≤ n),

and Πk = −θ ∧ Λk , It is an exercise using the structure equations to show that dΛk = Πk+1. Although these forms are initially deﬁned up on F, it is easily veriﬁed that they are pull-backs of forms on M , which we denote by the same name. It can be proved using the ﬁrst fundamental theorem of orthogonal invariants that any Euclidean-invariant Poincar´-Cartan form is a linear combination of e Π0 , . . . , Πn. Note that such a Poincar´-Cartan form is induced by a Euclideane invariant functional if and only if Π0 is not involved.

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´ CHAPTER 1. LAGRANGIANS AND POINCARE-CARTAN FORMS

We can geometrically interpret Λk |N for transverse Legendre submanifolds N as the sum of the k ×k minor determinants of the second fundamental form IIN , times the hypersurface area form of N . In case k = n we have dΛn = Πn+1 = 0, reﬂecting the fact that the functional Λn =

N N

K dA

is variationally trivial, where K is the Gauss-Kronecker curvature. Contact Equivalence of Linear Weingarten Equations for Surfaces The Euclidean-invariant Poincar´-Cartan forms for surfaces in E3 give rise to e the linear Weingarten equations, of the form aK + bH + c = 0 for constants a, b, c. Although these second-order PDEs are inequivalent under point-transformations for non-proportional choices of a, b, c, we will show that under contact transformations there are only ﬁve distinct equivalence classes of linear Weingarten equations. To study surfaces, we work on the unit sphere bundle π : M 5 → E3 , and recall the formula for the contact form θ(x,e0 ) (v) = π∗ (v), e0 , ϕt(x, e0 ) = ψs(x, e0 ) = v ∈ T(x,e0 ) M.

We deﬁne two 1-parameter groups of diﬀeomorphisms of M as follows: (x + te0 , e0), (exp(s)x, e0 ).

It is not hard to see geometrically that these deﬁne contact transformations on M , although this result will also come out of the following calculations. We will carry out calculations on the full Euclidean frame bundle F → E3 , where 1 there is a basis of 1-forms ω 1 , ω2 , θ, π1 , π2, ω2 satisfying structure equations presented earlier. ∂ To study ϕt we use its generating vector ﬁeld v = ∂θ , which is the dual of the 1-form θ with respect to the preceding basis. We can easily compute Lie derivatives Lv ω1 = −π1 , Lv ω2 = −π2 , Lv θ = 0, Lv π1 = 0, Lv π2 = 0.

Now, the ﬁbers of F → M have tangent spaces given by {ω 1, ω2 , θ, π1, π2}⊥ , and this distribution is evidently preserved by the ﬂow along v. This implies that v induces a vector ﬁeld downstairs on M , whose ﬂow is easily seen to be ϕt . The fact that Lv θ = 0 conﬁrms that ϕt is a contact transformation. We can now examine the eﬀect of ϕt on the invariant Euler-Lagrange systems corresponding to linear Weingarten equations by introducing Ψ2 = π 1 ∧ π2 , Ψ1 = π1 ∧ ω 2 − π2 ∧ ω 1 , Ψ0 = ω 1 ∧ ω 2 .

1.4. HYPERSURFACES IN EUCLIDEAN SPACE

27

Restricted to a transverse Legendre submanifold over a surface N ⊂ E3 , these give K dA, H dA, and the area form dA of N , respectively. Linear Weingarten surfaces are integral manifolds of a Monge-Ampere system {θ, dθ, Ψ(a, b, c) = aΨ2 + bΨ1 + cΨ0 }. Our previous Lie derivative computations may be used to compute Ψ0 0 −1 0 Ψ0 Lv Ψ1 = 0 0 −2 Ψ1 . Ψ2 0 0 0 Ψ2 Exponentiate this to see ϕ∗ Ψ(a, b, c) = Ψ(a − 2bt + ct2, b − ct, c). t (1.18) def This describes how the 1-parameter group ϕt acts on the collection of linear Weingarten equations. Similar calculations show that the 1-parameter group ψs introduced earlier consists of contact transformations, and acts on linear Weingarten equations as

∗ ψs Ψ(a, b, c) = Ψ(a, exp(s)b, exp(2s)c).

(1.19)

It is reasonable to regard the coeﬃcients (a, b, c) which specify a particular linear Weingarten equation as a point [a : b : c] in the real projective plane RP2 , and it is an easy exercise to determine the orbits in RP2 of the group action generated by (1.18) and (1.19). There are ﬁve orbits, represented by the points [1 : 0 : 0], [0 : 1 : 0], [1 : 0 : 1], [1 : 0 : −1], [0 : 0 : 1]. The special case

1 ϕ∗1 Ψ(0, 1, A) = Ψ(− A , 0, A)

A

gives the classically known fact that to every surface of non-zero constant mean curvature −A, there is a (possibly singular) parallel surface of constant positive Gauss curvature A2 . Note ﬁnally that the Monge-Ampere system corresponding to [0 : 0 : 1] has for integral manifolds those non-transverse Legendre submanifolds of M which project to curves in E3, instead of surfaces.

1.4.3

Conservation Laws for Minimal Hypersurfaces

In Chapter 3, we will be concerned with conservation laws for various EulerLagrange equations arising in conformal geometry. We will emphasize two questions: how are conservation laws found, and how can they be used? In this section, we will explore these two questions in the case of the minimal hypersurface equation H = 0, regarding conservation laws arising from Euclidean symmetries. We compute these conservation laws ﬁrst for the translations, and then for the rotations. The results of these computations will be the two vector-valued conservation laws d(∗dx) = 0, d(∗(x ∧ dx)) = 0.

28

´ CHAPTER 1. LAGRANGIANS AND POINCARE-CARTAN FORMS

The notation will be explained in the course of the calculation. These may be thought of as analogs of the conservation of linear and angular momentum that are ubiquitous in physics. To carry out the computation, note that the prescription for Noether’s theorem given in (1.13, 1.14, 1.15) is particularly simple for the case of the functional Λ = ω1 ∧ · · · ∧ ωn on the contact manifold M 2n+1. This is because ﬁrst, dΛ = Π already, so no correction term is needed, and second, the inﬁnitesimal Euclidean symmetries (prolonged to act on M ) actually preserve Λ, and not merely the equivalence class [Λ]. Consequently, the Noether prescription is (with a sign change) η(v) = v Λ.

This v Λ is an (n − 1)-form on M which is closed modulo the Monge-Ampere system EΛ . Proceeding, we can suppose that our translation vector ﬁeld is written up on the Euclidean frame bundle as vF = Ae0 + Ai ei , where the coeﬃcients are such that the equation dv = 0 holds; that is, the functions A and Ai are the coeﬃcients of a ﬁxed vector with respect to a varying oriented orthonormal frame. We easily ﬁnd n ϕv = v F

Λ= i=1 Ai ω(i).

This, then, is the formula for an (n − 1)-form on F which is well-deﬁned on the contact manifold M and is closed when restricted to integral manifolds of the Monge-Ampere system EΛ. To see it in another form, observe that if we restrict our (n − 1)-form to a transverse Legendre submanifold N , ϕ v |N = Ai (∗ωi ) = ∗ v, dx .

Here and throughout, the star operator ∗ = ∗N is deﬁned with respect to the induced metric and orientation on N , and the last equality follows from the equation of En+1 -valued 1-forms dx = e0 θ + ei ωi , where θ|N = 0. We now have a linear map from Rn+1, regarded as the space of translation vectors v, to the space of closed (n−1)-forms on any minimal hypersurface N . Tautologically, such a map may be regarded as one closed (Rn+1)∗ -valued (n − 1)-form on N . Using the metric to identify (Rn+1)∗ ∼ Rn+1, this may be written as = ϕtrans = ∗dx. This is the meaning of the conservation law stated at the beginning of this section. Note that each component d(∗dxa) = 0 of this conservation law is

1.4. HYPERSURFACES IN EUCLIDEAN SPACE

29

equivalent to the claim that the coordinate function xa of the immersion x : N → En+1 is a harmonic function with respect to the induced metric on N . Turning to the rotation vector ﬁelds, we ﬁrst write such a vector ﬁeld on En+1 as n+1 ∂ Ra + Rb = 0. v= xa R b b , b a a ∂x a,b=1 It is not hard to verify that this vector ﬁeld lifts naturally to the frame bundle F as ∂ ∂ vF = xa R b A b c + Ab Rb Aa c , a c c a d ∂ω ∂ωd

∂ ∂ Ab ∂xb , and the where the coeﬃcients Aa are deﬁned by the equation ∂ωa = a b ∂ ∂ a a tangent vectors ∂ωa , ∂ωa are dual to the canonical coframing ω , ωb of F. b We can now compute (restricted to N , for convenience)

(vF

Λ)|N

= = =

b xa Rb Ai ω(i) a b ∗(xa Rb Ac ωc ) a

∗ R · x, dx .

Reformulating the Noether map in a manner analogous to that used previously, 2 Rn+1 ∼ so(n + 1, R)∗ -valued (n − 1)-form on N we can deﬁne a = ϕrot = ∗(x ∧ dx). Once again, ϕrot is a conservation law by virtue of the fact that it is closed if N is a minimal hypersurface. It is interesting to note that the conservation law for rotation symmetry is a consequence of that for translation symmetry. This is because we have from d(∗dx) = 0 that d(x ∧ ∗dx) = dx ∧ ∗dx = 0. The last equation holds because the exterior multiplication ∧ refers to the En+1 where the forms take values, not the exterior algebra in which their components live. It is an exercise to show that these translation conservation laws are equivalent to minimality of N . Another worthwhile exercise is to show that all of the classical conservation laws for the H = 0 system arise from inﬁnitesimal Euclidean symmetries. In the next chapter, we will see directly that the group of symmetries of the Poincar´e Cartan form for this system equals the group of Euclidean motions, giving a more illuminating proof of this fact. At the end of this section, we will consider a dilation vector ﬁeld which preserves the minimal surface system E, but not the Poincar´-Cartan form, and use it to compute an “almost-conservation law”. e By contrast, in this case there is no discrepancy between g[Λ] and gΠ. To see this, ﬁrst note that by Noether’s theorem 1.3, gΠ is identiﬁed with H n (EΛ ),

30

´ CHAPTER 1. LAGRANGIANS AND POINCARE-CARTAN FORMS

and g[Λ] ⊆ gΠ is identiﬁed with the image of the connecting map δ in the long exact sequence n · · · → H n−1(Ω∗ /EΛ) → H n(EΛ ) → HdR (M ) → · · · . n ∼ With M = En+1 × S n , we have the isomorphism HdR (M ) ∼ R obtained by = n+1 integrating an n-form along a ﬁber of M → E , and it is not hard to see that any n-form in EΛ must vanish when restricted to such a ﬁber. Therefore the map ι is identically 0, so δ is onto, and that proves our claim. δ ι

Interpreting the Conservation Laws for H = 0 To understand the meaning of the conservation law ϕtrans, we convert the equation dϕtrans|N = 0 to integral form. For a smoothly bounded, oriented neighborhood U ⊂ N ⊂ En+1 with N minimal, we have by Stokes’ theorem

∂U

∗dx = 0.

To interpret this condition on U , we take an oriented orthonormal frame ﬁeld (e0 , . . . , en ) along U ∪ ∂U , such that along the boundary ∂U the following hold: e0 is the oriented normal to N, en is the outward normal to ∂U in N, (1.20) e1 , . . . , en−1 are tangent to ∂U. Calculations will be much easier in this adapted frame ﬁeld. The dual coframe ωa for En+1 along U ∪ ∂U satisﬁes n−1 dx = e0 ω0 + i=1 ei ω i + e n ω n .

Now, the ﬁrst term vanishes when restricted to N . The last term vanishes when restricted to ∂U , but cannot be discarded because it will aﬀect ∗N dx, which we are trying to compute. Consequently, n−1 ∗N dx = Now we restrict to ∂U , and ﬁnd ∗N dx|∂U = =

ei ω(i) + en ω(n). i=1 (−1)n−1en ω1 ∧ · · · ∧ ωn−1 (−1)n−1n dσ.

Here we use n to denote the normal to ∂U in N and dσ to denote the area measure induced on ∂U . Our conservation law therefore reads n dσ = 0.

∂U

1.4. HYPERSURFACES IN EUCLIDEAN SPACE

31

In other words, in a minimal hypersurface the average of the exterior unit normal vectors over the smooth boundary of any oriented neighborhood must vanish. One consequence of this is that a minimal surface can never be locally convex; that is, a neighborhood of a point can never lie on one side of the tangent plane at that point. This is intuitively reasonable from the notion of minimality. Similar calculations give an analogous formulation for the rotation conservation law:

∂U

(x ∧ n)dσ = 0.

These interpretations have relevance to the classical Plateau problem, which asks whether a given simple closed curve γ in E3 bounds a minimal surface. The answer to this is aﬃrmative, with the caveat that such a surface is not necessarily unique and may not be smooth at the boundary. A more well-posed version gives not only a simple closed curve γ ⊂ E3 , but a strip, which is a curve γ (1) ⊂ M consisting of a base curve γ ⊂ E3 along with a ﬁeld of tangent planes along γ containing the tangent lines to γ. Such a strip is the same as a curve in M along which the contact 1-form vanishes. Asking for a minimal surface whose boundary and boundary-tangent planes are described by a given γ (1) is the same as asking for a transverse integral manifold of EΛ having boundary γ (1) ⊂ M . The use of our two conservation laws in this context comes from the fact that γ (1) determines the vector-valued form ∗N dx along ∂N for any possible solution to this initial value problem. The conservation laws give integral constraints, often called moment conditions, on the values of ∗N dx, and hence constrain the possible strips γ (1) for which our problem has an aﬃrmative answer. However, the moment conditions on a strip γ (1) are not suﬃcient for there to exist a minimal surface with that boundary data. We will discuss additional constraints which have the feel of “hidden conservation laws” after a digression on similar moment conditions that arise for boundaries of holomorphic curves. It is natural to ask whether a given real, simple, closed curve γC in complex space Cn (always n ≥ 2) is the boundary of some holomorphic disc. There is a diﬀerential ideal J ⊂ Ω∗ (Cn ) whose integral manifolds are precisely holomorR phic curves, deﬁned by J = {(Ω2,0(Cn ) + Ω0,2(Cn )) ∩ Ω2 (Cn )}. R In other words, J is algebraically generated by real 2-forms which, when regarded as complex 2-forms, have no part of type (1, 1). It is elementary to see that in degree k ≥ 3, J k = Ωk (Cn ), and that the integral 2-planes in T Cn R are exactly the complex 1-dimensional subspaces. This implies our claim that integral manifolds of J are holomorphic curves. Now, J has many conservation laws. Namely, for any holomorphic 1-form ϕ ∈ Ω1,0 (Cn ), we ﬁnd that hol dϕ + dϕ ∈ J , ¯

32

´ CHAPTER 1. LAGRANGIANS AND POINCARE-CARTAN FORMS

so that ϕ + ϕ is a conservation law for J . These give rise to inﬁnitely many ¯ moment conditions ϕ=0 γC which must be satisﬁed by γC , if it is to be the boundary of a holomorphic disc. It is a fact which we shall not prove here that every conservation law for J is of this form; trivial conservation laws clearly arise when ϕ = df for some holomorphic function f ∈ O(Cn ). Another fact, not to be proved here, is that these moment conditions are suﬃcient for γC to bound a (possibly branched) holomorphic disc. Returning to our discussion of minimal surfaces, suppose that x : U → E3 is a minimal immersion of a simply connected surface. Then ∗dx deﬁnes a closed, vector-valued 1-form on U , so there exists a vector-valued function y : U → E 3 satisfying dy = ∗dx. (1.21)

Note that our ability to integrate the conservation law to obtain a function relies essentially on the fact that we are in dimension n = 2. We can deﬁne z = (x + iy) : U → C3 , and (1.21) is essentially the Cauchy-Riemann equations, implying that z is a holomorphic curve, with the conformal structure induced from the immersion z. Furthermore, the complex derivative z is at each point of U a null vector for the complex bilinear inner-product (dz i)2 . This gives the classical Weierstrass representation of a minimal surface in E3 as locally the real part of a holomorphic null curve in C3 . We can now incorporate the result of our digression on conservation laws for holomorphic discs. Namely, given a strip γ (1) , the Euclidean moment condition ∗dx = 0 implies that there exists another real curve y so that dy = ∗dx (along γ γ). Then we can use z = (x + iy) : γ → C3 as initial data for the holomorphic disc problem, and all of the holomorphic moment conditions for that problem come into play. These are the additional hidden constraints needed to ﬁll the real curve γ with a (possibly branched) minimal surface. Conservation Laws for Constant Mean Curvature It is also a worthwhile exercise to determine the conservation laws corresponding to Euclidean motions for the constant mean curvature system when the constant H is non-zero. Recall that for that system the Poincar´-Cartan form e ΠH = −θ ∧ πi ∧ ω(i) − Hω

is invariant under the full Euclidean group, but that no particular Lagrangian Λ is so invariant; we will continue to work with the SO(n + 1, R)-invariant

1.4. HYPERSURFACES IN EUCLIDEAN SPACE Lagrangian ΛH = ω +

H x n+1

33

Ω, n dΛH = ΠH .

Fortunately, the equivalence class [Λ] ∈ H (Ω∗ (M )/I) is invariant under the Euclidean group, because as the reader can verify, the connecting map δ : H n(Ω∗ /I) → H n+1(I) taking [Λ] to Π is an isomorphism for this contact manifold. This means that, as in the case H = 0, we will ﬁnd conservation laws corresponding to the full Euclidean Lie algebra. Computing the conservation laws corresponding to translations requires the more complicated form of the Noether prescription, because it is the translation vector ﬁelds v ∈ Rn+1 which fail to preserve our ΛH . Instead, we have Lv ΛH = = =

H Ω) (because Lv ω = n+1 Lv (x H ((Lv x) Ω + x (Lv Ω)) n+1 H Ω + 0). n+1 (v

0),

In the last step, we have used Lv x = [v, x] = v (by a simple calculation), and Lv Ω = 0 (because the ambient volume Ω is translation invariant). To apply the Noether prescription, we need an anti-derivative of this last term, which we ﬁnd by experimenting: d(x (v Ω)) = = = Lx (v Ω) − x d(v Ω) ((Lx v) Ω + v (LxΩ)) − x −v Ω + (n + 1)v Ω,

0

where we have again used Lxv = [x, v] = −v, and LxΩ = (n + 1)Ω. Combining these two calculations, we have Lv ΛH =

H d(x n(n+1)

(v

Ω)).

The prescription (1.13, 1.14, 1.15) now gives ϕv = = −v −v ΛH + ω+

H x n(n+1)

(v Ω).

Ω)

H nx

(v

As in the case of minimal hypersurfaces, we consider the restriction of ϕv to an integral manifold N . From the previous case, we know that v ω restricts to ∗ v, dx , where ∗ = ∗N is the star operator of the metric on N and ·, · denotes the ambient inner-product. To express the restriction of the other term of ϕv , decompose x = xt + xν ν into tangential and normal parts along N (so xt is a vector and xν is a scalar), and a calculation gives

H nx

(v

Ω)|N

= =

− H (xν n

− H (xν v n

∗ v, dx − (v

ω − (v

θ)(xt

ω))

θ) ∗ ·, xt );

34

´ CHAPTER 1. LAGRANGIANS AND POINCARE-CARTAN FORMS

In the second term, the normal ν provides the “vector-valued” part (it replaced θ, to which it is dual), and ∗ ·, xt provides the “(n − 1)-form” part. Calculating the conservation laws for rotations is a similar process, simpliﬁed somewhat by the fact that Lv ΛH = 0; of course, the lifted rotation vector ﬁelds v are not so easy to work with as the translations. The resulting Λ2 Rn+1-valued (n − 1)-form is ϕrot = −(1 +

H n+1 xν )

the latter ∗ is being applied to the 1-form on N that is dual via the metric to the tangent vector xt . Again as in the H = 0 case, we can write these (n − 1)-forms ϕv , which depend linearly on v ∈ Rn+1 , as an (Rn+1 )∗ -valued (n − 1)-form on N . It is ϕtrans = −(1 + H xν ) ∗dx + H ν ∗ ·, xt . n n

∗(x ∧ dx) +

H n+1 (x

∧ ν) ∗ ·, xt .

These can be used to produce moment conditions, just as in the H = 0 case. We conclude with one more observation suggesting extensions of the notion of a conservation law. Recall that we showed in (1.7) that a Monge-Ampere system EΛ might have an inﬁnitesimal symmetry which scales the corresponding Poincar´-Cartan form Π. This is the case for the minimal surface system, which e is preserved by the dilation vector ﬁeld on En+1 x=

∂ xa ∂xa .

This induces a vector ﬁeld x on the contact manifold of tangent hyperplanes to En+1 where the functional Λ and Poincar´-Cartan form Π are deﬁned, and e there are various ways to calculate that Lx Λ = nΛ. If one tries to apply the Noether prescription to x by writing ϕdil = x the resulting form satisﬁes dϕdil = = LxΛ − x dΛ nΛ − x Π. Λ,

Restricted to a minimal surface N , we will then have dϕdil |N = nΛ. Because the right-hand side is not zero, we do not have a conservation law, but it is still reasonable to look for consequences of integrating on neighborhoods U in N , where we ﬁnd ϕ=n

∂U U

Λ.

(1.22)

1.4. HYPERSURFACES IN EUCLIDEAN SPACE

35

The right-hand side equals n times the area of U , and the left-hand side can be investigated by choosing an oriented orthonormal frame ﬁeld (e0 , . . . , en) along U ∪ ∂U satisfying the conditions (1.20) as before. We write the coeﬃcients x=

∂ xa ∂xa =

v a ea ,

and then restricted to ∂U , we have ϕ|∂U = x Λ = (−1)n−1vn ω1 ∧ · · · ∧ ωn−1.

Up to sign, the form ω 1 ∧ · · ·∧ ωn−1 along ∂U is exactly the (n − 1)-dimensional area form for ∂U . These interpretations of the two sides of (1.22) can be exploited by taking for U the family of neighborhoods Ur for r > 0, deﬁned as the intersection of N ⊂ En+1 with an origin-centered ball of radius r. In particular, along ∂Ur we will have ||x|| = r, so that v n ≤ r and r · Area(∂Ur ) ≥ n · Vol(Ur ). Observe that Area(∂Ur ) = (1.23)

d Vol(Ur ), dr and (1.23) is now a diﬀerential inequality for Vol(Ur ) which can be solved to give Vol(Ur ) ≥ Crn

for some constant C. This is a remarkable result about minimal hypersurfaces, and amply illustrates the power of “almost-conservation laws” like ϕdil .

36

´ CHAPTER 1. LAGRANGIANS AND POINCARE-CARTAN FORMS

Chapter 2

The Geometry of Poincar´-Cartan Forms e

In this chapter, we will study some of the geometry associated to Poincar´e ´ Cartan forms using E. Cartan’s method of equivalence. The idea is to identify such a Poincar´-Cartan form with a G-structure—that is, a subbundle of the e principal coframe bundle of a manifold—and then attempt to ﬁnd some canonically determined basis of 1-forms on the total space of that G-structure. The diﬀerential structure equations of these 1-forms will then exhibit associated geometric objects and invariants. e The pointwise linear algebra of a Poincar´-Cartan form in the case of n = 2 “independent variables” (that is, on a contact manifold of dimension 5) is quite diﬀerent from that of higher dimensional cases. Therefore, in the ﬁrst section we study only the former, which should serve as a good illustration of the method of equivalence for those not familiar with it. Actually, in case n = 2 we will study the coarser equivalence of Monge-Ampere systems rather than Poincar´e Cartan forms, and we will do this without restricting to those systems which are locally Euler-Lagrange. An extensive study of the geometry of Monge-Ampere systems in various low dimensions was carried out in [LRC93], with a viewpoint somewhat similar to ours. In the succeeding sections, we will ﬁrst identify in case n ≥ 3 a narrower class of Poincar´-Cartan forms, called neo-classical, which are of the same algebraic e type as those arising from classical variational problems. We will describe some of the geometry associated with neo-classical Poincar´-Cartan forms, consisting e of a ﬁeld of hypersurfaces in a vector bundle, well-deﬁned up to ﬁberwise aﬃne motions of the vector bundle. A digression on the local geometry of individual hypersurfaces in aﬃne space follows this. We then turn to the very rich equivalence problem for neo-classical Poincar´-Cartan forms; the diﬀerential invariants e that this uncovers include those of the various associated aﬃne hypersurfaces. In the last section of this chapter, we use these diﬀerential invariants to characterize systems locally contact-equivalent to those for prescribed mean curvature 37

38

´ CHAPTER 2. THE GEOMETRY OF POINCARE-CARTAN FORMS

hypersurfaces in Riemannian manifolds. In the next chapter, we will specialize to the study of those neo-classical Poincar´-Cartan forms whose primary diﬀerential invariants all vanish. These e correspond to interesting variational problems arising in conformal geometry. We begin with a few elementary notions used in the method of equivalence. On a manifold M of dimension n, a coframe at a point x ∈ M is a linear isomorphism ∼ ux : Tx M −→ Rn .

∗ This is equivalent to a choice of basis for the cotangent space Tx M , and we will not maintain any distinction between these two notions. The set of all coframes for M has the structure of a principal GL(n, R)-bundle π : F(M ) → M , with right-action

where the right-hand side denotes composition of ux with multiplication by g −1 . A local section of π : F(M ) → M is called a coframing, or coframe ﬁeld. On the total space F(M ), there is an Rn-valued tautological 1-form ω, given at u ∈ F(M ) by ωu (v) = u(π∗ v) ∈ Rn , v ∈ Tu F(M ). (2.1)

ux · g = g−1 ux ,

def

g ∈ GL(n, R),

The n components ω i of this Rn-valued 1-form give a global basis for the semibasic 1-forms of F(M ) → M . In terms of coordinates x = (x1, . . . , xn) on M , there is a trivialization ∼ M × GL(n, R) = F(M ) given by (x, g) ↔ (x, g−1dx), where on the right-hand side, dx is a column of 1-forms regarded as a coframe at x, and g−1 dx is the composition of that coframe with multiplication by g−1 ∈ GL(n, R). In this trivialization, we can express the tautological 1-form as ω = g−1 dx, where again the right-hand side represents the product of a GL(n, R)-valued ﬁber coordinate and an Rn-valued semibasic 1-form. The geometric setting of the equivalence method is the following. Deﬁnition 2.1 Let G ⊂ GL(n, R) be a subgroup. A G-structure on the nmanifold M is a principal subbundle of the coframe bundle F(M ) → M , having structure group G. We will associate to a hyperbolic Monge-Ampere system (to be deﬁned, in case n = 2), or to a neo-classical Poincar´-Cartan form (in case n ≥ 3), a succession e of G-structures on the contact manifold M , which carry increasingly detailed information about the geometry of the system or form, respectively.

2.1. THE EQUIVALENCE PROBLEM FOR N = 2

39

2.1

The Equivalence Problem for n = 2

In this section, we will study the equivalence problem for certain Monge-Ampere systems on contact manifolds of dimension 5. We will give criteria in terms of the diﬀerential invariants thus obtained for a given system to be locally equivalent the system associated to the linear homogeneous wave equation. We will also give the weaker criteria for a given system to be locally equivalent to an Euler-Lagrange system, as in the previously discussed inverse problem. Unless otherwise noted, we use the index ranges 0 ≤ a, b, c ≤ 4, 1 ≤ i, j, k ≤ 4. We assume given a 5-dimensional contact manifold (M, I) and a MongeAmpere system E, locally algebraically generated as E = {θ, dθ, Ψ}, where 0 = θ ∈ Γ(I) is a contact form, and Ψ ∈ Ω2(M ) is some 2-form. As noted previously, E determines I and I. We assume that Ψx ∈ Ix for all / x ∈ M . Recall from the discussion in §1.2.3 that given E, the generator Ψ may be uniquely chosen modulo {I} (and modulo multiplication by functions) by the condition of primitivity; that is, we may assume dθ ∧ Ψ ≡ 0 (mod {I}). The assumption Ψx ∈ Ix means that this primitive form is non-zero everywhere. / We do not necessarily assume that E is Euler-Lagrange. On the contact manifold M , one can locally ﬁnd a coframing η = (η a ) such that η0 ∈ Γ(I), dη0 ≡ η1 ∧ η2 + η3 ∧ η4 (mod {I}). (2.2)

1 Then we can write Ψ ≡ 2 bij ηi ∧ ηj (mod {I}), where the functions bij depend on the choice of coframing and on the choice of Ψ. The assumption that Ψ is primitive means that in terms of a coframing satisfying (2.2),

b12 + b34 = 0. We now ask what further conditions may be imposed on the coframing η = (η a ) while preserving (2.2). To investigate this, we ﬁrst consider changes of coframe that ﬁx η 0 ; we will later take into account non-trivial rescalings of η 0 . In this case, an element of GL(5, R) preserves the condition (2.2) if and only if it acts as a ﬁberwise sympletic transformation, modulo the contact line bundle I. Working modulo I, we can split 2 (T ∗ M/I) ∼ (R · dη0) ⊕ P 2(T ∗ M/I), =

where P 2 (T ∗ M/I) is the 5-dimensional space of 2-forms that are primitive with respect to the symplectic structure on I ⊥ induced by dη0. The key observation is that the action of the symplectic group Sp(2, R) on P 2(R4 ) is equivalent to

40

´ CHAPTER 2. THE GEOMETRY OF POINCARE-CARTAN FORMS

the standard action of the group SO(3, 2) on R5. This is because the symmetric bilinear form ·, · on P 2(T ∗ M/I) deﬁned by ψ1 ∧ ψ2 = ψ1 , ψ2 (dη0 )2 has signature (3, 2) and symmetry group Sp(2, R). Therefore, the orbit decomposition of the space of primitive forms Ψ modulo {I} under admissible changes of coframe will be a reﬁnement of the standard orbit decomposition under SO(3, 2). To incorporate rescaling of η 0 into our admissible changes of coframe, note that a rescaling of η 0 requires via (2.2) the same rescaling of the symplectic form η1 ∧ η2 + η3 ∧ η4, so we should actually allow changes by elements of GL(5, R) inducing the standard action of CSp(2, R); this is the group that preserves the standard symplectic form up to scale. This in turn corresponds to the split-signature conformal group CO(3, 2), which acts on R5 with three non-zero orbits: a negative space, a null space, and a positive space. The three orbits of this representation correspond to three types of MongeAmpere systems: • If Ψ ∧ Ψ is a negative multiple of dη 0 ∧ dη0, then the local coframing η may be chosen so that in addition to (2.2), Ψ ≡ η1 ∧ η2 − η3 ∧ η4 (mod {I});

for a classical variational problem, this occurs when the Euler-Lagrange PDE is hyperbolic. • If Ψ ∧ Ψ = 0, then η may be chosen so that Ψ ≡ η1 ∧ η3 (mod {I}); for a classical variational problem, this occurs when the Euler-Lagrange PDE is parabolic. • If Ψ ∧ Ψ is a positive multiple of dη 0 ∧ dη0 , then η may be chosen so that Ψ ≡ η1 ∧ η4 − η3 ∧ η2 (mod {I}); for a classical variational problem, this occurs when the Euler-Lagrange PDE is elliptic. The equivalence problem for elliptic Monge-Ampere systems in case n = 2 develops in analogy with that for hyperbolic systems; we will present the hyperbolic case. The conclusion will be: Associated to a hyperbolic Monge-Ampere system (M 5, E) is a canonical subbundle B1 → M of the coframe bundle of M carrying a pair of 2 × 2-matrix-valued functions S1 and S2 , involving up to second derivatives of the given system. (M, E) is locally of Euler-Lagrange type if and only if S2 vanishes identically, while it is equivalent to the system associated to the homogeneous wave equation zxy = 0 if and only if S1 and S2 both vanish identically.

2.1. THE EQUIVALENCE PROBLEM FOR n = 2

41

An example of a hyperbolic Monge-Ampere system, to be studied in more detail in Chapter 4, is the linear Weingarten system for surfaces in E3 with Gauss curvature K = −1. To begin, assume that (M 5 , E) is a hyperbolic Monge-Ampere system. A coframing η = (η a ) of M is said to be 0-adapted to E if E = {η0, η1 ∧ η2 + η3 ∧ η4 , η1 ∧ η2 − η3 ∧ η4} and also dη0 ≡ η1 ∧ η2 + η3 ∧ η4 (mod {I}). (2.4) (2.3)

According to the following proposition, a hyperbolic Monge-Ampere system is equivalent to a certain type of G-structure, and it is the latter to which the equivalence method directly applies. Proposition 2.1 The 0-adapted coframings for a hyperbolic Monge-Ampere system (M 5, E) are the sections of a G0-structure on M , where G0 ⊂ GL(5, R) is the (disconnected) subgroup generated by all matrices of the form (displayed in blocks of size 1, 2, 2) a 0 0 (2.5) g0 = C A 0 , D 0 B with a = det(A) = det(B) = 0, along with the matrix 1 0 0 J = 0 0 I2 . 0 I2 0

(2.6)

Proof. The content of this proposition is that any two 0-adapted coframes diﬀer by multiplication by an element of G0 . To see why this is so, note that the 2-forms η1 ∧ η2 + η3 ∧ η4 and η1 ∧ η2 − η3 ∧ η4

have, up to scaling, exactly 2 decomposable linear combinations, η 1 ∧ η2 and η3 ∧ η4 . These must be either preserved or exchanged by any change of coframe preserving their span modulo {I}, and this accounts for both the block form (2.5) and the matrix J. The condition on determinants then corresponds to (2.4).

Although not every G0-structure on a 5-manifold M is induced by a hyperbolic Monge-Ampere system E, it is easy to see that those that do determine E uniquely. We therefore make a digression to describe the ﬁrst steps of the equivalence method, by which one investigates the local geometry of a general G-structure. This will be followed by application to the case at hand of a G0 structure, then a digression on the next general steps, and application to the

42

´ CHAPTER 2. THE GEOMETRY OF POINCARE-CARTAN FORMS

case at hand, and so on. One major step, that of prolongation, will not appear in this chapter but will be discussed in the study of conformal geometry in Chapter 3. Fix a subgroup G ⊂ GL(n, R). Two G-structures Bi → Mi , i = 1, 2, are equivalent if there is a diﬀeomorphism M1 → M2 such that under the induced isomorphism of principal coframe bundles F(M1 ) → F(M2 ), the subbundle B1 ⊂ F(M1 ) is mapped to B2 ⊂ F(M2 ). One is typically interested only in those properties of a G-structure which are preserved under this notion of equivalence. For instance, if one has a pair of 5-manifolds with hyperbolic Monge-Ampere systems, then a diﬀeomorphism of the 5-manifolds carries one of these systems to the other if and only if it induces an equivalence of the associated G0-structures. It is easy to see that a diﬀeomorphism F : B1 → B2 between the total spaces of two G-structures Bi → Mi is an equivalence in the above sense if and only if F ∗ (ω2 ) = ω1 , where ωi is the restriction of the tautological Rn-valued form (2.1) on F(Mi ) ⊇ Bi . The ﬁrst step in investigating the geometry of a G-structure B → M is therefore to understand the local behavior of this tautological form. To do this, we seek an expression for its exterior derivative, and to understand what such an expression should look like, we proceed as follows. Consider a local trivialization B ∼ M × G, induced by a choice of section η = of B → M whose image is identiﬁed with M × {e} ⊂ M × G. The section η is in particular an Rn -valued 1-form on M , and the tautological 1-form is ω = g−1 η ∈ Ω1(B) ⊗ Rn. The exterior derivative of this equation is dω = −g−1 dg ∧ ω + g−1 dη. (2.7)

Note that the last term in this equation is semibasic for B → M , and that the matrix 1-form g −1 dg takes values in the Lie algebra g of G. Of course, these pieces g−1 dη and g−1 dg each depend on the choice of trivialization. To better understand the pointwise linear algebra of (2.7), we introduce the following notion. Deﬁnition 2.2 A pseudo-connection in the G-structure B → M is a g-valued 1-form on B whose restriction to the ﬁber tangent spaces Vb ⊂ Tb B equals the ∼ identiﬁcation Vb = g induced by the right G-action on B. This diﬀers from the deﬁnition of a connection in the principal bundle B → M by omission of an equivariance requirement. In terms of our trivialization above, a pseudo-connection on M × G is any g-valued 1-form of the form g−1 dg + (semibasic g-valued 1-form); in particular, every G-structure carries a pseudo-connection. A consequence of (2.7) is that any pseudo-connection ϕ ∈ Ω1 (B) ⊗ g satisﬁes a structure equation

2.1. THE EQUIVALENCE PROBLEM FOR n = 2 that is fundamental for the equivalence method: dω = −ϕ ∧ ω + τ,

43

(2.8)

1 i where τ = ( 2 Tjk ωj ∧ωk ) is a semibasic Rn-valued 2-form on B, called the torsion of the pseudo-connection ϕ. It is natural to consider exactly how a diﬀerent choice of pseudo-connection—remember that any two diﬀer by an arbitrary semibasic g-valued 1-form—yields a diﬀerent torsion form. We will pursue this after considering the situation for our hyperbolic Monge-Ampere systems.

Let B0 ⊂ F(M ) be the G0-bundle of 0-adapted coframes for a hyperbolic Monge-Ampere system E. A local section η corresponds to an R5 -valued 1-form ∼ (ηa ) satisfying (2.3, 2.4). In terms of the trivialization B0 = M × G0 induced by −1 5 η, the tautological R -valued 1-form is ω = g0 η. Locally (over neighborhoods in M ), there is a structure equation (2.8), in which 0 0 ϕ0 0 0 0 0 ω ϕ1 ϕ1 ϕ1 0 ω1 0 1 2 0 2 2 ω and ϕ = ϕ2 ϕ2 ϕ2 0 0 ω= 0 1 3 ϕ0 0 ω3 0 ϕ 3 ϕ3 3 4 ϕ4 0 0 ϕ 4 ϕ4 ω4 0 3 4 are the tautological R5 -valued 1-form and the pseudo-connection form, respectively; note that the condition for ϕ to be g0-valued includes the condition ϕ0 = ϕ 1 + ϕ 2 = ϕ 3 + ϕ 4 . 4 2 3 0 1 The torsion τ of ϕ is an R5-valued 2-form, semibasic for B0 → M and depending on a choice of pseudo-connection. Returning to the general situation of a G-structure B → M , our goal is to understand how diﬀerent choices of pseudo-connection in (2.8) yield diﬀerent torsion forms. We will use this to restrict attention to those pseudo-connections whose torsion is in some normal form. The linear-algebraic machinery for this is as follows. Associated to the linear Lie algebra g ⊂ gl(n, R) is a map of G-modules δ : g ⊗ (Rn )∗ → Rn ⊗ deﬁned as the restriction to g ⊗ (Rn)∗ ⊂ (Rn ⊗ (Rn )∗ ) ⊗ (Rn)∗ of the surjective skew-symmetrization map Rn ⊗ (Rn)∗ ⊗ (Rn )∗ → Rn ⊗ The cokernel of δ H 0,1(g) = (Rn ⊗ def 2 2 2

(Rn )∗ ,

(2.9)

(Rn)∗ .

(Rn )∗ )/δ(g ⊗ (Rn)∗ )

(2.10)

44

´ CHAPTER 2. THE GEOMETRY OF POINCARE-CARTAN FORMS

is one of the Spencer cohomology groups of g ⊂ gl(n, R). Note that to each b ∈ B ∼ is associated an isomorphism Tπ(b) M → Rn , and consequently an identiﬁcation of semibasic 1-forms at b ∈ B with (Rn )∗ . Now, given a pseudo-connection in i the G-structure, the semibasic Rn -valued torsion 2-form ( 1 Tjk ωj ∧ ωk ) at b ∈ B 2 n 2 n ∗ can be identiﬁed with an element τb ∈ R ⊗ Λ (R ) . Similarly, a permissible change at b ∈ B of the pseudo-connection—that is, a semibasic g-valued 1form—can be identiﬁed with an element ϕb ∈ g ⊗ (Rn )∗ . Under these two identiﬁcations, the map δ associates to a change ϕb the corresponding change in the torsion ϕb ∧ ωb, where in this expression we have contracted the middle factor of (Rn )∗ in ϕb (see (2.9)) with the values of the Rn-valued 1-form ωb. Therefore, diﬀerent choices of pseudo-connection yield torsion maps diﬀering by elements of Im(δ), so what is determined by the G-structure alone, independent of a choice of pseudo-connection, is a map τ : B → H 0,1(g), called the intrinsic ¯ torsion of B → M . This suggests a major step in the equivalence method, called absorption of torsion, which one implements by choosing a (vector space) splitting of the projection Rn ⊗

2

(Rn )∗ → H 0,1(g) → 0.

(2.11)

As there may be no G-equivariant splitting, one is merely choosing some vector 2 subspace T ⊂ Rn ⊗ (Rn )∗ which complements the kernel δ(g⊗(Rn )∗ ). Fixing a choice of T , it holds by construction that any G-structure B → M locally has pseudo-connections whose torsion at each b ∈ B corresponds to a tensor lying in T . We will see from our example of hyperbolic Monge-Ampere systems that this is not as complicated as it may seem. Denote the semibasic 2-form components of the R5 -valued torsion by 0 τ τ1 τ = τ2 . 3 τ τ4

0 τ 0 = dω0 + ϕ0 ∧ ω0 = ω1 ∧ ω2 + ω3 ∧ ω4 + σ ∧ ω0 def

We know from the condition (2.4) in the deﬁnition of 0-adapted that

for some semibasic 1-form σ. We may now replace ϕ0 by ϕ0 − σ in our pseudo0 0 connection, eliminating the term σ ∧ ω 0 from the torsion. We then rename this altered pseudo-connection entry again as ϕ0 ; to keep the pseudo-connection g0 0 valued, we have to make a similar change in ϕ1 + ϕ2 and ϕ3 + ϕ4 . What we 1 2 3 4 have just shown is that given an arbitrary pseudo-connection in a G0 -structure B0 → M , there is another pseudo-connection whose torsion satisﬁes (using 0 0 obvious coordinates on Rn ⊗ 2(Rn )∗ ) T0a = Ta0 = 0. By choosing this latter pseudo-connection, we are absorbing the corresponding torsion components into

2.1. THE EQUIVALENCE PROBLEM FOR n = 2

45

ϕ. Furthermore, the fact that our G0-structure is not arbitrary, but comes from a hyperbolic Monge-Ampere system, gave us the additional information that 0 0 0 T12 = T34 = 1, and all other independent Tij = 0. Note incidentally that our decision to use a pseudo-connection giving σ = 0 determines ϕ0 uniquely, 0 up to addition of multiples of ω 0 ; this uniqueness applies also to ϕ1 + ϕ2 and 1 2 ϕ3 +ϕ4 . The eﬀort to uniquely determine pseudo-connection forms should guide 3 4 the choices one makes in the equivalence method. Other torsion terms may be absorbed using similar methods. Using the index range 1 ≤ i, j, k ≤ 4, we write i i τ i = Tj0ωj ∧ ω0 + 1 Tjk ωj ∧ ωk 2 i i i for functions Tj0 and Tjk = −Tkj . First, by altering the nilpotent part ϕi , we 0 i can arrange that all Tj0 = 0. Second, by altering the oﬀ-diagonal terms ϕ1, ϕ2 , 2 1 4 ϕ3 , ϕ3 , we can arrange that 4 1 2 3 4 T2j = T1j = T4j = T3j = 0.

Third, by altering the traceless diagonal parts ϕ1 − ϕ2 and ϕ3 − ϕ4 , we can 1 2 3 4 arrange that

1 2 1 2 3 4 3 4 T13 = T23, T14 = T24, T13 = T14, T23 = T24.

We summarize this by renaming τ1 τ2 τ3 τ4 = = = = (V3 ω3 + V4 ω4 ) ∧ ω1 + U 1 ω3 ∧ ω4 , (V3 ω3 + V4 ω4 ) ∧ ω2 + U 2 ω3 ∧ ω4 , (V1 ω1 + V2 ω2 ) ∧ ω3 + U 3 ω1 ∧ ω2 ,

(V1 ω1 + V2 ω2 ) ∧ ω4 + U 4 ω1 ∧ ω2 ,

a for 8 torsion functions Ui , Vi on B0 . The collection of torsion tensors (Tbc ) 0 0 taking this form, and satisfying T0a = Ta0 = 0, constitutes the splitting of (2.11) given in the general discussion, to which we will return shortly. At this point, we can uncover more consequences of the fact that we are dealing not with an arbitrary G0-structure on a 5-manifold, but a special one induced by a hyperbolic Monge-Ampere system. We already found as one consequence the fact that

τ 0 ≡ ω1 ∧ ω2 + ω3 ∧ ω4

(mod {ω0 }),

which has nothing to do with our choices in absorbing torsion; absorbing torsion allowed us to render this congruence into an equality. Similarly, we now obtain pointwise relations among other torsion coeﬃcients by computing, modulo {I}

46

´ CHAPTER 2. THE GEOMETRY OF POINCARE-CARTAN FORMS

(which in this case means ignoring all ω 0 terms after diﬀerentiating), 0 ≡ ≡ d(dω0) ϕ0 ∧ (ω1 ∧ ω2 + ω3 ∧ ω4 ) 0

≡ so that

(U 1 + 2V2 )ω2 ∧ ω3 ∧ ω4 − (U 2 − 2V1)ω1 ∧ ω3 ∧ ω4 +(U 3 + 2V4 )ω1 ∧ ω2 ∧ ω4 − (U 4 − 2V3 )ω1 ∧ ω2 ∧ ω3 , U 1 = −2V2 , U 2 = 2V1 , U 3 = −2V4 , U 4 = 2V3.

+((−ϕ1 + V3 ω3 + V4 ω4 ) ∧ ω1 + U 1 ω3 ∧ ω4 ) ∧ ω2 1 −ω1 ∧ ((−ϕ2 + V3 ω3 + V4ω4 ) ∧ ω2 + U 2 ω3 ∧ ω4 ) 2 +((−ϕ3 + V1 ω1 + V2 ω2 ) ∧ ω3 + U 3 ω1 ∧ ω2 ) ∧ ω4 3 4 −ω3 ∧ ((−ϕ4 + V1 ω1 + V2ω2 ) ∧ ω4 + U 4 ω1 ∧ ω2 )

These are pointwise linear-algebraic relation among our 8 torsion functions. In the general study of G-structures B → M , we now have to consider the group action in more detail. Speciﬁcally, H 0,1(g) is the cokernel of a map of Gmodules, so it inherits a G-action as well, and it is easy to see that the intrinsic torsion τ : B → H 0,1(g) is equivariant for this action. Therefore, there is an ¯ induced map [¯] : M → H 0,1(g)/G, τ which is an invariant of the equivalence class of the G-structure B → M ; that is, under a diﬀeomorphism M1 → M2 inducing an equivalence of G-structures, [τ2] must pull back to [τ1]. Now, H 0,1(g)/G typically has a complicated topology, and is rarely a manifold. However, in many cases of interest one can ﬁnd a slice W ⊂ H 0,1(g), a submanifold whose points all have the same stabilizer G1 ⊂ G, and which is a cross-section of the orbits which W itself intersects. If the intrinsic torsion τ : B → H 0,1(g) of a G-structure takes values in a union of ¯ orbits represented by such a slice, then the set B1 = τ −1(W ) ¯ is a smooth principal subbundle of B → M having structure group G1 ⊂ G. The process of reducing to a subbundle deﬁned as the locus where intrinsic torsion lies in a slice is called normalizing the torsion. If G1 is a proper subgroup of G, then we can essentially start the process over, starting with an arbitrary pseudo-connection, absorbing torsion, and so on. Typically, one inherits from B → M some information about the torsion of the subbundle B1 → M , because the original structure equations restrict to the submanifold B1 ⊂ B. We will see an example of this below. In practice, one typically studies the G-action on H 0,1(g) by transporting it 2 to the representing vector space T ⊂ Rn ⊗ (Rn)∗ . If T is not an invariant 2 subspace of Rn ⊗ (Rn)∗ , then typically G will act by aﬃne-linear motions on def 2.1. THE EQUIVALENCE PROBLEM FOR n = 2

47

T . This is the case in the next step of our equivalence problem for hyperbolic Monge-Ampere systems. We have represented the intrinsic torsion of a G0 -structure B0 → M corresponding to a hyperbolic Monge-Ampere system by 4 independent functions on B; that is, our torsion takes values in a 4-dimensional subspace of the lift T of H 0,1(g0 ). The next step is to determine how the independent torsion functions vary along the ﬁbers of B0 → M . This will be expressed inﬁnitesimally, in an equation for the exterior derivative of the torsion functions, modulo the space of forms that are semibasic for B0 → M ; the expressions will be in terms of the pseudo-connection forms which parallelize the ﬁbers. They are obtained as follows. We ﬁrst consider the equations for dω 1 , dω2 . Taking the exterior derivative of each, modulo the algebraic ideal {ω 0, ω1 , ω2}, yields equivalences of 3-forms that do not involve derivatives of any psuedo-connection forms, but do involve dU 1, dU 2. From each of these can be factored the 2-form ω 3 ∧ ω4 , yielding a pair of equivalences modulo {ω 0 , . . . , ω4}, expressible in matrix form as 0≡d U1 U2 + ϕ1 0 ϕ2 0 + ϕ1 1 ϕ2 1 ϕ1 2 ϕ2 2 · U1 U2 − ϕ0 · 0 U1 U2 .

A similar procedure applied to the equations for dω 3 , dω4 yields the pair 0≡d U3 U4 + ϕ3 0 ϕ4 0 + ϕ3 3 ϕ4 3 ϕ3 4 ϕ4 4 · U3 U4 − ϕ0 · 0 U3 U4 .

These describe the derivatives of the functions U i along the ﬁbers of B0 → M . They are to be interpreted as giving d dt t=0 Ui (u

· gt ),

where gt is a path in G0 passing through the identity matrix at t = 0. Exponentiated, we see that the vector-valued functions (U1 , U2 ) and (U3 , U4) on B0 each transform by an aﬃne-linear action of G0 along the ﬁbers; that is, they vary by a linear representation composed with a translation.1 It is the “nilpoi tent” part of the group, with components g0, which gives rise to the translation. Speciﬁcally, we have for g0 as in (2.5) U 1(u · g0) U 2(u · g0) = = aA−1 aB −1 U 1 (u) U 2 (u) U 3(u) U 4(u) − A−1C, − B −1 D. (2.12) (2.13)

U 3(u · g0) U 4(u · g0)

1 Strictly speaking, we have only shown that the torsion function (U i ) varies by an aﬃnelinear action under the identity component of G0 . What will be important, however, is that if u ∈ B0 satisﬁes U i (u) = 0, then U i (u · J) = 0 as well, and likewise for some matrix in each component where a < 0. These claims can be veriﬁed directly.

48

´ CHAPTER 2. THE GEOMETRY OF POINCARE-CARTAN FORMS

with a = det(A) = det(B) = 0. The structure equation (2.8) on B0 still holds when restricted to B1 , with τ i|B1 = 0; but the pseudo-connection forms ϕi |B1 0 are semibasic for B1 → M , and their contribution should be regarded as torsion. With everything now restricted to B1 , we write i ϕi = P0 ω0 + Pjiωj 0

Now deﬁne a 1-adapted coframe to be a 0-adapted coframe u ∈ B0 satisfying U i (u) = 0 for 1 ≤ i ≤ 4. It then follows from the above reasoning that the subset B1 ⊂ B0 of 1-adapted coframes is a G1-subbudle of B0 , where the subgroup G1 ⊂ G0 is generated by the matrix J of (2.6), and by matrices of the form (again, in blocks of size 1, 2, 2) a 0 0 g1 = 0 A 0 , (2.14) 0 0 B

and then have dω = −ϕ ∧ ω + τ with ϕ=

0 ϕ0 0 0 0 0

0 ϕ1 1 ϕ2 1 0 0

0 ϕ1 2 ϕ2 2 0 0

0 0 0 3 ϕ3 4 ϕ3

0 0 0 ϕ3 4 ϕ4 4

As before, we can absorb some of this torsion into the pseudo-connection form, respecting the constraint ϕ0 = ϕ1 + ϕ2 = ϕ3 + ϕ4 , until the torsion is of the 0 1 2 3 4 form ω1 ∧ ω2 + ω3 ∧ ω4 1 1 −(P ω1 + P3 ω3 + P4 ω4 ) ∧ ω0 2 2 −(P ω2 + P3 ω3 + P4 ω4 ) ∧ ω0 . (2.15) dω + ϕ ∧ ω = 3 3 −(Qω3 + P1 ω1 + P2 ω2 ) ∧ ω0 4 4 −(Qω4 + P1 ω1 + P2 ω2 ) ∧ ω0 We can go further: recall that ϕ0 was uniquely determined up to addition of 0 a multiple of ω0 . We now exploit this, and take the unique choice of ϕ0 = 0 ϕ1 + ϕ2 = ϕ3 + ϕ4 that yields a torsion vector of the form (2.15), with 1 2 3 4 P + Q = 0. Now that ϕ0 is uniquely determined, it is reasonable to try to get information 0 about its exterior derivative. To do this, we diﬀerentiate the equation

0 dω0 = −ϕ0 ∧ ω0 + ω1 ∧ ω2 + ω3 ∧ ω4 ,

and τ =

ω1 ∧ ω2 + ω3 ∧ ω4 −Pj1ωj ∧ ω0 −Pj2ωj ∧ ω0 −Pj3ωj ∧ ω0 −Pj4ωj ∧ ω0

.

2.1. THE EQUIVALENCE PROBLEM FOR n = 2 which simpliﬁes to 0 =

0 (−dϕ0 + 2P ω1 ∧ ω2 + 2Qω3 ∧ ω4 2 4 1 4 +(P3 − P1 )ω1 ∧ ω3 − (P3 + P2 )ω2 ∧ ω3 2 3 1 3 +(P4 + P1 )ω1 ∧ ω4 − (P4 − P2 )ω2 ∧ ω4 ) ∧ ω0 .

49

This tells us the derivative of ϕ0 modulo the algebraic ideal {ω 0} = {I}, which 0 we now use in a somewhat unintuitive way. We easily compute that d(ω0 ∧ ω1 ∧ ω2 ) = −2ϕ0 ∧ ω0 ∧ ω1 ∧ ω2 + ω1 ∧ ω2 ∧ ω3 ∧ ω4 . 0

With knowledge of dϕ0 ∧ ω0 from above, we can diﬀerentiate this equation to 0 ﬁnd 0 = 2(P − Q)ω 0 ∧ ω1 ∧ ω2 ∧ ω3 ∧ ω4 . This implies that P − Q = 0, and combined with our normalization P + Q = 0, we have P = Q = 0,

2 4 1 4 2 3 1 3 dϕ0 ≡ (P3 − P1 )ω1 ∧ ω3 − (P3 + P2 )ω2 ∧ ω3 + (P4 + P1 )ω1 ∧ ω4 − (P4 − P2 )ω2 ∧ ω4 . 0 (2.16) 1 1 2 2 As before, the next step is to study the 8 torsion coeﬃcients P3 , P4 , P3 , P4 , 4 4 3 P2 , P1 , P2 . We can again obtain a description of how they vary along the connected components of the ﬁbers using inﬁnitesimal methods, and then get a full description of their variation along ﬁbers by explicitly calculating how they transform under one representative of each component of the structure group G1 . We state only the result of this calculation. The torsion in each ﬁber transforms by an 8-dimensional linear representation of the group G1 , which decomposes as the direct sum of two 4-dimensional representations. Motivated by (2.16), we deﬁne a pair of 2 × 2 matrix-valued functions on B1 3 P1 ,

which somewhat simpliﬁes our structure equations (2.15). In particular, we have modulo {I}

S1 (u) =

1 4 P3 − P 2 2 4 P3 + P 1

1 3 P4 + P 2 3 2 P4 − P 1

(u),

S2 (u) =

1 4 P3 + P 2 2 4 P3 − P 1

1 3 P4 − P 2 3 2 P4 + P 1

(u).

Now, for g1 ∈ G1 as in (2.14), one ﬁnds that S1 (u · g1 ) = aA−1 S1 (u)B,

S2 (u · g1) = aA−1 S2 (u)B.

In particular, the two summand representations for our torsion are the same, when restricted to the components of G1 of (2.14). However, one may also verify that S1 (u · J) S2 (u · J) = − = 0 1 −1 0 t S1 (u) t S2 (u)

0 −1 1 0 0 −1 1 0 .

,

0 1 −1 0

50

´ CHAPTER 2. THE GEOMETRY OF POINCARE-CARTAN FORMS

An immediate conclusion to be drawn from this is that if S1 (u) = 0 at some point, then S1 (u) = 0 everywhere on the same ﬁber of B1 → M , and likewise for S2 . If the torsion vector takes its values in a union of non-trivial orbits having conjugate stabilizers, then we can try to make a further reduction to the subbundle consisting of those coframes on which the torsion lies in a family of normal forms. However, it is usually interesting in equivalence problems to consider the case when no further reduction is possible; in the present situation, this occurs when all of the invariants vanish identically. We ﬁrst claim that S2 = 0 identically if and only if the uniquely determined form ϕ0 is closed. To see this, note ﬁrst that from (2.16) we have S2 = 0 if and 0 only if dϕ0 = µ ∧ ω0 0 for some 1-form µ. We diﬀerentiate modulo {ω 0} to obtain 0 ≡ −µ ∧ dω0 (mod {ω0})

which by symplectic linear algebra implies that µ ≡ 0 (mod {ω0 }). But then dϕ0 = 0, as claimed. Conversely, if dϕ0 = 0, then obviously S2 = 0. 0 0 Now suppose that S1 = S2 = 0 identically. Then because dϕ0 = 0, we can 0 locally ﬁnd a function λ > 0 satisfying ϕ0 = λ−1 dλ. 0 We can also compute in case S1 = S2 = 0 that d(ω1 ∧ ω2 ) = −ϕ0 ∧ ω1 ∧ ω2, 0 so that d(λ ω1 ∧ ω2 ) = 0. Now, by a variant of the Darboux theorem, this implies that there are locally deﬁned functions p, x such that −dp ∧ dx = λ ω1 ∧ ω2 . Similar reasoning gives locally deﬁned functions q, y such that −dq ∧ dy = λ ω 3 ∧ ω4 . In terms of these functions, note that d(λ ω0 ) = λ(ω1 ∧ ω2 + ω3 ∧ ω4 ) = −dp ∧ dx − dq ∧ dy, which by the Poincar´ lemma implies that there is another locally deﬁned funce tion z such that λω0 = dz − p dx − q dy.

2.1. THE EQUIVALENCE PROBLEM FOR n = 2

51

The linear independence of ω 0 , . . . , ω4 implies that pulled back by any 1-adapted coframe (that is, any section of B1 ), the functions x, y, z, p, q form local coordinates on M . In terms of these local coordinates, our hyperbolic Monge-Ampere system is E = = {ω0 , ω1 ∧ ω2 + ω3 ∧ ω4 , ω1 ∧ ω2 − ω3 ∧ ω4 } (2.17)

{dz − p dx − q dy, dp ∧ dx + dq ∧ dy, dp ∧ dx − dq ∧ dy}. (2.18)

In an obvious way, transverse local integral surfaces of E are in one-to-one correspondence with solutions to the wave equation for z(x, y) ∂2z = 0. ∂x ∂y This establishes the following. Theorem 2.1 A hyperbolic Monge-Ampere system (M 5 , E) satisﬁes S1 = S2 = 0 if and only if it is locally equivalent to the Monge-Ampere system (2.18) for the linear homogeneous wave equation. This gives us an easily computable method for determining when a given secondorder scalar Monge-Ampere equation in two variables is contact-equivalent to this wave equation. Looking at the equation (2.16) for dϕ0 (mod {I}), it is natural to ask about 0 the situation in which S2 = 0, but possibly S1 = 0. This gives an alternative version of the solution to the inverse problem discussed in the previous chapter. Theorem 2.2 A hyperbolic Monge-Ampere system (M 5, E) is locally equivalent to an Euler-Lagrange system if and only if its invariant S2 vanishes identically. Proof. The condition for our E to contain a Poincar´-Cartan form e Π = λ ω0 ∧ (ω1 ∧ ω2 − ω3 ∧ ω4 ) is that this Π be closed for some function λ on B1 , which we can assume satisﬁes λ > 0. Diﬀerentiating then gives 0 = (dλ − 2λϕ0 ) ∧ ω0 ∧ (ω1 ∧ ω2 − ω3 ∧ ω4 ). 0 Exterior algebra shows that this is equivalent to dλ − 2λϕ0 being a multiple of 0 ω0 , say dλ − 2λϕ0 = σ λ ω0 0 d(log λ) − 2ϕ0 = σ ω0 . 0 Such an equation can be satisﬁed if and only if dϕ0 is equivalent modulo {I} to 0 a multiple of dω0. But we know that dω0 ≡ ω1 ∧ ω2 + ω3 ∧ ω4 ,

for some function σ, or in other words,

52

´ CHAPTER 2. THE GEOMETRY OF POINCARE-CARTAN FORMS

and from (2.16) we see that dϕ0 is a multiple of this just in case S2 = 0. 0 This result may be thought of as follows. For any hyperbolic Monge-Ampere system, dϕ0 ∈ Ω2 (B1 ) is both closed and semibasic for B1 → M . This means 0 that it is the pullback of a 2-form on M canonically associated with E.2 We showed that this 2-form vanishes if and only if S2 = 0, which is equivalent to E being locally Euler-Lagrange. This condition is reminiscent of the vanishing of a curvature, when ϕ0 is viewed as a connection in the contact line bundle I. 0

2.2

Neo-Classical Poincar´-Cartan Forms e

We now turn to the geometry of Poincar´-Cartan forms in case n ≥ 3. In the e preceding section, we emphasized the corresponding Monge-Ampere system; from now on, we will instead emphasize the more specialized Poincar´-Cartan e form. Let M 2n+1 be a manifold with contact line bundle I, locally generated by a 1-form θ. Let Π ∈ Ωn+1 (M ) be a closed (n + 1)-form locally expressible as Π = θ ∧ Ψ, ¯ where Ψ ∈ P n(T ∗ M/I) is primitive modulo {I}. As in the preceding section, the pointwise linear algebra of this data involves the action of the conformal n R2n. When n = 2, symplectic group CSp(n, R) on the space P n(R2n) ⊂ there are four orbits (including {0}) for this action, but for n > 2, the situation is more complicated. For example, when n = 3, the space of primitive 3-forms on R6 has two open orbits and many degenerate orbits, while for n = 4 there are no open orbits. Which orbits contain the Poincar´-Cartan forms of most interest to us? e Consider the classical case, in which M = J 1(Rn , R), θ = dz − pi dxi, and Λ = L(x, z, p)dx. We have already seen that Π = = = d(L dx + θ ∧ Lpi dx(i)) (2.19) (2.20) (2.21)

−θ ∧ (d(Lpi ) ∧ dx(i) − Lz dx) −θ ∧ (Lpi pj dpj ∧ dx(i) + (Lpi z pi + Lpi xi − Lz )dx).

This suggests the following deﬁnition, which singles out Poincar´-Cartan forms e of a particular algebraic type; it is these—with a slight reﬁnement in the case n = 3, to be introduced below—whose geometry we will study. Note that non-degeneracy of the functional is built in to the deﬁnition. Deﬁnition 2.3 A closed (n + 1)-form Π on a contact manifold (M 2n+1, I) is almost-classical if it can locally be expressed as Π = −θ ∧ (H ij πi ∧ ω(j) − Kω) (2.22)

2 This statement also requires one to verify that dϕ0 is invariant under the action of some 0 element of each connected component of G1 ; this is easily done.

´ 2.2. NEO-CLASSICAL POINCARE-CARTAN FORMS

53

for some coframing (θ, ω i , πi) of M with θ ∈ Γ(I), some invertible matrix of functions (H ij ), and some function K. Later, we will see the extent to which this deﬁnition generalizes the classical case. We remark that the almost-classical forms Π = θ ∧ Ψ are those for which ¯ the primitive Ψ lies in the tangent variety of the cone of totally decomposable3 n n-forms in P (T ∗ M/I), but not in the cone itself. Applying the equivalence method will yield diﬀerential invariants and geometric structures intrinsically associated to our Poincar´-Cartan forms. This e will be carried out in §2.4, but prior to this, it is best to directly look for some naturally associated geometry. The preview that this provides will make easier the task of interpreting the results of the equivalence method. First note that the local coframings and functions appearing in the deﬁnition of an almost-classical form are not uniquely determined by Π. The extent of the non-uniqueness of the coframings is described in the following lemma, which prepares us for the equivalence method. Lemma 2.1 If (θ, ωi , πi) is a coframing adapted to an almost-classical form ¯ ¯ ¯ Π as in Deﬁnition 2.3, then (θ, ωi, πi) is another if and only if the transition matrix is of the form (in blocks of size 1, n, n) ¯ a 0 0 θ θ ω i = C i Ai 0 ωj . ¯ j j πj πi ¯ Di Eij Bi Proof. That the ﬁrst row of the matrix must be as shown is clear from the ¯ requirement that θ, θ ∈ Γ(I). The real content of the lemma is that Pfaﬃan system def JΠ = Span{θ, ω1 , . . . , ωn} is uniquely determined by Π. This follows from the claim that JΠ is characterized as the set of 1-forms ξ such that ξ ∧ Π is totally decomposable; this claim we leave as an exercise for the reader. The Pfaﬃan system JΠ = {θ, ω1 , . . . , ωn } associated to Π is crucial for all that follows. It is canonical in the sense that any local diﬀeomorphism of M preserving Π also preserves JΠ . In the classical case described previously we have JΠ = {dz, dx1, . . . , dxn}, which is integrable and has leaf space J 0 (Rn , R). Proposition 2.2 If n ≥ 4, then for any almost-classical form Π on a contact manifold (M 2n+1, I), the Pfaﬃan system JΠ is integrable. Proof. We need to show that dθ, dω i ≡ 0 (mod {JΠ}), for some (equivalently, any) coframing (θ, ω i , πi) adapted to Π as in the deﬁnition. We write Π = −θ ∧ (H ij πi ∧ ω(j) − Kω),

3A

n-form is totally decomposable if it is equal to the exterior product of n 1-forms.

54 and

´ CHAPTER 2. THE GEOMETRY OF POINCARE-CARTAN FORMS

Then taking those terms of the equation dΠ ≡ 0 (mod {I}) that are cubic in πi , we ﬁnd aij πi ∧ πj ∧ H kl πl = 0.

dθ ≡ aij πi ∧ πj

(mod {JΠ }).

Then the 2-form aij πi ∧πj has at least n ≥ 3 linearly independent 1-forms as divisors, which is impossible unless aij πi ∧ πj = 0. Therefore, dθ ≡ 0 (mod {JΠ }) (with only the hypothesis n ≥ 3). For the next step, it is useful to work with the 1-forms πi = H ij πj , def and write i dωi ≡ Pjkπj ∧ πk

(mod {JΠ }), 0 = ωi ∧ ωj ∧ Π

i i Pjk + Pkj = 0.

From the form of Π (2.22), we have

for any pair of indices 1 ≤ i, j ≤ n. Diﬀerentiating, we obtain 0 = = = (dωi ∧ ωj − ωi ∧ dωj ) ∧ Π j i (Pkl πk ∧ πl ∧ πj − Pklπk ∧ πl ∧ πi ) ∧ θ ∧ ω j j i i (δm Pkl − δm Pkl )πk ∧ πl ∧ πm ∧ θ ∧ ω.

It is now an exercise in linear algebra to show that if n ≥ 4, then this implies i i i Pjk = 0. The hypotheses are that Pjk = −Pkj and j j i j j i i i j i (δm Pkl − δm Pkl ) + (δk Plm − δk Plm ) + (δlj Pmk − δli Pmk ) = 0.

(2.23)

By contracting ﬁrst on jk and then on il, one ﬁnds that for n = 2 the contraction i i Pik vanishes. Contracting (2.23) only on jk and using Pik = 0, one ﬁnds that i for n = 3, all Pjk vanish. There do exist counterexamples in case n = 3, for which (2.23) implies only that dωi = P ij π(j), P ij = P ji. For example, if we ﬁx constants P ij = P ji also satisfying P ii = 0, then there is a unique simply connected, 7-dimensional Lie group G having a basis of leftinvariant 1-forms (ω i , θ, πi ) satisfying structure equations dωi = P ij π(j), dθ = −πi ∧ ωi , dπi = 0.

In this case, θ generates a homogeneous contact structure on G, and the form Π = −θ ∧ πi ∧ ω(i) def ´ 2.2. NEO-CLASSICAL POINCARE-CARTAN FORMS

55

is closed, giving an almost-classical form for which JΠ is not integrable. These counterexamples cannot arise from classical cases, however, and this suggests that we consider the following narrower class of Poincar´-Cartan forms. e Deﬁnition 2.4 An almost-classical Poincar´-Cartan form Π is neo-classical if e its associated Pfaﬃan system JΠ is integrable. So the preceding Proposition states that in case n ≥ 4, every almost-classical Poincar´-Cartan form is neo-classical, and we have narrowed the deﬁnition only e in case n = 3. The foliation corresponding to the integrable Pfaﬃan system JΠ is the bee ginning of the very rich geometry associated to a neo-classical Poincar´-Cartan form. Before investigating it further, we justify the study of this class of objects with the following. Proposition 2.3 Every neo-classical Poincar´-Cartan form Π on a contact e manifold (M, I) is locally equivalent to that arising from some classical variational problem. More precisely, given such (M, I, Π), there are local coordinates (xi , z, pi) on M with respect to which the contact system I is generated by e dz − pi dxi , and there is a Lagrangian of the form L(xi , z, pi)dx whose Poincar´Cartan form is Π. Note that we have already observed the converse, that those non-degenerate Poincar´-Cartan forms arising form classical variational problems (in case n ≥ 3) e are neo-classical. Proof. We ﬁx a coframing (θ, ω i , πi) as in the deﬁnition of an almost-classical form. Using the Frobenius theorem, we take independent functions (xi, z) on M so that JΠ = {ωi, θ} = {dxi, dz}. By relabelling if necessary, we may assume θ ∈ {dxi}, and we ﬁnd that there / are functions pi so that θ ∈ R · (dz − pi dxi).

The fact that θ ∧ (dθ)n = 0 implies that (xi , z, pi) are local coordinates on M . We now introduce a technical device that is often useful in the study of exterior diﬀerential systems. Let F p Ωq ⊂ Ωp+q (M ) be the collection of (p+q)-forms with at least p factors in JΠ ; this is well-deﬁned. With this notation, the fact that JΠ is integrable may be expressed as d(F p Ωq ) ⊂ F p Ωq+1 . e There is a version of the Poincar´ lemma that can be applied to each leaf of the foliation determined by JΠ, with smooth dependence on the leaves’ parameters; it says precisely that the complex F p Ω0 −→ F p Ω1 −→ · · · d d

56

´ CHAPTER 2. THE GEOMETRY OF POINCARE-CARTAN FORMS

is locally exact for each p. Now, any almost-classical form Π lies in F n Ω1; so not only is the closed form Π locally equal to dΛ for some Λ ∈ Ωn (M ), we can actually choose Λ to lie in F n Ω0. In other words, we can locally ﬁnd a Lagrangian Λ of the form Λ = L0 (x, z, p)dx + Li (x, z, p)dz ∧ dx(i) for some functions L0 , Li . This may be rewritten as Λ = (L0 + pi Li )dx + θ ∧ (Li dx(i)), and then the condition θ ∧ dΛ = 0 (recall that this was part of the construction of the Poincar´-Cartan form associated to any class in H n (Ω∗ /I)) gives the e relation ∂L Li (x, z, p) = (x, z, p). ∂pi This is exactly the condition for Λ to locally be a classical Lagrangian. Returning to the geometry associated to a neo-classical Poincar´-Cartan e form Π, we have found (or in case n = 3, postulated) an integrable Pfaﬃan system JΠ which is invariant under contact transformations preserving Π. Locally in M , the induced foliation has a smooth “leaf-space” Q of dimension n + 1, and there is a smooth submersion q : M → Q whose ﬁbers are n-dimensional integral manifolds of JΠ . On such a neighborhood, the foliation will be called simple, and as we are only going to consider the local geometry of Π in this section, we assume that the foliation is simple on all of M . We may restrict to smaller neighborhoods as needed in the following. To explore the geometry of the situation, we ask what the data (M 2n+1, I, Π) look like from the point of view of Qn+1. The ﬁrst observation is that we can locally identify M , as a contact manifold, with the standard contact manifold Gn (T Q), the Grassmannian bundle parameterizing n-dimensional subspaces of ﬁbers of T Q. This is easily seen in coordinates as follows. If, as in the preceding proof, we integrate JΠ as JΠ = {dz, dxi} for some local functions z, xi on M , then the same functions z, xi may be regarded as coordinates on Q. With the assumption that θ ∈ {dxi} (on M , again), / we must have dz − pidxi ∈ Γ(I) for some local functions pi on M , which by the non-degeneracy condition for I make (xi , z, pi) local coordinates on M . These pi can also thought of as local ﬁber coordinates for M → Q, and we can map M → Gn(T Q) by (xi, z, pi) → ((xi , z); {dz − pi dxi}⊥ ). The latter notation refers to a hyperplane in the tangent space of Q at (xi, z). Under this map, the standard contact system on Gn(T Q) evidently pulls back to I, so we have a local contact diﬀeomorphism commuting with projections to

´ 2.2. NEO-CLASSICAL POINCARE-CARTAN FORMS

57

Q. Every point transformation of Q prolongs to give a contact transformation of Gn(T Q), hence of M as well. Conversely, every contact transformation of M that preserves Π is the prolongation of a point transformation of Q, because the foliation by integral manifolds of JΠ deﬁning Q is associated to Π in a contact-invariant manner.4 In this sense, studying the geometry of a neoclassical Poincar´-Cartan form (in case n ≥ 3) under contact transformations is e locally no diﬀerent than studying the geometry of an equivalence class of classical non-degenerate ﬁrst-order scalar Lagrangians under point transformations. We have now interpreted (M, I) as a natural object in terms of Q, but our real interest lies in Π. What kind of geometry does Π deﬁne in terms of Q? We will answer this question in terms of the following notion. Deﬁnition 2.5 A Lagrangian potential for a neo-classical Poincar´-Cartan e form Π on M is an n-form Λ ∈ F nΩ0 (that is, Λ is semibasic for M → Q) such that dΛ = Π. We saw in the proof of Proposition 2.3 that locally a Lagrangian potential Λ exists. Such Λ are not unique, but are determined only up to addition of closed forms in F n Ω0. It will be important below to note that a closed form in F n Ω0 must actually be basic for M → Q; that is, it must be locally the pull-back of a (closed) n-form on Q. In particular, the diﬀerence between any two Lagrangian potentials for a give neo-classical form Π must be basic. Consider one such Lagrangian potential Λ, semibasic over Q. Then at each n ∗ point m ∈ M , one may regard Λm as an element of (Tq(m) Q), an n-form at the corresponding point of Q. This deﬁnes a map ν:M → n (T ∗ Q),

commuting with the natural projections to Q. Counting dimensions shows that n if ν is an immersion, then we actually obtain a hypersurface in (T ∗ Q); to be more precise, we have a smoothly varying ﬁeld of hypersurfaces in the vector bundle n (T ∗ Q) → Q. It is not hard to see that ν is an immersion if the Poincar´-Cartan form Π is non-degenerate, which is a standing hypothesis. We e n (T ∗ Q) over an can work backwards, as well: given a hypersurface M → (n + 1)-dimensional manifold Q, we may restrict to M the tautological n-form n (T ∗ Q) to obtain a form Λ ∈ Ωn (M ). Under mild technical hypotheses on on the hypersurface M , the form dΛ ∈ Ωn+1(M ) will be a neo-classical Poincar´e Cartan form. So we have associated to a Poincar´-Cartan form Π, and a choice of Lae n grangian potential Λ ∈ F nΩ0 , a ﬁeld of hypersurfaces in (T ∗ Q) → Q. However, we noted that Λ was not canonically deﬁned in terms of Π, so neither are these hypersurfaces. As we have seen, the ambiguity in Λ is that another ˜ admissible Λ may diﬀer from Λ by a form that is basic over Q. This means ˜ that Λ − Λ does not depend on the ﬁber-coordinate for M → Q, and therefore

4 This statement is only valid in case the foliation by integral manifolds of J is simple; in Π other cases, only a cumbersome local version of the statement holds.

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the two corresponding immersions ν, ν diﬀer in each ﬁber Mq (q ∈ Q) only by ˜ ∗ ∗ a translation in n(Tq Q). Consequently, we have in each n (Tq Q) a hypersurface well-deﬁned up to translation. A contact transformation of M which preserves Π will therefore carry the ﬁeld of hypersurfaces for a particular choice of Λ to a ﬁeld of hypersurfaces diﬀering by (a ﬁeld of) aﬃne transformations. To summarize, one can canonically associate to any neo-classical Poincar´-Cartan e n form (M, Π) a ﬁeld of hypersurfaces in the bundle (T ∗ Q) → Q, regarded as a bundle of aﬃne spaces. We expect the diﬀerential invariants of Π to include information about the geometry of each of these aﬃne hypersurfaces, and this will turn out to be the case.

2.3

Digression on Aﬃne Geometry of Hypersurfaces

Let An+1 denote (n+1)-dimensional aﬃne space, which is simply Rn+1 regarded as a homogeneous space of the group A(n + 1) of aﬃne transformations x → g · x + v, g ∈ GL(n + 1, R), v ∈ Rn+1 .

Let x : F → An+1 denote the principal GL(n + 1, R)-bundle of aﬃne frames; that is, F = {f = (x, (e0, . . . , en))}, where x ∈ An+1 is a point, and (e0 , . . . , en) is a basis for the tangent space Tx An+1. The action is given by a b b (x, (e0, . . . , en)) · (gb ) = (x, (ebg0 , . . . , eb gn)). def

(2.24)

These equations implicitly use a trivialization of T An+1 that commutes with aﬃne transformations. Diﬀerentiating, we obtain the structure equations for F: dωa = −ϕa ∧ ωb , b dϕa = −ϕa ∧ ϕc . b c b

For this section, we adopt the index ranges 0 ≤ a, b, c ≤ n, 1 ≤ i, j, k ≤ n, and always assume n ≥ 2. There is a basis of 1-forms ω a , ϕa on F deﬁned by decomposing the An+1 b valued 1-forms dx = ea · ωa , dea = eb · ϕb . a

(2.25)

Choosing a reference frame f0 ∈ F determines an identiﬁcation F ∼ A(n + 1), = and under this identiﬁcation the 1-forms ω a , ϕa on F correspond to a basis b of left-invariant 1-forms on the Lie group A(n + 1). The structure equations (2.25) on F then correspond to the usual Maurer-Cartan structure equations for left-invariant 1-forms on a Lie group.

2.3. DIGRESSION ON AFFINE GEOMETRY OF HYPERSURFACES

59

In this section, we will study the geometry of smooth hypersurfaces M n ⊂ A , to be called aﬃne hypersurfaces, using the method of moving frames; no previous knowledge of this method is assumed. In particular, we give constructions that associate to M geometric objects in a manner invariant under aﬃne transformations of the ambient An+1 . Among these objects are tensor ﬁelds Hij , U ij , and Tijk on M , called the aﬃne ﬁrst and second fundamental forms and the aﬃne cubic form of the hypersurface. We will classify those nondegenerate (to be deﬁned) hypersurfaces for which Tijk = 0 everywhere. This is of interest because the particular neo-classical Poincar´-Cartan forms that we e study later induce ﬁelds of aﬃne hypersurfaces of this type. n+1 Suppose given a smooth aﬃne hypersurface M ⊂ An+1 . We deﬁne the collection of 0-adapted frames along M by F0 (M ) = {(x, (e0, . . . , en )) ∈ F : x ∈ M, e1 , . . . , en span Tx M } ⊂ F. This is a principal subbundle of F|M whose structure group is5 G0 = def g0 =

a v

0 A

: a ∈ R∗ , A ∈ GL(n, R), v ∈ Rn .

(2.26)

Restricting forms on F to F0 (M ) (but supressing notation), we have ω0 = 0, ω1 ∧ · · · ∧ ωn = 0.

Diﬀerentiating the ﬁrst of these gives 0 = dω0 = −ϕ0 ∧ ωi , i and we apply the Cartan lemma to obtain ϕ0 = Hij ωj for some functions Hij = Hji. i One way to understand the meaning of these functions Hij , which constitute the ﬁrst fundamental form of M ⊂ An+1 , is as follows. At any given point of M ⊂ An+1 , one can ﬁnd an aﬃne frame and associated coordinates with respect to which M is locally a graph ¯ x0 = 1 Hij (x1, . . . , xn)xi xj 2 ¯ for some functions Hij . Restricted to the 0-adapted frame ﬁeld deﬁned by e0 (x) = ¯ ∂ ∂ ∂ 1 ¯ ¯ , ei (x) = (Hij (x)xj + 2 ∂i Hjk(x)xj xk ) 0 + ¯ , 0 ∂x ∂x ∂xi

¯ one ﬁnds that the values over 0 ∈ M of the functions Hij equal Hij (0). Loosely speaking, the functions Hij express the second derivatives of a deﬁning function for M .

5 Here and throughout, R∗ denotes the connected group of positive real numbers under multiplication.

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Returning to the general situation, we calculate as follows. We substitute 0 the expression ϕ0 = Hij ωj into the structure equation dϕi = −ϕ0 ∧ ϕb , collect i i b terms, and conclude 0 = (dHij + Hij ϕ0 − Hkj ϕk − Hikϕk ) ∧ ωj . 0 i j Using the Cartan lemma, we have k dHij = −Hij ϕ0 + Hkj ϕk + Hik ϕj + Tijk ωk 0 i

for some functions Tijk = Tikj = Tkji. This inﬁnitesimally describes how the functions Hij vary along the ﬁbers of F0(M ), on which ωj = 0. In particular, as a matrix-valued function H = (Hij ) on F0 (M ), it transforms by a linear representation of the structure group: H(f · g0) = (a−1) tAH(f)A, where g0 ∈ G0 is as in (2.26).6 Now we consider the quantity ∆(f) = det(Hij (f)), which vanishes at some point of F0 (M ) if and only if it vanishes on the entire ﬁber containing that point. We will say that M ⊂ An+1 is non-degenerate if ∆ = 0 everywhere on F0 (M ). Also note that the absolute signature of Hij is well-deﬁned at each point of M . It is easy to see that Hij is deﬁnite if and only if M ⊂ An+1 is convex. In what follows, we will assume that M is a non-degenerate hypersurface, but not necessarily that it is convex. It turns out that T = (Tijk ), which one would like to regard as a sort of covariant derivative of H = (Hij ), is not a tensor; that is, it does not transform by a linear representation along the ﬁbers of F0(M ) → M . We will exploit this below to reduce the principal bundle F0 (M ) → M to a subbundle of frames satisfying a higher-order adaptivity condition. Namely, F1(M ) ⊂ F0(M ) will consist of those frames where Tijk is traceless with respect to the non-degenerate symmetric bilinear form Hij , meaning H jk Tijk = 0, where (H ij ) is the matrix inverse of (Hij ). Geometrically, the reduction will amount to a canonical choice of line ﬁeld Re0 transverse to M , which we will think of as giving at each point of M a canonical aﬃne normal line. To justify this, we let (H ij ) denote the matrix inverse of (Hij ), and let Ci = H jk Tijk be the vector of traces of T with respect to H. We compute d(log ∆) = = = = ∆−1d∆ Tr(H −1 dH) H ij dHij −nϕ0 + 2ϕi + Ciωi . 0 i def def

6 As usual, our argument only proves this claim for g in the identity component of G , 0 0 but it may be checked directly for representative elements of each of the other components.

2.3. DIGRESSION ON AFFINE GEOMETRY OF HYPERSURFACES Now diﬀerentiate again and collect terms to ﬁnd 0 = (dCi − Cj ϕj − (n + 2)Hij ϕj ) ∧ ωi . i 0 Therefore, we have dCi ≡ Cj ϕj + (n + 2)Hij ϕj 0 i (mod {ω1 , . . . , ωn }),

61

(2.27)

(2.28)

which expresses how the traces Ci vary along the ﬁbers of F0 (M ) → M . In particular, if the matrix (Hij ) is non-singular, as we are assuming, then the action of the structure group on the values of the vector (Ci ) ∈ Rn is transitive; that is, every value in Rn is taken by (Ci) in each ﬁber. Therefore, the set of 0-adapted frames f ∈ F0 (M ) where each Ci (f) = 0 is a principal subbundle F1 (M ) ⊂ F0 (M ), whose structure group is the stabilizer of 0 ∈ Rn under the action. This stabilizer is G1 = def g1 =

a 0 0 A

: a ∈ R∗ , A ∈ GL(n, R) .

Comparing to the full action (2.24) of the aﬃne group A(n + 1) on F, we see that along each ﬁber of F1(M ), the direction Re0 is ﬁxed. Thus, we have uniquely chosen the direction of e0 at each point of M by the condition Ci = 0 for i = 1, . . . , n. A more concrete explanation of what we have done is seen by locally presenting our hypersurface in the form x0 = 1¯ 1¯ Hij (0)xixj + Tijk (x1, . . . , xn)xixj xk . 2 6

An aﬃne change of coordinates that will preserve this form is the addition of a multiple of x0 to each xi; the n choices that this entails can be uniquely made ¯ ¯ so that Tijk(0) is traceless with respect to Hij (0). Once such choices are ﬁxed, ∂ then so is the direction of ∂x0 , and this gives the canonical aﬃne normal line at x = 0. There is a remarkable interpretation of the aﬃne normal direction at a point where Hij is positive-deﬁnite (see [Bla67]). Consider the 1-parameter family of hyperplanes parallel to the tangent plane at the given point. For those planes suﬃciently near the tangent plane, the intersection with a ﬁxed neighborhood in the surface is a closed submanifold of dimension n − 2 in M , having an aﬃneinvariant center-of-mass. These centers-of-mass form a curve in aﬃne space, passing through the point of interest; this curve’s tangent line at that point is the aﬃne normal direction. We can see from (2.28) that on F1 (M ), where Tijk is traceless, the forms ϕj 0 are semibasic over M . It is less convenient to express these in terms of the basis ωi than to instead use ϕ0 = Hij ωj , assuming that M ⊂ An+1 is non-degenerate. i On F1 (M ) we write ϕj = U jk ϕ0 . 0 k Now (2.27), restricted to F1 (M ) where Ci = 0, implies that U ij = U ji.

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The reader may carry out computations similar to those above to show that the (U ij ) and (Tijk ) are tensors; that is, they transform along the ﬁbers of F1 (M ) by a linear representation of G1. For example, if Tijk = 0 at some point of F1(M ), then Tijk = 0 everywhere along the same ﬁber of F1 (M ) → M . Furthermore, the transformation law for U ij is such that if U ij = λH ij at some point, for some λ, then the same is true—with possibly varying λ—everywhere on the same ﬁber. We can now give some additional interpretations of the simplest cases of the aﬃne second fundamental form U ij and the aﬃne cubic form Tijk. The following theorem is the main purpose of this digression. Theorem 2.3 (1) If U ij = λH ij everywhere on F1 (M )—that is, if the second fundamental form is a scalar multiple of the ﬁrst fundamental form—then either λ = 0 everywhere or λ = 0 everywhere. In the ﬁrst case, the aﬃne normal lines of M are all parallel, and in the second case, the aﬃne normal lines of M are all concurrent. (2) If Tijk = 0 everywhere on F1(M ), then U ij = λH ij everywhere. In this case, if λ = 0, then M is a paraboloid, while if λ = 0, then M is a non-degenerate quadric. Proof. Suppose ﬁrst that U ij = λH ij on F1 (M ) for some function λ. This is same as writing ϕj = λH jl ϕ0 = λωj . l 0 We diﬀerentiate this equation (substituting itself), and obtain (dλ − λϕ0 ) ∧ ωj = 0 for each j. 0 Under the standing assumption n > 1, this means that dλ = λϕ0 . 0 So assuming that M is connected, we have the ﬁrst statement of (1). We will describe the geometric consequences of each of the two possibilities. First, suppose that λ = 0, so that U ij = 0, and then ϕj = 0 throughout 0 F1 (M ). Then the deﬁnition of our original basis of 1-forms gives de0 = ea ϕa = e0 ϕ0 , 0 0 meaning that the direction in An+1 of e0 is ﬁxed throughout F1 (M ), or equivalently, all of the aﬃne normals of M are parallel. Next, suppose λ = 0, and assume for simplicity that λ < 0. The diﬀerential equation dλ = λϕ0 implies that we can restrict to the principal subbundle 0 F2 (M ) where λ = −1. This amounts to a choice of a particular vector ﬁeld e0 along the aﬃne normal line ﬁeld already deﬁned. Note that on F2 (M ), we have ϕj = −ωj = −H jk ϕ0 , k 0 ϕ0 = 0. 0 (2.29)

As a result, the structure equations dx = ei ωi and de0 = ea ϕa = −ei ωi imply 0 that d(x + e0 ) = 0,

2.3. DIGRESSION ON AFFINE GEOMETRY OF HYPERSURFACES

63

so that x + e0 is a constant element of An+1. In particular, all of the aﬃne normal lines of M pass through this point. This completes the proof of (1). Now assume that Tijk = 0 identically; this will be satisﬁed by each member of the ﬁelds of aﬃne hypersurfaces associated to certain neo-classical Poincar´e Cartan forms of interest. Our ﬁrst claim is that U ij = λH ij for some function λ on F1 (M ). To see this, note that our hypothesis means dHij = −Hij ϕ0 + Hkj ϕk + Hik ϕk . 0 i j We diﬀerentiate this, using the structure equations in the simpliﬁed form that deﬁned the reduction to F1 (M ), and obtain 0 = −Hkj ϕk ∧ ϕ0 − Hik ϕk ∧ ϕ0 . 0 i 0 j If we use Hij to raise and lower indices and deﬁne Uij = Hik Hlj U kl , then the preceding equation may be written as 0 = −(Ujl Hik + Uil Hjk)ωl ∧ ωk . The coeﬃcients of this vanishing 2-form then satisfy 0 = Ujl Hik + Uil Hjk − Ujk Hil − Uik Hjl ; we multiply by H ik (and sum over i, k) to conclude Ujl = This proves that U ij = λH ij ,

1 with λ = n H kl Ukl . We now return to the possibilities λ = 0, λ = 0 under the stronger hypothesis Tijk = 0. In the ﬁrst case, note that with the condition ϕj = 0 on F1(M ), we have 0 that the Pfaﬃan system generated by ϕ0 and ϕi (for 1 ≤ i, j ≤ n) is integrable. 0 j ˜ ˜ Let M be any leaf of this system. Restricted to M , we have

1 ik (H Uik )Hjl . n

dHij = 0, so that the functions Hij are constants. Furthermore, the linearly independent ˜ 1-forms ωi on M are each closed, so that (at least locally, or else on a simply ˜ connected cover) there are coordinates ui on M with ωi = dui. Substituting all of this into the structure equations, we have:

64

´ CHAPTER 2. THE GEOMETRY OF POINCARE-CARTAN FORMS ˜ • de0 = 0, so that e0 is a constant element of An+1 on M; • dei = e0 ϕ0 = e0 Hij ωj = d(e0 Hij uj ), so that i ei = ei + e0 Hij uj ¯ for some constant ei ∈ An+1 ; ¯

1 • dx = ei ωi = (¯i + e0 Hij uj )dui = d(ui ei + 2 e0 Hij uiuj ), so that e ¯

1 x = x + uiei + Hij uiuj e0 ¯ ¯ 2 for some constant x ∈ An+1. ¯

˜ The conclusion is that as the coordinates ui vary on M, the An+1 -valued func˜ tion x on M traces out a paraboloid, with vertex at x and axis along the direction ¯ of e0 . Turning to the case λ = 0, recall that under the assumption λ < 0, we can reduce to a subbundle F2 (M ) ⊂ F1 (M ) on which λ = −1. We use the diﬀerential equation dHij = Hik ϕk + Hkj ϕk j i ¯ to reduce again to a subbundle F3 (M ) ⊂ F2 (M ) on which Hij = Hij is some i constant matrix. On F3 (M ), the forms ϕj satisfy linear algebraic relations ¯ ¯ 0 = Hik ϕk + Hkj ϕk . j i Our assumption λ = −1 allows us to combine these with the relations (2.29) by deﬁning 0 ϕ0 1 0 j , H= , Φ= ¯ 0 H ϕi ϕi 0 j and then HΦ + t ΦH = 0. In other words, the matrix-valued 1-form Φ on F3 (M ) takes values in the Lie algebra of the stabilizer of the bilinear form H. For instance, if our hypersurface M is convex, so that (Hij ) is deﬁnite everywhere, then we could have ¯ chosen Hij = δij , and then Φ would take values in the Lie algebra so(n + 1, R). Whatever the signature of Hij , let the stabilizer of H be denoted by O(H) ⊂ GL(n + 1, R), with Lie algebra so(H). Then the structure equation dΦ + Φ ∧ Φ = 0 implies that there is locally (alternatively, on a simply connected cover) a map g : F3 (M ) → O(H) such that Φ = g−1 dg.

2.4. THE EQUIVALENCE PROBLEM FOR N ≥ 3 Using the structure equations dea = eb ϕb , this implies a d(ea · (g−1 )a ) = 0, b so that ea = e b ga ¯ b

65

for some ﬁxed aﬃne frame (¯b ). In particular, the An+1 -valued function e0 e on F3 (M ) takes as its values precisely the points of a level surface of a nondegenerate quadratic form, deﬁned by H. Recalling from the ﬁrst part of the proof that x + e0 is constant on An+1, this means that the hypersurface M , thought of as the image of the map x : F3 (M ) → An+1 , is a constant translate of a non-degenerate quadric hypersurface. The signature of the quadric is (p, q), where (p − 1, q) is the signature of the ﬁrst fundamental form (Hij ). The case λ > 0 instead of λ < 0 is quite similar, but M is a quadric of signature (p, q) when (Hij ) has signature (p, q − 1).

2.4

The Equivalence Problem for n ≥ 3

Π = −θ ∧ (H ij πi ∧ ω(j) − Kω). (2.30)

We now consider a contact manifold (M, I) with a closed, almost-classical form

We will shortly specialize to the case in which Π is neo-classical. The coframes in which Π takes the form (2.30), for some functions H ij and K, constitute a Gstructure as described in Lemma 2.1. The purpose of this section is to describe a canonical reduction of this G-structure to one carrying a pseudo-connection satisfying structure equations of a prescribed form, as summarized in (2.47– 2.48), at least in case the matrix (H ij ) is either positive- or negative-deﬁnite everywhere. This application of the equivalence method involves no techniques beyond those introduced in §2.1, but some of the linear-algebraic computations are more involved. We begin by reﬁning our initial G-structure as follows. Lemma 2.2 Let (M, I) be a contact manifold with almost-classical form Π. (1) There exist local coframings (θ, ω i , πi) on M such that Π has the form (2.30) and such that dθ ≡ −πi ∧ ωi (mod {I}). (2) Local coframings as in (1) are the sections of a G0-structure B0 → M , where G0 is the group of matrices of the form (in blocks of size 1, n, n) a g0 = C i Di 0 Ai j Sik Ak j 0 , 0 a(A−1 )j i

A ∈ GL(n, R), Sij = Sji.

(2.31)

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¯ ¯ ¯ ¯¯ and if Π = −θ ∧ (H ij πi ∧ ω(j) − Kω) = −θ ∧ (H ij πi ∧ ω(j) − K ω) are the ¯ expressions for Π with respect to these coframings, then H K = = ¯ a2(det A)A−1 H tA−1 , ¯ − T r(HS)). ¯ a(det A)(K (2.32) (2.33)

(3) If two local coframings as in (1) are related as ¯ θ θ −1 ω i = g0 · ω j , ¯ πj ¯ πi

Proof. (1) First observe that in any coframing, we may write dθ ≡ aij πi ∧ πj + bj πj ∧ ωi + cij ωi ∧ ωj i (mod {I}).

We will deal with each of the three coeﬃcient matrices (aij ), (bj ), (cij ) to obtain i the desired condition dθ ≡ − πi ∧ ωi . • The proof of Proposition 2.2 showed for n ≥ 3 that 0 ≡ dθ ≡ aij πi ∧ πj (mod {JΠ }),

which implies aij πi ∧ πj = 0. This followed from calculating 0 = dΠ modulo {I}. • From the fact that θ is a contact form, we have 0 = θ ∧ (dθ)n = ±det(bj )θ ∧ ω ∧ π, i so that (bj ) is an invertible matrix. Therefore, we may apply the matrix i j j −(bi )−1 to the 1-forms πj to obtain a new basis in which we have bj = −δi , i so that dθ ≡ −πi ∧ ωi + cij ωi ∧ ωj (mod {I}). Note that this coframe change is of the type admitted by Lemma 2.1, preserving the form (2.30). • Finally, we can replace πi by πi + cij ωj to have the desired dθ ≡ −πi ∧ ωi . This coframe change also preserves the form (2.30). (2) We already know that any matrix as in Lemma 2.1 will preserve the form (2.30). We write the action of such a matrix as ¯ θ ωi ¯ πi ¯ = = = aθ i C i θ + Aj ω j j Di θ + Sik Ak ωj + Bi πj . j

2.4. THE EQUIVALENCE PROBLEM FOR n ≥ 3

67

It is easily veriﬁed that the condition dθ ≡ −πi ∧ ωi implies the analogous ¯ condition dθ ≡ −¯i ∧ ωi if and only if π ¯ j j Bi Ai = aδk , k Sjk = Skj .

This is what we wanted to prove. ¯ ¯ ¯ (3) These formulae are seen by substituting the formulae for (θ, ωi, πi) into the equation for the two expressions for Π, and comparing terms. One uses the following fact from linear algebra: if ω i ≡ Ai ω j ¯ j then ω(j) ≡ (det A)(A−1 )i ω(i) ¯ j (mod {I}); that is, the coeﬃcients of ω(j) in terms of ω(i) are the cofactors of the coeﬃcient ¯ matrix of ωi in terms of ωj . ¯ We can see from (2.32) that the matrix H = (H ij ) transforms under coframe changes like a bilinear form, up to scaling, and in particular that its absolute signature is ﬁxed at each point of M . To proceed, we have to assume that this signature is constant throughout M . In particular, we shall from now on assume that H is positive or negative deﬁnite everywhere, and refer to almost-classical forms Π with this property as deﬁnite. Cases of diﬀerent constant signature are of interest, but can be easily reconstructed by the reader in analogy with the deﬁnite case examined below. Once we assume that the matrix-valued function H on B0 is deﬁnite, the following is an easy consequence of the preceding lemma. Lemma 2.3 Given a deﬁnite, almost-classical Poincar´-Cartan form Π on a e contact manifold (M, I), there are 0-adapted local coframings (θ, ω i , πi) for which Π = −θ ∧ (δ ij πi ∧ ω(j)), and these form a G1 -structure B1 ⊂ B0 → M , where G1 is the group of matrices g1 of the form (2.31) with det A > 0, a(det A) 2 A ∈ O(n, R),

1

(mod {I}),

Sii = 0.

¯ ¯ This follows from imposing the conditions H = H = In , K = K = 0 in the previous lemma. Unfortunately, it is diﬃcult to give a general expression in coordinates for such a 1-adapted coframing in the classical case, because such an expression requires that we normalize the Hessian matrix (Lpi pj ). In practice, however, such a coframing is usually easy to compute.

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where A = (Ai ) ∈ SO(n, R), r > 0, Sij = Sji, Sii = 0. Also, now that j the orthogonal group has appeared, some of the representations occuring in the sequel are isomorphic to their duals, for which it may be unuseful and sometimes confusing to maintain the usual summation convention, in which one only contracts a pair of indices in which one index is raised and the other lowered. Therefore, we will now sum any index occuring twice in a single term, regardless of its positions. We now assume that we have a deﬁnite, neo-classical Poincar´-Cartan form e Π with associated G1-structure B1 → M , and we begin searching for diﬀerential invariants. There are local pseudo-connection 1-forms ρ, γ i , δi, αi , σij deﬁned so j that equations of the following form hold: (n − 2)ρ 0 0 θ θ Θ i ∧ ω j + Ωi , γi −2ρδj + αi 0 d ωi = − j j j πi πj Πi δi σij nρδi − αi where θ, ωi , πi are the tautological 1-forms on B1 , the torsion 2-forms Θ, Ωi, Πi are semibasic for B1 → M , and the psuedo-connection 1-forms satisfy j αi + αi = 0, j

It is convenient for later purposes to use a diﬀerent parameterization of our group G1. Namely, an arbitrary element will be written as ±rn−2 0 0 , r−2Ai 0 (2.34) g1 = C i j Di Sik Ak ±rn (A−1 )j j i

σij = σji,

σii = 0.

or equivalently, Θ = −πi ∧ ωi . Second, our assumption that Π is neo-classical means that the Pfaﬃan system JΠ = {θ, ωi } is integrable (even up on B1 ). In the structure equation i dωi = −γ i ∧ θ − (−2ρδj + αi ) ∧ ωj + Ωi , j

These last conditions mean that the psuedo-connection matrix takes values in the Lie algebra g1 ⊂ gl(2n + 1, R) of G1 . The psuedo-connection 1-forms are not uniquely determined, and our next step is to exploit this indeterminacy to try to absorb components of the torsion. First, we know that dθ ≡ −πi ∧ ωi (mod {I}). The diﬀerence between Θ = dθ + (n − 2)ρ ∧ θ and −πi ∧ ωi is therefore a semibasic multiple of θ, which can be absorbed by a semibasic change in ρ. We can therefore simply assume that dθ = −(n − 2)ρ ∧ θ − πi ∧ ωi ,

(2.35)

this means that Ωi ≡ 0 (mod {JΠ}). Also, Ωi is semibasic over M , so we can write 1 i Ωi ≡ T ijkπj ∧ ωk + Pjkωj ∧ ωk 2 (mod {I}). (2.36)

2.4. THE EQUIVALENCE PROBLEM FOR n ≥ 3

69

Now, adding semibasic 1-forms to γ i allows us to preserve the equation (2.35) while also making (2.36) an equality, and not merely a congruence. A little linear algebra shows that there is a unique linear combination of the ω i that can j i be added to αi , preserving αi + αi = 0, to absorb the term 1 Pjkωj ∧ ωk . This j j 2 leaves us only with Ωi = T ijkπj ∧ ωk .

i As in the elimination of the Pjk , we can add a combination of the πi to αi to j arrange T ijk = T kji.

To investigate the third torsion term Πi , we use an alternate derivation of the equation for dπi. Namely, we diﬀerentiate the equation dθ = −(n − 2)ρ ∧ θ − πi ∧ ωi , and take the result only modulo {I} to avoid the unknown quantity dρ. This eventually yields 0 ≡ −(Πk − T ijkπi ∧ πj ) ∧ ωk (mod {I}).

As before, multiples of θ may be absorbed by redeﬁning δi , so that we can assume this congruence is an equality. Reasoning similar to that which proves the Cartan lemma gives Πk − T ijkπi ∧ πj = νkl ∧ ωl for some semibasic 1-forms νkl = νlk . Now, most of these forms νkl can be subtracted from the psuedo-connection forms σkl , simplifying the torsion; but the condition σii = 0 prevents us from completely absorbing them. Instead, the trace remains, and we have Πk = δkl ν ∧ ωl + T ijkπi ∧ πj . We can learn more about ν using the integrability condition dΠ = 0, taken modulo terms quadratic in the πi : 0 = dΠ ≡ nθ ∧ ν ∧ ω. A consequence is that ν ≡ 0 (mod {θ, ω i }); in other words, ν has no πi-terms, and may be written (using again a change in δi ) as ν= Then replacing σij by σij + n n+2 (δik Nj 2 + δjk Ni − n δij Nk )ωk

Ni ω i .

yields new psuedo-connection forms, for which the third torsion term is simply Πk = T ijkπi ∧ πj .

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This completes the major step of absorbing torsion by altering the pseudoconnection. Before proceeding to the next major step, we look for linear-algebraic conditions on the torsion which may simplify later calculations. In particular, we made only very coarse use of dΠ = 0 above. Now we compute more carefully 0 = dΠ = −θ ∧ (2T ijk + δ ij T lkl )πk ∧ πj ∧ ω(i), so we must have 2T ijk + δ ij T lkl = 2T ikj + δ ik T ljl . (2.37)

The next major step is a reduction of our G1-structure. We will examine def the variation of the functions T j = T iji along ﬁbers of B1 → M , and observe j that the zero-locus {T = 0} deﬁnes a G2-structure for a certain codimension-n subgroup G2 ⊂ G1 . As usual, the variation of T j will be described inﬁnitesimally. To study dT j without knowledge of the traceless part of dT ijk , we exploit the exterior algebra, writing d(θ ∧ ω1 ∧ · · · ∧ ωn ) = ((n + 2)ρ + T k πk ) ∧ θ ∧ ω. (2.38)

We will diﬀerentiate this for information about dT k , but in doing so we will need information about dρ as well. Fortunately, this is available by diﬀerentiating the ﬁrst structure equation dθ = −(n − 2)ρ ∧ θ − πk ∧ ωk , yielding (n − 2)dρ ≡ γ k ∧ πk (mod {θ, ωi }). (mod {θ, ωi , πi}).

Now we return to diﬀerentiating (2.38) and eventually ﬁnd k dT k ≡ − n+2 γ k + (nδj ρ − αk )T j j n−2

This means that along ﬁbers of B1 → M , the vector-valued function T (u) = (T j (u)), u ∈ B1 , is orthogonally rotated (inﬁnitesimally, by αk ), scaled (by ρ), j and translated (by γ i ). In fact, for g1 ∈ G1 as in (2.34), T (u · g1) = ±r2A−1 (rn−2T (u) − Now the set def n+2 n−2 C).

B2 = {u ∈ B1 : T (u) = 0} ⊂ B1 is a G2-subbundle of B1 → M , where G2 consists of matrices as in (2.34) with T i = 0. On the submanifold B2 ⊂ B1 , we have from (2.37) the symmetry T ijk = T kji = T ikj .

2.4. THE EQUIVALENCE PROBLEM FOR n ≥ 3 As a consequence, the torsion Πk restricts to Πk = T ijkπj ∧ πk = 0.

71

The previous structure equations continue to hold, but the forms γ i |B2 should not be regarded as part of the psuedo-connection, as they are now semibasic over M . We therefore write (n − 2)ρ 0 0 θ θ Θ i ∧ ω j + Ωi , 0 −2ρδj + αi 0 d ωi = − j j j πi πj Πi δi σij nρδi − αi (2.39) where still αi + αj = 0, σij = σji, σii = 0, j i and now Θ = −πi ∧ ωi , i Ωi = −(Sj ωj + U ij πj ) ∧ θ + T ijk πj ∧ ωk , Πi = 0. T ijk = T kji = T ikj , T iik = 0. (2.41) (2.40)

i Here we have denoted γ i ≡ Sj ωj + U ij πj (mod {I}). Also, we still have

(2.42)

Notice that we can alter αi and ρ to assume that j j i Sj = S i , i Si = 0,

(2.43)

where we also have to add combinations of ω i to δi to preserve Πi = 0. In fact, these assumptions uniquely determine αi and ρ, although δi and σij still admit j some ambiguity. Equations (2.39–2.43) summarize the results of the equivalence method carried out to this point. We have uncovered the primary diﬀerential invariants of i a deﬁnite neo-classical Poincar´-Cartan form: they are the functions T ijk , Sj e and U ij . Their properties are central in what follows. For example, note that the rank-n Pfaﬃan system {ω i } on B2 is invariant under the action of the structure group G2, and therefore it is the pullback of a Pfaﬃan system (also to be denoted {ω i }) down on M . Testing its integrability, we ﬁnd dωi ≡ −U ij πj ∧ θ (mod {ωi }). (2.44)

We will see shortly that the matrix-valued function (U ij ) varies along the ﬁbers of B2 → M by a linear representation of G2, so that it is plausible to ask about those Poincar´-Cartan forms for which U ij = 0; (2.44) shows that this e

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is equivalent to the integrability of {ω i }. In this case, in addition to the local ﬁbration M → Q whose ﬁbers are leaves of JΠ , we have Qn+1 → N n, where N is the locally-deﬁned n-dimensional “leaf space” for {ω i }. Coordinates on N —equivalently, functions on M whose diﬀerentials lie in {ω i}—may be thought of as “preferred independent variables” for the contact-equivalence class of our Euler-Lagrange equation, canonical in the sense that every symmetry of M preserving the Poincar´-Cartan form preserves the ﬁbration M → N and e therefore acts on N . Note that even if an (M, Π) satisfying U ij = 0 came to us from a classical Lagrangian with independent variables (xi ), we need not have {ωi } = {dxi}. This is not to say that the case U ij = 0 is uninteresting. In the next section, we will see an important family of examples from Riemannian geometry with i U ij = λδj . To obtain preliminary information about U ij in a manner that will i not require much knowledge of Sj or T ijk , we start with the equation d(ω1 ∧ · · · ∧ ωn ) = 2nρ ∧ (ω1 ∧ · · · ∧ ωn ) + U ij θ ∧ πj ∧ ω(i) . (2.45)

We will diﬀerentiate again, but we need more reﬁned information about dρ; this is obtained from 0 = d2θ = −((n − 2)dρ + δi ∧ ωi + πi ∧ γ i ) ∧ θ. i Keep in mind that γ i = Sj ωj + U ij πj (mod {I}) on this reduced bundle. We can now write

(n − 2)dρ + δi ∧ ωi + πi ∧ γ i = τ ∧ θ 0≡

2n n−2 (−πi

(2.46)

for some unknown 1-form τ . Returning to the derivative of (2.45), we ﬁnd ∧ U ij πj ) ∧ ω + U ij πi ∧ πj ∧ ω U ij = U ji . We will need an even more reﬁned version of the equation (2.46) for dρ. In the preceding paragraph, we substituted that equation into the equation for 0 ≡ d2(ω1 ∧ · · · ∧ ωn) (mod {I}). Now, we substitute it instead into 0 ≡ ≡ d2(ω1 ∧ · · · ∧ ωn ) (mod {π1, . . . , πn})

2n n−2 τ

(mod {I}).

This implies that U ij πi ∧ πj ∧ ω = 0, so that we have

− U ij σij ∧ θ ∧ ω1 ∧ · · · ∧ ωn.

2n i This means that n−2 τ − U ij σij lies in {θ, ωi , πi}. Recall that also γ i = Sj ωj + U ij πj + V i θ for some functions V i , and we can put this back into (2.46) to ﬁnally obtain i (n − 2)dρ = −δi ∧ ωi − Sj πi ∧ ωj + n−2 2n

U ij σij ∧ θ + (si ωi − ti πi) ∧ θ,

2.4. THE EQUIVALENCE PROBLEM FOR n ≥ 3

73

for some functions si , ti . Furthermore, we can replace each δi by δi − si θ, preserving previous equations, to assume that si = 0. This gives i (n − 2)dρ = −δi ∧ ωi − Sj πi ∧ ωj + n−2 2n

U ij σij ∧ θ − ti πi ∧ θ,

which will be used in later sections. The last formulae that we will need are those for the transformation rules for i T ijk , U ij , Sj along ﬁbers of B → M . These are obtained by computations quite similar to those carried out above, and we only state the results here, which are: • dT ijk ≡ nρT ijk − αi T ljk − αj T ilk − αk T ijl , l l l i • dU ij ≡ 2nρU ij − αl U lj − αj U il , l 1 i i i i l • dSj ≡ (n − 2)ρSj − αl Sj + Sli αl + 1 (U il σlj + U jl σli ) − n δj U kl σkl + T ijk δk , j 2

all modulo {θ, ωi , πi}. Notice in particular that T ijk and U ij transform by a combination of rescaling and a standard representation of SO(n). However, i (Sj ) is only a tensor when the tensors (T ijk ) and (U ij ) both vanish. We will consider this situation in the next chapter. An interpretation of the ﬁrst two transformation rules is that the objects T = T ijk (πi ◦ πj ◦ πk ) ⊗ |π1 ∧ · · · ∧ πn |− n , U ij πi ◦ πj

2

U =

are invariant modulo JΠ = {θ, ωi } under ﬂows along ﬁbers over M ; that is, when restricted to a ﬁber of B2 → Q, they actually descend to well-deﬁned objects on the smaller ﬁber of M → Q. The restriction to ﬁbers suggests our next result, which nicely relates the diﬀerential invariants of the Poincar´-Cartan form with e the aﬃne geometry of hypersurfaces discussed in the preceding section. Theorem 2.4 The functions T ijk and U ij are coeﬃcients of the aﬃne cubic form and aﬃne second fundamental form for the ﬁberwise aﬃne hypersurfaces n in (T ∗ Q) induced by a semibasic Lagrangian potential Λ of Π. Proving this is a matter of identifying the bundles where the two sets of invariants are deﬁned, and unwinding the deﬁnitions. In the next section, we will brieﬂy build on the preceding results in the case where T ijk = 0 and U ij = 0, showing that these conditions roughly characterize those deﬁnite neo-classical Poincar´-Cartan forms appearing in the problem of e ﬁnding prescribed mean curvature hypersurfaces, in Riemannian or Lorentzian manifolds. In the next chapter, we will extensively consider the case T ijk = 0, U ij = 0, which includes remarkable Poincar´-Cartan forms arising in conformal e geometry. About the case for which T ijk = 0, nothing is known. For reference, we summarize the results of the equivalence method that will be used below. Associated to a deﬁnite, neo-classical Poincar´-Cartan form Π e

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on a contact manifold (M, I) is a G-structure B → M , where n−2 ±r 0 G= Di 0 r−2 Ai j Sik Ak j

0 (Ai ) ∈ SO(n, R), r > 0, j : 0 . Sij = Sji, Sii = 0 ±rn (A−1 )j i (2.47) 0 , 0 j j nρδi − αi

B → M supports a pseudo-connection (not uniquely determined) (n − 2)ρ 0 ϕ = − δi 0 i −2ρδj + αi j σij

with αi + αj = 0, σij = σji, σii = 0, such that in the structure equation j i θ θ d ωi = −ϕ ∧ ωj + τ, πj πi

the torsion is of the form with

−πi ∧ ωi i τ = −(Sj ωj + U ij πj ) ∧ θ + T ijk πj ∧ ωk , 0

j i i T ijk = T jik = T kji, T iik = 0; U ij = U ji ; Sj = Si , Si = 0.

In terms of any section of B → M , the Poincar´-Cartan form is e Π = −θ ∧ πi ∧ ω(i). One further structure equation is i (n − 2)dρ = −δi ∧ ωi − Sj πi ∧ ωj + n−2 2n

U ij σij ∧ θ − ti πi ∧ θ.

(2.48)

2.5

The Prescribed Mean Curvature System

In this section, we will give an application of the part of the equivalence method completed so far. We will show that a deﬁnite, neo-classical Poincar´-Cartan e form with T ijk = 0, and satisfying an additional open condition speciﬁed below, is locally equivalent to that which arises in the problem of ﬁnding in a given Riemannian manifold a hypersurface whose mean curvature coincides with a prescribed background function. This conclusion is presented as Theorem 2.5.

2.5. THE PRESCRIBED MEAN CURVATURE SYSTEM

75

To obtain this result, we continue applying the equivalence method where we left oﬀ in the preceding section, and take up the case T ijk = 0. From our calculations in aﬃne hypersurface geometry, we know that this implies that U ij = λδ ij , for some function λ on the principal bundle B → M ; alternatively, this can be shown by computations continuing those of the preceding section. We will show that under the hypothesis λ < 0, the Poincar´-Cartan form Π is locally e equivalent to that occuring in a prescribed mean curvature system. We have in general on B that dU ij ≡ 2nρU ij − αi U kj − αj U ik k k (mod {θ, ωi , πi}).

Then for our U ij = λδ ij , the function λ scales positively along ﬁbers of B → M , so under our assumption λ < 0 we may make a reduction to B1 = {u ∈ B : λ(u) = −1} ⊂ B; this deﬁnes a subbundle of B of codimension 1, on which ρ is semibasic over M .7 In particular, on B1 we may write ρ=− H θ + E i ω i + F i πi 2n

for some functions H, Ei , F i. The reason for the normalization of the θcoeﬃcient will appear shortly. We claim that F i = 0. To see this, start from the equation (2.48) for dρ, which on B1 reads i (n − 2)dρ = −δi ∧ ωi − ti πi ∧ θ − Sj πi ∧ ωj .

Then, as we have done so often, we compute d2ω, where ω = ω1 ∧ · · · ∧ ωn and i dωi = 2ρ ∧ ωi − αi ∧ ωj + πi ∧ θ − Sj ωj ∧ θ. j

We ﬁnd and the next step is simpliﬁed by knowing dΠ = 0: 0 = = = d2 ω 2n dρ ∧ ω − 2nρ ∧ dω −

2n n−2

dω = 2nρ ∧ ω − θ ∧ πi ∧ ω(i) = 2nρ ∧ ω + Π,

(ti πi ∧ θ) ∧ ω ti + E i πi ∧ ω . n−2

+2n(Ej ωj + F j πj ) ∧ θ ∧ πi ∧ ω(i) = 2nθ ∧ F j πi ∧ πj ∧ ω(i) +

7 In this section, we will denote by B , B , etc., successive reductions of the G-structure 1 2 B → M which was constructed in the preceding section. These are not the same as the bundles of the same name used in constructing B, which are no longer needed.

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´ CHAPTER 2. THE GEOMETRY OF POINCARE-CARTAN FORMS

This gives our claim F i = 0, as well as (n − 2)E i = −ti . For our next reduction, we will show that we can deﬁne a principal subbundle B2 = {u ∈ B1 : E i (u) = 0} ⊂ B1 , having structure group deﬁned by the condition Di = 0, r = 1 in (2.47). This 2 follows by computing modulo {θ, ωi , πi}: (n − 2)dρ ≡ −δi ∧ ωi , and also

1 dρ ≡ − 2n dH ∧ θ + dEi ∧ ωi − Ej αj ∧ ωi . i

Comparing these, we obtain dEi − Ej αj + i This implies that dEi − Ej αj + i

1 n−2 δi 1 n−2 δi

∧ ωi −

1 2n dH

∧ θ ≡ 0.

≡ 0 (mod {θ, ωi , πi}),

justifying the described reduction to B2 → M , on which ρ and δi are semibasic. Finally, a third reduction is made possible by the general equation i i k i 1 i dSj ≡ (n − 2)ρSj − αi Sj + Sk αk + 1 (U il σlj + U jl σli ) − n δj U kl σkl, k j 2

modulo {θ, ωi , πi}. On B2 , where in particular λ = −1 and ρ is semibasic, we have i k i dSj ≡ −αi Sj + Sk αk − σij (mod {θ, ωi , πi}). k j i This means that the torsion matrix (Sj ) can undergo translation by an arbitrary traceless symmetric matrix along the ﬁbers of B2 → M , so the locus i B3 = {u ∈ B2 : Sj (u) = 0} ⊂ B2

is a subbundle, whose structure group is SO(n, R) with Lie algebra represented by matrices of the form 0 0 0 i 0 , a2 = 0 α j αi + αj = 0. j i j 0 0 −αi This is all the reduction that we shall need. On B3 , we have equations H ρ = − 2n θ, dθ = −πi ∧ ωi (because ρ ∧ θ = 0 on B3 ), (n − 2)dρ = −δi ∧ ωi (because ti = −(n − 2)E i = 0 on B3 ).

2.5. THE PRESCRIBED MEAN CURVATURE SYSTEM

77

The δi appearing the third equation are semibasic over M , and the three equations together imply that dH ≡ 0 (mod {θ, ω i }). This last observation is quite important. Recall the integrable Pfaﬃan system JΠ = {θ, ωi }, assumed to have a well-deﬁned leaf-space Qn+1 with submersion M → Q. The last equation shows that H is locally constant along the ﬁbers of M → Q, and may therefore be thought of as a function on Q. Now, considering the two structure equations dθ = −πi ∧ ωi , dωi = 2ρ ∧ ωi − αi ∧ ωj + πi ∧ θ, j it is tempting to deﬁne πi = πi + ˜ and rewrite them as d θ ωi =− 0 −˜i π πj ˜ αi j ∧ θ ωj .

H i nω ,

Observe that this looks exactly like the structure equation characterizing the Levi-Civita connection of a Riemannian metric. We justify and use this as follows. Consider the quadratic form on B3 θ2 + (ωi )2 .

An easy computation shows that for any vertical vector ﬁeld v ∈ Ker(π∗ ) for π : B3 → Q, (ωi )2 = 0. Lv θ 2 + This means that our quadratic form is the pullback of a quadratic form on Q, which deﬁnes there a Riemannian metric ds2 . There is locally a bundle isomorphism over Q B3 → F(Q, ds2)

from B3 , which was constructed from the neo-classical Poincar´-Cartan form e Π, to the orthonormal frame bundle of this Riemannian metric. Under this isomorphism, the Q-semibasic forms θ, ω i correspond to the tautological semibasic forms on F(Q, ds2), while the matrix 0 −˜i π πj ˜ αi j

corresponds to the Levi-Civita connection matrix. The contact manifold M , as a quotient of B3 , may be then identiﬁed with the manifold of tangent hyperplanes to Q; and the Poincar´-Cartan form is e Π = = −θ ∧ (πi ∧ ω(i) ) −θ ∧ (˜i ∧ ω(i) − Hω). π

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We recognize this as exactly the Poincar´-Cartan form for the prescribed mean e curvature H = H(q) system, in an arbitrary (n + 1)-dimensional Riemannian manifold. The following is what we have shown. Theorem 2.5 A deﬁnite neo-classical Poincar´-Cartan form (M, Π) whose dife i ferential invariants satisfy T ijk = 0 and U ij = λδj with λ < 0 is locally equivalent to the Poincar´-Cartan of the prescribed mean curvature system on some e Riemannian manifold (Qn+1 , ds2). We will consider these Poincar´-Cartan forms further in §4.1, when we discuss e the formula for the second variation of a Lagrangian functional FΛ. At that time, we will also see an interpretation of the partial reduction B2 ⊃ B3 in terms of the Riemannian geometry. Note that it is easy, given (M, Π) as in the proposition, to determine the prescribed function H(q) by carrying out the reductions described above, and to determine the Riemann curvature of the ambient (n + 1)-manifold in terms of the connection 1-forms πi, αi . The ˜ j Euclidean minimal surface system discussed in §1.4 is the case H = 0, Rijkl = 0. The fact that such an (M, Π) canonically determines (Q, ds2) implies the following.8 Corollary 2.1 The symmetry group of (M, Π) is equal to the group of isometries of (Q, ds2) that preserve the function H. A consequence of this is the fact, claimed in §1.4, that all symmetries of the minimal surface Poincar´-Cartan form—and hence, all classical conservation e laws for the Euler-Lagrange equation—are induced by Euclidean motions. i Finally, in case T ijk = 0 and U ij = λδj with λ > 0 instead of λ < 0, one can carry out similar reductions, eventually producing on the quotient space Qn+1 a Lorentz metric ds2 = −θ2 + (ωi )2 ; the Poincar´-Cartan form is then e equivalent to that for prescribed mean curvature of space-like hypersurfaces.

8 As usual, this assumes that the foliation associated to J Π is simple; otherwise, only a local reformulation holds.

Chapter 3

Conformally Invariant Systems

Among non-linear Euler-Lagrange equations on Rn , the largest symmetry group that seems to occur is the (n+1)(n+2) -dimensional conformal group. This consists 2 of diﬀeomorphisms of the n-sphere that preserve its standard conformal structure, represented by the Euclidean metric under stereographic projection to Rn . These maximally symmetric equations have a number of special properties, including of course an abundance of classical conservation laws as predicted by e Noether’s theorem. This chapter concerns the geometry of the Poincar´-Cartan forms associated to these equations, and that of the corresponding conservation laws. We will begin by presenting background material on conformal geometry. This includes a discussion of the ﬂat conformal structure on the n-sphere and its symmetry group, a construction of a canonical parallelized principal bundle over a manifold with conformal structure, and the deﬁnition of the conformal Laplacian, a second-order diﬀerential operator associated to a conformal structure. This material will provide the framework for understanding the geometry of non-linear Poisson equations, in particular the maximally symmetric nonlinear example n+2 ∆u = Cu n−2 , C = 0. After developing the geometric context for this equation, we will continue the equivalence problem for Poincar´-Cartan forms, pursuing the branch in which e these Euler-Lagrange equations occur. We then turn to conservation laws for these conformally invariant equations. The elaborate geometric structure allows several approaches to computing these conservation laws, and we will carry out one of them in detail. The analogous development for non-linear wave equations involves conformal structures with Lorentz signature, and the conserved quantities for maximally symmetric EulerLagrange equations in this case give rise to integral identities that have been very useful in analysis. 79

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3.1

Background Material on Conformal Geometry

In this section, we discuss some of the less widely known aspects of conformal geometry. In the ﬁrst subsection, we deﬁne a ﬂat model for conformal geometry which is characterized by its large symmetry group, and we give structure equations in terms of the Maurer-Cartan form of this group. In the second subsection, we give Cartan’s solution to the local equivalence problem for general conformal structures on manifolds. This consists of an algorithm by which one associates to any conformal structure (N, [ds2 ]) a parallelized principal bundle P → N having structure equations of a speciﬁc algebraic form. In the third subsection, we introduce a second-order diﬀerential operator ∆, called the conformal Laplacian, which is associated to any conformal structure and which appears in the Euler-Lagrange equations of conformal geometry that we study in the remainder of the chapter. The fundamental deﬁnition is the following. Deﬁnition 3.1 A conformal inner-product at a point p ∈ N is an equivalence class of positive inner-products on Tp N , where two such inner-products are equivalent if one is a positive scalar multiple of the other. A conformal structure, or conformal metric, on N consists of a conformal inner-product at each point p ∈ N , varying smoothly in an obvious sense. Note that this emphasizes the pointwise data of the conformal structure, unlike the usual deﬁnition of a conformal structure as an equivalence class of global Riemannian metrics. An easy topological argument shows that these notions are equivalent.

3.1.1

Flat Conformal Space

We start with oriented Lorentz space Ln+2 , with coordinates x = (x0 , . . . , xn+1), orientation dx0 ∧ · · · ∧ dxn+1 > 0, and inner-product x, y = −(x0yn+1 + xn+1y0 ) + xi y i . i Throughout this section, we use the index ranges 0 ≤ a, b ≤ n + 1 and 1 ≤ i, j ≤ n. A non-zero vector x ∈ Ln+2 is null if x, x = 0. A null vector x is positive if x0 > 0 or xn+1 > 0; this designation is often called a “time-orientation” for Ln+2 . The symmetries of Lorentz space are the linear transformations of Ln+2 preserving the inner-product, the orientation, and the time-orientation, and they constitute a connected Lie group SO o (n + 1, 1). We denote the space of positive null vectors by Q = {x ∈ Ln+2 : x, x = 0, and x0 > 0 or xn+1 > 0},

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81

which is one half of the familiar light-cone, with axis {xi = x0 − xn+1 = 0}. We now deﬁne ﬂat conformal space R to be the space of null lines in Ln+2. As a manifold, R is a non-singular quadric in the projective space P(Ln+2), which is preserved by the natural action of the symmetry group SO o (n + 1, 1) of Ln+2. We will describe the ﬂat conformal structure on R below, in terms of the Maurer-Cartan form of the group. Note that the obvious map Q → R, which we will write as x → [x], gives a principal bundle with structure group R∗ . In the literature, R is usually deﬁned as Rn with a point added at inﬁnity to form a topological sphere. To make this identiﬁcation, note that for x, y ∈ Q, we have x, y ≤ 0, with equality if and only if [x] = [y]. We then claim that is diﬀeomorphic to both Rn and R\[y]; this is easily proved for y = (0, . . . , 0, 1), for instance, where the map Rn → Hy is given by (x1, . . . , xn) → (1, x1, . . . , xn, 1 ||x||2). 2 (3.1) The classical description of the conformal structure on R is obtained by transporting the Euclidean metric on Rn to Hy , and noting that for y = y with [y] = [y ], this gives unequal but conformally equivalent metrics on R\[y]. The fact that SOo (n + 1, 1) acts transitively on R then implies that for [x] = [y] the conformal structures obtained on R\[x] and R\[y] are the same. A Lorentz frame is a positively oriented basis f = (e0 , . . . , en+1) of Ln+2, in which e0 and en+1 positive null vectors, and for which the inner-product is (in blocks of size 1, n, 1, like most matrices in this section) 0 0 −1 e a , eb = 0 I n 0 . −1 0 0 Hy = {x ∈ Q : x, y = −1} def We let P denote the set of all Lorentz frames. There is a standard simply transitive right-action of SO o (n + 1, 1) on P , by which we can identify the two spaces in a way that depends on a choice of basepoint in P ; this gives P the structure of a smooth manifold. Because we have used the right-action, the pullback to P of any left-invariant 1-form on SO o (n + 1, 1) is independent of this choice of basepoint. These pullbacks can be intrinsically described on P as follows. We view each ea as a map P → Ln+2, and we deﬁne 1-forms ρ, ω i , βj , αi on P by decomposing the Ln+2 -valued 1-forms dea in terms of the bases j {eb }: de0 = 2e0 ρ + ei ωi , dej = e0 βj + ei αi + en+1 ωj , j den+1 = ei βi − 2en+1ρ. Equivalently, d e0 ej en+1 = e0 ei en+1 2ρ βj ω i αi j 0 ωj 0 βi . −2ρ

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All of the local geometry of R that is invariant under SO o (n + 1, 1) can be expressed in terms of these Maurer-Cartan forms. In particular, the ﬁbers of the map πR : P → R given by πR : (e0 , . . . , en+1) → [e0] are the integral manifolds of the integrable Pfaﬃan system IR = {ω1 , . . . , ωn }.

These forms satisfy αi + αj = 0 but are otherwise linearly independent, and j i they span the left-invariant 1-forms on SO o (n + 1, 1) under the preceding identiﬁcation with P . Decomposing the exterior derivatives of these equations gives the Maurer-Cartan equations, expressed in matrix form as 2ρ βj 0 2ρ βj 0 2ρ βk 0 k i βi + ω i α k βk = 0. (3.2) d ω i αi βi ∧ ω k α j j j k j 0 ω −2ρ 0 ω −2ρ 0 ω −2ρ

This ﬁbration has the structure of a principal bundle, whose structure group consists of matrices in SO o (n + 1, 1) of the form 2 r bj 1 r−2 b2 j 2 r−2 ai bk , (3.3) g = 0 ai j k 0 0 r−2 where r > 0, ai aj = δ ij . Now, the symmetric diﬀerential form on P given by k k q= (ωi )2

is semibasic for πR : P → R, and a Lie derivative computation using the structure equations (3.2) gives, for any vertical vector ﬁeld v ∈ Ker (πR )∗ , Lv q = 4(v ρ)q. This implies that there is a unique conformal structure [ds2] on R whose rep∗ resentative metrics pull back under πR to multiples of q. By construction, this conformal structure is invariant under the action of SO o (n + 1, 1), and one can verify that it gives the same structure as the classical construction described above. In §3.1.2, we will follow Cartan in showing that associated to any conformal structure (N, [ds2]) is a principal bundle P → N with 1-forms αi = −αj , ρ, j i ωi , and βj , satisfying structure equations like (3.2) but with generally non-zero curvature terms on the right-hand side. Before doing this, however, we point out a few more structures in the ﬂat model which will have useful generalizations. These correspond to Pfaﬃan systems IR = {ωi }, IQ = {ωi , ρ}, IM = {ωi , ρ, βj }, IP0 = {ωi , ρ, αi }, j

3.1. BACKGROUND MATERIAL ON CONFORMAL GEOMETRY

83

each of which is integrable, and in fact has a global quotient; that is, there are manifolds R, Q, M , and P0, and surjective submersions from P to each of these, whose leaves are the integral manifolds of IR , IQ , IM , and IP0 , respectively: P P0 Q ↓ R. We have already seen that the leaves of the system IR are ﬁbers of the map πR : P → R. Similarly, the leaves of IQ are ﬁbers of the map πQ : P → Q given by πQ : (e0 , . . . , en+1) → e0 . To understand the leaves of IM , we let M be the set of ordered pairs (e, e ) of positive null vectors satisfying e, e = −1. We then have a surjective submersion πM : P → M deﬁned by πM : (e0 , . . . , en+1) → (e0 , en+1), and the ﬁbers of this map are the leaves of the Pfaﬃan system IM . Note that the 1-form ρ and its exterior derivative are semibasic for πM : P → M , and this means that there is a 1-form (also called ρ) on M which pulls back to ρ ∈ Ω1(P ). In fact, the equation for dρ in (3.2) shows that on P , ρ ∧ (dρ)n = 0, so the same is true on M . Therefore, ρ deﬁnes an SO o (n+1, 1)-invariant contact structure on M . The reader can verify that M has the structure of an R∗ -bundle over the space G(1,1)(Ln+2 ) parameterizing those oriented 2-planes in Ln+2 on which the Lorentz metric has signature (1, 1). In this context, 2ρ ∈ Ω1(M ) can be interpreted as a connection 1-form. Finally, to understand the leaves of IP0 , we proceed as follows. Deﬁne a conformal frame for (R, [ds2]) at a point [x] ∈ R to be a positive basis (v1, . . . , vn) for T[x] R normalizing the conformal inner-product as ds2 (vi , vj ) = λδij , for some λ ∈ R∗ not depending on i, j. The set of conformal frames for (R, [ds2]) is the total space of a principal bundle P0 → R, and there is a surjective submersion P → P0. This last is induced by the maps ei : P → T R associating to a Lorentz frame f = (e0 , . . . , en+1 ) an obvious tangent vector ei to R at [e0]. The reader can verify that the ﬁbers of the map P → P0 are the leaves of the Pfaﬃan system IP0 . Each of the surjective submersions P → R, P → Q, P → M , P → P0 has the structure of a principal bundle, deﬁned as a quotient of P by a subgroup of M

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SOo (n + 1, 1). Additionally, the spaces P , R, Q, M , and P0 are homogeneous spaces of SOo (n + 1, 1), induced by the standard left-action on Ln+2 . We conclude with a brief description of the geometry of SO 0 (n + 1, 1) acting on ﬂat conformal space R. This will be useful later in understanding the space of conservation laws of conformally invariant Euler-Lagrange equations. There are four main types of motions. • The translations are deﬁned as motions of R induced by left-multiplication by matrices of the form 1 0 0 wi In 0 . (3.4) ||w||2 wj 1 2 In the standard coordinates on R\{∞} described in (3.1), this is simply translation by the vector (w i ).

• The rotations are deﬁned as motions of R induced by matrices of the form 1 0 0 0 ai 0 , j 0 0 1

where (ai ) ∈ SO(n, R). In the standard coordinates, this is the usual j rotation action of the matrix (ai ). j

• The dilations are deﬁned as motions of R induced by matrices of the form 2 r 0 0 0 I 0 . 0 0 r−2

In the standard coordinates, this is dilation about the origin by a factor of r−2.

• The inversions are deﬁned as motions of R induced by matrices of the form 2 1 bj ||b|| 2 0 I bi . 0 0 1

Now, J itself is not in SO o (n + 1, 1), but it still acts in an obvious way on R; in standard coordinates, it gives the familiar inversion in the sphere of √ radius 2. So the inversions can be thought of as conjugates of translation by the standard sphere-inversion, or alternatively, as “translations with the origin ﬁxed”.

Note that these are exactly conjugates of the translation matrices (3.4) by the matrix 0 0 1 J = 0 I 0 . 1 0 0

3.1. BACKGROUND MATERIAL ON CONFORMAL GEOMETRY

85

These four subgroups generate SO o (n + 1, 1). Although the conformal isometry group of R has more that this one component, the others do not appear in the Lie algebra, so they do not play a role in calculating conservation laws for conformally invariant Euler-Lagrange equations.

3.1.2

The Conformal Equivalence Problem

We will now apply the method of equivalence to conformal structures of dimension n ≥ 3. This will involve some of the ideas used in the equivalence problem for deﬁnite Poincar´-Cartan forms discussed in the preceding chapter, but we e we will also encounter the new concept of prolongation. This is the step that one takes when the usual process of absorbing and normalizing the torsion in a G-structure does not uniquely determine a pseudo-connection. Let (N, [ds2]) be an oriented conformal manifold of dimension n ≥ 3, and let P0 → N be the bundle of 0th -order oriented conformal coframes ω = (ω1 , . . . , ωn), which by deﬁnition satisfy [ds2] = (ωi )2 , ω1 ∧ · · · ∧ ωn > 0.

This is a principal bundle with structure group CO(n, R) = {A ∈ GL+ (n, R) : A tA = λI, for some λ ∈ R∗ }, having Lie algebra co(n, R) = = {a ∈ gl(n, R) : a + ta = λI, for some λ ∈ R} i {(−2rδj + ai ) : ai + aj = 0, ai , r ∈ R}. j j j i

We will describe a principal bundle P → P0 , called the prolongation of P0 → N , whose sections correspond to torsion-free pseudo-connections in P0 → N , and construct a canonical parallelism of P which deﬁnes a Cartan connection in P → N . In case (N, [ds2 ]) is isomorphic to an open subset of ﬂat conformal space, this will correspond to the restriction of the Lorentz frame bundle P → R to that open subset, with parallelism given by the Maurer-Cartan forms of SOo (n + 1, 1) ∼ P . = Recall that a pseudo-connection in P0 → N is a co(n, R)-valued 1-form i ϕ = (ϕi ) = (−2ρδj + αi ), j j

αi + αj = 0, j i

whose restriction to each tangent space of a ﬁber of P0 → N gives the canonical identiﬁcation with co(n, R) induced by the group action. As discussed previously (see §2.1), this last requirement means that ϕ satisﬁes a structure equation

1 i dωi = −ϕi ∧ ωj + 2 Tjk ωj ∧ ωk , j i i Tjk + Tkj = 0,

(3.5)

where ωi are the components of the tautological Rn-valued 1-form on P0, and 1 i T ωj ∧ωk is the semibasic Rn -valued torsion 2-form. We also noted previously 2 jk

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that a psuedo-connection ϕ is a genuine connection if and only if it is Adequivariant for the action of CO(n, R) on P0 , meaning that R∗ ϕ = Adg−1 (ϕ), g where Rg : P0 → P0 is the right-action of g ∈ CO(n, R) and Adg−1 is the adjoint action on co(n, R), where ϕ takes its values. However, completing this equivalence problem requires us to consider the more general notion of a pseudoconnection. Although the parallelism that we eventually construct is sometimes called the “conformal connection”, there is no canonical way (that is, no way that is invariant under all conformal automorphisms) to associate to a conformal structure a linear connection in the usual sense. What we seek instead is a psuedo-connection ϕi for which the torsion vanj i ishes, Tjk = 0. We know from the fundamental lemma of Riemannian geometry, which guarantees a unique torsion-free connection in the orthonormal frame bundle of any Riemannian manifold, that whatever structure equation (3.5) we have with some initial pseudo-connection, we can alter the pseudo-connectioni forms αi = −αj to arrange that Tjk = 0. Speciﬁcally, we replace i j j i k 1 αi ; αi + 2 (Tjk − Tik − Tij )ωk . j j i So we can assume that Tjk = 0, and we have simply i dωi = −ϕi ∧ ωj = −(−2ρδj + αi ) ∧ ωj , j j i with αj + αj = 0. However, in contrast to Riemannian geometry, this condition i on the torsion does not uniquely determine the psuedo-connection forms ρ, αi . j If we write down an undetermined semibasic change in psuedo-connection

ρ ; ρ + tk ω k , αi ; α i + t i ω k , j j jk

ti + tj = 0, jk ik

then the condition that the new pseudo-connection be torsion-free is that i (2δj tk − ti )ωj ∧ ωk = 0. jk

This boils down eventually to the condition j i ti = 2(δk ti − δk tj ). jk

Therefore, given one torsion-free pseudo-connection ϕi in P0 → N , the most j general is obtained by adding j i i 2(δj tk − δk ti + δk tj )ωk ,

(3.6)

where t = (tk ) ∈ Rn is arbitrary. This fact is needed for the next step of the equivalence method, which consists of prolonging our CO(n, R)-structure. We

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87

now digress to explain this general concept, starting with the abstract machinery underlying the preceding calculation. We begin by amplifying the discussion of normalizing torsion in §2.1. Associated to any linear Lie algebra g ⊂ gl(n, R) is an exact sequence of g-modules 0 → g(1) → g ⊗ (Rn)∗ → Rn ⊗ δ 2

(Rn)∗ → H 0,1(g) → 0.

(3.7)

Here, the map δ is the restriction to the subspace g ⊗ (Rn)∗ ⊂ (Rn ⊗ (Rn )∗ ) ⊗ (Rn)∗ of the surjective skew-symmetrization map Rn ⊗ (Rn)∗ ⊗ (Rn )∗ → Rn ⊗

2

(Rn)∗ .

The space g(1) is the kernel of this restriction, and is called the prolongation of g; the cokernel H 0,1(g), a Spencer cohomology group of g, was encountered in §2.1. Note that g(1) and H 0,1(g) depend on the representation g → gl(n, R), and not just on the abstract Lie algebra g. Recall from §2.1 that the intrinsic torsion of a G-structure vanishes if and only if there exist (locally) torsion-free pseudo-connections in that G-structure. This is a situation in which further canonical reduction of the structure group is not generally possible. In particular, this will always occur for G-structures with H 0,1(g) = 0. In this situation, the torsion-free pseudo-connection is unique if and only if g(1) = 0. For example, when g = so(n, R), both g(1) = 0 and H 0,1(g) = 0, which accounts for the existence and uniqueness of a torsion-free, metric-preserving connection on any Riemannian manifold. In this favorable situation, we have essentially completed the method of equivalence, because the tautological form and the unique torsion-free pseudo-connection constitute a canonical, global coframing for the total space of our G-structure. Equivalences of G-structures correspond to isomorphisms of the associated coframings, and there is a systematic procedure for determining when two parallelized manifolds are locally isomorphic. However, one frequently works with a structure group for which g(1) = 0. The observation that allows us to proceed in this case is that any pseudoconnection ϕ in a G-structure P → N deﬁnes a particular type of g⊕Rn -valued coframing ϕ ⊕ ω : T P → g ⊕ Rn (3.8)

of the total space P . Our previous discussion implies that given some torsion-free pseudo-connection ϕ, any change ϕ lying in g(1) ⊂ g ⊗ (Rn )∗ yields a pseudoconnection ϕ + ϕ which is also torsion-free. This means that the coframings of P as in (3.8), with ϕ torsion-free, are exactly the sections of a g(1)-structure P (1) → P , where we regard g(1) as an abelian Lie group. This P (1) → P

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is by deﬁnition the prolongation of the G-structure P → N , and diﬀerential invariants of the former are also diﬀerential invariants of the latter.1 The next natural step in studying P → N is therefore to start over with P (1) → P , by choosing a pseudo-connection, absorbing and normalizing its torsion, and so forth. In practice, completely starting over would be wasteful. The total space P (1) supports tautological forms ϕ and ω, valued in g and Rn , respectively; and the equation dω + ϕ ∧ ω = 0 satisﬁed by any particular torsion-free psuedoconnection ϕ on P still holds on P (1) with ϕ replaced by a tautological form. We can therefore diﬀerentiate this equation and try to extract results about the algebraic form of dϕ. These results can be interpreted as statements about the intrinsic torsion of P (1) → P . Only then do we return to the usual normalization process. We will now illustrate this, returning to our situation in the conformal structure equivalence problem. We have shown the existence of torsion-free pseudo-connections ϕi in the j CO(n, R)-structure P0 → N , so the intrinsic torsion of P0 → N vanishes.2 We also have that such ϕi are unique modulo addition of a semibasic co(n, R)j valued 1-form linearly depending on an arbitrary choice of (tk ) ∈ Rn. Therefore co(n, R)(1) ∼ Rn , and the inclusion co(n, R)(1) → co(n, R) ⊗ (Rn)∗ is described = def by (3.6). As explained above, we have an Rn -structure P = (P0 )(1) → P0, whose sections correspond to torsion-free pseudo-connections in P0 → N . Any choice of the latter trivializes P → P0, and then (tk ) ∈ Rn is a ﬁber coordinate. We now search for structure equations on P , with the goal of identifying a canonical Rn-valued pseudo-connection form for P → P0. The ﬁrst structure equation is still dωi = −ϕi ∧ ωj , j

where i ϕi = −2δj ρ + αi , j j

αi + αj = 0, j i

i and αj , ρ are tautological forms on P . Diﬀerentiating this gives i i (dϕj + ϕk ∧ ϕk ) ∧ ωj = 0, j

(3.9)

so Taking the trace of this equation of matrix 2-forms shows that dρ ≡ 0, so guided by the ﬂat model (3.2), we write dρ = − 1 βi ∧ ωi 2 (3.10) dϕi + ϕi ∧ ϕk ≡ 0 j k j (mod {ω1 , . . . , ωn}).

1 Situations with non-unique torsion-free pseudo-connections are not the only ones that call for prolongation; sometimes one ﬁnds intrinsic torsion lying in the ﬁxed set of H 0,1 (g), and essentially the same process being described here must be used. However, we will not face such a situation. 2 In fact, what we proved is that δ is surjective for g = co(n, R), so H 0,1 (co(n, R)) = 0.

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89

for some 1-forms βi which are not uniquely determined. We will recognize these below as pseudo-connection forms in P → P0, to be uniquely determined by conditions on the torsion which we will uncover shortly. Substituting (3.10) back into (3.9), we have (dαi + αi ∧ αk − βj ∧ ωi + βi ∧ ωj ) ∧ ωj = 0, j k j and we set i Aj = dαi + αi ∧ αk − βj ∧ ωi + βi ∧ ωj . j k j

Note that Ai + Aj = 0. We can write j i

i Ai = ψjk ∧ ωk j i i for some 1-forms ψjk = ψkj , in terms of which the condition Ai + Aj = 0 is j i j i (ψjk + ψik ) ∧ ωk = 0,

which implies

Now computing modulo {ω 1 , . . . , ωn } as in Riemannian geometry, we have j j i k k i i ψjk ≡ −ψik ≡ −ψki ≡ ψji ≡ ψij ≡ −ψkj ≡ −ψjk , i so ψjk ≡ 0. We can now write i 1 Ai = ψjk ∧ ωk = 2 Ai ωk ∧ ωl , jkl j i and forget about the ψjk , as our real interest is in dαi . We can assume that j j Ai + Ai = 0, and we necessarily have Ai + Aikl = 0. Substituting once jlk jkl jkl more into Ai ∧ ωj = 0, we ﬁnd that j

j i ψjk + ψik ≡ 0

(mod {ω1 , . . . , ωn}).

(3.11)

Ai + Ai + Ai = 0. jkl klj ljk In summary, we have dαi + αi ∧ αk − βj ∧ ωi + βi ∧ ωj = 1 Ai ωk ∧ ωl , j k j 2 jkl where Ai has the symmetries of the Riemann curvature tensor. jkl In particular, we need only n new 1-forms βi to express the derivatives of dρ, i dαj . The βi are pseudo-connection forms for the prolonged co(n, R)(1)-bundle P → P0 , chosen to eliminate torsion in the equation for dρ, while the functions Ai constitute the torsion in the equations for dαi . Some of this torsion will j jkl now be absorbed in the usual manner, by making a uniquely determined choice of βi . Notice that the equation (3.10) for dρ is preserved exactly under substitutions of the form βi ; βi + sij ωj , sij = sji.

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CHAPTER 3. CONFORMALLY INVARIANT SYSTEMS

This substitution will induce a change j i i Ajkl ; Ai + (−δli sjk + δlj sik + δk sjl − δk sil ). jkl

Now, we know from the symmetries of the Riemann curvature tensor that Al = A l , jkl kjl and on this contraction (the “Ricci” component) our substitution will induce the change Al ; Al − (n − 2)sjk − δjk sll . jkl jkl As we are assuming n ≥ 3, there is a unique choice of sij which yields l Ajkl = 0.

It is not diﬃcult to compute that the appropriate sij is given by sij = In summary, On P , there is a unique coframing ω i , ρ, βj , αi = −αj , where ωi are j i the tautological forms over N , and such that the following structure equations are satisﬁed: dωi dαi j = = i (2δj ρ − αi ) ∧ ωj , j k −αi ∧ αj + βj ∧ ωi − βi ∧ ωj + 1 Ai ωk ∧ ωl , k 2 jkl 1 n−2

Al − ijl

1 δ Al 2n−2 ij kkl

.

dρ =

− 1 βi ∧ ω i , 2

with Al = 0. jkl

We now seek structure equations for dβj . We start by diﬀerentiating the 1 simplest equation in which βj appears, which is dρ = − 2 βj ∧ ωj , and this gives (dβj + 2ρ ∧ βj + βk ∧ αk ) ∧ ωj = 0. j We write dβj + 2ρ ∧ βj + βk ∧ αk = Bjk ∧ ωk j (3.12)

for some 1-forms Bjk = Bkj . Because the equation for dρ did not determine βj uniquely, we cannot expect to use it to completely determine expressions for dβj ; we need to diﬀerentiate the equations for dαi , substituting (3.12). This j gives j i (DAi − Bik δlj + Bjk δli + Bil δk − Bjl δk ) ∧ ωk ∧ ωl = 0. jkl

3.1. BACKGROUND MATERIAL ON CONFORMAL GEOMETRY Here we have deﬁned for convenience the “covariant derivative” m DAi = dAi + 4ρAi + αi Ajkl − Ai αm − Ai αm − Ai αm . jkl jkl jkl m mkl j jml k jkm l

91

(3.13)

Now we can write j i DAi − Bik δlj + Bjk δli + Bil δk − Bjl δk ≡ 0 (mod {ω1, . . . , ωn }), jkl

and contracting on il gives Bjk ≡ 0 (mod {ω1 , . . . , ωn}). This allows us to write simply dβi + 2ρ ∧ βi + βj ∧ αj = 1 Bijk ωj ∧ ωk , i 2 for some functions Bijk = −Bikj . Returning to the equation 0 = d2ρ = − 1 d(βj ∧ ωj ) 2 now yields the cyclic symmetry Bijk + Bjki + Bkij = 0. We now have complete structure equations, which can be summarized in the matrix form suggested by the ﬂat model (3.2): 2ρ βj φ = ω i αi j 0 ωj def

0 βi , −2ρ

0 Bj Φ = dφ + φ ∧ φ = 0 Ai j 0 0 def

0 Bi , 0

(3.14)

where Ai j = k l 1 i 2 Ajkl ω ∧ ω , Ai + A j = jkl ikl i Ai + Aklj + jkl k l 1 2 Bjkl ω ∧ ω ,

Ai + Ai = 0, jkl jlk Ai = Al = 0, ljk jkl

Bj

=

Bjkl + Bjlk = Bjkl + Bklj + Bljk = 0.

Furthermore, the action of Rn on P → P0 and that of CO(n, R) on P0 → N may be combined, to realize P → N as a principal bundle having structure group G ⊂ SOo (n + 1, 1) consisting of matrices of the form (3.3). The matrix 1-form φ in (3.14) deﬁnes an so(n+1, 1)-valued parallelism on P , under which the tangent spaces of ﬁbers of P → N are carried to the Lie algebra g ⊂ so(n+1, 1) of G, and φ is equivariant with respect to the adjoint action of G on so(n+1, 1). The data of (P → N, φ) is often called a Cartan connection modelled on g → so(n + 1, 1).

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CHAPTER 3. CONFORMALLY INVARIANT SYSTEMS

The components of this matrix equation yield linear-algebraic consequences about the derivatives of Ai , Bjkl . First, one ﬁnds that jkl i 1 2 DAjkl

We conclude this discussion by describing some properties of the functions Ai , Bjkl on P . Diﬀerentiating the deﬁnition of Φ (3.14) yields the Bianchi jkl identity dΦ = Φ ∧ ϕ − ϕ ∧ Φ.

∧ ωk ∧ ωl = 1 Bikl ωj ∧ ωk ∧ ωl − 1 Bjkl ωi ∧ ωk ∧ ωl . 2 2

(3.15)

In particular, referring to the deﬁnition (3.13), this shows that the collection of functions (Ai ) vary along the ﬁbers of P → N by a linear representation jkl of the structure group G. In other words, they correspond to a section of an associated vector bundle over N . Speciﬁcally, we can see that the expression A = def 1 i A (ωi 4 jkl

Detailed information can be obtained from this equation, but note immediately the fact that DAi ≡ 0 (mod {ωi}). jkl

∧ ωj ⊗ ωk ∧ ωl ) ⊗ (ω1 ∧ · · · ∧ ωn )−2/n

2

on P is invariant under the group action, so A deﬁnes a section of Sym2 ( T ∗ N ) ⊗ D−2/n ,

where D is the density line bundle for the conformal structure, to be deﬁned shortly. This section is called the Weyl tensor of the conformal structure. Something diﬀerent happens with Bjkl . Namely, the Bianchi identity for dBjkl yields DBjkl def =

≡

m dBjkl + 6ρBjkl − Bmkl αm − Bjml αk − Bjkmαm l j

−βi Ai jkl

(mod {ωi}).

In particular, the collection (Bjkl ) transforms by a representation of G if and only if the Weyl tensor A = 0. In case n = 3, the symmetry identities of Ai imply that A = 0 automatically; there is no Weyl tensor in 3-dimensional jkl conformal geometry. In this case, (Bjkl ) deﬁnes a section of the vector bun2 ∗ dle T ∗ N ⊗ T N , which actually lies in a subbundle, consisting of traceless elements of the kernel of T ∗N ⊗

2

T ∗N →

3

T ∗ N → 0.

This section is called the Cotten tensor of the 3-dimensional conformal structure. If the Cotten tensor vanishes, then the conformal structure is locally equivalent to the ﬂat conformal structure on the 3-sphere. In case n > 3, from (3.15) one can show that the functions Bjkl can be expressed as linear combinations of the covariant derivatives of Ai . In particular, jkl if the Weyl tensor A vanishes, then so do all of the Bjkl , and the conformal structure of N is locally equivalent to the ﬂat conformal structure on the n-sphere.

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93

3.1.3

The Conformal Laplacian

To every conformal manifold (N n , [ds2]) is canonically associated a linear diﬀerential operator ∆, called the conformal Laplacian. In this section, we deﬁne this operator and discuss its elementary properties. One subtlety is that ∆ does not act on functions, but on sections of a certain density line bundle, and our ﬁrst task is to deﬁne this. We will use the parallelized principal bundle π : P → N canonically associated to [ds2 ] as in the preceding discussion. To begin, note that any n-form σ on N pulls back to P to give a closed n-form π∗ σ = u ω1 ∧ · · · ∧ ωn ∈ Ωn (P ),

where u is a function on P whose values on a ﬁber π −1 (x) give the coeﬃcient of n ∗ σx ∈ (Tx N ) with respect to various conformal coframes at x ∈ N . Among all n-forms on P of the form u ω 1 ∧ · · · ∧ ωn , those that are locally pullbacks from N are characterized by the property of being closed. Using the structure equations, we ﬁnd that this is equivalent to (du + 2nuρ) ∧ ω 1 ∧ · · · ∧ ωn = 0, or du ≡ −2nuρ (mod {ω 1, . . . , ωn }). (3.16) This is the inﬁnitesimal form of the relation u(p · g) = r−2nu(p),

for p ∈ P and g ∈ G as in (3.3). This is in turn the same as saying that the function u on P deﬁnes a section of the oriented line bundle D → N associated to the 1-dimensional representation g → r 2n of the structure group.3 Positive sections of D correspond to oriented volume forms on N , which in an obvious way correspond to Riemannian metrics representing the conformal class [ds2]. Because so many of the PDEs studied in the conformal geometry literature describe conditions on such a metric, we should expect our study of EulerLagrange equations in conformal geometry to involve this density bundle. s In analogy with this, we deﬁne for any positive real number s the degree- n s/n 2s density bundle D associated to the 1-dimensional representation g → r ; the degree-1 density bundle is the preceding D. Sections are represented by functions u on P satisfying u(p · g) = r−2su(p), or inﬁnitesimally, du ≡ −2suρ (mod {ω 1 , . . . , ωn }). (3.18) (3.17)

3 That is, D is the quotient of P × R by the equivalence relation (p, u) ∼ (p · g, r −2n u) for p ∈ P , u ∈ R, g ∈ G; a series of elementary exercises shows that this is naturally a line bundle over N , whose sections correspond to functions u(p) satisfying (3.16).

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CHAPTER 3. CONFORMALLY INVARIANT SYSTEMS

Summarizing, we will say that any function u on P satisfying (3.18) deﬁnes a s section of the degree- n density bundle, and write u ∈ Γ(Ds/n ). We further investigate the local behavior of u ∈ Γ(D s/n ), writing du + 2suρ = uiωi for some “ﬁrst covariant derivative” functions ui . Diﬀerentiating again, and applying the Cartan lemma, we obtain dui + suβi + 2(s + 1)uiρ − uj αj = uij ωj , i (3.19)

for some “second covariant derivatives” uij = uji; this is the inﬁnitesimal form of the transformation rule ui(p · g) = r−2(s+1)(uj (p)aj − sbi u(p)). i (3.20)

Note that unless s = 0 (so that u is actually a function on N ), the vector-valued function (ui) on P does not represent a section of any associated vector bundle. Diﬀerentiating again, and factoring out ω k , we obtain modulo {ω 1 , . . . , ωn } k duij ≡ δij uk βk − (s + 1)(uj βi + uiβj ) − 2(s + 2)uij ρ + ukj αk + uik αj , i

so once again, uij is not a section of any associated vector bundle. However, we can take the trace duii ≡ (n − 2s − 2)βk uk − 2(s + 2)ρuii and we see that in case s = summarize, n−2 2 ,

(mod {ω1, . . . , ωn}), s+2 n

the function uii on P is a section of D

. To

the map u → uii deﬁnes a second-order linear diﬀerential operator, called the conformal Laplacian, ∆ : Γ(D n−2 2n

) → Γ(D

n+2 2n

).

Note that for sections u, v ∈ Γ(D 2n ), the quantity u∆v ∈ Γ(D 1 ) can be thought of as an n-form on N , and integrated. Furthermore, the reader can compute that (u∆v − v∆u)ω = d((uvi − vui )ω(i) ). We interpret this as saying that u∆v − v∆u is canonically a divergence, so that n−2 (·, ∆·) is a symmetric bilinear form on Γo (D 2n ), where (·, ·) : Γo (D n−2 2n )

n−2

× Γo (D

n+2 2n )

→R

is given by integration on N of the product.

3.1. BACKGROUND MATERIAL ON CONFORMAL GEOMETRY

95

To clarify the meaning of ∆, we can choose a particular Riemannian metric g representing the conformal structure, and compare the second covariant s derivatives of an n -density u taken in the conformal sense with those derivatives taken in the usual sense of Riemannian geometry. By construction of P , the pulled-back quadratic form π ∗ g ∈ Sym2 (T ∗ P ) may be expressed as π∗ g = λ((ω1 )2 + · · · + (ωn )2 ) for some function λ > 0 on P . Proceeding in a manner similar to the preceding, we note that Lv (π∗ g) = 0

for any vector ﬁeld v that is vertical for P → N . Knowing the derivatives of ω i quite explicitly, we can then calculate that dλ = −4λρ + λi ωi for some functions λi . Diﬀerentiating again, we ﬁnd dλi = −2λβi − 6λi ρ + λj αj + λij ωj , i

(3.21)

(3.22)

for some functions λij = λji . Now we can reduce our bundle P → N to a subbundle Pg ⊂ P , deﬁned by Pg = {p ∈ P : λ(p) = 1, λ1 (p) = · · · = λn (p) = 0}. Equations (3.21, 3.22) imply that Pg has structure group SO(n, R) ⊂ G, and using bars to denote restrictions to Pg , we have for the pseudo-connection forms ρ = 0, ¯

1¯ ¯ βi = 2 λij ωj , ¯

d¯ i = −¯ i ∧ ωj . ω αj ¯

The last of these means that if we identify Pg with the usual orthonormal frame bundle of (N, g), then αi gives the Levi-Civita connection. The curvature is by ¯j deﬁnition ¯ d¯ i + αi ∧ αk = 1 Ri ωk ∧ ωl , αj ¯ k ¯ j 2 jkl ¯ but we have an expression for the left-hand side coming from the conformal geometry; namely, ¯ ¯ ¯ d¯ i + αi ∧ αk = −βi ∧ ωj + βj ∧ ωi + 1 Ai ωk ∧ ωl . αj ¯ k ¯ j ¯ ¯ 2 jkl ¯ ¯ ¯ ¯ Substituting βi = 1 λij ωj and comparing these two expressions gives 2 j¯ i¯ ¯ ¯ Ri = 1 (δli λjk − δlj λik − δk λjl + δk λil ) + Ai . jkl jkl 2

From this we ﬁnd the other components of curvature i 1 ¯ ¯ Ricjl = Rjil = 2 ((2 − n)λjl − δlj λii ), ¯ R = Ricll = (1 − n)λii .

96

CHAPTER 3. CONFORMALLY INVARIANT SYSTEMS n−2 Now we will compute the conformal Laplacian of u ∈ Γ(D 2n ), but restrict the computation to Pg . Note that the choice of g amounts to a trivialization of D and of all of its powers, so in this setting it is correct to think of u as a function. We have d¯ = u d¯i u = = = −(n − 2)¯u + uiωi ρ¯ ¯ ¯

ui ω i , ¯ ¯ ¯ u ¯ ¯ ¯i ¯ ¯ − n−2 uβi − n¯iρ + uj αj + uij ωj 2 ¯ j ¯¯ ω uj α + (¯ij − n−2 uλij )¯ j . ¯ ¯ u i 4

Denoting by ∆g the Riemannian Laplacian, we now have ∆g u = ¯ = ¯ uii − n−2 uλii ¯ 4 ¯ n−2 ∆¯ + 4(n−1) R¯. u u

This is the more familiar expression for the conformal Laplacian, deﬁned in terms of the Riemannian Laplacian of some representative metric. In the case of the ﬂat model of conformal geometry, if one uses standard coordinates on Rn = R\{∞}, then the Euclidean metric represents the conformal class, and ∂ we can use the ordinary Laplacian ∆ = ( ∂xi )2 . Its transformation properties, often stated and proved with tedious calculations, can be easily derived from the present viewpoint. Of particular interest to us will be non-linear Poisson equations, of the form ∆u = f(xi , u), (3.23)

where we will have an interpretation of ∆ as the conformal Laplacian on a conformal manifold with coordinates xi. We will therefore want to interpret n−2 the unknown u as a section of the density bundle D 2n , and we will want to interpret f(x, u) as a (0th -order) bundle map f :D n−2 2n

→D

n+2 2n

.

Certain obvious bundle maps f come to mind. One kind is given by multiplication by any section λ ∈ Γ(D 2/n ); this would make (3.23) a linear equation. Another is the appropriate power map u → u n−2 . This yields a non-linear Poisson equation, and we will examine it quite closely in what follows. We conclude this discussion with an alternate perspective on the density bundles Ds/n. First, note that for any conformal manifold (N, [ds2]) with its associated parallelized bundle P → N , the Pfaﬃan system IQ = {ρ, ω1 , . . . , ωn} n+2 ´ 3.2. CONFORMALLY INVARIANT POINCARE-CARTAN

FORMS

97

is integrable, and its associated foliation is simple. The leaf space of this foliation is just the quotient Q of P by the action of a subgroup of its structure group, and this Q is also a ﬁber-bundle over N , with ﬁber R∗ . This generalizes the space Q of positive null vectors in Ln+2 which appeared in the discussion of the ﬂat model. Now, the density bundles D s/n are all canonically oriented, and we claim that Q is canonically identiﬁed with the positive elements of D s/n , for any s. To see this, note that any positive u ∈ D s/n, over x ∈ N , is deﬁned as a positive function on the ﬁber Px ⊂ P satisfying (3.17). It is not hard to see that the locus {p ∈ Px : u(p) = 1} ⊂ Px is a leaf of the foliation deﬁned by IQ . Conversely, let LQ ⊂ P be a leaf of the foliation deﬁned by IQ . Then LQ lies completely in some ﬁber Px of P → N , and we can deﬁne a function u on Px by setting u = 1 on LQ , and extending to Px by the rule (3.17). These are clearly inverse processes. s/n We can extend the identiﬁcation as follows. Let J 1 (N, D+ ) be the space of 1-jets of positive sections of D; it is a contact manifold, in the usual manner. Let M be the leaf space of the simple foliation associated to the integrable Pfaﬃan system on P IM = {ρ, ω1, . . . , ωn, β1 , . . . , βn}. This M is also a contact manifold, a with global contact form pulling back to ρ ∈ Ω1 (P ), and it generalizes the contact manifold M mentioned in our discussion of the ﬂat model. We claim that there is a canonical contact isomorphism between s/n J 1 (N, D+ ) and M . To see this, note that a 1-jet at x ∈ N of a positive section of D s/n is speciﬁed by n + 1 functions (u, u1, . . . , un) on the ﬁber Px satisfying (3.17, 3.20). It is then not hard to see that the locus {p ∈ Px : u(p) = 1, ui (p) = 0} ⊂ Px is a leaf of the foliation deﬁned by IM . Conversely, let LM ⊂ P be a leaf of the foliation deﬁned by IM . Then LM lies completely in some ﬁber Px of P → N , and we can deﬁne n + 1 functions (u, u1, . . . , un) on Px by setting u = 1 and ui = 0 on LM , and extending to Px by the rules (3.17, 3.20). These are again inverse processes, and we leave it to the reader to investigate the correspondence between contact structures.

3.2

Conformally Invariant Poincar´-Cartan e Forms

e In this section, we identify the Poincar´-Cartan forms on the contact manifold M over ﬂat conformal space R that are invariant under the action of the conformal group SO o (n + 1, 1). We then specialize to one that is neo-classical, and determine expressions for the corresponding Euler-Lagrange equation in coordinates; it turns out to be the non-linear Poisson equation with critical exponent ∆u = Cu n−2 . n+2 98

CHAPTER 3. CONFORMALLY INVARIANT SYSTEMS

For easy reference we recall the structure equations for Lorentz frames de0 = 2e0 ρ + ei ωi , de = e0 βj + ei αi + en+1 ωj , αi + αj = 0, i j j j den+1 = ei βi − 2en+1ρ; i 1 dρ + 2 βi ∧ ω = 0, dωi − 2ρ ∧ ωi + αi ∧ ωj = 0, dβi + 2ρ ∧ βi + βj ∧ αj = 0, i k dαi + αi ∧ αj + βi ∧ ωj − βj ∧ ωi = 0. j k j The calculation should clarify some of the more abstract constructions of the preceding section. It will also be helpful in understanding the branch of the equivalence problem in which this Poincar´-Cartan form appears, which is the e topic of the next section. We denote by P the set of Lorentz frames for Ln+2 , by M the set of pairs (e, e ) of positive null vectors with e, e = −1, by Q the space of positive null vectors, and by R the ﬂat conformal space of null lines. There are SO o (n+1, 1)equivariant maps πM : P → M, (e0 , . . . , en+1) → (e0 , en+1 ), πQ : P → Q, (e0 , . . . , en+1) → e0 , πR : P → R, (e0 , . . . , en+1) → [e0 ]. (3.24)

(3.25)

We noted in the previous section that M has a contact 1-form which pulls back to ρ, and this is the setting for our Poincar´-Cartan forms. e Proposition 3.1 The SO o (n + 1, 1)-invariant Poincar´-Cartan forms on M , e pulled back to P , are constant linear combinations of Πk = ρ ∧ def |I|=k

βI ∧ ω(I),

where 0 ≤ k ≤ n. Those that are neo-classical with respect to Q are of the form Π = c 1 Π1 + c 0 Π0 , c0 , c1 ∈ R. (3.26)

Proof. In this setting, an invariant Poincar´-Cartan form on M , pulled back e to P , is an (n + 1)-form that is a multiple of ρ, semibasic over M , invariant under the left-action of SO o (n + 1, 1), invariant under the right-action of the isotropy subgroup SO(n, R) of M , and closed. That Π must be semibasic and SOo (n + 1, 1)-invariant forces it to be a constant linear combination of exterior products of ρ, βi , ωi. It is then a consequence of the Weyl’s theory of vector invariants that the further conditions of being a multiple of ρ and SO(n, R)invariant force Π to be a linear combination of the given Πk . It follows from the structure equations of P that dΠk = 0, so each Πk is in fact the pullback of a Poincar´-Cartan form. e

´ 3.2. CONFORMALLY INVARIANT POINCARE-CARTAN FORMS We note that for the n-form Λk deﬁned by Λk = we have def |I|=k

99

βI ∧ ω(I)

This means that for k = n , the Poincar´-Cartan form Πk is associated to an e 2 SOo (n + 1, 1)-invariant functional, which in the standard coordinates discussed below is second-order. For the exceptional case n = 2k, there is no invariant functional corresponding to Πk , but in the neo-classical case k ≤ 1 with n ≥ 3, this is not an issue. We now focus on the neo-classical case (3.26), for which it will be convenient to rescale and study Π = ρ ∧ βi ∧ ω(i) −

2C ω n−2

dΛk = 2(n − 2k)Πk .

,

(3.27)

where C is a constant. This is the exterior derivative of the Lagrangian Λ=

1 β 2(n−2) i

∧ ω(i) −

C ω, n(n−2)

and our Monge-Ampere diﬀerential system is generated by ρ and the n-form Ψ = βi ∧ ω(i) −

2C n−2 ω.

Proposition 3.2 The Euler-Lagrange equation corresponding to the Poincar´e Cartan form (3.27) is locally equivalent to ∆u = Cu n−2 . n+2 (3.28)

The meaning of “locally equivalent” will come out in the proof. It includes an explicit and computable correspondence between integral manifolds of the Monge-Ampere system and solutions to the PDE. We remark that the PDEs corresponding to higher Poincar´-Cartan forms e Πk , with k > 1, have been computed and analyzed by J. Viaclovsky in [Via00]. Proof. We begin by deﬁning a map σ : J 1 (Rn , R) → P , which can be projected to M to give an open inclusion of contact manifolds with dense image. This map will be expressed in terms of the usual contact coordinates (xi , z, pi) on J 1 (Rn, R), except that z is replaced by u = eλz for some undetermined constant λ = 0, so that in particular, dz − pi dxi = (λu)−1du − pi dxi.

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CHAPTER 3. CONFORMALLY INVARIANT SYSTEMS

When using coordinates (xi , u, pi), we denote our jet space by J 1 (Rn, R+ ). We then pull back Ψ via σ, and consider its restriction to a transverse Legendre submanifold. With a convenient choice of λ, we will obtain a non-zero multiple n+2 of ∆u − Cu n−2 , implying the Proposition. We deﬁne σ as a lift of the following map Rn → P , to be extended to 1 J (Rn , R+) shortly: 1 x1 . . . 0 . . . 1i , en+1 (x) = ¯ ei (x) = ¯ . . . xi 0 0 . . .

e0 (x) = ¯ xn

||x||2 2

,

. 0 1

(3.29)

It is easy to verify that this does take values in P . Also, note that the composition Rn → P → R gives standard (stereographic) coordinates on R\{∞}. This partly indicates the notion of “locally equivalent” used in this Proposition. We now let e0 (x, u, p) = ei (x, u, p) = en+1 (x, u, p) = u2k e0 (x), ¯ ei (x) + pi e0 (x), ¯ ¯ u−2k (¯n+1 (x) + pj ej (x) + e ¯ (3.30)

||p||2 e0 (x)), ¯ 2

for some constant k = 0 to be determined shortly. Our use of the dependent variable u as a scaling factor for e0 reﬂects the fact that we expect u to represent a section of some density line bundle. The formula for en+1 is chosen just so that our map takes values in P . Now we can compute directly de0 = = 2ku−1e0 du + u2k ei dxi ¯

1 2(ku−1du − 2 pidxi)e0 + (u2k dxi)ei ,

so by comparison with the expression in (3.24) we obtain some of the pulled-back Maurer-Cartan forms: σ∗ ωi σ∗ ρ Similarly, we have βi = = = − en+1 , dei = = u2k dxi, 1 ku−1 du − 2 pi dxi.

− u−2k (¯n+1 + pj ej + e ¯ en+1 dxi + ¯

||p||2 ¯ 2 e0 ), pi ek dxk + e0 dpi ¯ ¯ ||p||2 i 2 dx

u−2k dpi − pi pj dxj +

.

´ 3.2. CONFORMALLY INVARIANT POINCARE-CARTAN FORMS

101

Because we want the projection to M of σ : J 1 (Rn , R+) → P to be a contact mapping, we need σ∗ ρ to be a multiple of dz − pi dxi = (λu)−1 du − pidxi, which holds if we choose 1 k = 2λ . Now, λ is still undetermined, but it will shortly be chosen to simplify the expression for the restriction of Π to a transverse Legendre submanifold. Namely, we ﬁnd that n−2 βi ∧ ω(i) = u λ dpi ∧ dx(i) + n−2 ||p||2dx , 2 and also ω = u λ dx. On transverse Legendre submanifolds, we have du = λeλz dz = λupi dxi, so that pi = Diﬀerentiating, we obtain dpi = 1 λ 1 ∂2u 1 ∂u ∂u − 2 i j u ∂xi ∂xj u ∂x ∂x dxj , 1 ∂u . λu ∂xi n so that on transverse Legendre submanifolds, Ψ=u n−2 λ

1 ∆u n − 2 − 2λ || u||2 + − λ u 2λ2 u2 n−2 , 2 n+2

2C n−2

u2/λ dx = 0.

We can eliminate the ﬁrst-order term by choosing λ= and then Ψ= which is the desired result. 2u n−2

∆u − Cu n−2

dx,

Note that z = λ−1 log u satisﬁes a PDE that is slightly more complicated, but equivalent under a classical transformation. Also, note that (3.28) is usually given as the Euler-Lagrange equation of the functional

1 || 2

u||2 +

2n n−2 Cu n−2 2n

dx,

which has the advantage of being ﬁrst-order, but the disadvantage of not being preserved by the full conformal group SO o (n+1, 1). In contrast, our Lagrangian Λ restricts to transverse Legendre submanifolds (in the coordinates of the preceding proof) as the variationally equivalent integrand Λ=

2n C 1 u∆u − n(n−2) u n−2 (n−2)2

dx.

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CHAPTER 3. CONFORMALLY INVARIANT SYSTEMS

3.3

The Conformal Branch of the Equivalence Problem

where ai + aj = 0 and sij = sji, sii = 0. In this section, we show how to j i uniquely characterize in terms of the invariants of the G-structure those (M, Π) which are locally equivalent to the Poincar´-Cartan form for the equation e ∆u = Cu n−2 , n+2 Let (M 2n+1, Π) be a manifold with a non-degenerate Poincar´-Cartan form; e that is, Π ∈ Ωn+1 (M ) is closed, and has a linear divisor that is unique modulo scaling and deﬁnes a contact structure. We also assume that n ≥ 3 and that Π is neo-classical and deﬁnite. Then as discussed in §2.4 we may associate to (M, Π) a G-structure B → M , where G is a subgroup of GL(2n + 1, R) whose Lie algebra consists of matrices of the form (n − 2)r 0 0 i , 0 −2rδj + ai 0 (3.31) j j j sij nrδi − ai di

C = 0.

(3.32)

on ﬂat conformal space. The result may be loosely summarized as follows. i The vanishing of the primary invariants T ijk, U ij , Sj is equivalent to the existence of a foliation B → N over a conformal manifold (N, [ds2 ]), for which [ds2 ] pulls back to the invariant [ (ωi )2 ]. In this case, under open conditions on further invariants, three successive reductions of B → M yield a subbundle which is naturally identiﬁed with the conformal bundle over N . The Poincar´-Cartan e form can then be identiﬁed with that associated to a non-linear Poisson equation. In case a further invariant is constant, this equation is equivalent to (3.32).

We ﬁnd these conditions by continuing to apply the equivalence method begun in §2.4, pursuing the case in which all of the non-constant torsion vanishes. One corollary of the discussion is a characterization of Poincar´-Cartan forms e locally equivalent to those for general non-linear Poisson equations of the form ∆u = f(x, u), x ∈ N, (3.33)

on an n-dimensional conformal manifold (N, [ds2 ]); here and in the following, ∆ is the conformal Laplacian. The condition that (3.33) be non-linear can be characterized in terms of the geometric invariants associated to (M, Π), as can the condition that (N, [ds2 ]) be conformally ﬂat. The characterization of (3.32) will imply that this equation has maximal symmetry group among non-linear Euler-Lagrange equations satisfying certain geometric conditions on the torsion. We will not actually prove the characterization result for general Poisson equations (3.33), but we will use these equations (in the conformally ﬂat case, with ∂ 2 ) as an example at each stage of the following calculations. ∆= ∂xi

3.3. CONFORMAL BRANCH OF THE EQUIVALENCE PROBLEM

103

We ﬁrst recall the structure equations of the G-structure B → M , associated to a neo-classical, deﬁnite Poincar´-Cartan form e Π = −θ ∧ (πi ∧ ω(i) ). There is a pseudo-connection (n − 2)ρ 0 i 0 −2ρδj + αi ϕ= j σij δi 0 , with 0 j j nρδi − αi

αi + αj = 0, j i σij = σji, σii = 0, (3.34)

having torsion θ θ −πi ∧ ωi i d ωi + ϕ ∧ ωj = −(Sj ωj + U ij πj ) ∧ θ + T ijk πj ∧ ωk , πi πj 0 (3.35) where enough torsion has been absorbed so that j i i T ijk = T jik = T kji, T iik = 0; U ij = U ji; Sj = Si , Si = 0.

(3.36)

We also recall the structure equation (2.48) i (n − 2)dρ = −δi ∧ ωi − Sj πi ∧ ωj + n−2 2n

U ij σij ∧ θ − ti πi ∧ θ.

(3.37)

The equations (3.34, 3.35, 3.36, 3.37) uniquely determine the forms ρ, αi , and j we are still free to alter our pseudo-connection by δi ; δi + bi θ + tij ωj , with tij = tji and tii = 0, σij ; σij + tij θ + tijk ωk , with tijk = tjik = tkji and tiik = 0, requiring also We set up our example (3.33) by taking coordinates (xi , u, qi) on M = J (Rn, R), with contact form

1

(3.38)

2nbi + (n − 2)U jk tijk = 0.

˜ def θ = du − qidxi. Then transverse Legendre submanifolds which are also integral manifolds of ˜ def Ψ = −dqi ∧ dx(i) + f(x, u)dx correspond locally to solutions of (3.33). One can verify that the form ˜ def ˜ ˜ Π = θ∧Ψ

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CHAPTER 3. CONFORMALLY INVARIANT SYSTEMS

is closed, so in particular our Poisson equation is an Euler-Lagrange equation. We ﬁnd a particular 1-adapted coframing of J 1 (Rn, R) as in Lemma 2.3 by writing ˜ ˜ Π = −θ ∧ (dqi − f dxi ) ∧ dx(i) , n and then setting

It turns out that this coframing is actually a section of B → J 1 (Rn, R), as one discovers by setting 1 ˜ ˜ ρ = 0, αi = 0, δi = − n fu ωi , ˜ ˜j and noting that the structure equations (3.34, 3.35) hold (with some complicated choice of σij which we will not need). In fact, (3.35) holds with torsion ˜ i coeﬃcients Sj , U ij , T ijk all vanishing, and we will see the signiﬁcance of this presently. In the general setting, we seek conditions under which the quadratic form on B def q = (ωi )2 can be regarded as deﬁning a conformal structure on some quotient of B. For the appropriate quotient to exist, at least locally, the necessary and suﬃcient condition is that the Pfaﬃan system I = {ω 1 , . . . , ωn } be integrable; it is easily seen from the structure equations (and we noted in §2.4) that this is equivalent to the condition U ij = 0. We assume this in what follows, and for convenience assume further that the foliation of B by leaves of I is simple; that is, there is a smooth manifold N and a surjective submersion B → N whose ﬁbers are the leaves of I. Coordinates on N may be thought of as “preferred independent variables” for the contactequivalence class of our Euler-Lagrange PDE, as indicated in §2.4. We can now compute the Lie derivative of q under a vector ﬁeld v which is vertical for B → N , satisfying v ω i = 0; using the hypothesis U ij = 0 and the structure equations, we ﬁnd Lv q = 2 T ijk(v i πj )ωi ωk + Sj (v

˜ du − qidxi θ ωi = . dxi ˜ f i πi ˜ dqi − n dx

θ)ωi ωj + 4(v

ρ)q.

i It follows that if T ijk = 0 and Sj = 0, then there is a quadratic form on N which pulls back to a non-zero multiple of q on B. A short calculation shows that the converse as true as well, so we have the following. i Proposition 3.3 The conditions U ij = T ijk = Sj = 0 are necessary and sufﬁcient for there to exist (locally) a conformal manifold (N, [ds 2]) and a map B → N such that the pullback to B of [ds2] is equal to [q] = [ (ωi )2 ].

3.3. CONFORMAL BRANCH OF THE EQUIVALENCE PROBLEM

105

i From now on, we assume U ij = Sj = T ijk = 0. From the discussion of the conformal equivalence problem in §3.1.2, we know that associated to (N, [ds2]) is the second-order conformal frame bundle P → N with global coframing ω i , ρ, αi , βi satisfying structure equations ¯ ¯ ¯j ¯

Our goal is to directly relate the principal bundle B → M associated to the Poincar´-Cartan form Π on M to the principal bundle P → N associated to the e induced conformal geometry on N . We shall eventually ﬁnd that under some further conditions stated below, the main one of which reﬂects the non-linearity of the Euler-Lagrange system associated to Π, there is a canonical reduction B3 → M of the G-structure B → M such that locally B3 ∼ P as parallelized = manifolds.4 Because the canonical coframings on B3 and P determine the bundle structure of each, we will then have shown that the subbundle B3 → N of B → N can be locally identiﬁed with the bundle P → N associated to the conformal structure (N, [ds2]). In the special case of our Poisson equation, we have ω i = dxi as part of a ˜ section of B → M , so we can already see that our quotient space N ∼ Rn is = conformally ﬂat. This reﬂects the fact that the diﬀerential operator ∆ in (3.33) is the conformal Laplacian for ﬂat conformal space. We return to the general case, and make the simplifying observation that under our hypotheses,

2n 0 = d2(ω1 ∧ · · · ∧ ωn ) = − n−2 tiπi ∧ θ ∧ ω1 ∧ · · · ∧ ωn ,

i i i ¯j d¯ − 2¯ ∧ ω + αj ∧ ω = 0, ω 1 ρ ¯i ¯ d¯ + β ∧ ω = 0, ¯i ¯ ρ 2 1 ¯ ¯ ¯ ¯ ¯ ¯ d¯ i + αi ∧ αk + βi ∧ ωj − βj ∧ ωi = 2 Ai ωk ∧ ωl , jkl ¯ αj ¯ k ¯ j ¯ j 1 ¯ j ¯ dβi + 2¯ ∧ βi + βj ∧ αi = 2 Bijk ω ∧ ωk . ρ ¯ ¯ ¯ ¯

(3.39)

so that ti = 0 in the equation (3.37) for dρ. We now have on B the equations dωi dρ = = 2ρ ∧ ωi − αi ∧ ωj , j ∧ω . i (3.40) (3.41)

1 − n−2 δi

With the goal of making our structure equations on B resemble the conformal structure equations (3.39), we deﬁne βi = def 2 n−2

δi .

The equations for dω i and dρ are now formally identical to those for d¯ i and d¯, ω ρ and computing exactly as in the conformal equivalence problem, we ﬁnd that dαi + αi ∧ αk + βi ∧ ωj − βj ∧ ωi = 1 Ai ωk ∧ ωl , j k j 2 jkl

4 As in the characterization in §2.5 of prescribed mean curvature systems, we will denote by B1 , B2 , etc., reductions of the bundle B → M associated to Π, and these are unrelated to the bundles of the same names used in the construction of B.

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CHAPTER 3. CONFORMALLY INVARIANT SYSTEMS

for some functions Ai on B having the symmetries of the Riemann curvature jkl tensor. ¯ Of course, we want Ai to correspond to the Weyl tensor Ai of (N, [ds2]), jkl jkl so we would like to alter our pseudo-connection forms (3.34) in a way that will give Al = 0. jkl Again, reasoning exactly as we did in the conformal equivalence problem, we know that there are uniquely determined functions tij = tji such that replacing βi ; βi + tij ωj accomplishes this goal. However, these may have tii = 0, meaning that we cannot make the compensating change in σij (see (3.38)) without introducing torsion in the equation for dπi. We proceed anyway, and now have structure equations (n − 2)ρ 0 0 θ θ −πi ∧ ωi i ∧ ωj = 0 αi −2ρδj 0 d ωi + 0 j j j i n−2 πi πj Aω ∧ θ βi σij nρδi −αi 2 (3.42) where A = n−2 tii is a component of the original Ai , analogous to scalar jkl 2n curvature in the Riemannian setting. Also, we have dρ = − 1 βi ∧ ωi , 2 i 1 dαi + αk ∧ αk + βi ∧ ωj − βj ∧ ωi = 2 Ai ωk ∧ ωl , jkl j j

(3.43) (3.44)

with Al = 0. These uniquely determine the pseudo-connection forms ρ, αi , j jkl βi , and leave σij determined only up to addition of terms of the form tijkωk , with tijk totally symmetric and trace-free. Now that βi is uniquely determined, we can once again mimic calculations from the conformal equivalence problem, deducing from (3.40, 3.43, 3.44) that

1 dβi + 2ρ ∧ βi + βj ∧ αj = 2 Bijk ωj ∧ ωk , i

(3.45)

with Bijk + Bikj = 0, Bijk + Bjki + Bkij = 0. In the case of our non-linear Poisson equation (3.33), a calculation shows that 2 ˜ ˜ the modiﬁcation of βi = n−2 δi = 0 is not necessary, and that with everything deﬁned as before, we have not only (3.42), but also (3.43, 3.44, 3.45) with Ai = Bijk = 0. This gives us another way of seeing that the conformal jkl structure associated to (3.33) is ﬂat. What will be important for us, however, is the fact that along this section of B → J 1 (Rn, R), the torsion function A is ˜ A=

1 f (x, u). n u

This comes out of the calculations alluded to above.

3.3. CONFORMAL BRANCH OF THE EQUIVALENCE PROBLEM

107

We now begin to reduce B → M , as promised. To get information about the derivative of the torsion coeﬃcient A without knowing anything about dσ ij , we consider 0 = = = d2(πi ∧ ω(i) )

d((n − 2)ρ ∧ πi ∧ ω(i) + (dA + 4ρA) ∧ θ ∧ ω.

n−2 2 θ

∧ βi ∧ ω(i) + Aθ ∧ ω)

This describes the variation of the function A along the ﬁbers of B → Q, where we recall that Q is the leaf space of the integrable Pfaﬃan system JΠ = {θ, ωi }. In particular, we can write dA + 4ρA = A0θ + Ai ωi , (3.46)

for some functions A0, Ai on B. We see that on each ﬁber of B → Q, either A vanishes identically or A never vanishes, and we assume that the latter holds throughout B. This is motivated by the case of the Poisson equation (3.33), 1 ˜ for which A = n fu (so we are assuming in particular that the zero-order term f(x, u) depends on u). Because the sign of A is ﬁxed, we assume A > 0 in what follows. The case A < 0 is similar, but the case A = 0 is quite diﬀerent. For the ﬁrst reduction of B → M , we deﬁne

1 B1 = {b ∈ B : A(b) = 4 } ⊂ B.

From equation (3.46) with the assumption A > 0 everywhere, it is clear that B1 → M is a principal subbundle of B, whose structure group’s Lie algebra consists of matrices (3.31) with r = 0. Furthermore, restricted to B1 there is a relation ρ = A0 θ + Ai ω i .

1

(3.47)

˜ ˜ ˜ In the case of a Poisson equation (3.33), our section (θ, ωi , πi) of B → J (Rn, R) is generally not a section of B1 ⊂ B, because we have along this ˜ section that A = fu . However, (3.46) guides us in ﬁnding a section of B1 . n Namely, we deﬁne a function r(x, u, q) > 0 on M by ˜ r 4 = 4A =

4 n fu ,

(3.48)

one has the structure equation (3.42), with

−1 ρ = r−1 dr = 1 fu dfu , ˆ 4

and then one can verify that for the coframing 2−n ˜ ˆ r 0 0 θ θ def i ωj , ωi = 0 0 r 2 δj ˜ ˆ −n j πi ˆ πj ˜ 0 0 r δi ˆ βi = αi = 0, ˆj ˆ 1 A = 4.

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CHAPTER 3. CONFORMALLY INVARIANT SYSTEMS

Again, we won’t have any need for σij . Observe that along this section of B1 , ˆ ˆ ˆ ˆ ˆ ρ = 4 fu dfu = A0 θ + Ai ωi , so that ˆ 1 −1

−1 ˆ A0 = 1 rn−2fu fuu, 4 −1 ˆ Ai = 1 r−2 fu (fuxi + fuuqi ), 4

(3.49)

with r given by (3.48). Returning to the general situation on B1 , we diﬀerentiate (3.47) and ﬁnd

1 (dA0 − (n − 2)ρA0 ) ∧ θ + (dAi + 2ρAi − Aj αj + 2 βi − A0πi ) ∧ ωi = 0, i

and the Cartan lemma then gives dA0 − (n − 2)ρA0 = A00θ + A0iωi , 1 dAi + 2ρAi − Aj αj + 2 βi − A0 πi = Ai0θ + Aij ωj , i (3.50) (3.51)

with A0i = Ai0 and Aij = Aji. We interpret (3.50) as saying that if A0 vanishes at one point of a ﬁber of B1 → Q, then it vanishes everywhere on that ﬁber. We make the assumption that A0 = 0; the other extreme case, where A0 = 0 everywhere, gives a diﬀerent branch of the equivalence problem. Note that in the case of a Poisson equation (3.33), the condition A0 = 0 implies by (3.49) that the equation is everywhere non-linear. This justiﬁes our decision to pursue, among the many branches of the equivalence problem within the larger conformal branch, the case A > 0, A0 = 0. This justiﬁcation was our main reason to carry along the example of the Poisson equation, and we will not mention it again. General calculations involving it become rather messy at this stage, but how to continue should be clear from the preceding. Returning to the general setting, our second reduction uses (3.51), which tells us that the locus B2 = {b ∈ B1 : Ai (b) = 0} ⊂ B1 is a principal subbundle of B1 → M , whose structure group’s Lie algebra consists of matrices (3.31) with r = di = 0. Furthermore, restricted to B2 there are relations βi = 2(Ai0θ + Aij ωj + A0 πi), and also ρ = A0 θ. With the Ai out of the way we diﬀerentiate once more, and applying the Cartan lemma ﬁnd that on B2 , modulo {θ, ωi , πi}, dA00 dA0i dAij ≡ ≡ ≡ 0, A0j αj , i Akj αk + Aik αk + A0 σij . i j (3.52) (3.53) (3.54)

3.3. CONFORMAL BRANCH OF THE EQUIVALENCE PROBLEM

109

We interpret (3.52) as saying that A00 descends to a well-deﬁned function on M . We interpret (3.53) as saying that the vector valued function (A0i) represents a section a vector bundle associated to B2 → M . We interpret (3.54) as saying that if A0 = 0, then the matrix (Aij ) represents a section of a vector bundle associated to B2 → M . However, we have already made the assumption that A0 = 0 everywhere. In some examples of interest, most notably for n+2 the equation ∆u = Cu n−2 , the section (A0i) vanishes; for a general non-linear Poisson equation, this vanishing loosely corresponds to the non-linearity being translation-invariant on ﬂat conformal space. We will not need to make any assumptions about this quantity. This allows us to make a third reduction. With A0 = 0, (3.54) tells us that the locus where the trace-free part of Aij vanishes,

1 B3 = {b ∈ B2 : A0 (b) = Aij (b) − n δij Akk(b) = 0}, ij def

is a subbundle B3 → M of B2 → M . In terms of (3.31), the Lie algebra of the structure group of B3 is deﬁned by r = di = sij = 0. Let us summarize what we have done. Starting from the structure equations (3.34, 3.35, 3.36, 3.37) on B → M for a deﬁnite, neo-classical Poincar´-Cartan e form with n ≥ 3, we specialized to the case where the torsion satisﬁes i U ij = Sj = T ijk = 0.

In this case, we found that the leaf space N of the Pfaﬃan system {ω 1 , . . . , ωn } (ωi )2 . We replaced each pseudohas a conformal structure pulling back to connection form δi by its multiple βi , and guided by computations from conformal geometry, we determined the torsion in the equation for dαi , which rej sembled a Riemann curvature tensor. This torsion’s analog of scalar curvature provided our fundamental invariant A, which had ﬁrst “covariant derivatives” A0 , Ai , and second “covariant derivatives” A00, Ai0 = A0i, Aij = Aji. With the assumptions A = 0, A0 = 0, we were able to make successive reductions by passing to the loci where

1 A = 4,

Ai = 0,

Aij =

1 n δij Akk .

This leaves us on a bundle B3 → M with a coframing ω i , ρ, βi, αi , satisj fying structure equations exactly like those on the conformal bundle P → N associated with (N, [ds2]). From here, a standard result shows that there is a local diﬀeomorphism B3 → P under which the two coframings correspond. In particular, the invariants Ai and Bjkl remaining in the bundle B3 equal the jkl invariants named similarly in the conformal structure, so we can tell for example if the conformal structure associated to our Poincar´-Cartan form is ﬂat. e We now write the restricted Poincar´-Cartan form, e Π = = = −θ ∧ (πi ∧ ω(i) )

1 − A0 ρ ∧

0

1 A0

1 β 2 i

1 − 2A2 ρ ∧ (βi ∧ ω(i) − 2Akkω).

1 − n Akk ωi ∧ ω(i)

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CHAPTER 3. CONFORMALLY INVARIANT SYSTEMS

We can see from previous equations that Akk is constant on ﬁbers of B3 → M . Therefore, it makes sense to say that Akk is or is not constant on B3 . If it is constant, and if the conformal structure on N is ﬂat (that is, Ai = 0 if jkl n ≥ 4, or Bjkl = 0 if n = 3), then our Poincar´-Cartan form is equivalent to e that associated to the non-linear Poisson equation ∆u = Cu n−2 , where C = (n − 2)Akk . This completes the characterization of Poincar´-Cartan e forms equivalent to that of this equation. Our next goal is to determine the conservation laws associated to this Poincar´-Cartan form. e n+2 3.4

Conservation Laws for ∆u = Cu n−2

n+2

In this section, we will determine the classical conservation laws for the conformally invariant non-linear Poisson equation ∆u = Cu n−2 . n+2 (3.55)

Recall that from Λ0 = ω and Λ1 = βi ∧ ω(i) we constructed the functional Λ=

1 2(n−2) Λ1

−

C n(n−2) Λ0

having the Poincare-Cartan form Π = dΛ = ρ ∧ (βi ∧ ω(i) −

2C ω), n−2

and that under a certain embedding σ : J 1(Rn , R+) → P , the Euler-Lagrange system of Π restricted to a transverse Legendre submanifold is generated by Ψ = ∆u − Cu n−2 dx, for coordinates on J 1 (Rn, R+ ) described in the proof of Proposition 3.2. We also proved that the composition of σ : J 1 (Rn, R+ ) → P with the projection P → M gives an open contact embedding of J 1 (Rn, R+ ) as a dense subset of M . Our invariant forms on P pull back via σ to give the following forms on J 1 (Rn , R+), expressed in terms of the canonical coordinates (xi, u, pi): −1 i 1 1 ρ = n−2 u du − 2 pi dx , 2 i ω = u n−2 dxi, (3.56) ||p||2 − 2 i j βi = u n−2 dpi − pi pj dx + 2 dx , i αj = pj dxi − pidxj . To describe the conservation laws, we ﬁrst calculate for symmetry vector ﬁelds V ∈ gΠ the expression ϕV = V Λ ∈ Ωn−1(P ) n+2 at points of J 1 (Rn, R+ ) ⊂ P , and then restrict this (n − 1)-form to that submanifold, where it will be a conserved integrand for the equation.

3.4. CONSERVATION LAWS FOR ∆u = Cu n−2

n+2

111

3.4.1

The Lie Algebra of Inﬁnitesimal Symmetries

We know that the Poincar´-Cartan forms e Πk = ρ ∧

|I|=k

on P are invariant under the simple, transitive left-action of the conformal group SOo (n+1, 1). The inﬁnitesimal generators of this action are the vector ﬁelds on P corresponding under the identiﬁcation P ∼ SOo (n + 1, 1) to right-invariant = vector ﬁelds. Our ﬁrst task is to determine the right-invariant vector ﬁelds in terms of the basis ∂ ∂ ∂ ∂ , , , ∂ρ ∂ωi ∂βi ∂αi j of left-invariant vector ﬁelds dual to the basis of left-invariant 1-forms used previously; this is because the Maurer-Cartan equation in our setup only allows us to compute in terms of left-invariant objects. For an unknown vector ﬁeld V =g ∂ ∂ ∂ ∂ + V i i + Vi + Vji i ∂ρ ∂ω ∂βi ∂αj (Vji + Vij = 0) (3.57)

βI ∧ ω(I)

to be right-invariant is equivalent to the conditions LV ρ = LV ωi = LV βi = LV αi = 0; j (3.58)

that is, the ﬂow of V should preserve all left-invariant 1-forms. We will solve the system (3.58) of ﬁrst-order diﬀerential equations for V along the submanifold J 1 (Rn, R+ ) ⊂ P . Such V are not generally tangent to J 1 (Rn, R+ ), but the calculation of conservation laws as V Λ is still valid, as J 1 (Rn , R+) is being used only as a slice of the foliation P → M . The solution will give the coeﬃcient functions g, V i , Vi of (3.57) in terms of the coordinates (xi, u, pi) of J 1(Rn , R+ ). We will not need the coeﬃcients Vji , because they do not appear in ϕV = V Λ; in fact, we compute g = V ρ only because it simpliﬁes the rest of the solution. First, we use the equation LV ρ = 0, which gives 0 = = d(V ρ) + V dρ 1 dg − 2 (Vi ωi − V i βi ).

We have the formulae (3.56) for the restrictions of ω i and βi to J 1 (Rn, R+ ), by which the last condition becomes dg =

2 2 1 Vi u n−2 dxi − V i u− n−2 (dpi − pipj dxj + 2

||p||2 i 2 dx )

.

This suggests that we replace the unknowns Vi , V i in our PDE system (3.58) with vi = def 1 i − 2 V u n−2 , 2

vi =

2 def 1 V u n−2 2 i

1 − 2 V j u− n−2 (−pj pi + δij ||p|| ). 2

2

2

112

CHAPTER 3. CONFORMALLY INVARIANT SYSTEMS

Then we have the result ∂g = vi , ∂xi ∂g = −vi , ∂pi ∂g = 0. ∂u (3.59)

In particular, we now need to determine only the function g. For this, we use the equation LV ωi = 0, which gives 0 = = d(V ωi ) + V dωi dV i − 2ρV i + αi V j + 2gωi − Vji ωj . j

When we restrict to J 1 (Rn, R+ ) using (3.56) and use our new dependent variables vi , vi , this gives dvi = (pi dxj − pj dxi)vj − (pj dxj )vi − g dxi + 1 Vji dxj . 2 (3.60)

This says in particular that v i (x, u, p) is a function of the variables xi alone, so along with (3.59) we ﬁnd that g(x, u, p) = f(x) + f i (x)pi, for some functions f(x), f i (x). Substituting this back into (3.60), we have

1 i df i = (pi f j − pj f i − 2 Vji + δj f)dxj .

This is a PDE system ∂f i i 1 = pif j − pj f i − 2 Vji + δj f ∂xj for the unknowns f i (x), and it can be solved in the following elementary way. We ﬁrst let 1 hi = pif j − pj f i − 2 Vji = −hj j i so that our equation is ∂f i i i = hj + δj f. ∂xj (3.61)

Diﬀerentiating this with respect to xk and equating mixed partials implies that the expression ∂hi i ∂f k ∂f k − δj k + δj i ∂xj ∂x ∂x (3.62)

is symmetric in j, k. It is also clearly skew-symmetric in i, k, and therefore equals zero (as in (3.11)). Now we can equate mixed partials of hi to obtain k i δk 2 ∂2f ∂2f ∂2f j ∂ f − δk i l = δli j k − δlj i k . ∂xj xl ∂x x ∂x x ∂x x

3.4. CONSERVATION LAWS FOR ∆u = Cu n−2

n+2

113

With the standing assumption n ≥ 3, this implies that all of these second partial derivatives of f are zero, and we can ﬁnally write f(x) = r + 1 bk xk , 2 for some constants r, bk . The reasons for our labelling of these and the following constants of integration will be indicated below. Because the expressions (3.62) vanish, we can integrate to obtain hi = − 1 a i + 1 b j xi − 1 b i xj j 2 j 2 2 for some constants ai = −aj , and then integrate (3.61) to ﬁnd j i i f i (x) = − 1 wi + (δj r − 1 ai )xj − 1 bi||x||2 + 2 2 j 4 1 2

b, x xi,

where we have written b, x = discussion in the following.

bk xk and ||x||2 =

(xk )2 . We summarize the

Proposition 3.4 The coeﬃcients of the vector ﬁelds (3.57) on P preserving the left-invariant 1-forms ρ, ω i along J 1 (Rn , R+) are of the form g vi vi = def r+

=

def

=

i b, x (1 + p, x ) + − 1 wi + δj r − 1 ai xj − 1 bi ||x||2 pi , 2 2 j 4 2 ∂g 1 i − n−2 , = − 2V u ∂pi 2 2 ∂g ||p||2 1 n−2 − 1 V j u− n−2 (−p p + δ , j i ij 2 ) = 2 Vi u 2 ∂xi 1 2

where r, bi , wi , ai = −aj are constants. j i It is easy to verify that such g, V i , Vi uniquely determine Vji = −Vij such that the vector ﬁeld (3.57) preserves βi and αi as well, but we will not need this fact. j Note that the number of constants in the Proposition equals the dimension of the Lie algebra so(n + 1, 1), as expected. The reader may be aware that one should not have to solve diﬀerential equations to determine right-invariant vector ﬁelds in terms of left-invariant vector ﬁelds. In fact, an algebraic calculation will suﬃce, which in this case would consist of writing an arbitrary Lie algebra element 2r bj 0 bi gL = w i ai j 0 wj −2r

interpreted as a left-invariant vector ﬁeld, and conjugating by σ(x, u, p) ∈ P regarded as a matrix with columns e0 (x, u, p), ej (x, u, p), en+1 (x, u, p) given by (3.29, 3.30). The resulting so(n + 1, 1)-valued function on J 1(Rn , R+ ) then has entries which are the coeﬃcients of a right-invariant vector ﬁeld V . The calculation is tedious, but of course the vector ﬁelds so obtained are as in Proposition 3.4.

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CHAPTER 3. CONFORMALLY INVARIANT SYSTEMS

3.4.2

Calculation of Conservation Laws

We can now use the formulae for the inﬁnitesimal symmetries derived above to calculate the conservation laws for Π, which are (n − 1)-forms on J 1 (Rn , R+) that are closed when restricted to integral submanifolds of the Euler-Lagrange system. The Noether prescription is particularly simple in this case, because the equations LV Λ = 0 and dΛ = Π mean that there are no compensating terms, and we can take for the conserved integrand just ϕV = V Λ. This is straightforward in principle, but there are some delicate issues of signs and constants. We ﬁnd that for V as in Proposition 3.4, V Λ0 = V i ω(i),

and restricting to J 1(Rn , R+ ), using vi instead of V i , we obtain (V Λ0 )|J 1 (Rn ,R+ ) = 2u n−2 vi dx(i). Λ1 is a little more complicated and gives n−2 i v ||p||2)dx(i) ). 2

2n

The analogous computation for V (V

Λ1)J 1 (Rn ,R+ ) = 2u2(−vj dpi ∧ dx(ij) + (vi +

On a transverse Legendre submanifold S of J 1(Rn , R+ ), we can use the condition ρ = 0 from (3.56) to write pi =

−1 ∂u 2 n−2 u ∂xi ,

(3.63)

and if we compute dpi and ||p||2 for such a submanifold, then we can substitute and obtain (V Λ1)|S =

4 n−2

−uuxi xj vj + uuxj xj vi + uxi uxj vj +

n−2 2 2 u vi

dx(i).

We summarize with the following. Proposition 3.5 The restriction of V equals ϕV =

2 n−2

Λ to the 1-jet graph of u(x1, . . . , xn)

uuxj xj vi − uuxi xj vj + uxi uxj vj −

2n 2C n−2 i v n u

+ u2vi dx(i).

We now have a representative for each of the classical conservation laws corresponding to a conformal symmetry of our equation ∆u = Cu n−2 . n+2 (3.64)

3.4. CONSERVATION LAWS FOR ∆u = Cu n−2

n+2

115

We say “representative” because a conservation law is actually an equivalence class of (n − 1)-forms. In fact, our ϕV is not the (n − 1)-form classically taken to represent the conservation law corresponding to V ; our ϕV involves second derivatives of the unknown u(x), while the classical expressions are all ﬁrstorder. We can ﬁnd the ﬁrst-order expressions by adding to ϕV a suitable exact (n − 1)-form, obtaining ϕg def =

ϕV +

2 j i n−2 d(uux v dx(ij))

=

j 4 i j n−2 ux ux v

−

2 n−2 ||

u||2 +

2n 2C n−2 n u

vi

+u2vi +

j 2 u(uxi vxj n−2

i − uxj vxj ) dx(i).

This turns out to give the classical expressions for the conservation laws associated to our equation (3.64), up to multiplicative constants. It could have been obtained more directly using the methods of Section 1.3. For this, one would work on the usual J 1 (Rn, R), with standard coordinates (xi, u, qi) in which the contact structure is generated by θ = du − qidxi, and then consider the Monge-Ampere system generated by θ and Ψ = −dqi ∧ dx(i) + Cu n−2 dx. A little experimenting yields a Lagrangian density L dx = so the functional Λ = L dy + θ ∧ Lqi dy(i) induces the Poincar´-Cartan form e Π = θ ∧ Ψ = dΛ. One can then determine the Lie algebra of the symmetry group of Π by solving an elementary PDE system, with a result closely resembling that of Proposition 3.4. Applying the Noether prescription to these vector ﬁelds and this Λ yields (n − 1)-forms which restrict to transverse Legendre submanifolds to give ϕg above. Returning to our original situation, we now compute ϕg explicitly for various choices of g as in Proposition 3.4. These choices of g correspond to subgroups of the conformal group.

||q||2 2 n+2 +

2n n−2 n−2 2n Cu

dx,

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CHAPTER 3. CONFORMALLY INVARIANT SYSTEMS

Translation: g = w ipi . In this case, we have v i = −wi , vi = 0, so we ﬁnd on a transverse Legendre submanifold of J 1 (Rn, R+ ) that ϕg =

2 || n−2

u||2wi −

4 u i u j wj n−2 x x

+

2n 2C n−2 i u w n

dx(i).

The typical use of a conservation law involves its integration along the smooth (n − 1)-dimensional boundary of a region Ω ⊂ Rn. To make more sense of the preceding expression, we take such a region to have unit normal ν and area element dσ (with respect to the Euclidean metric), and using the fact that ∂ qi dx(i)|∂Ω = q, ν dσ for a vector q = q i ∂xi , we have ϕg |∂Ω =

2 n−2 ||

u||2w −

4 n−2

u, w

u+

2n 2C n−2 w, ν n u

dσ.

∂ Here, we have let w = w i ∂xi be the translation vector ﬁeld induced on ﬂat n conformal space R = R ∪ {∞} by the right-invariant vector ﬁeld on P which gives this conservation law.

Rotation: g = ai pixj , ai + aj = 0. j j i In this case, we have v i = −ai xj , vi = aj pj . On a transverse Legendre subj i 2 manifold of J 1(Rn , R+ ), we have from (3.63) that pi = n−2 u−1uxi , and we ﬁnd that ϕg =

2 n−2 ||

u||2 +

2n 2C n−2 n u

a i xj − j

k j 4 i n−2 ux uxk aj x

+

j 2 j n−2 uux ai

dx(i).

In this formula, the last term represents a trivial conservation law; that is, d(uuxj aj dx(i)) = 0 on any transverse Legendre submanifold, so it will be ignored i below. Restricting as in the preceding case to the smooth boundary of Ω ⊂ Rn with unit normal ν and area element dσ, this is ϕg |∂Ω =

2 || n−2

u||2 +

2n 2C n−2 u n

a−

4 n−2

u, a

u, ν dσ.

∂ Here, we have let a = ai xj ∂xi be the rotation vector ﬁeld induced on ﬂat j conformal space R by the right-invariant vector ﬁeld on P which gives this conservation law.

Dilation: g = 1 + xipi . This generating function gives the right-invariant vector ﬁeld whose value at the identity is the Lie algebra element (in blocks of size 1, n, 1) 2 0 0 0 0 0 , 0 0 −2

3.4. CONSERVATION LAWS FOR ∆u = Cu n−2

n+2

117

which generates a 1-parameter group of dilations about the origin in ﬂat conformal space R. In this case, we have v i = −xi , vi = pi , and on a transverse 2 Legendre submanifold with pi = n−2 u−1 uxi , we ﬁnd that ϕg =

2 || n−2

u||2 +

2n 2C n−2 u n

xi −

4 u i u j xj n−2 x x

− 2uuxi dx(i).

(3.65)

For this conservation law, it is instructive to take for Ω ⊂ Rn the open ball of radius r > 0 centered at the origin, and then ϕg |∂Ω = r

2 n−2 ||

u||2 +

2n 2C n−2 n u

−

4 n−2

u, ν

2

− 2u

u, ν

dσ.

(3.66)

A simple consequence of this conservation law is the following uniqueness theorem.5 ¯ Theorem 3.1 (Pohoˇaev) If u(x) ∈ C 2(Ω) is a solution to ∆u = Cu n−2 in z the ball Ω of radius r, with u ≥ 0 in Ω and u = 0 on ∂Ω, then u = 0. Proof. We will ﬁrst use the conservation law to show that u = 0 everywhere on ∂Ω. If we decompose u = uτ + uν ν into tangential and normal components along ∂Ω, so that in particular uτ = 0 by hypothesis, then the conserved integrand (3.66) is 2r ϕg |∂Ω = − n−2 u2 dσ, ν so the conservation law ∂Ω ϕg = 0 implies that uν = 0 on ∂Ω. Now with u = 0 on ∂Ω, we can compute 0 =

∂Ω n+2

∗du

=

Ω

d ∗ du ∆u dx.

=

Ω

But it is clear from the PDE that ∆u cannot change sign, so it must vanish identically, and this implies that u = 0 throughout Ω. In fact, looking at the expression (3.65) for ϕg for a more general region, it is not hard to see that the same proof applies whenever Ω ⊂ Rn is bounded and star-shaped. Inversion: g = − 1 pj bj ||x||2 + bj xj (1 + pi xi ). 2 This is the generating function for the vector ﬁeld in R = Rn ∪{∞} which is the conjugate of a translation vector ﬁeld by inversion in an origin-centered sphere.

5 See [Poh65], where a non-existence theorem is proved for a more general class of equations, for which dilation gives an integral identity instead of a conservation law.

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CHAPTER 3. CONFORMALLY INVARIANT SYSTEMS

1 In this case, we have v i = 2 bi||x||2 −bj xj xi, vi = bixj pj −bj xipj +bj xj pi +bi , and on a transverse Legendre submanifold, we ﬁnd after some tedious calculation that

ϕg

=

2 ( n−2 || u||2 +

2n 2C n−2 )δij n u

−

4 n−2 uxi uxj

(bk xk xj − 1 bj ||x||2) 2

−2ubj xj uxi + u2bi dx(i). Again taking Ω ⊂ Rn to be the open ball of radius r > 0 centered at the origin, we have (n − 2)ϕg |∂Ω = (r2 (−4 u, ν

2

+ || u||2 +

2n C n−2 ) nu

+ u2)b

+2(r2 b,

u − (n − 2)ru b, ν ) u, ν dσ,

∂ where b = bi ∂xi is the vector ﬁeld whose conjugate by a sphere-inversion is the vector ﬁeld generating the conservation law.

3.5

Conservation Laws for Wave Equations z = f(z), (3.67)

In this section, we will consider non-linear wave equations

which are hyperbolic analogs of the non-linear Poisson equations considered previously. Here, we are working in Minkowski space Ln+1 with coordinates (t, y1 , . . . , yn ), and the wave operator is =− ∂ ∂t

2

+

∂ ∂yi

2

.

It is in this hyperbolic case that conservation laws have been most eﬀectively used. Everything developed previously in this chapter for the Laplace operator and Poisson equations on Riemannian manifolds has an analog for the wave operator and wave equations on Lorentzian manifolds, which by deﬁnition carry a metric of signature (n, 1). Indeed, even the coordinate formulae for conservation laws that we derived in the preceding section are easily altered by a sign change to give corresponding conservation laws for the wave equation. Our goal in this section is to see how certain analytic conclusions can be drawn from these conservation laws. Before doing this, we will illustrate the usefulness of understanding the wave operator geometrically, by presenting a result of Christodoulou asserting that the Cauchy problem for the non-linear hyperbolic equation z = z n−1 n+3 (3.68)

3.5. CONSERVATION LAWS FOR WAVE EQUATIONS

119

evidently diﬀeomorphic to S 1 × S n . The (n + 1, 2)-inner-product induces a Lorentz metric on this hypersurface, well-deﬁned up to scaling, and its conformal isometry group has identity component SO o (n + 1, 2), which we will revisit in considering conservation laws. What is important for us now is that among the representative Lorentz metrics for this conformal structure one ﬁnds g = −dT 2 + dS 2 , where T is a coordinate on S 1 , and dS 2 is the standard metric on S n . In certain spherical coordinates (“usual” spherical coordinates applied to Rn, after stereographic projection, this may be written g = −dT 2 + dR2 + (sin2 R) dZ 2,

has solutions for all time, given suﬃciently small initial data.6 The proof exploits conformal invariance of the equation in an interesting way, and this is what we want to explain. Note that (3.68) is the hyperbolic analog of the maximally n+2 symmetric non-linear Poisson equation ∆z = z n−2 considered previously; the change in exponent reﬂects the fact that the number of independent variables is now n+1, instead of n. This equation will be of special interest in our discussion of conservation laws, as well. The idea for proving the long-time existence result is to map Minkowski space Ln+1, which is the domain for the unknown z in (3.68), to a bounded domain, in such a way that the equation (3.68) corresponds to an equation for which short-time existence of the Cauchy problem is already known. With suﬃciently small initial data, the “short-time” will cover the bounded domain, and back on Ln+1 we will have a global solution. The domain to which we will map Ln+1 is actually part of a conformal compactiﬁcation of Minkowski space, analogous to the conformal compactiﬁcation of Euclidean space constructed in §3.1.1. This compactiﬁcation is diﬀeomorphic to a product S 1 × S n , and topologically may be thought of as the result of adding a point at spatial-inﬁnity for each time, and a time-at-inﬁnity for each spatial point. Formally, one can begin with a vector space with inner-product of signature (n + 1, 2), and consider the projectivized null-cone; it is a smooth, real quadric hypersurface in Pn+2, which in certain homogeneous coordinates is given by 2 2 2 2 ξ1 + ξ 2 = η 1 + · · · + η n ,

where R ∈ [0, π], and dZ 2 is the standard metric on the unit (n − 1)-sphere. Now we will conformally embed Minkowski space Ln+1 as a bounded domain in the ﬁnite part R × Rn of S 1 × S n , the latter having coordinates (T, R, Z). The map ϕ(t, r, z) = (T, R, Z) is given by arctan(t + r) + arctan(t − r) T R = arctan(t + r) − arctan(t − r) , z Z

6 See

[Chr86]; what is proved there is somewhat more general.

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CHAPTER 3. CONFORMALLY INVARIANT SYSTEMS

and one can easily check that ϕ∗ (−dT 2 + dR2 + (sin2 R) dZ 2) = Ω2 (−dt2 + dr2 + r2 dz 2), where dZ 2 and dz 2 are both the standard metric on the unit (n − 1)-sphere; the conformal factor is Ω = 2(1 + (t + r)2 )− 2 (1 + (t − r)2 )− 2 , and the right-hand side is a multiple of the ﬂat Minkowski metric. The image of ϕ is the “diamond” D = {(T, R, Z) : R − π < T < π − R, R ≥ 0}. Note that the initial hyperplane {t = 0} corresponds to {T = 0}, and that with ﬁxed (R, Z), as T → π − R, t → ∞. Consequently, the long-time Cauchy problem for the invariant wave equation (3.68) corresponds to a short-time Cauchy problem on the bounded domain D for some other equation. We can see what this other equation is without carrying out tedious calculations by considering the conformally invariant wave operator, an analog of the conformal Laplacian discussed in §3.1.3. This is a diﬀerential operator c 1 1

: Γ(D 2(n+1) ) → Γ(D 2(n+1) )

n−1

n+3

between certain density line bundles over a manifold with Lorentz metric. With n−1 a choice of Lorentz metric g representing the conformal class, u ∈ Γ(D 2(n+1) ) is represented a function ug , and the density c u is represented by the function ( c u)g

=

g (ug )

+

n−1 4n Rg ug ,

where Rg is the scalar curvature and g is the wave operator associated to g. We interpret our wave equation (3.68) as a condition on a density represented in the ﬂat (Minkowski) metric g0 by the function u0 , and the equation transformed by the map ϕ introduced above should express the same condition represented in the new metric g. The representative functions are related by ug = Ω − n−1 2

u0 ,

(

c u)g

= Ω−

n+3 2

(

c u)0 ,

so the condition (3.68) becomes g (ug )

+

n−1 R g ug 4n

= = = =

( Ω Ω

c u)g − n+3 2 − n+3 2

( u0

c u)0 n+3 n−1

ug

n+3 n−1

.

The scalar curvature is just that of the round metric on the n-sphere, Rg = n(n − 1), so letting u = u0, U = ug , the equation (3.68) is transformed into gU +

(n−1)2 U 4

= U n−1 .

n+3

(3.69)

3.5. CONSERVATION LAWS FOR WAVE EQUATIONS

121

Finally, suppose given compactly supported initial data u(0, x) = u0 (x) and ut (0, x) = u1(x) for (3.68). These correspond to initial data U0 (X) and U1(X) for (3.69), supported in the ball of radius π. The standard result on local existence implies that the latter Cauchy problem can be solved for all X, in some time interval T ∈ [0, T0], with a lower-bound for T0 determined by the size of the initial data. Therefore, with suﬃciently small initial data, we can arrange T0 ≥ π, and translated back to the original coordinates, this corresponds to a global solution of (3.68). We now turn to more general wave equations (3.67), where conservation laws have been most eﬀectively used.7 Equation (3.67) is the Euler-Lagrange equation for the action functional

2 1 2 (−zt

R

Rn

+ || z||2) + F (z) dy dt,

where F (z) = f(z), and the gradient z is with respect to the “space” variables y1 , . . . , yn . Rather than redevelop the machinery of conformal geometry in the Lorentz case, we work in the classical setting, on J 1 (Ln+1 , R) with coordinates t, y i , z, pa (as usual, 1 ≤ i ≤ n, 0 ≤ a ≤ n), contact form θ = dz − p0 dt − pidyi , Lorentz inner-product ds2 = −dt2 + (dyi )2 , and Lagrangian L(t, y, z, p) = 1 (−p2 + 0 2 |pi|2) + F (z).

A normalized representative functional is then Λ = L dt ∧ dy + θ ∧ (−p0 dy − pi dt ∧ dy(i) ), satisfying dΛ = Π = θ ∧ (dp0 ∧ dy + dpi ∧ dt ∧ dy(i) + f(z)dt ∧ dy). This is the example discussed at the end of §1.3. As mentioned there, the invariance of the equation under time-translation gives an important conservation law, and its uses will be our ﬁrst topic below. In fact, there are conservation laws associated to space-translations and Lorentz rotations, the latter generated by ∂ ∂ ∂ ordinary spatial rotations bi yj ∂yj (bi + bj = 0) and Lorentz boosts y i ∂t + t ∂yi ; j j i however, these seem to have been used less widely in the analysis of (3.67). Especially interesting is the case of (3.68), which is preserved under a certain action of the conformal Lorentz group SO 0 (n + 1, 2) on J 1 (Ln+1, R). In particular, there are extensions to J 1(Ln+1 , R) of the dilation and inversion vector ﬁelds on Ln+1 , and these give rise to more conservation laws. We will consider these after discussing uses of the time-translation conservation law for the more general wave equations (3.67).

7 This

(3.70)

material and much more may be found in [Str89].

122

CHAPTER 3. CONFORMALLY INVARIANT SYSTEMS

3.5.1

Energy Density

∂ The time-translation vector ﬁeld ∂t on Ln+1 lifts to J 1 (Ln+1 , R) to a symmetry ∂ of Λ having the same expression, V = ∂t . The Noether prescription gives

ϕt = V

Λ=

1 2 (p 2 0

+

p2 ) + F (z) dy + p0 pi dt ∧ dy(i) , i

as calculated in Section 1.3. The coeﬃcient of dy here is the energy density e = def 1 2 (p 2 0

+ |pi|2) + F (z),

and it appears whenever we integrate ϕt along a constant-time level surface Rn = {t} × Rn . The energy function t E(t) = def Rn t

e dy ≥ 0,

is constant by virtue of the equation (3.67), assuming suﬃcient decay of z and its derivatives in the space variables for the integral to make sense. A more substantial application involves a region Ω ⊂ Ln+1 of the form Ω= t∈(t0 ,t1 )

{||y|| < r0 − (t − t0 )},

a union of open balls in space, with initial radius r0 decreasing with speed 1. The boundary ∂Ω is T − B + K, where • B = {t0 } × {||y|| ≤ r0} is the initial disc, • T = {t1} × {||y|| ≤ r0 − (t1 − t0 )} is the ﬁnal disc, and • K = ∪t∈[t0 ,t1] {||y|| = r0 − (t − t0)} is part of a null cone. The conservation of ϕt on ∂Ω reads 0=

∂Ω

ϕt =

T −B

e dy +

K

ϕt .

(3.71)

The term K ϕt describes the ﬂow of energy across part of the null cone; we will compute the integrand more explicitly in terms of the area form dK induced from an ambient Euclidean metric dt2 + (dyi )2 , with the goal of showing that K ϕt ≥ 0. This area form is the contraction of the outward unit normal

1 √ ( ∂ 2 ∂t

+

yi ∂ ||y|| ∂y i )

with the ambient Euclidean volume form dt ∧ dy, giving dK =

1 √ (dy 2

−

yi dt ∧ ||y||

dy(i) )|K =

√ 2 dy|K .

It is easy to calculate that the restriction to K of ϕt is ϕ t |K =

1 √ (e 2

−

y i pi p )dK. ||y|| 0

3.5. CONSERVATION LAWS FOR WAVE EQUATIONS Separating the radial and tangential space derivatives pr = we can rewrite this as ϕ t |K = =

1 √ ( 1 (p2 2 2 0 1 √ ( 1 (p0 2 2 def

123

y i pi , ||y||

pτ =

def

p2 − p 2 , r i

+ p2 + p2 ) + F (z) − pr p0)dK r τ − pr )2 + 1 p2 ) + F (z))dK. 2 τ

(3.72) (3.73)

In the region of J 1 (Ln+1, R) where F (z) ≥ 0 this integrand is positive, and from (3.71) we obtain the bound e dy ≤ e dy.

B

T

This says that “energy travels with at most unit speed”—no more energy can end up in T than was already present in B. If F (z) ≥ 0 everywhere, then one can obtain another consequence of the expression (3.73) by writing ϕt =

K B

ϕt −

T

ϕt ≤

B

ϕt ≤ E(t0 ) = E.

This gives an upper bound for ||(dz|K )||2 2 = L √ ((p0 − pr )2 + p2 )dK ≤ 2 2E τ

K

which holds for the entire backward null cone; that is, our bound is independent of t0. Here the L2 -norm is with respect to Euclidean measure. We should also mention that the spatial-translation and Lorentz rotations give rise to conserved quantities that may be thought of as linear and angular momenta, respectively. The uses of these are similar to, though not as extensive as, the uses of the conserved energy.

3.5.2

The Conformally Invariant Wave Equation

We now determine some additional conservation laws for the conformally invariant wave equation (3.68). Again, we could duplicate the process used for Poisson equations by calculating restrictions of the right-invariant vector ﬁelds of SOo (n + 1, 2) to the image of an embedding J 1 (Ln+1, R+ ) → SOo (n + 1, 2), and contracting with the left-invariant Lagrangian. Instead, we will illustrate the more concrete, coordinate-based approach, though we will still make some use of the geometry. The Dilation Conservation Law To ﬁnd the conservation law corresponding to dilation symmetry of (3.68), we have to ﬁrst determine a formula for this symmetry on J 1 (Ln+1, R+ ), and then

124

CHAPTER 3. CONFORMALLY INVARIANT SYSTEMS

and the dilation matrix (in blocks of size 1, 1, n, 1) acts projectively on this slice of the null-cone by −1 1 1 r 0 0 0 2 def 0 n−1 t tr 1 0 0 n−1 . · z 2 = z r i 0 yi yr 0 In 0 1 2 2 1 0 0 0 r 2 ||(tr , yr )|| ) 2 ||(t, y)|| Taking the derivative with respect to r and setting r = 1 gives the vector ﬁeld

∂ ∂ ∂ ¯ Vdil = − n−1 z ∂z + t ∂t + yi ∂yi . 2

apply the Noether prescription. For this, we will ﬁrst determine the vector ﬁeld’s action on J 0 (Ln+1, R+ ); the lift of this action to J 1(Ln+1 , R+) is determined by the requirement that it preserve the contact line bundle. By analogy with (3.30), we have an embedding of J 0(Ln+1 , R+ ) into the null-cone of Ln+1,2 given by 1 2 t , (t, yi , z) → z n−1 i y 2 2 1 2 (−t + |y| )

The scaling in the z-coordinate reﬂects an interpretation of the unknown z(t, y) ¯ as a section of a certain density line bundle. We then ﬁnd the lift from Vdil ∈ V(J 0 (Ln+1 , R+)) to Vdil ∈ V(J 1 (Ln+1, R+ )) by the requirement that the contact form θ = dz − p0 dt − pi dyi be preserved up to scaling; that is, ¯ Vdil = Vdil + v0 p0 + vi pi must satisfy where v0 and vi are the unknown coeﬃcients of the lift. This simple calculation yields ∂ ∂ ∂ ∂ Vdil = − n−1 z ∂z + t ∂t + yi ∂yi − n+1 pa ∂pa . 2 2 Then one can compute even for the general wave equation (3.67) that LVdil (L(t, y, z, p)dt ∧ dy) = ((n + 1)F (z) − n−1 2 zf(z))dt

LVdil θ ≡ 0 (mod {θ}),

∧ dy.

Tentatively following the Noether prescription for the general wave equation, we set ϕdil = Vdil Λ, where Λ is given by (3.70), and because Λ ≡ L dt ∧ dy modulo {I}, we can calculate dϕdil = ≡ = LVdil Λ − Vdil Π LVdil (L dt ∧ dy) − 0 (mod EΛ ) ((n + 1)F (z) − n−1 zf(z))dt ∧ dy 2

(mod EΛ ).

3.5. CONSERVATION LAWS FOR WAVE EQUATIONS

125

The condition on the equation (3.67) that ϕdil be a conservation law is therefore F (z) = Cz so the PDE is

2(n+1) n−1

,

z = C z n−1 , n−1 as expected (cf. (3.68)); we work with C = 1, C = 2(n+1) . Now, one can calculate that restricted to any Legendre submanifold the conserved density is

n+3

ϕdil = L(t, y, z, p)(t dy − y i dt ∧ dy(i)) + ( n−1 z + tp0 + yi pi )(p0 dy + pj dt ∧ dy(j) ). 2 Typically, one considers the restriction of this form to the constant-time hyperplanes Rn = {t} × Rn , which is t ϕdil ≡ (te + rp0 pr + d dt n−1 2 zp0 )dy

(mod {dt}).

For example, we ﬁnd that for solutions to (3.68) with compact support in y i , (te + rp0 pr + n−1 2 zp0 )dy

= 0.

(3.74)

Rn t

For more general wave equations (3.67), an identity like (3.74) holds, but with a non-zero right-hand side; our conservation law is a special case of this. The general dilation identity is of considerable use in the analysis of non-linear wave equations. It is analogous to the “almost-conservation law” derived from scaling symmetry used to obtain lower bounds on the area growth of minimal surfaces, as discussed in §1.4.3. An Inversion Conservation Law We now consider the inversion symmetry corresponding to the conjugate of time-translation by inversion in a unit (Minkowski) “sphere”. We will follow the same procedure as for dilation symmetry, ﬁrst determining a vector ﬁeld on J 0 (Ln+1, R+ ) generating this inversion symmetry, then lifting it to a contactpreserving vector ﬁeld on J 1 (Ln+1, R+ ), and then applying the Noether prescription to obtain the conserved density. The conjugate by sphere-inversion of a time-translation in SO o (n + 1, 2) is the matrix 1 b 0 1 b2 2 0 1 0 −b , 0 0 In 0 0 0 0 1 and diﬀerentiating its projective linear action on 1 2 t z n−1 i y 2 1 2 ||(t, y)||

126

CHAPTER 3. CONFORMALLY INVARIANT SYSTEMS

yields the vector ﬁeld ¯ Vinv = n−1 ∂ tz ∂z 2 ∂ ∂ − 1 (t2 + |y|2) ∂t − tyi ∂yi . 2

∂ ∂ Again, the coeﬃcients of ∂t and ∂yi describe an inﬁnitesimal conformal motion of Minkowski space—representing an element of the Lie algebra of the Lorentz ∂ conformal group—and the coeﬃcient of ∂z gives the induced action on a density ∂ ∂ ¯ line bundle. We now look for coeﬃcients v0 ∂p0 + vi ∂pi to add to Vinv so that the new vector ﬁeld will preserve θ up to scaling, and the unique solution is

¯ Vinv = Vinv +

n−1 2 z

+ pi y i +

n+1 2 tp0

∂ ∂p0

+ p0 y i +

n+1 2 tpi

∂ ∂pi .

In applying the Noether prescription to Λ and Vinv , it will turn out that we need a compensating term, because LVinv Λ ≡ 0 modulo EΛ . However, instead of performing this tedious calculation, we can simply test ˜ ϕinv def = ≡

−Vinv Λ L(t, y, z, p)( 1 (t2 + |y|2 )dy − tyi dt ∧ dy(i) ) 2

+( n−1 tz + 1 p0 (t2 + |y|2 ) + tpiyi ) ∧ (p0 dy + pj dt ∧ dy(j) ), 2 2 dϕinv = ˜ n−1 2 p0 z

We ﬁnd that on solutions to (3.68), dt ∧ dy = d( n−1 z 2 dy), 4 n−1 2 4 z dy.

and we therefore set ϕinv = ϕinv − ˜

As usual, we consider the restriction of this form to a hyperplane Rn = {t}×Rn, t which gives ϕinv ≡

1 2 (L

+ p2 )(t2 + |y|2 ) + 0

n−1 2 p0 tz

+ tp0pi yi −

n−1 2 4 z

dy,

modulo {dt, θ}. Again, for more general wave equations (3.67), this quantity gives not a conservation law, but a useful integral identity. The usefulness of the integrand follows largely from the fact that after adding an exact n-form, the coeﬃcient of dy is positive. One notices this by expanding in terms of radial and tangential derivatives ϕinv = 1 (p2 + p2 + p2)(t2 + r2) + 2(n − 1)p0tz + 4trp0pr − (n − 1)z 2 0 τ r 4 1 + (t2 + r2)F (z) dy, 2 (3.75) 1 |p0y + tp|2 + (tp0 + rpr + (n − 1)z)2 + (rpτ )2 4 1 n−1 + (t2 + r2)F (z) − (nz 2 + 2rzpr ) dy. 2 4

which suggests completing squares: ϕinv =

(3.76)

3.5. CONSERVATION LAWS FOR WAVE EQUATIONS

127

The last term is the divergence d( n−1 yi z 2 dy(i) ) modulo {dt, θ}, and the remain4 ing terms are positive. The positive expression ec = def 1 2

|p0y + tp|2 + (tp0 + rpr + (n − 1)z)2 + (rpτ )2 + (t2 + r2)F (z)

is sometimes called the conformal energy of the solution z(t, y). It is also sometimes convenient to set ed = rp0pr + so that ϕdil ≡ (ed + te)dy and also ϕinv ≡ (ted + 1 (t2 + |y|2 )e − 2 n−1 2 4 z )dy n−1 zp0 2

(mod {dt}), (mod {dt}).

The fact that the integrals of these quantities are constant in t yields results about growth of solutions. In fact, in the analysis of wave equations that are perturbations of the conformally invariant equation (3.68), the most eﬀective estimates pertain to the quantities appearing in these conservation laws. One can think of the conservation laws as holding for our “ﬂat” non-linear wave equation (3.68), and then the estimates are their analogs in the “curved” setting.

3.5.3

Energy in Three Space Dimensions

We conclude this section by discussing a few more properties that involve the energy in three space dimensions. The fact that the energy E(t) is constant implies in particular that Rn zt dy t is bounded with respect to t. This allows us to consider the evolution of the spatial L2 -norm of a solution to z = z 3 as follows: d2 dt2 z 2 dy

R3 t

= = = ≤

2

R3 t

2 (z ztt dy + zt )dy 2 (z∆z − z 4 + zt )dy

2

R3 t

−2 4E,

(|

R3 t

y z|

2

+ z 4)dy + 2

Rn t

2 zt dy

where the second equality follows from the diﬀerential equation and the third from Green’s theorem (integration by parts). The conclusion is that ||z||2 2(Rn ) L t grows at most quadratically, z 2dy ≤ 2Et2 + C1 t + C2 , (3.77)

R3 t

128 and in particular

CHAPTER 3. CONFORMALLY INVARIANT SYSTEMS z 2 dy = O(t2 ).

The energy plays another interesting role in the equation z = −z 3 Here, it is possible for the energy to be negative: E=

R3 t 1 2 (z 2 t

(3.78)

+ || z||2) −

z4 4

dy.

(3.79)

We will prove that a solution to (3.78), with compactly supported initial data z(0, y), zt (0, y) satisfying E < 0, must blow up in ﬁnite time ([Lev74]). Notice that any non-trivial compactly supported initial data may be scaled up to achieve E < 0, and may be scaled down to achieve E > 0. The idea is to show that the quantity I(t) = becomes unbounded as t its derivatives I (t) I (t) = = = def R3 t 1 2 z dy 2

T for some ﬁnite time T > 0. We start by computing

zzt dy,

2 zt dy + 2 zt dy −

zztt dy | z|2dy + z 4 dy.

The last step uses Green’s theorem, requiring the solution to have compact ysupport for each t ≥ 0. To dispose of the z 4 term, we add 4E to each side using (3.79): I (t) + 4E = 3

2 zt dy +

| z|2dy.

We can discard from the right-hand side the positive gradient term, and from the left-hand side the negative energy term, to obtain I (t) > 3

2 zt dy.

To obtain a second-order diﬀerential inequality for I, we multiply the last inequality by I(t) to obtain I(t)I (t) > ≥ 3 2

3 I 2

z 2dy (t)2 .

2 zt dy

3.5. CONSERVATION LAWS FOR WAVE EQUATIONS

129

The last step follows from the Cauchy-Schwarz inequality, and says that I(t)−1/2 has negative second derivative. We would like to use this to conclude that I −1/2 vanishes for some T > 0 (which would imply that I blows up), but for this we would need to know that (I −1/2 ) (0) < 0, or equivalently I (0) > 0, which may not hold. To rectify this, we shift I to J(t) = I(t) − 1 E(t + τ )2 , 2 with τ > 0 chosen so that J (0) > 0. We now mimic the previous reasoning to show that (J −1/2) (t) < 0. We have J (t) J (t) = = = 3 > 3 From this we obtain J(t)J (t) − 3 J (t)2 > 2

3 2

zzt dy − E(t + τ ),

2 zt dy + 2 zt dy +

zztt dy − E || z||2dy − 5E

2 zt dy − E .

z 2 − E(t + τ )2

2 zt − E −

zzt − E(t + τ )

2

,

which is positive, again by the Cauchy-Schwarz inequality. This means that (J −1/2 ) (t) < 0. Along with J −1/2(0) > 0 and (J −1/2 ) (0) < 0, this implies that for some T > 0, J −1/2(T ) = 0, so J(t) blows up. We conclude by noting that the qualitative behavior of solutions of z = f(z) depends quite sensitively on the choice of non-linear term f(z). In contrast to the results for z = ±z 3 described above, we have for the equation z = −z 2 (n = 3),

that every solution must blow up in ﬁnite time ([Joh79]). We will outline the proof in case the initial data are compactly supported and satisfy u(0, t)dy > 0, ut(0, t)dy > 0.

Note that replacing z by −z gives the equation z = z 2 , which therefore has the same behavior. This proof is fairly similar to the previous one; we will derive diﬀerential inequalities for J(t) = def Rn t

z dy

130

CHAPTER 3. CONFORMALLY INVARIANT SYSTEMS

which imply that this quantity blows up. We use integration by parts to obtain J (t) = ztt dy = (∆z + z 2)dy = z2,

and using H¨lder’s inequality on Supp z ⊂ {|y| ≤ R0 + t} in the form ||z||L1 ≤ o ||z||L2 ||1||L2 , this gives

2

J (t) ≥ C

z dy

(R0 + t)−3 ≥ C(1 + t)−3 J(t)2 .

(3.80)

This is the ﬁrst ingredient. Next, we use the fact that if z0 (y, t) is the free solution to the homogeneous wave equation z0 = 0, with the same initial data as our z, then z(y, t) ≥ z0 (y, t) for t ≥ 0; this follows from a certain explicit integral expression for the solution. Note that if we set J0(t) = Rn z0 , then it follows from the equation alone that t J0 (t) = 0, and by the hypotheses on our initial data we have J0 (t) = C0 + C1 t with C0, C1 > 0. Another property of the free solution is that its support at time t lies in the annulus At = {t − R0 ≤ |y| ≤ t + R0}. Now C0 + C 1 t ≤ ≤ ≤ ≤ This gives J (t) = z 2 dy ≥ C0 + C 1 t C(1 + t)

1/2

z dy

At

||z||L1(A) ||z||L2(A) ||1||L2(A) C(1 + t) z 2dy

2

.

,

and in particular, J > 0. With the assumptions on the initial data, this gives J (t) J(t) > 0 ≥ C(1 + t)2.

We can use (3.80, 3.81, 3.81) to conclude that J must blow up at some ﬁnite time. This follows by writing J (t) ≥ ≥ C(1 + t)−3 J 3/2J 1/2 C(1 + t)−2 J 3/2.

Multiply by J and integrate to obtain J (t) ≥ C(1 + t)−1 J(t)5/4 .

3.5. CONSERVATION LAWS FOR WAVE EQUATIONS Integrating again, we have J(t) ≥ J(0)−1/4 − 1 C ln(1 + t) 4

−4

131

,

and because J(0) > 0 and C > 0, J(t) must blow up in ﬁnite time.

132

CHAPTER 3. CONFORMALLY INVARIANT SYSTEMS

Chapter 4

Additional Topics

4.1 The Second Variation

In this section, we will discuss the second variation of the Lagrangian functionals considered in the preceding chapters. We begin by giving an invariant, coordinate-free calculation of the formula (4.8) for the second derivative of a functional under ﬁxed-boundary variations. This formula has an interpretation in terms of conformal structures induced on integral manifolds of the EulerLagrange system, which we will describe. The role played by conformal geometry here is not to be confused with that in the previous chapter, although both situations seem to reﬂect the increasing importance of variational equations in conformal geometry. The usual integration by parts that one uses to establish local minimality of a solution to the Euler-Lagrange equations cannot generally be done in an invariant manner, and we discuss a condition under which this diﬃculty can be overcome. We considered in §2.5 the example of prescribed mean curvature hypersurfaces in Euclidean space; we will give an invariant calculation of the second variation formula and the integration by parts for this example. We conclude by discussing various classical conditions under which an integral manifold of an Euler-Lagrange system is locally minimizing, using the Poincar´-Cartan e form to express and prove some of these results in a coordinate-free manner.

4.1.1

A Formula for the Second Variation

We start by reconsidering the situation of §1.2, in which we calculated the ﬁrst variation of a Lagrangian Λ on a contact manifold. This amounted to taking the ﬁrst derivative, at some ﬁxed time, of the values of the functional FΛ on a 1-parameter family of Legendre submanifolds. Our goal is to extend the calculation to give the second variation of Λ, or equivalently, the second derivative of FΛ on a 1-parameter family at a Legendre submanifold for which the ﬁrst variation vanishes. The result appears in formula (4.5) below, and in a more geometric form in (4.8). This process is formally analogous to computing 133

134

CHAPTER 4. ADDITIONAL TOPICS

the Hessian matrix of a smooth function f : Rn → R at a critical point, which is typically done with the goal of identifying local extrema. Let (M, I) be a contact manifold, with contact form θ ∈ Γ(I), and Lagrangian Λ ∈ Ωn (M ) normalized so that the Poincar´-Cartan form is given by e Π = dΛ = θ ∧ Ψ. Fix a compact manifold N n with boundary ∂N , and a smooth map F : N × [0, 1] → M which is a Legendre submanifold Ft for each ﬁxed t ∈ [0, 1] and is independent of t on ∂N × [0, 1]. Two observations will be important: • F ∗ θ = G dt for some function G on N × [0, 1], depending on the choice of generator θ ∈ Γ(I). This holds because each Ft is a Legendre submanifold, meaning that Ft∗ θ = 0. • For every form α ∈ Ω∗ (M ), and every boundary point p ∈ ∂N , we have

∂ ∂t

F ∗α (p, t) = 0.

This is equivalent to the ﬁxed-boundary condition; at each p ∈ ∂N , we ∂ have F∗ ( ∂t ) = 0. We previously calculated the ﬁrst variation as (see §1.2.2) d dt

Nt

Ft∗ Λ

=

Nt

G · Ft∗ Ψ,

where Π = dΛ = θ ∧ Ψ is the Poincar´-Cartan form for Λ. This holds for each e t ∈ [0, 1]. We now assume that F0 is stationary for Λ; that is, F is a Legendre variation of an integral manifold F0 : N → M of the Euler-Lagrange system EΛ = {θ, dθ, Ψ}. This is the situation in which we want to calculate the second derivative: δ 2 (FΛ)N0 (g) = = =

N0

d2 dt2 t=0 d dt t=0

∂t

Nt

Ft∗ Λ

Nt

G Ft∗Ψ

L ∂ (G F ∗Ψ) g L ∂ (F ∗ Ψ),

∂t

=

N0

∗ where g = G|t=0, and the last step uses the fact that F0 Ψ = 0. ∗ To better understand the Lie derivative L ∂ (F Ψ), we use the results ob∂t tained via the equivalence method in §2.4. This means that we are restricting

4.1. THE SECOND VARIATION

135

our attention to the case n ≥ 3, with a Poincar´-Cartan form that is neoe classical and deﬁnite. In this situation, we have a G-structure B → M , where G ⊂ GL(2n + 1, R) has Lie algebra 0 0 (n − 2)r j i : ai + ai = sij − sji = sii = 0 . 0 −2rδj + ai 0 g= j j j di sij nrδi − aj i (4.1) The sections of B → M are local coframings (θ, ω i , πi) of M for which: • θ generates the contact line bundle I, • the Poincar´-Cartan form is Π = −θ ∧ πi ∧ ω(i), e • there exists a g-valued 1-form (n − 2)ρ 0 i −2ρδj + αi 0 ϕ= j δi σij 0 , 0 j j nρδi − αi

(4.2)

where

satisfying a structure equation −πi ∧ ωi θ θ Ωi d ωi = −ϕ ∧ ωj + 0 πj πi i Ωi = T ijkπj ∧ ωk − (Sj ωj + U ij πj ) ∧ θ,

(4.3)

(4.4)

j i i with T ijk = T jik = T kji, T iik = 0; U ij = U ji; Sj = Si , Si = 0. The pseudo-connection form ϕ may be chosen so that also (cf. (2.48)) i (n − 2)dρ = −δi ∧ ωi − Sj πi ∧ ωj + ( n−2 U ij σij − ti πi) ∧ θ 2n

for some functions ti. For any point p ∈ N , we consider a neighborhood U ⊂ M of F0 (p) on which we can ﬁx onesuch coframing (θ, ω i , πi) with pseudo-connection ϕ. All of the forms and functions may be pulled back to W = F −1(U ) ⊂ N × [0, 1], which is the setting for the calculation of L ∂ (F ∗ Ψ). From now on, we drop all F ∗s. ∂t We have Ψ = −πi ∧ ω(i) , and we now have the structure equations needed to diﬀerentiate Ψ, but it will simplify matters if we further adapt the forms (θ, ωi , πi) on W in a way that does not alter the structure equations. Note ﬁrst that restricted to each Wt = W ∩ Nt , we have ωi = 0, so (ω1 , . . . , ωn, dt) forms a coframing on W . We can therefore write πi = sij ωj + gi dt for some sij , gi , and because each Wt is Legendre, we must have sij = sji . The structure

136

CHAPTER 4. ADDITIONAL TOPICS

group of B → M admits addition of a traceless, symmetric combination of the ωj to the πi , so we replace πi ; π i − s o ω j , ij

1 i where s0 = sij − n δj skk is the traceless part. Now we have ij

πi = sωi + gi dt for some functions s, gi on W , so that Ψ = −πi ∧ ω(i) = −ns ω − gidt ∧ ω(i) . Note that because F0 is assumed integral for the Euler-Lagrange system, we have s = 0 everywhere on W0 ⊂ W ; in particular, πi |W0 = 0. With our choice of πi , recalling that along W0, θ = g dt for some function g on W0 , and keeping in mind that restricted to W0 we have πi = 0 and s = 0, we can calculate L∂ Ψ

∂t

t=0

= =

∂ − ∂t ∂ ∂t

∂ d(πi ∧ ω(i) ) − d( ∂t

(πi ∧ ω(i) ))

j (δi ∧ θ + (nρδi − αj ) ∧ πj ) ∧ ω(i) i

=

−(gδi + dgi + nρgi −

∂ −d (gi + s ∂t

+(sωi + gi dt) ∧ dω(i) g j αj ) i

∂ ωi )ω(i) − πi ∧ ( ∂t

ω(i))

∧ ω(i) .

This gives our desired formula: d2 dt2 Λ t=0 Nt

=−

N0

g(dgi + nρgi − gj αj + gδi ) ∧ ω(i) . i

(4.5)

Unfortunately, in its present form this is not very enlightening, and our next task is to give a geometric interpretation of the formula.

4.1.2

Relative Conformal Geometry

It is natural to ask what kind of geometric structure is induced on an integral manifold f : N → M of an Euler-Lagrange system EΛ . What we ﬁnd is: If Π = dΛ is a deﬁnite, neo-classical Poincar´-Cartan form, then an e integral manifold N of its Euler-Lagrange system EΛ has a natural conformal structure, invariant under symmetries of (M, Π, N ), even though there may be no invariant conformal structure on the ambient M. This is a simple pointwise phenomenon, in the sense that any n-plane V n ⊂ Tp M 2n+1 on which ωi = 0 has a canonical conformal inner-product deﬁned as follows. Taking any section (θ, ω i , πi) of B → M , we can restrict the induced

4.1. THE SECOND VARIATION

137

quadratic form (ωi )2 on T M to V ⊂ Tp M , where it is positive deﬁnite, and then the action of the structure group (4.1) on (ω i ) shows that up to scaling, this quadratic form is independent of our choice of section. Alternatively, one can show this inﬁnitesimally by using the structure equations to compute on B the Lie derivative of (ωi )2 along a vector ﬁeld that is vertical for B → M ; this Lie derivative is itself multiple of (ωi )2 . Note that we have not restricted to the conformal branch of the equivalence problem, characterized by T ijk = i U ij = Sj = 0 and discussed in §3.3. In particular, any integral manifold f : N → M for the Euler-Lagrange system EΛ inherits a canonical conformal structure [ds2]f . We now want to develop the conformal structure equations for (N, [ds2 ]f ), in terms of the structure equations on B → M , and our procedure will work only for integral manifolds of EΛ . We ﬁrst note that along our integral manifold N we can choose local sections (θ, ωi , πi) of BN which are adapted to N in the sense that Tp N = {θ, π1, · · · , πn}⊥ ⊂ Tp M, for each p ∈ N . In fact, such sections deﬁne a reduction Bf → N of the principal bundle BN → N , having Lie algebra deﬁned as in (4.1) by sij = 0. Restricted to Bf , we have the same structure equations as on B, but with θ = πi = 0, and dθ = dπi = 0. Now observe that two of our structure equations restrict to give dωi = i (2ρδj − αi ) ∧ ωj , j

dρ =

1 − n−2 δi ∧ ωi .

We therefore have a situation similar to that in §3.3 (cf. (3.40, 3.41)), with equations formally like those in the conformal geometry equivalence problem of §3.1.2. We can mimic the derivation of conformal structure equations in the 2 present case by ﬁrst setting βi = n−2 δi , and then we know that this results in an equation

1 dαi + αi ∧ αk + βi ∧ ωj − βj ∧ ωi = 2 Ai ωk ∧ ωl , jkl j k j

(4.6)

where the quantity (Ai ) has the symmetries of the Riemann curvature tensor. jkl Furthermore, we know that there are unique functions tij = tji such that replacing βi ; βi + tij ωj will yield the preceding equation with Al = 0. However, jkl it will simplify matters later if we go back and replace instead δi ; δ i + n−2 o j 2 tij ω ,

1 where to = tij − n δij tkk is the traceless part (note that only a traceless addition ij to δi will preserve the structure equations on B → M ). In terms of the new δi , we deﬁne

βi =

def

2 (δ n−2 i

1 − n Kωi ),

(4.7)

138 where

CHAPTER 4. ADDITIONAL TOPICS

This K was chosen so that deﬁning βi by (4.7) gives the correct conformal structure equation (4.6) with Al = 0; it reﬂects the diﬀerence between the jkl pseudo-connection forms for the Poincar´-Cartan form and the Cartan cone nection forms for the induced conformal geometry on the submanifold. One interpretation is the following. The function K on Bf is a fundamental invariant of a stationary submanifold (N, [ds2]f ) → (M, Π) of an Euler-Lagrange system, and may be thought of as an extrinsic curvature depending on up to third derivatives of the immersion f. In the classical setting, N is already the 1-jet graph of a solution z(x) of an EulerLagrange equation, so an expression for K(x) depends on fourth derivatives of z(x). Now suppose that our integral manifold f : N → M , with δi , βi , and K as above, is the initial manifold F0 = f in a Legendre variation F : N ×[0, 1] → M . Then we can rewrite our formula (4.5) for the second variation as δ 2 (FΛ )N0 (g) = − g(dgi + nρgi − gj αj + i n−2 2 gβi )

K = − n−2 tjj . 2

def

N0

∧ ω(i) + g2 Kω .

Part of this integrand closely resembles the expression (3.19) for the second covariant derivative of a section of a density line bundle, discussed in constructing the conformal Laplacian in §3.1.3. This suggests the following computations. First, consider the structure equation dθ = −(n − 2)ρ ∧ θ − πi ∧ ωi . Along W0 (but not yet restricted to W0 ), where θ = g dt and πi = gidt, this reads dg ∧ dt = −(n − 2)ρ ∧ g dt − gidt ∧ ωi , so that restricted to W0 , we have dg + (n − 2)ρg = giωi . This equation should be interpreted on Bf , which is identiﬁed with the principal bundle associated to the conformal structure [ds2]f . It says that g is a section n−2 of the density bundle D 2n , and that gi are the components of its covariant derivative (see (3.18) ﬀ.). We can now write dgi + nρgi − gj αj + i and by deﬁnition where ∆f is the conformal Laplacian on N induced by f = F0 : N → M . We now have a more promising version of the second variation formula: δ 2 (FΛ )N0 (g) = − (g∆f g + g2 K)ω.

N0 n−2 gβi 2 n+2 2n

= gij ωj , ),

∆f g = gii ∈ Γ(D

(4.8)

4.1. THE SECOND VARIATION

139

It is worth noting that the sign of this integrand does not depend on the sign of the variation’s generating function g, and if we ﬁx an orientation of N , then the integrand Kω on Bf has a well-deﬁned sign at each point of N .

4.1.3

Intrinsic Integration by Parts

In order to detect local minima using the second variation formula (4.8), it is often helpful to convert an integral like g∆g dx into one like − || g||2dx. In the Euclidean setting, with either compact supports or with boundary terms, this is done with integration by parts and is straightforward; the two integrands diﬀer by the divergence of g g, whose integral depends only on boundary data. We would like to perform a calculation like this on N , for an arbitrary Legendre variation g, using only intrinsic data. In other words, we would like to associate to any N and g some ξ ∈ Ωn−1(N ) such that dξ is the diﬀerence between (g∆f g)ω and some quadratic expression Q( g, g)ω, possibly with some additional zero-order terms. A natural expression to consider, motivated by the ﬂat case, is ξ = ggi ω(i) . (4.9)

Here, in order for gi to make sense, we are assuming that we have a local coframing on M adapted to the integral manifold N , so that at points of N , θ = g dt, πi = gi dt, and so that restricted to N , θ|N = πi |N = 0. We can then compute: dξ = dg ∧ gi ω(i) + g dgi ∧ ω(i) + ggi dωj ∧ ω(ij) = (gi )2 + g∆f g ω − n−2 2 2 g βi

(4.10) g j αj ) i ∧ ω(i) (4.11) (4.12)

= (dg + (n − 2)ρg) ∧ giω(i) + g(dgi + nρgi − ∧ ω(i) .

Now, the ﬁrst term is exactly what we are looking for, and second is fairly harmless because it is of order zero in the variation g, and in practice contributes only terms similar to g 2 Kω. The problem is that ξ (4.9) is deﬁned on the total space Bf → N , and although semibasic for this bundle, it is not basic; that is, there is no form on N that pulls back to Bf to equal ξ, even locally. The criterion for ξ to be basic is that dξ be semibasic, and this fails because of the appearance of βi in (4.12). But suppose that we can ﬁnd some canonical reduction of the ambient B → M to a subbundle on which δi becomes semibasic over M ; in terms of the Lie algebra (4.1), this means that we can reduce to the subgroup having Lie algebra 2 given by {di = 0}. In this case, each βi = n−2 (δi − K ωi ) is a linear combination n i of θ, ω , πi , and is in particular semibasic over N . Consequently, ξ is basic over N , and we can perform the integration by parts in an invariant manner. Unfortunately, there are cases in which no such canonical reduction of B is possible. An example is the homogeneous Laplace equation on Rn , ∆z = 0,

140

CHAPTER 4. ADDITIONAL TOPICS

which is preserved under an action of the conformal group SO o (n + 1, 1). The associated conformal geometry on the trivial solution z = 0 is ﬂat, and our second variation formula reads δ 2 (FΛ )0 (g) =

Ω ∞ for a variation g ∈ C0 (Ω), Ω ⊂ Rn . It follows from our construction that this integrand is invariant under a suitable action of the conformal group. However, the tempting integration by parts

g∆g dx

Ω

g∆g dx = −

Ω

|| g||2dx

leaves us with an integrand which is not conformally invariant. It is this phenomenon that we would like to avoid. To get a sense of when one might be able to ﬁnd a canonical subbundle of B → M on which the δ i are semibasic, recall that (n − 2)dρ ≡ −δi ∧ ωi (mod {θ, πi}).

Working modulo {θ, πi} essentially amounts to restricting to integral manifolds of the Euler-Lagrange system. The preceding then says that a choice of subbundle B ⊂ B on which δi are semibasic gives a subbundle of each conformal bundle Bf → N on which dρ is semibasic over N . Now, typically the role of ρ in the Cartan connection for a conformal structure is as a psuedo-connection in the density line bundle D; a special reduction is required for ρ to be a genuine connection, and the latter requirement is equivalent to dρ being semibasic. In other words, being able to integrate by parts in an invariant manner as described above is equivalent to having a connection in D represented by the pseudo-connection ρ. One way to ﬁnd a connection in D is to suppose that D has somehow been trivialized, and this is equivalent to choosing a Riemannian metric representing the conformal class. This suggests that Euler-Lagrange systems whose integral manifolds have canonical Riemannian metrics will have canonical reductions of this type. In fact, we have seen an example of a Poincar´-Cartan form whose geometry e B → M displays this behavior. This is the system for Riemannian hypersurfaces having prescribed mean curvature, characterized in §2.5 in terms of diﬀerential invariants of its neo-classical, deﬁnite Poincar´-Cartan form. To illustrate the e preceding discussion, we calculate the second variation formula for this system. The reader should note especially how use of the geometry of the Poincar´e Cartan form gives a somewhat simpler derivation of the formula than one ﬁnds in standard sources.1

1 See for example pp. 513–539 of [Spi75], where the calculation is prefaced by a colorful warning about its diﬃculty.

4.1. THE SECOND VARIATION

141

4.1.4

Prescribed Mean Curvature, Revisited

In §2.5, we considered a deﬁnite, neo-classical Poincar´-Cartan form (M, Π) e whose associated geometry (B → M, ϕ) had invariants satisfying T ijk = 0, We further assumed the open condition λ < 0, and this led to a series of reductions of B → M , resulting in a principal subbundle B3 → M , having structure group with Lie algebra 0 0 0 0 : ai + a j = 0 , g3 = 0 a i j j i 0 0 −aj i U ij = λδ ij .

on which the original structure equations (4.2, 4.3, 4.4) hold, with U ij = −δ ij , 1 H i Sj = T ijk = 0, ρ = − 2n θ for a function H on B3 , and dρ = − n−2 δi ∧ ωj , where δi ≡ 0 (mod {θ, ωj , πj }). In this case we computed that dH ≡ 0 (mod {θ, ω i }), so that H deﬁnes a function on the local quotient space Qn+1, which also inherits a Riemannian metric (ωi )2 . The contact manifold M can be locally identiﬁed with the bundle of tangent hyperplanes of Q, and the integral manifolds of the Euler-Lagrange diﬀerential system EΛ are the tangent loci of hypersurfaces in Q whose mean curvature coincides with the background function H. In this case B3 → M → Q is locally identiﬁed with the orthonormal frame bundle of the Riemannian manifold Q. An important point here is that the Riemannian geometry associated to Π only appears after reducing to B3 → M . However, if our goal is to see the formula for the second variation, then we face the following diﬃculty. That formula required the use of coframes of M adapted to a stationary submanifold N → M in a certain way, but while adapted coframes can always be found in B → M , there is no guarantee that they can be found in the subbundle B3 → M , where the Riemannian geometry is visible. We will overcome these diﬃculties and illustrate the invariant calculation of the second variation by starting only with the Riemannian geometry of (Q, ds 2). This is expressed in the Levi-Civita connection in the orthonormal frame bundle, where we can also give the Poincar´-Cartan form and Euler-Lagrange system e for prescribed mean curvature. We then introduce higher-order data on a larger bundle, which allows us to study the second fundamental form. In fact, this larger bundle corresponds to the partial reduction B2 → M on which ρ and δi are semibasic, but σij is not. The end result of our calculation is formula (4.16). In the following discussion, index ranges are 0 ≤ a, b, c ≤ n and 1 ≤ i, j, k ≤ n. We begin with a generalization of the discussion in §1.4 of constant mean curvature hypersurfaces in Euclidean space. Let (Q, ds2) be an oriented Riemannian manifold of dimension n + 1. A frame for Q is a pair f = (q, e) consisting

142

CHAPTER 4. ADDITIONAL TOPICS

of a point q ∈ Q and a positively-oriented orthonormal basis e = (e0 , . . . , en) for Tq Q. The set F of all such frames is a manifold, and the right SO(n + 1, R)action a a a (q, (e0 , . . . , en )) · (gb ) = (q, ( ea g0 , . . . , ea gn )) gives the basepoint map q:F →Q the structure of a principal bundle. The unit sphere bundle M 2n+1 = {(q, e0) : q ∈ Q, e0 ∈ Tq Q, ||e0|| = 1} is identiﬁed with the Grassmannian bundle of oriented tangent n-planes in T Q, and it has a contact structure generated by the 1-form θ(q,e0 ) (v) = ds2 (e0 , q∗(v)), v ∈ T(q,e0 ) M, (4.13)

where q : M → Q is the projection. An immersed oriented hypersurface in Q has a unit normal vector ﬁeld, which may be thought of as a 1-jet lift of the submanifold to M . The submanifold of M thus obtained is easily seen to be a Legendre submanifold for this contact structure, and the transverse Legendre submanifold is locally of this form. To carry out calculations on M , and even to verify the non-degeneracy of θ, we will use the projection F → M deﬁned by (q, (e0, . . . , en)) → (q, e0 ). Calculations can then be carried out using structure equations for the canonical parallelization of F, which we now introduce. First, there are the n + 1 tautological 1-forms

2 1 ϕa (q,e) = ds (ea , q∗ (·)) ∈ Ω (F),

which form a basis for the semibasic 1-forms over Q. Next, there are globally deﬁned, uniquely determined Levi-Civita connection forms ϕa = −ϕb ∈ Ω1(F) a b satisfying dϕa = −ϕa ∧ ϕb , b c dϕa = −ϕa ∧ ϕb + 1 Ra ϕc ∧ ϕd . c b 2 bcd (4.14)

The functions Ra on F are the components of the Riemann curvature tensor bcd with respect to diﬀerent orthonormal frames, and satisfy Ra + Ra = Ra + Rb = Ra + Ra + Ra = 0. dbc cdb bcd acd bcd bdc bcd We now distinguish the 1-form θ = ϕ0 ∈ Ω1(F), which is the pullback via F → M of the contact 1-form given the same name in (4.13). One of our structure equations now reads dθ = −ϕ0 ∧ ϕi . i (4.15) def 4.1. THE SECOND VARIATION

143

This implies that the original θ ∈ Ω1(M ) is actually a contact form, and also that (θ, ϕi , ϕ0 ) is a basis for the semibasic 1-forms for F → M . i At this point, we can give the Poincar´-Cartan form for the prescribed mean e curvature system. Namely, let H ∈ C ∞ (Q) be a smooth function, and deﬁne on F the (n + 1)-form Π = −θ ∧ (ϕ0 ∧ ϕ(i) − Hϕ). i Because H is the pullback of a function on Q, its derivative is of the form dH = Hν θ + Hiϕi , and using this and the structure equations (4.14), one can verify that Π is closed. Because Π is semibasic over M and closed, it is the pullback of a closed form on M , which is then a deﬁnite, neo-classical Poincar´-Cartan form. The associated e Euler-Lagrange diﬀerential system then pulls back to F as EH = {θ, dθ, ϕ0 ∧ ϕ(i) − Hϕ}. i While (4.15) shows that generic Legendre n-planes in M are deﬁned by equations θ = 0, ϕ0 − hij ϕj = 0, i def with hij = hji, integral n-planes in M for EH are deﬁned by the same equations, plus hii = H. The functions hij describing the tangent locus of a transverse Legendre submanifold of M are of course the coeﬃcients of the second fundamental form of the corresponding submanifold of Q. Therefore, the transverse integral manifolds of EH correspond locally to hypersurfaces in Q whose mean curvature hii equals the background function H. This will appear quite explicitly in what follows. To investigate these integral manifolds, we employ the following apparatus. First consider the product F × Rn(n+1)/2, where Rn(n+1)/2 has coordinates hij = hji, and inside this product deﬁne the locus def F (1) = (f, h) ∈ F × Rn(n+1)/2 : hii = H .

To perform calculations, we want to extend our parallelization of F to F (1) . With a view toward reconstructing some of the bundle B → M associated to the Poincar´-Cartan form Π, we do this in a way that is as well-adapted to Π e as possible. On F (1), we continue to work with θ, and deﬁne ω i = ϕi , πi = ϕ0 − hij ϕj . i def def

144 With these deﬁnitions, we have

CHAPTER 4. ADDITIONAL TOPICS

dθ = −πi ∧ ωi , Π = −θ ∧ πi ∧ ω(i). Motivated by Riemannian geometry, we set Dhij = dhij − hkj αk − hik αk , i j so that in particular Dhij = Dhji and Dhii = dH. We also deﬁne the traceless part def 1 Dh0 = Dhij − n δij dH. ij def Direct computations show that we will have exactly the structure equations (4.2, 4.3, 4.4) if we deﬁne αi j ρ δi σij ∧ ωj = ϕi , j

1 = − 2n Hθ, 1 = − 1 Hπi + hij πj + (hik hkj − R0 − n δij Hν )ωj , ij0 2 1 = (Dh0 + n δij Hk ωk + 1 R0 ωk ) ∧ ωj , ij 2 ijk

with σij = σji, σii = 0. The last item requires some comment. Some linear algebra involving a Koszul complex shows that for any tensor Vijk with Vijk = 1 −Vikj (this will be applied to 2n (δij Hk − δik Hj ) + 1 R0 ), there is another 2 ijk tensor Wijk , not unique, satisfying Wijk = Wjik, Wiik = 0, and 1 (Wijk − 2 Wikj ) = Vijk. This justiﬁes the existence of σij satisfying our requirements. The structure equations (4.2, 4.3, 4.4) resulting from these assignments have torsion coeﬃcients T ijk = 0, U ij = −δ ij , i 1 i Sj = −hij + n δj H.

The general calculations of §4.1.1 for the second variation can now be applied; note that we have the freedom to adapt coframes to a single integral submanifold in M of EH . Repeating those calculations verbatim leads to d2 dt2 Λ t=0 Nt

=−

N0

g(dgi + nρgi − gj αj + gδi ) ∧ ω(i) , i

where F : N × [0, 1] → M is a Legendre variation in M , F0 is an integral manifold of EH , and the forms are all pullbacks of forms on F ∗ (F (1)) by a section of F ∗ (F (1) ) → N × [0, 1], adapted along N0 in the sense that θN0 = g dt, (πi )N0 = gidt. These imply that restricted to N0 , we have θ|N0 = πi |N0 = 0, and the preceding formula becomes δ 2 (FΛ )N0 (g) = −

N0 1 g(dgi − gj αj + g(hik hkj − n δij Hν − R0 )ωj ) ∧ ω(i) . ij0 i

4.1. THE SECOND VARIATION

145

Recognizing that g can be thought of as a section of the normal bundle of the hypersurface N → M → Q, and that in this case gi are the coeﬃcients of its covariant derivative, this can be rewritten as δ 2 (FΛ)N0 (g) = − (g∆g + g2 (||h||2 − Hν − R0 ))ω, ii0 (4.16)

F0 q

N0

where ∆ is the Riemannian Laplacian, and ||h||2 = T r(h∗ h). Here, the extrinsic curvature function K appears as the quantity ||h||2 − Hν − R0 . Notice that ii0 we have actually calculated this second variation without ever determining the functional Λ. In case the ambient manifold Q is ﬂat Euclidean space, if the background function H is a constant and the variation g is compactly supported in the interior of N , this simpliﬁes to d2 dt2 Λ t=0 Nt

= − =

N0

(g∆g + g2 ||h||2)ω

N0

(|| g||2 − g2 ||h||2)ω.

Even for the minimal surface equation H = 0, we cannot conclude from this formula alone that a solution locally minimizes area.

4.1.5

Conditions for a Local Minimum

We now discuss some conditions under which an integral manifold N → M of an Euler-Lagrange system EΛ ⊂ Ω∗ (M ) is a local minimum for the functional FΛ , in the sense that FΛ (N ) < FΛ(N ) for all Legendre submanifolds N near N . However, there are two natural meanings for “near” in this context, and this will yield two notions of local minimum. Namely, we will say that FΛ has a strong local minimum at N if the preceding inequality holds whenever N is C 0-close to N , while FΛ has a weak local minimum at N if the preceding inequality holds only among the narrower class of N which are C 1-close to N .2 Our goal is to illustrate how the Poincar´-Cartan form may be used to une derstand in a simple geometric manner some classical conditions on extrema. Speciﬁcally, we will introduce the notion of a calibration for an integral manifold of the Euler-Lagrange system; its existence (under mild topological hypotheses) implies that the integral manifold is a strong local minimum. Under certain classical conditions for a local minimum, we will use the Poincar´-Cartan form e to construct an analogous weak calibration. Finally, our geometric description of the second variation formula highlights the Jacobi operator Jg = −∆c g + Kgω, and some linear analysis shows that the positivity of the ﬁrst eigenvalue of J implies the classical conditions. Let Π be a neo-classical Poincar´-Cartan form Π on a contact manifold e (M, I). There is a local foliation M → Q, and we can choose coordinates to

2A

thorough, coordinate-based discussion of the relevant analysis can be found in [GH96].

146

CHAPTER 4. ADDITIONAL TOPICS

have (xi, z) ∈ Q ⊂ Rn × R, (xi, z, pi) ∈ M ⊂ J 1 (Rn, R), θ = dz − pi dxi ∈ Γ(I), and a Lagrangian potential Λ = L(x, z, p)dx + θ ∧ Lpi dx(i) ∈ Ωn (M ) whose Poincar´-Cartan form is e Π = dΛ = θ ∧ (−dLpi ∧ dx(i) + Lz dx). We will conﬁne our discussion to a domain where this classical description holds. We may regard an integral manifold of the Euler-Lagrange system EΛ as a submanifold N0 → Q given by the graph {(x, z0(x)) : x ∈ U } of a solution to the Euler-Lagrange equation over some open U ⊂ Rn. It has a natural 1-jet (1) extension N0 → M , equal to {(x, z0(x), z0(x)) : x ∈ U }, which is an integral manifold of EΛ in the sense discussed previously. We deﬁne a strong neighborhood of N0 to be the collection of hypersurfaces in Q lying in some open neighborhood of N0 in Q, and a weak neighborhood of N0 to be the collection of hypersurfaces (1) N → Q whose 1-jet prolongations N (1) lie in some open neighborhood of N0 in M . Whether or not a given stationary submanifold N0 is minimal depends on which of these two classes of competing submanifolds one studies. Starting with strong neighborhoods, we ﬁx a neighborhood W ⊂ Q of a stationary submanifold N0 ⊂ Q for Λ, and introduce the following useful notion. ˜ Deﬁnition 4.1 A calibration for (Λ, N0) is an n-form Λ ∈ Ωn(W ) satisfying ˜ • dΛ = 0; ˜ • Λ|N0 = Λ|N (1) ;

0

˜ • Λ|E ≤ ΛE for each n-plane E n ⊂ Tq W . In the right-hand side of the last inequality, we are regarding the n-plane E as specifying a point of M ⊂ Gn(T Q) over q ∈ Q, and evaluating Λ(q,E) on any tangent n-plane E ⊂ T(q,E) M projecting one-to-one into Tq Q; the value is independent of the choice of E , because Λ is semibasic over Q. In particular, ˜ the third condition says that the integral of Λ on any N ⊂ W ⊂ Q will not exceed the integral of Λ on N (1) ⊂ M . In both the strong and weak settings, we will only have N0 compete against submanifolds having the same boundary. For this reason, we assume that W ⊃ N0 is chosen so that ∂N0 = N0 ∩ ∂W , and that (N0 , ∂N0 ) generates the relative homology Hn (W, ∂W ; Z). ˜ Proposition 4.1 If there exists a calibration Λ for (Λ, N0), then FΛ (N0 ) ≤ FΛ (N ) for every hypersurface N → W ⊂ Q satisfying ∂N = ∂N0 . We then say that N0 is a strong (but not strict!) local minimum for FΛ .

4.1. THE SECOND VARIATION Proof. We simply calculate FΛ (N0 ) = =

N0

(1)

147

Λ

N0

˜ Λ ˜ Λ

N

= ≤ =

Λ

N (1)

FΛ (N ).

The third equality uses Stokes’ theorem, which applies because our topological hypothesis on W implies that the cycle N − N0 in W is a boundary. The question of when one can ﬁnd a calibration naturally arises. For this, we use the following classical concept. Deﬁnition 4.2 A ﬁeld for (Λ, N0 ) is a neighborhood W ⊂ Q of N0 with a smooth foliation by a 1-parameter family F : N × (−ε, ε) → W of integral manifolds of EΛ . This family does not have a ﬁxed boundary. We retain the topological hypotheses on W used in Proposition 4.1, and have the following. Proposition 4.2 If there exists a ﬁeld for (Λ, N0), then there exists a closed ˜ ˜ form Λ ∈ Ωn (W ) such that Λ|N0 = Λ|N (1) .

0

˜ ˜ Λ is then a calibration if it additionally satisﬁes the third condition, Λ|E ≤ ΛE . ˜ using the Poincar´-Cartan form. In the proof, we will explicitly construct Λ e Proof. The ﬁeld F : N × (−ε, ε) → Q may be thought of as a family of graphs Nt = {(x, z(x, t))}, where each z(·, t) is a solution of the Euler-Lagrange equations, and the domain of z(·, t) may depend on t ∈ (−ε, ε). Because each point of W lies on exactly one of these graphs, we can deﬁne a 1-jet lift F : W → M , given by (x, z) = (x, z(x, t)) → (x, z(x, t), xz(x, t)).

˜ ˜ Let Λ = (F )∗ Λ ∈ Ωn (W ). Then it is clear that Λ|N0 = Λ|N (1) , and to show that 0 ˜ ˜ Λ is closed, we need to see that (F )∗ Π = dΛ = 0. This holds because Π = θ ∧ Ψ (1) is quadratic in an ideal of forms vanishing on each leaf Ft : N → M ; more concretely, each of (F )∗ θ and (F )∗ Ψ must be a multiple of dt, so their product vanishes.

148

CHAPTER 4. ADDITIONAL TOPICS

˜ General conditions for Λ = (F )∗ Λ to be a calibration, and for N0 to therefore be a strong local minimum, are not clear. However, we can still use the preceding to detect weak local minima. Proposition 4.3 Under the hypotheses of Propositions 4.1 and 4.2, if Lpi pj (x, z0 (x), z0(x))ξi ξj ≥ c||ξ||2, (4.17)

˜ for some constant c > 0 and all (ξi ), then Λ|E ≤ ΛE for all E ⊂ Tq Q suﬃciently near Tq N0 , with equality if and only if E = Tq N0 . Furthermore, FΛ(N0 ) < FΛ (N ) for all N = N0 in a weak neighborhood of N0 . ˜ The ﬁrst statement allows us to think of Λ as a weak calibration for (Λ, N0). The proof of the second statement from the ﬁrst will use Stokes’ theorem in exactly the manner of Proposition 4.1. Proof. The positivity of the Weierstrass excess function def z-Hessian of L suggests that we deﬁne the

E(x, z, p, q) = L(x, z, p) − L(x, z, q) −

(pi − qi )Lqi (x, z, q),

which is the second-order remainder in a Taylor series expansion for L. This ˜ function will appear in a more detailed expression for π ∗ Λ = (F ◦ π)∗ Λ ∈ Ωn (M ), computed modulo {I}. We write F (xi , z) = (xi, z, qi(x, z)) ∈ M, where (xi , z, pi) are the usual coordinates on M , and the functions qi(x, z) are the partial derivatives of the ﬁeld elements z(x, t). We have ˜ π∗ Λ = = ≡ ≡ π∗ ◦ F ∗ (L dx + θ ∧ Lpi dx(i))

L(xi , z, qi(x, z))dx + (dz − qi(x, z)dxi) ∧ Lpi (xi, z, qi(x, z))dx(i)

Λ − E(xi, z, pi, qi(x, z))dx (mod {I}).

L(xi , z, qi(x, z)) + (pi − qi (x, z))Lpi (xi , z, qi(x, z)) dx (mod {I})

The hypothesis (4.17) on the Hessian Fpi pj implies that for each (xi, z), and pi suﬃciently close to qi (x, z), the second-order remainder satisﬁes E(xi, z, pi, qi(x, z)) ≥ 0, ˜ with equality if and only if pi = qi(x, z). The congruence of π ∗ Λ and Λ − E dx modulo {I} then implies our ﬁrst statement.

4.1. THE SECOND VARIATION For the second statement, we use the Stokes’ theorem argument:

(1)

149

Λ =

N0

˜ Λ ˜ Λ

N

N0

= =

N (1)

(Λ − E(xi , z, pi, qi(x, z))dx) Λ,

≤

N (1)

with equality in the last step if and only if N0 = N . This proof shows additionally that if the Weiestrass excess function satisﬁes E(xi , z, pi, qi) > 0 for all p = q, then N0 is a strong (and strict) local minimum for FΛ. So far, we have shown that if we can cover some neighborhood of a sta˜ tionary submanifold N0 with a ﬁeld, then we can construct an n-form Λ, whose calibration properties imply extremal properties of N0. It is therefore natural to ask when there exists such a ﬁeld, and the answer to this involves some analysis of the Jacobi operator. We will describe the operator, and hint at the analysis. The Jacobi operator acts on sections of a density line bundle on a given integral manifold N0 of the Euler-Lagrange system, with its induced conformal structure. Speciﬁcally, n+2 n−2 J : D 2n → D 2n is the diﬀerential operator given by Jg = −∆c g − gK, where ∆c is the conformal Laplacian, and K is the curvature invariant introduced in §4.1.2. The second variation formula (4.8) then reads δ 2 (FΛ)N0 (g) =

N0

g Jg ω.

The main geometric fact is: The Jacobi operator gives the (linear) variational equations for integral manifolds of the Euler-Lagrange system EΛ . This means the following. Let F : N ×[0, 1] → M be a Legendre variation of the Λ-stationary submanifold F0—not necessarily having ﬁxed boundary—and let θ, as usual. Then our previous calculations imply that Jg = 0 g = ∂F |t=0 ∂t if and only if L ∂ (F ∗ Ψ)|t=0 = 0.

∂t

We might express condition by saying that Ft is an integral manifold for EΛ = I + {Ψ} modulo O(t2 ).

150

CHAPTER 4. ADDITIONAL TOPICS

We now indicate how a condition on the Jacobi operator of N0 can imply the existence of a ﬁeld near N0 . Consider the eigenvalue problem Jg = −∆c g − gK = λg,

∞ g ∈ C0 (N ),

for smooth, ﬁxed boundary variations. It is well-known J has a discrete spectrum bounded from below, λ1 < λ2 < · · · , with λk → ∞ and with ﬁnitedimensional eigenspaces. We consider the consequences of the assumption λ1 > 0. Because λ1 = inf{ to

N

g Jg ω : ||g||L2 = 1}, the assumption λ1 > 0 is equivalent δ 2 (FΛ )N (g) > 0, for g = 0.

The main analytic result is the following. Proposition 4.4 If λ1 > 0, then given g0 ∈ C ∞ (∂N ), there is a unique solution g ∈ C ∞ (N ) to the boundary value problem Jg = 0, g|∂N = g0.

Furthermore, if g0 > 0 on ∂N , then this solution satisﬁes g > 0 on N . The existence and uniqueness statements follow from standard elliptic theory. The point is that we can compare the second variation δ 2(FΛ )N (g) to the 2 Sobolev norm ||g||1 = N (|| g||2+|g|2 )ω, and if λ1 > 0, then there are constants c1 , c2 > 0 such that c1||g||2 ≤ 1

N

g Jg ω ≤ c2 ||g||2. 1

The Schauder theory gives existence and uniqueness in this situation. Less standard is the positivity of the solution g under the assumption that g|∂N > 0, and this is crucial for the existence of a ﬁeld. Namely, a further implicit function argument using elliptic theory guarantees that the variation g is tangent to an arc of integral manifolds of E, and the fact that g = 0 implies that near the initial N , this arc deﬁnes a ﬁeld. For the proof of the positivity of g, and details of all of the analysis, see Giaquinta & Hildebrandt (cit. p. 145n).

4.2

Euler-Lagrange PDE Systems

Up to this point, we have studied geometric aspects of ﬁrst-order Lagrangian functionals FL (z) =

∂z L xi , z, ∂xi dx,

Ω

Ω ⊂ Rn ,

(4.18)

4.2. EULER-LAGRANGE PDE SYSTEMS

151

where x = (x1, . . . , xn) and z = z(x) is a scalar function. In this section, we consider the more general situation of functionals FL(z) = L xi , z α(x), ∂z i (x) dx, ∂x α Ω

Ω ⊂ Rn ,

(4.19)

where now z(x) = (z 1 (x), . . . , z s(x)) is an Rs-valued function of x = (xi ), and L = L(xi , z α, pα) is a smooth function on Rn+s+ns. The Euler-Lagrange i equations describing maps z : Ω → Rs which are stationary for FL under all ﬁxed-boundary variations form a PDE system ∂L − ∂z α d dxi ∂L ∂pα i = 0, α = 1, . . . , s. (4.20)

i

In the scalar case s = 1, we have examined the geometry of the equivalence class of FL under contact transformations and found the canonically deﬁned Poincar´-Cartan form to be of considerable use. In this section, we describe e a generalization of the Poincar´-Cartan form for s ≥ 1. Geometrically, we e study functionals on the space of compact submanifolds of codimension s, in an (n + s)-dimensional manifold with local coordinates (xi , z α). An immediate diﬀerence between the cases s = 1 and s ≥ 2 is that in the latter case, there are no proper contact transformations of Rn+s+ns; that is, the only smooth maps x = x (x, z, p), z = z (x, z, p), p = p (x, z, p) for which {dz α − pαdxi } = {dz α − i pα dxi } i

are point transformations x = x (x, z), z = z (x, z), with p = p (x, z, p) determined by the chain rule. We will explain why this is so, and later, we will see that in case s = 1 our original contact-invariant Poincar´-Cartan form still e appears naturally in the more limited context of point transformations. Our ﬁrst task, however, is to introduce the geometric setting for studying functionals (4.19) subject to point transformations, analogous to our use of contact manifolds for (4.18). Throughout this section, we have as always n ≥ 2 and we use the index ranges 1 ≤ i, j ≤ n, 1 ≤ α, β ≤ s.

4.2.1

Multi-contact Geometry

Having decided to apply point transformations to the functional (4.19), we interpret z(x) = (z α (xi )) as corresponding to an n-dimensional submanifold of α Rn+s . The ﬁrst derivatives pα = ∂z i specify the tangent n-planes of this subi ∂x manifold. This suggests our ﬁrst level of geometric generalization. π Let X be a manifold of dimension n + s, and let Gn (T X) → X be the Grassmannian bundle of n-dimensional subspaces of tangent spaces of X; that is, a point of Gn(T X) is of the form m = (p, E), p ∈ X, E n ⊂ Tp X.

152

CHAPTER 4. ADDITIONAL TOPICS

Any diﬀeomorphism of X induces a diﬀeomorphism of Gn(T X), and either of these diﬀeomorphisms will be called a point transformation. We can deﬁne on Gn(T X) two Pfaﬃan systems I ⊂ J ⊂ T ∗ (Gn(T X)), of ranks s and n + s, respectively, which are canonical in the sense that they are preserved by any point transformation. First, J = π ∗ (T ∗ X) consists of all forms that are semibasic over X; J is integrable, and its maximal integral submanifolds are the ﬁbers of Gn (T X) → X. Second, we deﬁne I at a point (p, E) ∈ Gn(T X) to be ∗ I(p,E) = πp (E ⊥ ),

∗ where E ⊥ ⊂ Tp X is the s-dimensional annihilator of the subspace E ⊂ Tp X. I is not integrable, and to understand its integral submanifolds, note that any n-dimensional immersion ι : N → X has a 1-jet lift ι(1) : N → Gn (T X). In fact, such lifts are the transverse integral submanifolds of the Pfaﬃan system I ⊂ T ∗ (Gn (T X)). To see this explicitly, choose local coordinates (xi , z α) on U ⊂ X n+s . These induce local coordinates (xi , z α, pα) corresponding to the n-plane E ⊂ T(xi ,zα ) U i deﬁned as E = {dz 1 − p1dxi, . . . , dz s − psdxi}⊥ . i i

These coordinates are deﬁned on a dense open subset of π −1 (U ) ⊂ Gn(T X), consisting of n-planes E ⊂ T X for which dx1 ∧ · · · ∧ dxn|E = 0. In terms of these local coordinates on Gn(T X), our Pfaﬃan systems are J I = = {dxi, dz α}, {dz α − pαdxi}. i

An immersed submanifold N n → U for which dx1 ∧ · · · ∧ dxn|N = 0 may be regarded as a graph N = {(xi , z α) : z α = f α (x1 , . . . , xn)}. Its lift to N → Gn(T X) lies in the domain of the coordinates (xi , z α, pα), and i equals the 1-jet graph α N (1) = {(xi, z α , pi ) : z α = f α (x1 , . . . , xn), pα = i ∂f α 1 ∂xi (x , . . .

, xn)}.

(4.21)

Clearly this lift is an integral submanifold of I. Conversely, a submanifold N (1) → π−1 (U ) ⊂ Gn (T X) on which dz α − pαdxi = 0 and dx1 ∧· · ·∧dxn = 0 i is necessarily given locally by a graph of the form (4.21). The manifold M = Gn (T X) with its Pfaﬃan systems I ⊂ J is our standard example of a multicontact manifold. This notion will be deﬁned shortly, in terms of the following structural properties of the Pfaﬃan systems. Consider on Gn (T X) the diﬀerential ideal I = {I, dI} ⊂ Ω∗ (Gn(T X)) generated by I ⊂ T ∗ (Gn(T X)). If we set ¯ θα = dz α − pα dxi, i ωi = dxi, ¯ πi = dpα, ¯α i

4.2. EULER-LAGRANGE PDE SYSTEMS then we have the structure equations ¯ dθα ≡ −¯i ∧ ωi πα ¯ (mod {I}), 1 ≤ α ≤ s.

153

(4.22)

α It is not diﬃcult to verify that the set of all coframings (θ α , ωi , πi ) on Gn(T X) for which

• θ1 , . . . , θs generate I, • θ1 , . . . , θs , ω1, . . . , ωn generate J, and α • dθα ≡ −πi ∧ ωi (mod {I})

where (aα ) ∈ GL(s, R), (bi ) ∈ GL(n, R), and eα = eα . From these properties j ij ji β we make our deﬁnition.

are the local sections of a G-structure on Gn(T X). Here G ⊂ GL(n + s + ns, R) α may be represented as acting on (θ α , ωi , πi ) by ¯α α β θ = aβ θ , ω i = ci θ β + bi ω j , ¯ (4.23) j β α β πi = dα θβ + eα (b−1)k ωj + aα πj (b−1 )j , ¯ i i iβ kj β

Deﬁnition 4.3 A multi-contact manifold is a manifold M n+s+ns, with a Gα structure as in (4.23), whose sections (θ α , ωi, πi ) satisfy dθα dωi α ≡ −πi ∧ ωi (mod {θ1 , . . . , θs }), ≡ 0 (mod {θ1 , . . . , θs , ω1, . . . , ωn}).

(4.24) (4.25)

Note that the G-structure determines Pfaﬃan systems I = {θ 1 , . . . , θs} and J = {θ1 , . . . , θs , ω1, . . . , ωn }, and we may often refer to (M, I, J) as a multicontact manifold, implicitly assuming that J is integrable and that there are coframings for which the structure equations (4.22) hold. The integrability of J implies that locally in M one can deﬁne a smooth leaf space X n+s and a surjective submersion M → X whose ﬁbers are integral manifolds of J. When working locally in a multi-contact manifold, we will often make reference to this quotient X. It is not diﬃcult to show that any multi-contact structure (M, I, J) is locally equivalent to that of Gn(T X) for a manifold X n+s . The integrability of J α implies that there are local coordinates (xi, z α , qi ) for which dxi, dz α generate i α α J. We can relabel the x , z to assume that dz − pαdxi generate I for some i functions pα(x, z, q). The structure equations then imply that dxi, dz α, dpα i i are linearly independent, so on a possibly smaller neighborhood in M , we can α replace the coordinates qi by pα, and this exhibits our structure as equivalent i to that of Gn(T X). We will see below that if s ≥ 2, then the Pfaﬃan system I of a multi-contact manifold uniquely determines the larger system J. Also, if s ≥ 3, the hypothesis

154

CHAPTER 4. ADDITIONAL TOPICS

This is a pointwise construction. We apply it to each element of the vector space {λαdθα : (λα ) ∈ Rs},

(4.25) that J = {θ α , ωi} is integrable is not necessary; it is easily seen to be a consequence of the structure equation (4.24). However, in the case s = 2, J is determined by I but is not necessarily integrable; our study of Euler-Lagrange systems will not involve this exceptional situation, so we have ruled it out in our deﬁnition. It is not at all obvious how one can determine, given a Pfaﬃan system I of rank s on a manifold M of dimension n + s + ns, whether I comes from a multicontact structure; deciding whether structure equations (4.24) can be satisﬁed for some generators of I is a diﬃcult problem. Bryant has given easily evaluated intrinsic criteria characterizing such I, generalizing the Pfaﬀ theorem’s normal form for contact manifolds, but we shall not need this here (see Ch. II, §4 of [B+ 91]). Aside from those of the form Gn(T X), there are two other kinds of multicontact manifolds in common use. One is J 1 (Y n , Z s), the space of 1-jets of 1 maps from an n-manifold Y to an s-manifold Z. The other is JΓ (E n+s , Y n ), the space of 1-jets of sections of a ﬁber bundle E → Y with base of dimension n and ﬁber Z of dimension s. These are distinguished by the kinds of coordinate changes considered admissible in each case; to the space J 1 (Y n , Z s), one would 1 apply prolonged classical transformations x (x), z (z), while to JΓ (E n+s , Y n ), one would apply prolonged gauge transformations x (x), z (x, z). These are both smaller classes than the point transformations x (x, z), z (x, z) that we apply to Gn (T X), the space of 1-jets of n-submanifolds in X n+s . Recall our claim that in the multi-contact case s ≥ 2, every contact transformation is a prolonged point transformation. This is the same as saying that any local diﬀeomorphism of M which preserves the Pfaﬃan system I also preserves J; for a local diﬀeomorphism of M preserving J must induce a diﬀeomorphism of the local quotient space X, which in turn uniquely determines the original local diﬀeomorphism of M . To see why a local diﬀeomorphism preserving I must preserve J, we will give an intrinsic construction of J in terms of I alone, 2 ∗ (Tm M ) the for the local model Gn(T X). First, deﬁne for any 2-form Ψ ∈ space of 1-forms C(Ψ) = {V Ψ : V ∈ Tm M }.

intrinsically given as the quotient of I2, the degree-2 part of the multi-contact diﬀerential ideal, by the subspace {I}2, the degree-2 part of the algebraic ideal {I}. For example, α C(dθα ) ≡ Span{πi , ωi } (mod {I}).

The intersection

Θ∈I2 /{I}2

C(Θ)

is a well-deﬁned subbundle of T ∗ M/I. If s ≥ 2, then its preimage in T ∗ M is ¯ ¯ J = {θα , ωi}, as is easily seen using the structure equations (4.22). Any local

4.2. EULER-LAGRANGE PDE SYSTEMS

155

diﬀeomorphism of M preserving I therefore preserves I2 , I2/{I}2, C(Θ), and ﬁnally J, which is what we wanted to prove. Note that in the contact case s = 1, C(Θ) ≡ C(dθ) ≡ {πi, ωi} modulo {I}, so instead of this construction giving J, it gives all of T ∗ M . In this case, introducing J in the deﬁnition of a multi-contact manifold reduces our pseudogroup from contact transformations to point transformations. We have given a generalization of the notion of a contact manifold to accomodate the study of submanifolds of codimension greater than one. There is a further generalization to higher-order contact geometry which is the correct setting for studying higher-order Lagrangian functionals, and we will consider it brieﬂy in the next section. In what follows, we will carry out the discussion of functionals modelled on (4.19) on a general multi-contact manifold (M, I, J), but the reader can concentrate on the case M = Gn(T X).

4.2.2

Functionals on Submanifolds of Higher Codimension

Returning to our functional (4.19), we think of the integrand L(xi , z α, pi )dx as α an n-form on a dense open subset of the multi-contact manifold Gn(T Rn+s ). Note that this n-form is semibasic for the projection Gn(T Rn+s ) → Rn+s , and that any n-form congruent to L(xi, z α , pi )dx modulo {dz α − pαdxi} gives the α i same classical functional. This suggests the following. Deﬁnition 4.4 A Lagrangian on a multi-contact manifold (M, I, J) is a smooth n section Λ ∈ Γ(M, J) ⊂ Ωn(M ). Two Lagrangians are equivalent if they are congruent modulo {I}. An equivalence class [Λ] of Lagrangians corresponds to a section of the vector n (J/I). It also deﬁnes a functional on the space of compact integral bundle manifolds (possibly with boundary) of the Pfaﬃan system I by FΛ (N ) = Λ,

N

where Λ is any representative of the class. The notion of divergence equivalence of Lagrangians will appear later. In the discussion in Chapter 1 of the scalar case s = 1, we combined these two types of equivalence by emphasizing a characteristic cohomology class in H n(Ω∗ (M )/I), and used facts about symplectic linear algebra to investigate these classes. However, the analogous “multi-symplectic” linear algebra that is appropriate for the study of multi-contact geometry is still poorly understood.3 n Our goal is to associate to any functional [Λ] ∈ Γ(M, (J/I)) a Lagrangian n n Λ ∈ Γ(M, J) ⊂ Ω (M ), not necessarily uniquely determined, whose exterior derivative Π = dΛ has certain favorable properties and is uniquely determined by [Λ]. Among these properties are:

3 The recent work [Gra00] of M. Grassi may illuminate this issue, along with some others that will come up in the following discussion.

156 • Π ≡ 0 (mod {I});

CHAPTER 4. ADDITIONAL TOPICS

• Π is preserved under any diﬀeomorphism of M preserving I, J, and [Λ]; • Π depends only on the divergence-equivalence class of [Λ]; • Π = 0 if and only if the Euler-Lagrange equations for [Λ] are trivial. Triviality of the Euler-Lagrange equations means that every compact integral manifold of I ⊂ Ω∗ (M ) is stationary for FΛ under ﬁxed-boundary variations. Some less obvious ways in which such Π could be useful are the following, based on our experience in the scalar case s = 1: • in Noether’s theorem, where one would hope for v → v Π to give an isomorphism from a Lie algebra of symmetries to a space of conservation laws; • in the inverse problem, where one can try to detect equations that are locally of Euler-Lagrange type not by ﬁnding a Lagrangian, but by ﬁnding a Poincar´-Cartan form inducing the equations; e • in the study of local minimization, where it could help one obtain a calibration in terms of a ﬁeld of stationary submanifolds. Recall that in the case of a contact manifold, we replaced any Lagrangian Λ ∈ Ωn (M ) by Λ − θ ∧ β, the unique form congruent to Λ (mod {I}) with the property that dΛ ≡ 0 (mod {I}). What happens in the multi-contact case? Any Lagrangian Λ0 ∈ Γ(M, congruent modulo {I} to a form (in local coordinates) L(xi , z α, pα)dx, i and motivated by the scalar case, we consider the equivalent form Λ = L dx + θα ∧ which has exterior derivative dΛ = θα ∧

∂L ∂z α dx − ∂L ∂pα dx(i) , i n

J) is

(4.26)

d

∂L ∂pα i

∧ dx(i) .

(4.27)

This suggests the following deﬁnition. Deﬁnition 4.5 An admissible lifting of a functional [Λ] ∈ Γ(M, n(J/I)) is a n Lagrangian Λ ∈ Γ(M, J) representing the class [Λ] and satisfying dΛ ∈ {I}.

4.2. EULER-LAGRANGE PDE SYSTEMS

157

The preceding calculation shows that locally, every functional [Λ] has an admissible lifting. Unfortunately, the admissible lifting is generally not unique. This will be addressed below, but ﬁrst we show that any admissible lifting is adequate for calculating the ﬁrst variation and the Euler-Lagrange system of the functional FΛ. We mimic the derivation in Chapter 1 of the Euler-Lagrange diﬀerential system in the scalar case s = 1. Suppose that we have a 1-parameter family {Nt } of integral manifolds of the multi-contact Pfaﬃan system I, given as a smooth map F : N × [0, 1] → M, for which each Ft = F |N ×{t} : N → M is an integral manifold of I and such that F |∂N ×[0,1] is independent of t. Then choosing generators θ α ∈ Γ(I), 1 ≤ α ≤ s, we have F ∗ θα = Gαdt

for some functions Gα on N × [0, 1]. As in the contact case, it is not diﬃcult to show that any collection of functions g α supported in the interior of N can be realized as Gα|t=0 for some 1-parameter family Nt . The hypothesis that Λ is an admissible lifting means that we can write dΛ = θ α ∧ Ψα L ∂ (F ∗ Λ)

∂t

for some Ψα ∈ Ωn(M ). Then we can proceed as in §1.2 to calculate d dt t=0 Nt

Ft∗ Λ

=

N0

=

N0

∂ ∂t

(θα ∧ Ψα ) +

N0

∂ d( ∂t

Λ)

=

N0

g α Ψα ,

When n ≥ 2, this implies that γα ≡ 0 (mod {I}), so while two general representatives of [Λ] need be congruent only modulo {I}, for admissible liftings we have the following.

where in the last step we used the ﬁxed-boundary condition, the vanishing of ∗ F0 θα , and the deﬁnition gα = Gα|N0 . Now the same reasoning as in §1.2 shows that F0 : N → M is stationary for FΛ under all ﬁxed-boundary variations if and only if Ψα |N0 = 0 for all α = 1, . . . , s. We now have a diﬀerential system {θ α , dθα, Ψα} whose integral manifolds are exactly the integral manifolds of I that are stationary for [Λ], but it is not clear that this system is uniquely determined by [Λ] alone; we might get diﬀerent systems for diﬀerent admissible liftings. To rule out this possibility, observe ﬁrst that if Λ, Λ are any two admissible liftings of [Λ], then the condition Λ − Λ ∈ {I} allows us to write Λ − Λ = θα ∧ γα , and then the fact that dΛ ≡ dΛ ≡ 0 (mod {I}) along with the structure equations (4.24) allows us to write α 0 ≡ dθα ∧ γα ≡ −πi ∧ ωi ∧ γα (mod {I}).

158

CHAPTER 4. ADDITIONAL TOPICS n Proposition 4.5 Two admissible liftings of the same [Λ] ∈ Γ( 2 congruent modulo { I}.

2

(J/I)) are

Of course, when s = 1, I = 0 and we have a unique lifting, whose derivative is the familiar Poincar´-Cartan form. This explains how the Poincar´-Cartan form e e occurs in the context of point transformation as well as contact transformations. We use the proposition as follows. If we take two admissible liftings Λ, Λ of the same functional [Λ], and write Λ − Λ = 1 θα ∧ θβ ∧ γαβ , 2 with γαβ + γβα = 0, then θα ∧ (Ψα − Ψα ) = d(Λ − Λ ) ≡ −θα ∧ dθβ ∧ γαβ (mod {

2

I}).

A consequence of this is that Ψα − Ψα ∈ I for each α, and we can therefore give the following. Deﬁnition 4.6 The Euler-Lagrange system EΛ of [Λ] ∈ Γ(M, (J/I)) is the diﬀerential ideal on M generated by I and the n-forms {Ψ1 , . . . , Ψs} ⊂ Ωn (M ), where Λ is any admissible lifting of [Λ] and dΛ = θα ∧ Ψα . A stationary Legendre submanifold of [Λ] is an integral manifold of EΛ . n 4.2.3

The Betounes and Poincar´-Cartan Forms e

For scalar variational problems, the Poincar´-Cartan form Π ∈ Ωn+1(M ) on e the contact manifold (M, I) is an object of central importance. Some of its key features were outlined above. Underlying its usefulness is the fact that we are associating to a Lagrangian functional—a certain equivalence class of diﬀerential forms—an object that is not merely an equivalence class, but an actual diﬀerential form with which we can carry out certain explicit computations. We would like to construct an analogous object in the multi-contact case. def We will do this by imposing pointwise algebraic conditions on Π = dΛ. Fix α i α an admissible coframing (θ , ω , πi ) on a multi-contact manifold as in Deﬁnition n 4.3. Then any admissible lifting Λ ∈ Γ( J) of a functional [Λ] has the form min (n,s)

Λ=

k=0

for some functions which are skew-symmetric with respect to each set of n indices. Because J is integrable, Π = dΛ lies in Ωn+1 (M )∩{ J}; the “highest weight” part can be written as min(n,s) A FI ,

(k!)−2

|A|=|I|=k

I FAθA ∧ ω(I) ,

Π=

k=1

(k!)−2

α,i |A|=|I|=k

iI α HαA πi ∧ θA ∧ ω(I)

(mod {

n+1

J}).

(4.28)

4.2. EULER-LAGRANGE PDE SYSTEMS

159

iI The functions HαA are skew-symmetric in the multi-indices I and A. Notice n+1 that the equation dΠ ≡ 0 (mod { J}) gives for the k = 1 term ij ji Hαβ = Hβα .

To understand the relevant linear algebra, suppose that V n is a vector space with basis {vi }, and that W s is a vector space with basis {wα} and dual basis {wα }. Then we have for k ≥ 2 the GL(W ) × GL(V )-equivariant exact sequence 0 → Uk → W ∗ ⊗ V ⊗ ( k W∗ ⊗

k

k V) →

τ

k+1

W∗ ⊗

k+1

V → 0.

Here the surjection is the obvious skew-symmetrization map, and Uk is by definition its kernel. The term k = 1 will be exceptional, and we instead deﬁne 0 → U1 → Sym2 (W ∗ ⊗ V ) → τ 2

W∗ ⊗

2

V → 0,

so that U1 = Sym2 W ∗ ⊗ Sym2 V . iI Now we can regard our coeﬃcients HαA, with |I| = |A| = k, at each point of M as coeﬃcients of an element iI Hk = HαAwα ⊗ vi ⊗ wA ⊗ vI ∈ W ∗ ⊗ V ⊗ ( k

W∗ ⊗

k

V ).

Deﬁnition 4.7 The form Π ∈ Ωn+1 (M )∩{ (4.28) has Hk ∈ Uk for all k ≥ 1. For k = 1 the condition is

n

J} is symmetric if its expansion

ij ji ij Hαβ = Hαβ = Hβα .

We ﬁrst need to show that the condition that a given Π be symmetric is independent of the choice of admissible coframe. Equivalently, we can show that the symmetry condition is preserved under the group of coframe changes of the form (4.23), and we will show this under three subgroups generating the group. First, it is obvious that a change ¯ θ α = aα θ β , β ω i = bi ω j , ¯ j β πi = aα πj (b−1)j , ¯α β i

preserves the symmetry condition, because of the equivariance of the preceding i exact sequences under (aα ) × (bj ) ∈ GL(W ) × GL(V ). Second, symmetry is β preserved under ¯ θα = θα , ωi = ωi , ¯ α πi = d α θ β + e α ω j + πi , ¯α iβ ij

with eα = eα , because such a change has no eﬀect on the expression for Π ij ji n+1 modulo { J}. Finally, consider a change of the form ¯ θα = θα , ω i = ci θ β + ω i , ¯ β α πi = πi . ¯α

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CHAPTER 4. ADDITIONAL TOPICS

We will prove the invariance of the symmetry condition inﬁnitesimally, writing instead of ωi the family ¯ i ωi (ε) = εcβ θβ + ωi .

(4.29)

This associates to each Hk a tensor Hk+1(ε) for each k ≥ 1, and we will show d that dε |ε=0Hk+1(ε) ∈ Uk+1 . This just amounts to looking at the terms linear in ε when (4.29) is substituted into (4.28). Noting that C = (ci ) ∈ W ∗ ⊗ V , we β consider the commutative diagram W∗ ⊗ V ⊗ W∗ ⊗ V ⊗ W∗ ⊗ V ↓ τk ⊗ 1 k+1 W ∗ ⊗ k+1 V ⊗ W ∗ ⊗ V k k

→ W∗ ⊗ V ⊗ →

σ

k+1

↓ τk+1 k+2 W∗ ⊗

W∗ ⊗ k+2 k+1

V

V,

where σ is skew-symmetrization with the latter W ∗ ⊗ V , and τk ⊗ 1 is an extension of the earlier skew-symmetrization. The point is that given Hk ⊗ C in the upper-left space of this diagram, d | H (ε) dε ε=0 k+1

= σ(Hk ⊗ C).

So if we assume that (Hk ) ∈ Uk , then τk (Hk ) = 0, so σ(Hk × C) ∈ Uk+1 , which is what we wanted to show. This proves that the condition that the symmetry condition on Π = dΛ is independent of the choice of adapted coframe. We can now state the following. Theorem 4.1 Given a functional [Λ] ∈ Γ(M, n(J/I)), there is a unique adn missible lifting Λ ∈ Γ(M, J) such that Π = dΛ ∈ Ωn+1(M ) is symmetric. Proof. We inductively construct Λ = Λ0 + Λ1 + · · · + Λmin(n,s), with each i Λi ∈ { I} chosen to eliminate the fully skew-symmetric part of Πi−1 = d(Λ0 + · · · + Λi−1). Initially, Λ0 = F ω is the prescribed X-semibasic n-form modulo {I}. We know from the existence of admissible liftings that there is some Λ1 ∈ {I} such that def def

Π1 = d(Λ0 + Λ1) ∈ {I}; and we know from Proposition 4.5 that Λ1 is uniquely 2 determined modulo { I}. Now let ij α Π1 ≡ Hαβ πi ∧ θβ ∧ ω(j)

(mod {

2

I} + {

n+1

J}).

If we add to Λ0 + Λ1 the I-quadratic term Λ2 = ij α 1 2!2 Fαβ θ

∧ θβ ∧ ω(ij),

then the structure equation (4.24) shows that this alters the I-linear term Π1 only by ij ij ij Hαβ ; Hαβ + Fαβ .

4.2. EULER-LAGRANGE PDE SYSTEMS

161

ij ji ij ij Because Fαβ = −Fαβ = −Fβα , we see that Fαβ may be uniquely chosen so that ij the new Hαβ lies in U1 . n The inductive step is similar. Suppose we have Λ0 + · · ·+ Λl ∈ Γ( J) such l−1 that Πl = d(Λ0 + · · · + Λl ) is symmetric modulo { I}. Then the term of I-degree l is of the form

Πl ≡

1 l!2

iI α HαAπi ∧ θA ∧ ω(I)

(mod {U1 } + · · · + {Ul−1 } + {

l+1

I}).

iI for some HαA. There is a unique skew-symmetric term

Λl+1 = which may be added so that

1 (l+1)!2 |I|=|A|=l+1

I FA θA ∧ ω(I)

Πl+1 ∈ {U1 } + · · · + {Ul } + {

l+1

I}.

We can continue in this manner, up to l = min(n, s). Deﬁnition 4.8 The unique Λ in the preceding theorem is called the Betounes form for the functional [Λ].4 Its derivative Π = dΛ is the Poincar´-Cartan form e for [Λ]. The unique determination of Π, along with the invariance of the symmetry condition under admissible coframe changes of M , implies that Π is globally deﬁned and invariant under symmetries of the functional [Λ] and the multicontact structure (M, I, J). It is instructive to see the ﬁrst step of the preceding construction in coordinates. If our initial Lagrangian is Λ0 = L(x, z, p)dx, then we have already seen in (4.26) that Λ0 + Λ1 = L dx + θα ∧ The H1-term of d(Λ0 + Λ1 ) (see (4.27)) is ∂2L ∂pα ∂pβ i j i j j i

∂L ∂pα dx(i) . i

dpβ ∧ θα ∧ dx(i). j

(4.30)

automatically. The proof shows that we can add Λ2 ∈ { includes

1 2 (Lpα pβ i j

4 It j i

Of course Lpα pβ = Lpβ pα , corresponding to the fact that H1 ∈ Sym2 (W ∗ ⊗ V )

2

I} so that Π2 instead

α + Lpα pβ )πi ∧ θβ ∧ ω(j) ∈ U1 = Sym2 (W ∗ ) ⊗ Sym2V.

was introduced in coordinates in [Bet84], and further discussed in [Bet87].

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CHAPTER 4. ADDITIONAL TOPICS

In fact, this corresponds to the principal symbol of the Euler-Lagrange PDE system (4.20), given by the symmetric s × s matrix Hαβ (ξ) = = ∂2L ∂pα ∂pβ i j

1 (Lpα pβ 2 i j

ξi ξj + Lpα pβ )ξi ξj , j i

ξ ∈ V ∗.

In light of this, it is not surprising to ﬁnd that only the symmetric part of (4.30) has invariant meaning. Note also that if Lagrangians Λ, Λ diﬀer by a divergence, Λ − Λ = dλ, λ ∈ Γ(M, n−1 J),

then the construction in the proof of Theorem 4.1 shows that the Poincar´e Cartan forms are equal, though the Betounes forms may not be. A related but more subtle property is the following. Theorem 4.2 For a functional [Λ] ∈ Γ(M, (J/I)), the Poincar´-Cartan e form Π = 0 if and only if the Euler-Lagrange system is trivial, EΛ = I. Proof. One direction is clear: if Π = 0, then the n-form generators Ψα for EΛ can be taken to be 0, so that EΛ = I. For the converse, we ﬁrst consider the I-linear term ij α Π1 ≡ Hαβ πi ∧ θβ ∧ ω(j) n

(mod {

2

I} + {

n+1

J}).

ij α EΛ is generated by I and Ψβ = Hαβ πi ∧ ω(j), 1 ≤ β ≤ s, and our assumption EΛ = I then implies that these Ψβ = 0; that is, ij α H1 = Hαβ πi ∧ θβ ∧ ω(j) = 0.

We will ﬁrst show that this implies H2 = H 3 = · · · = 0 as well, which will imply Π ∈ { J}. To see this, suppose Hl is the ﬁrst non-zero term, having I-degree l. Then we can consider 0 ≡ dΠ (mod { l n+1

I} + {

n+1

J}),

and using the structure equations (4.24), 0=

|I|=|A|=l−1 ijI β α HαβAπi ∧ πj .

Written out fully, this says that

2 l+1 1 l+1 Hα11 α2···αl+1 = Hα22 α1···αl+1 .

i i ···i

i i ···i

4.2. EULER-LAGRANGE PDE SYSTEMS

163

iI Also, HαA is fully skew-symmetric in I and A. But together, these imply in that Hl is fully skew-symmetric in all upper and all lower indices, for i 2 Hα11iαi3 ······ 2 α3

= = = = = =

i 1 −Hα21iαi3 ······ . 2 α3

i 1i Hα21iα33 ······ α2

i 2i Hα13iα13 ······ α2

i 2 −Hα13iαi3 ······ 2 α1

i 1 −Hα22iαi3 ······ 3 α1

i 1i Hα22iα13 ······ α3

This proves full skew-symmetry in the upper indices, and the proof for lower indices is similar. However, we constructed Π so that each Hk lies in the invariant complement of the fully skew-symmetric tensors, so we must have Hk = 0. Now we have shown that if the Euler-Lagrange equations of [Λ] are trivial, n+1 then Π ∈ { J}. But that means that the Betounes form Λ is not merely semibasic over the quotient space X, but actually basic. We can then compute the (assumed trivial) ﬁrst variation down in X instead of M , and ﬁnd that for any submanifold N → X, and any vector ﬁeld v along N vanishing at ∂N , 0=

N

v

dΛ.

But this implies that dΛ = 0, which is what we wanted to prove. The preceding results indicate that Π is a good generalization of the classical Poincar´-Cartan form for second-order, scalar Euler-Lagrange equations. We e note that for higher-order Lagrangian functionals on vector-valued functions of one variable (i.e., functionals on curves), such a generalization is known, and not diﬃcult; but for functionals of order k ≥ 2 on vector-valued functions of several variables, little is known.5 We want to brieﬂy mention a possible generalization to the multi-contact case of Noether’s theorem, which gives an isomorphism from a Lie algebra of symmetries to a space of conservation laws. To avoid distracting global considq erations, we will assume that HdR (M ) = 0 in all degrees q > 0. First, we have the space gΠ, consisting of vector ﬁelds on M which preserve I and Π, gΠ = {v ∈ V(M ) : Lv I ⊆ I, Lv Π = 0}. Second, we have the space of conservation laws C = H n−1(Ω∗ (M )/EΛ ); under our topological assumption, this is identiﬁed with H n(EΛ ), and we need not introduce a notion of “proper” conservation law as in §1.3. In this situation, Noether’s theorem says the following.

5 But

see Grassi, cit. p. 155n.

164

CHAPTER 4. ADDITIONAL TOPICS There is a map η : gΠ → H n (EΛ), deﬁned by v → [v Π], which is an isomorphism if Π is non-degenerate in a suitable sense.

The map is certainly well-deﬁned; that is, for any v ∈ gΠ , the form v closed section of EΛ. First, v so that v Π = (v θα )Ψα − θα ∧ (v Ψα ),

Π is a

Π is a section of EΛ ; and second, d(v Π) = Lv Π − v dΠ = 0,

so that v Π is closed. However, the proof that under the right conditions this map is an isomorphism involves some rather sophisticated commutative algebra, generalizing the symplectic linear algebra used in Chapter 1. This will not be presented here. As in the scalar case, a simple prescription for the conserved density in H n−1(Ω∗ (M )/EΛ) corresponding to v ∈ gΠ is available when also Lv Λ = 0. One virtue of the Betounes form is that this holds for inﬁnitesimal multi-contact symmetries of [Λ]. Assuming only that dΛ = Π and Lv Λ = 0, we can calculate that d(−v Λ) = −Lv Λ + v dΛ = v Π. (4.31)

Therefore, −v Λ ∈ Ωn−1(M ) represents a class in C = H n−1(Ω∗ (M )/EΛ) corresponding to η(v) ∈ H n (EΛ ). We will use this prescription in the following.

4.2.4

Harmonic Maps of Riemannian Manifolds

The most familiar variational PDE systems in diﬀerential geometry are those describing harmonic maps between Riemannian manifolds. Let P, Q be Riemannian manifolds of dimensions n, s. We will deﬁne a Lagrangian density on P , depending on a map P → Q and its ﬁrst derivatives, whose integral over P may be thought of as the energy of the map. The appropriate multi-contact manifold for this is the space of 1-jets of maps P → Q, M = J 1 (P, Q), whose multi-contact system will be described shortly. We may also think of M as Hom(T P, T Q), the total space of a rank-ns vector bundle over P × Q. To carry out computations, it will be most convenient to work on F = F(P ) × F(Q) × Rns, def 4.2. EULER-LAGRANGE PDE SYSTEMS

165

where F(P ), F(Q) are the orthonormal frame bundles. These are parallelized i in the usual manner by (ω i , ωj ), (ϕα , ϕα), respectively, with structure equations β i i i k dωi = −ωj ∧ ωj , dωj = −ωk ∧ ωj + Ωi , j α α β α α dϕ = −ϕβ ∧ ϕ , dϕβ = −ϕγ ∧ ϕγ + Φα . β β

These forms and structure equations will be considered pulled back to F. To complete a coframing of F, we take linear ﬁber coordinates pα on Rns, and i deﬁne j α πi = dpα + ϕα pβ − pαωi . i β i j The motivation here is that Hom(T P, T Q) → P ×Q is a vector bundle associated to the principal (O(n) × O(s))-bundle F(P ) × F(Q) → P × Q, with the data ((eP ), (eQ ), (pα)) ∈ F deﬁning the homomorphism eP → eQ pα. Furthermore, α α i i i i if a section σ ∈ Γ(Hom(T P, T Q)) is represented by an equivariant map (pα ) : i α F(P ) × F(Q) → Rns, then the Rns-valued 1-form (πi ) represents the covariant derivative of σ. For our purposes, note that M = Hom(T P, T Q) is the quotient of F under a certain action of O(n) × O(s), and that the forms semibasic for the projection α F → M are generated by ω i , ϕα, πi . A natural multi-contact system on M pulls back to F as the Pfaﬃan system I generated by θ α = ϕα − p α ω i , i and the associated integrable Pfaﬃan system on M pulls back to J = {ϕα, ωi } = {θα , ωi }. The structure equations on F adapted to these Pfaﬃan systems are i α β α α dθ = −πi ∧ ω − ϕβ ∧ θ , j i i dω = −ωj ∧ ω , β j α α α dπi = Φβ pβ − pαΩj − ϕα ∧ πi − πj ∧ ωi . j i β i ˜ Λ = 1 ||p||2ω ∈ Γ( 2 where the norm is ˜ Although this Λ is not an admissible lifting of its induced functional [Λ], a computation using the structure equations (4.32) shows that Λ = is admissible: dΛ = = j α −θα ∧ πi ∧ ω(i) − pα pβ ϕα ∧ ω + pαpα ωi ∧ ω i i β i j α α −θ ∧ πi ∧ ω(i), def 1 2 2 ||p|| ω n def

(4.32)

We now deﬁne the energy Lagrangian

J) ⊂ Ωn(F), (pα )2. i

||p||2 = Tr(p∗ p) =

+ pα θα ∧ ω(i) i

166

CHAPTER 4. ADDITIONAL TOPICS

j i where the last step uses ϕα + ϕβ = ωj + ωi = 0. Now we deﬁne α β α Π = −θα ∧ πi ∧ ω(i) ,

and note that Π is in fact the lift to F of a symmetric form on M , as deﬁned earlier. Therefore, we have found the Betounes form and the Poincar´-Cartan e form for the energy functional. The Euler-Lagrange system for [Λ], pulled back to F, is α α EΛ = {θα , πi ∧ ωi , πi ∧ ω(i) }.

A Legendre submanifold N → M = J 1 (P, Q) on which ωi = 0 is the 1-jet graph of a map f : P → Q. On the inverse image π −1(N ) ⊂ F, in addition to θα = 0, there are relations α πi = h α ω j , ij

hα = h α . ij ji

Diﬀerentiating this equation shows that the expression h = hα ω i ω j ⊗ e Q ij α is invariant along ﬁbers of π −1 (N ) → N , so it gives a well-deﬁned section of Sym2(T ∗ P ) ⊗ T Q; this is called the second fundamental form of the map f : P → Q. The condition for N to be an integral manifold of the EulerLagrange system is then Tr(h) = hα = 0 ∈ Γ(M, f ∗ T Q), ii

Deﬁnition 4.9 A map f : P → Q between Riemannian manifolds is harmonic if the trace of its second fundemental form vanishes. Expressed in coordinates on P and Q, this is a second-order PDE system for f : P → Q. We now consider conservation laws for the harmonic map system EΛ ⊂ Ω∗ (F) corresponding to inﬁnitesimal isometries (Killing vector ﬁelds of either P or Q. These are symmetries not only of Π but of the Lagrangian Λ, so we can use the simpliﬁed prescription (4.31) for a conserved (n − 1)-form. First, an inﬁnitesimal isometry of P induces a unique vector ﬁeld on M = J 1 (P, Q) preserving I and ﬁxing Q. This vector ﬁeld preserves Λ, Π, and EΛ , and has a natural lift to F which does the same. This vector ﬁeld v on F satisﬁes v ωi = vi , v ϕα = 0 ⇒ v θα = −pα vi , i for some functions v i . We can then calculate ϕv = v def Λ ≡ ( 1 ||p||2vi − pα pαvj )ω(i) i j 2

(mod {I}).

4.2. EULER-LAGRANGE PDE SYSTEMS

167

As in Chapter 3, it is useful to write this expression restricted to the 1-jet graph of a map f : P → Q, which is v Λ|N = ∗P ( 1 ||p||2vi − pα pαvj )ωi i j 2 = ∗P 1 v 2 ( 1 ||df||2 2 (ωi )2 − f ∗ (ϕα )2 ) ,

where we use f ∗ ϕα = pαωi , and ∗P ω(i) = ωi . One might recognize the stressi energy tensor S = 1 ||df||2ds2 − f ∗ ds2 , P Q 2 and write our conserved density as 2(v Λ) ≡ ∗P (v S) (mod {I}). (4.33)

In fact, S is traditionally deﬁned as the unique symmetric 2-tensor on P for which the preceding equation holds for arbitrary v ∈ V(P ) and f : P → Q; then (4.33) gives a conserved density when v is an inﬁnitesimal isometry and f is a harmonic map. In this case In fact, for any inﬁnitesimal isometry v, a calculation gives d(∗P (v S)) = (v div S)ω (4.34)

on the 1-jet graph of any map.6 Now consider an inﬁnitesimal isometry of Q, whose lift w ∈ V(F) satisﬁes w Then ωi = 0, w def ϕα = w i

⇒

w

θα = wα.

ϕw = w

Λ = wαpα ω(i). i

2 Given a map f : P → Q, we can use df ∈ Hom(T P, T Q) and dsQ ∈ Sym2(T ∗ Q) 2 to regard dsQ (df(·), w) as a 1-form on P , and then

ϕw = ∗P (ds2 (df(·), w)). Q Because this expression depends linearly on w, we can simplify further by letting a denote the Lie algebra of inﬁnitesimal symmetries of Q, and then the map w → w Λ is an element of a∗ ⊗ Ωn−1(P ). If we deﬁne an a∗ -valued 1-form on P by α(v) = ds2 (df(v), ·), v ∈ Tp P, Q then our conservation laws read d(∗P α) = 0 ∈ a∗ ⊗ Ωn (P ). (4.35)

The a∗ -valued (n − 1)-form ∗P α may be formed for any map f : P → Q, and it is closed if f is harmonic. In fact, if Q is locally homogeneous, meaning that

6 The divergence of a symmetric 2-form S is the 1-form div S = is the ei S(ei , ·), where Levi-Civita covariant derivative and (ei ) is any orthonormal frame. Equation (4.34) is true of any symmetric 2-form S and inﬁnitesimal isometry v.

168

CHAPTER 4. ADDITIONAL TOPICS

inﬁnitesimal isometries span each tangent space Tq Q, then (4.35) is equivalent to the harmonicity of f. An important special case of this last phenomenon is when Q itself is a Lie group G with bi-invariant metric ds2 . Examples are compact semisimple Lie G groups, such as O(N ) or SU (N ), with metric induced by the Killing form on the Lie algebra g. Now a map f : P → G is uniquely determined up to lefttranslation by the pullback f ∗ ϕ of the left-invariant g-valued Maurer-Cartan ∼ 1-form ϕ. Using the metric to identify g = g∗ , the conservation laws state that ∗ if f is harmonic, then d(∗P (f ϕ)) = 0. Conversely, if P is simply connected, then given a g-valued 1-form α on P satisfying dα + 1 [α, α] = 0, 2 d(∗P α) = 0, there is a harmonic map f : P → G with f ∗ ϕ = α, uniquely determined up to left-translation. This is the idea behind the gauge-theoretic reformulation of certain harmonic map systems, for which remarkable results have been obtained in the past decade.7 Quite generally, PDE systems that can be written as systems of conservation laws have special properties; one typically exploits such expressions to deﬁne weak solutions, derive integral identities, and prove regularity theorems.

4.3

Higher-Order Conservation Laws

One sometimes encounters a conservation law for a PDE that involves higherorder derivatives of the unknown function, but that cannot be expressed in terms of derivatives of ﬁrst-order conservation laws considered up to this point. An 2 example is the (1+1)-dimensional wave equation −ztt +zxx = 0, for which (ztt + 2 ztx )dt+2ztt ztx dx is closed on solutions, but cannot be obtained by diﬀerentiating any conservation law on J 1(R2 , R). In this section, we introduce the geometric framework in which such conservation laws may be found, and we propose a version of Noether’s theorem appropriate to this setting. While other general forms of Noether’s theorem have been stated and proved (e.g., see [Vin84] or [Olv93]), it is not clear how they relate to that conjectured here. We also discuss (independently from the preceding) the higher-order relationship between surfaces in Euclidean space with Gauss curvature K = −1 1 and the sine-Gordon equation ztx = 2 sin(2z), in terms of exterior diﬀerential systems.

4.3.1

The Inﬁnite Prolongation

We begin by deﬁning the prolongation of an exterior diﬀerential system (EDS). When this is applied to the EDS associated to a PDE system, it gives the EDS associated to the PDE system augmented by the ﬁrst derivatives of the original

7 The

literature on this subject is vast, but a good starting point is [Woo94].

4.3. HIGHER-ORDER CONSERVATION LAWS

169

equations. This construction then extends to that of the inﬁnite prolongation, an EDS on an inﬁnite-dimensional manifold which includes information about derivatives of all orders. The general deﬁnition of prolongation uses a construction introduced in §4.2, in the discussion of multi-contact manifolds. Let X n+s be a manifold, and π Gn (T X) → X the bundle of tangent n-planes of X; points of Gn(T X) are of the form (p, E), where p ∈ X and E ⊂ Tp X is a vector subspace of dimension n. As discussed previously, there is a canonical Pfaﬃan system I ⊂ T ∗ Gn(T X) of rank s, deﬁned at (p, E) by I(p,E) = π∗ (E ⊥ ). Given local coordinates (xi, z α ) on X, there are induced coordinates (xi , z α, pα) i on Gn(T X), in terms of which I is generated by the 1-forms θα = dz α − pα dxi. i (4.36) def We let I ⊂ Ω∗ (Gn(T X)) be the diﬀerential ideal generated by I. Now let (M, E) be an exterior diﬀerential system; that is, M is a manifold of dimension m+s and E ⊂ Ω∗ (M ) is a diﬀerential ideal for which we are interested in m-dimensional integral manifolds. We then deﬁne the locus M (1) ⊂ Gm (T M ) to consist of the integral elements of E ⊂ Ω∗ (M ); that is, (p, E) ∈ M (1) if and only if ∗ (E ∗ ) for all ϕ ∈ E. ϕE = 0 ∈ We will assume from now on that M (1) → Gm (T M ) is a smooth submanifold. Then we deﬁne def E (1) = ι∗ I ⊂ Ω∗ (M (1) ) ι as the restriction to M (1) of the multi-contact diﬀerential ideal. This is the same as the diﬀerential ideal generated by the Pfaﬃan system ι∗ I ⊂ T ∗ M (1), and the ﬁrst prolongation of (M, E) is deﬁned to be the exterior diﬀerential system (M (1), E (1)). Note that the ﬁrst prolongation is always a Pfaﬃan system. Furthermore, if π : M (1) → M is the obvious projection map, and assuming that E is a Pfaﬃan system, then one can show that π ∗ E ⊆ E (1) . However, the projection π could be quite complicated, and need not even be surjective. Finally, note that any integral manifold f : N → M of E lifts to an integral manifold f (1) : N → M (1) of E (1), and that the transverse integral manifold of E (1) is locally of this form. Inductively, the kth prolongation (M (k) , E (k)) of (M, E) is the ﬁrst prolongation of the (k − 1)st prolongation of (M, E). This gives rise to the prolongation tower · · · → M (k) → M (k−1) → · · · → M (1) → M.

An integral manifold of (M, E) lifts to an integral manifold of each (M (k), E (k)) in this tower. Two examples will help to clarify the construction. The ﬁrst is the prolongation tower of the multi-contact system (Gn(T X), I) itself, and this will

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give us more detailed information about the structure of the ideals E (k) for general (M, E). The second is the prolongation tower of the EDS associated to a ﬁrst-order PDE system, most of which we leave as an exercise. Example 1. Consider the multi-contact ideal I on Gn(T X), over a manifold X of dimension n + s with local coordinates (xi , z α). We can see from the coordinate expression (4.36) that its integral elements over the dense open subset where i dxi = 0 are exactly the n-planes of the form α Epα = {dz α − pαdxi , dpi − pα dxj }⊥ ⊂ T (Gn(T X)), i ij ij

for some constants pα = pα . These pα are local ﬁber coordinates for the proij ji ij longation (Gn (T X)(1) , I (1)). Furthermore, with respect to the full coordinates (xi , z α, pα, pα ) for Gn(T X)(1) ⊂ Gn(T Gn (T X)), the 1-jet graphs of integral ij i manifolds of I ⊂ Ω∗ (Gn(T X)) satisfy dz α − pα dxi = 0, dpα − pα dxj = 0. i i ij It is these s + ns 1-forms that diﬀerentially generate the prolonged Pfaﬃan system I (1). It is not diﬃcult to verify that we have globally Gn(T X)(1) = G2,n(X), the bundle of 2-jets of n-dimensional submanifolds of X, and that I (1) ⊂ Ω∗ (G2,n(X)) is the Pfaﬃan system whose transverse integral manifolds are 2-jet graphs of submanifolds of X. More generally, let Gk = Gk,n(X) → X be the bundle of k-jets of ndimensional submanifolds of X. Because a 1-jet of a submanifold is the same as a tangent plane, G1 = Gn (T X) is the original space whose prolongation tower we are describing. Gk carries a canonical Pfaﬃan system Ik ⊂ Ω∗ (Gk ), whose transverse integral manifolds are k-jet graphs f (k) : N → Gk of ndimensional submanifolds f : N → X. This is perhaps clearest in coordinates. Letting (xi , z α) be coordinates on X, Gk has induced local coordinates (xi , z α, pα, . . . , pα), |I| ≤ k, corresponding to the jet at (xi , z α) of the submanI i ifold {(¯i, z α) ∈ X : z α = z α + pα (¯i − xi ) + · · · + x ¯ ¯ i x

1 α x I! pI (¯

− x)I }.

In terms of these coordinates, the degree-1 part Ik ⊂ T ∗ (Gk ) of the Pfaﬃan system Ik is generated by θα = dz α − pα dxi, i α θi = dpα − pα dxj , i ij . . . α θI = dpα − pα dxj , I Ij

(4.37) |I| = k − 1.

It is not hard to see that the transverse integral manifolds of this Ik are as described above. The point here is that (Gk , Ik) is the ﬁrst prolongation of (Gk−1, Ik−1) for each k > 1, and is therefore the (k − 1)st prolongation of the original (G1 , I1) = (Gn(T X), I).

4.3. HIGHER-ORDER CONSERVATION LAWS For future reference, we note the structure equations α dθα = −θi ∧ dxi, α α dθi = −θij ∧ dxj , . . . α dθI α dθI

171

(4.38) |I| = k − 2, |I| = k − 1.

= =

α −θIj ∧ dxj , −dpα ∧ dxj , Ij

There is a tower · · · → Gk → Gk−1 → · · · → G1 , (4.39)

and one can pull back to Gk any functions or diﬀerential forms on Gk , with k < k. Under these maps, our diﬀerent uses of the coordinates pα and forms I α θI are consistent, and we can also write Ik ⊂ I k ⊂ T ∗ G k , for k < k.

None of the Ik is an integrable Pfaﬃan system. In fact, the ﬁltration on Gk Ik ⊃ Ik−1 ⊃ · · · ⊃ I1 ⊃ 0 coincides with the derived ﬂag of Ik ⊂ T ∗ Gk,n (cf. Ch. II, §4 of [B+ 91]). Example 2. Our second example of prolongation relates to a ﬁrst-order PDE α system F a(xi , z α(x), zxi (x)) = 0 for some unknown functions z α (x). The equaa i α α tions F (x , z , pi ) deﬁne a locus MF in the space J 1 (Rn , Rs) of 1-jets of maps z : Rn → Rs , and we will assume that this locus is a smooth submanifold which submersively surjects onto Rn. The restriction to MF ⊂ J 1 (Rn, Rs) of the multi-contact Pfaﬃan system I1 = {dz α − pα dxi} generates an EDS (MF , IF ). i Now, the set of integral elements for (MF , IF ) is a subset of the set of integral elements for I1 in J 1(Rn , Rs); it consists of those integral elements of I1 which are tangent to MF ⊂ J 1(Rn , Rs). Just as in the preceding example, the integral elements of I1 may be identiﬁed with elements of the space J 2(Rn , Rs) of 2-jets of maps. The collection of 2-jets which correspond to integral elements of (MF , IF ) are exactly the 2-jets satisfying the augmented PDE system 0 = 0 = α F a (xi, z α (x), zxi (x)), ∂F a ∂F a α ∂F a α z i j. + z + i α xi ∂x ∂z ∂pα x x j

Therefore, integral manifolds of the prolongation of the EDS associated to a PDE system correspond to solutions of this augmented system. For this reason, prolongation may generally be thought of as adjoining the derivatives of the original equations. It is important to note that as the ﬁrst prolongation of arbitrary (M, E) is embedded in the canonical multi-contact system (Gn(T M ), I), so can all higher prolongations (M (k), E (k)) be embedded in the prolongations (Gk,n(M ), Ik ).

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Among other things, this implies that E (k) is locally generated by forms like (4.37), satisfying structure equations (4.38), typically with additional linearalgebraic relations. Of most interest to us is the inﬁnite prolongation (M (∞) , E (∞) ) of an EDS (M, E). As a space, M (∞) is deﬁned as the inverse limit of

1 · · · → M (k) → · · · → M (0) = M ;

πk+1

πk

π

that is, M (∞) = {(p0, p1, . . . ) ∈ M (0) × M (1) × · · · : πk (pk ) = pk−1 for each k ≥ 1}. An element of M (∞) may be thought of as a Taylor series expansion for a possible integral manifold of (M, E). M (∞) is generally of inﬁnite dimension, but its presentation as an inverse limit will prevent us from facing analytic diﬃculties. In particular, smooth functions and diﬀerential forms are by deﬁnition the corresponding objects on some ﬁnite M (k), pulled up to M (∞) by the projections. It therefore makes sense to deﬁne E (∞) = E (k),

k>0

which gives an EDS on M (∞) whose transverse integral manifolds are the inﬁnite-jet graphs of integral manifolds of (M (0), E (0)). E (∞) is a Pfaﬃan system, diﬀerentially generated by its degree-1 part I (∞) = I (k) , where each I (k) is the degree-1 part of E (k). In fact, we can see from (4.38) that E (∞) is algebraically generated by I (∞) ; that is, I (∞) is a formally integrable Pfaﬃan system, although this is not true of any ﬁnite I (k) . However, there is no analog of the Frobenius theorem for the inﬁnite-dimensional M (∞) , so we must be cautious about how we use this fact. Vector ﬁelds on M (∞) are more subtle. By deﬁnition, V(M (∞) ) is the Lie algebra of derivations of the ring R(M (∞) ) of smooth functions on M (∞) . In case M (∞) = J ∞ (Rn, Rs), a vector ﬁeld is of the form α ∂ α ∂ ∂ v = vi ∂xi + v0 ∂zα + · · · + vI ∂pα + · · · .

I

α Each coeﬃcient vI is a function on some J k (Rn, Rs), possibly with k > |I|. Although v may have inﬁnitely many terms, only ﬁnitely many appear in its application to any particular f ∈ R(M (∞) ), so there are no issues of convergence.

4.3.2

Noether’s Theorem

To give the desired generalization of Noether’s theorem, we must ﬁrst discuss a generalization of the inﬁnitesimal symmetries used in the classical version. For convenience, we change notation and let (M, E) denote the inﬁnite prolongation of an exterior diﬀerential system (M (0) , E (0)).

4.3. HIGHER-ORDER CONSERVATION LAWS

173

Deﬁnition 4.10 A generalized symmetry of (M (0), E (0)) is a vector ﬁeld v ∈ V(M ) such that Lv E ⊆ E. A trivial generalized symmetry is a vector ﬁeld v ∈ V(M ) such that v E ⊆ E. The space g of proper generalized symmetries is the quotient of the space of generalized symmetries by the subspace of trivial generalized symmetries. Several remarks are in order. • The Lie derivative in the deﬁnition of generalized symmetry is deﬁned by the Cartan formula Lv ϕ = v dϕ + d(v ϕ). The usual deﬁnition involves a ﬂow along v, which may not exist in this setting. • A trivial generalized symmetry is in fact a generalized symmetry; this is an immediate consequence of the fact that E is diﬀerentially closed. • The space of generalized symmetries has the obvious structure of a Lie algebra. • Using the fact that E is a formally integrable Pfaﬃan system, it is easy to show that the condition v E ⊆ E for v to be a trivial generalized symmetry is equivalent to the condition v I = 0, where I = E ∩ Ω1(M ) is the degree-1 part of E. • The vector subspace of trivial generalized symmetries is an ideal in the Lie algebra of generalized symmetries, so g is a Lie algebra as well. The following proof of this fact uses the preceding characterization v I = 0 for trivial generalized symmetries: if Lv E ⊂ E, w I = 0, and θ ∈ Γ(I), then [v, w] θ = = = = −w −w −w 0. (v dθ) + v(w θ) − w(v θ) (Lv θ − d(v θ)) + 0 − w d(v Lv θ

θ)

The motivation for designating certain generalized symmetries as trivial comes from a formal calculation which shows that a trivial generalized symmetry is tangent to any integral manifold of the formally integrable Pfaﬃan system E. Thus, the “ﬂow” of a trivial generalized symmetry does not permute the integral manifolds of E, but instead acts by diﬀeomorphisms of each “leaf”. The following example is relevant to what follows. Let M (0) = J 1(Rn , R) be the standard contact manifold of 1-jets of functions, with global coordinates (xi , z, pi) and contact ideal E (0) = {dz − pi dxi, dpi ∧ dxi}. The inﬁnite prolongation of (M (0) , E (0)) is M = J ∞ (Rn , R), E = {θI : |I| ≥ 0},

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where M has coordinates (xi , z, pi, pij , . . . ), and θI = dpI − pIj dxj . (For the empty index I = ∅, we let p = z, so θ = dp − pi dxi is the original contact form.) Then the trivial generalized symmetries of (M, E) are the total derivative vector ﬁelds ∂ ∂ ∂ Di = ∂xi + pi ∂z + · · · + pIi ∂pI + · · · . We will determine the proper generalized symmetries of (M, E) shortly. There is another important feature of a vector ﬁeld on the inﬁnite prolongation (M, E) of (M (0) , E (0)), which is its order. To introduce this, ﬁrst note that any vector ﬁeld v0 ∈ V(M (0) ) on the original, ﬁnite-dimensional manifold induces a vector ﬁeld and a ﬂow on each ﬁnite prolongation M (k), and therefore induces on M itself a vector ﬁeld v ∈ V(M ) having a ﬂow. A further special property of v ∈ V(M ) induced by v0 ∈ V(M (0) ) is that Lv (Ik ) ⊆ Ik for each k ≥ 1. Though it is tempting to try to characterize those v ∈ V(M ) induced by such v0 using this last criterion, we ought not to do so, because this is not a criterion that can be inherited by proper generalized symmetries of (M, E). Speciﬁcally, an arbitrary trivial generalized symmetry v ∈ V(M ) only satisﬁes Lv (Ik ) ⊆ Ik+1, so a generalized symmetry v can be equivalent (modulo trivials) to one induced by a v0 ∈ V(M (0) ), without satisfying Lv (Ik ) ⊆ Ik . Instead, we have the following. Deﬁnition 4.11 For a vector ﬁeld v ∈ V(M ), the order of v, written o(v), is the minimal k ≥ 0 such that Lv (I0 ) ⊆ Ik+1. With the restriction o(V ) ≥ 0, the orders of equivalent generalized symmetries of E are equal. A vector ﬁeld induced by v0 ∈ V(M (0) ) has order 0. Further properties are: • o(v) = k if and only if for each l ≥ 0, Lv (Il ) ⊆ Il+k+1 ; • letting gk = {v : o(v) ≤ k}, we have [gk, gl ] ⊆ gk+l . We now investigate the generalized symmetries of the prolonged contact system on M = J ∞ (Rn, R). The conclusion will be that the proper generalized symmetries correspond to smooth functions on M ; this is analogous to the ﬁnite-dimensional contact case, in which we could locally associate to each contact symmetry its generating function, and conversely. Recall that we have a coframing (dxi, θI ) for M , satisfying dθI = −θIj ∧ dxj for all multi-indices I. To describe vector ﬁelds on M , we will work with the dual framing (Di , ∂/∂θI ), which in terms of the usual framing (∂/∂xi , ∂/∂pI ) is given by Di ∂ ∂θI = = ∂ + ∂xi ∂ . ∂pI pIi

|I|≥0

∂ , ∂pI

4.3. HIGHER-ORDER CONSERVATION LAWS

175

The vector ﬁelds Di may be thought of as “total derivative” operators, and applied to a function g(xi , z, pi, . . . , pI ) on some J k (Rn , R) give (Di g)(xj , z, pj , . . . , pI , pIj ) =

∂g ∂xi ∂g ∂g + pi ∂g + pij ∂pj + · · · + pIi ∂pI , ∂z

which will generally be deﬁned only on J k+1(Rn , R) rather than J k (Rn, R). These operators can be composed, and we set DI = D i 1 ◦ · · · ◦ D i k , I = (i1 , . . . , ik ).

We do this because the proper generalized symmetries of I = {θI : |I| ≥ 0} are uniquely represented by vector ﬁelds v=g where g = v θ and gI = DI g, |I| > 0. (4.41) ∂ ∂ ∂ + gi i + · · · + g I +··· , ∂θ ∂θ ∂θI (4.40)

To see this, ﬁrst note that any vector ﬁeld is congruent modulo trivial generalized symmetries to a unique one of the form (4.40). It then follows from a straightforward calculation that a vector ﬁeld of the form (4.40) is a generalized symmetry of I if and only if it satisﬁes (4.41). If one deﬁnes Rk ⊂ R(M ) to consist of functions pulled back from J k (Rn, R), then one can verify that for any proper generalized symmetry v ∈ g, o(v) ≤ k ⇐⇒ g=v θ ∈ Rk+1 . The general version of Noether’s theorem will involve proper generalized symmetries. However, recall that our ﬁrst-order version requires us to distinguish among the symmetries of an Euler-Lagrange system the symmetries of the original variational problem; only the latter give rise to conservation laws. We therefore have to give the appropriate corresponding notion for proper generalized symmetries. For this purpose, we introduce the following algebraic apparatus. We ﬁlter the diﬀerential forms Ω∗ (M ) on the inﬁnite prolongation (M, E) of an EulerLagrange system (M (0), E (0)) by letting I p Ωp+q (M ) = Image(E ⊗ · · · ⊗ E ⊗Ω∗ (M ) → Ω∗ (M )) ∩ Ωp+q (M ). p (4.42)

We deﬁne the associated graded objects Ωp,q (M ) = I p Ωp+q (M )/I p+1 Ωp+q (M ). Because E is formally integrable, the exterior derivative d preserves this ﬁltration and its associated graded objects: d : Ωp,q (M ) → Ωp,q+1 (M ).

176 We deﬁne the cohomology p,q HΛ (M ) =

CHAPTER 4. ADDITIONAL TOPICS

Ker(d : Ωp,q (M ) → Ωp,q+1 (M )) . Im(d : Ωp,q−1(M ) → Ωp,q (M ))

A simple diagram-chase shows that the exterior derivative operator d induces p+1,q p,q a map d1 : HΛ (M ) → HΛ (M ).8 Now, the Poincar´-Cartan form Π ∈ e n+1 (0) Ω (M ) pulls back to an element Π ∈ I 2 Ωn+1 (M ) which is closed and 2,n−1 therefore deﬁnes a class [Π] ∈ HΛ (M ). It follows from the deﬁnition that a generalized symmetry of E preserves the 2,n−1 ﬁltration I p Ωp+q (M ), and therefore acts on the cohomology group HΛ (M ). The generalized symmetries appropriate for Noether’s theorem are exactly those generalized symmetries v of E satisfying the additional condition

2,n−1 Lv [Π] = 0 ∈ HΛ (M ).

In other words, v is required to preserve Π modulo (a) forms in I 3 Ωn+1 (M ), and (b) derivatives of forms in I 2 Ωn (M ). We also need to verify that a trivial generalized symmetry v preserves the class [Π]; this follows from the fact that v I = 0, for then v Π ∈ I 2 Ωn (M ), so that Lv [Π] = [d(I 2 Ωn(M ))] = 0. We now have the Lie subalgebra g[Π] ⊆ g of proper generalized symmetries of the variational problem. It is worth noting that this requires only that we have Π deﬁned modulo I 3 Ωn+1(M ). A consequence is that even in the most general higher-order, multi-contact case where a canonical Poincar´-Cartan form is not e known to exist, there should be a version of Noether’s theorem that includes both the ﬁrst-order multi-contact version discussed in the previous section, and the higher-order scalar version discussed below. However, we will not pursue this. The other ingredient in Noether’s theorem is a space of conservation laws, deﬁned by analogy with previous cases as

0,n−1 ¯ C(E) = H n−1(Ω∗ /E, d) = HΛ (M ),

where the last notation refers to the cohomology just introduced. It is a substantial result (see [BG95a]) that over contractible subsets of M (0) we can use the exterior derivative to identify C(E) with def 1,n−1 2,n−1 ¯ C(E) = Ker(d1 : HΛ (M ) → HΛ (M )).

Now we can identify conservation laws as classes of n-forms, as in the previous case of Noether’s theorem, and we will do so without comment in the following.

8 Of course, H ∗,∗ (M ) is the E -term of a spectral sequence. Because we will not be using 1 Λ any of the higher terms, however, there is no reason to invoke this machinery. Most of this theory was introduced in [Vin84].

4.3. HIGHER-ORDER CONSERVATION LAWS ¯ We deﬁne a Noether map g[Π] → C(E) as v → v well-deﬁned, ﬁrst note that v Π ∈ I 1 Ωn(M ),

177 Π. To see that this is

so v Π represents an element of Ω1,n−1(M ), which we shall also denote as v Π. Furthermore, its exterior derivative is d(v Π) = Lv Π, (4.43)

and this lies in I 2 Ωn+1, simply because v preserves E and therefore also the ﬁltration (4.42). Consequently, v Π ∈ Ker(d : Ω1,n−1(M ) → Ω1,n(M )),

and we therefore have an element [v Finally, we need to verify that [v

1,n−1 2,n−1 Π] ∈ Ker(d1 : HΛ (M ) → HΛ (M )). 1,n−1 Π] ∈ HΛ (M ).

This follows from the hypothesis that v preserves not only the Euler-Lagrange system E and associated ﬁltration (4.42), but also the class [Π]. Speciﬁcally, the image 2,n−1 (M ) d1 ([v Π]) ∈ HΛ is represented by the class (see (4.43)) [d(v This proves that v → [v Π] deﬁnes a map between the appropriate spaces. We can now make the following proposal for a general form of Noether’s theorem. Conjecture 4.1 Let (M, E) be the inﬁnite prolongation of an Euler-Lagrange q system, and assume that the system is non-degenerate and that HdR (M ) = 0 for all q > 0. Then the map v → [v Π] induces an isomorphism

∼ ¯ g[Π] −→ C(E).

Π)] = [Lv Π] = Lv [Π] = 0.

It is quite possible that this is already essentially proved in [Vin84] or [Olv93], but we have not been able to determine the relationship between their statements and ours. In any case, it would be illuminating to have a proof of the present statement in a spirit similar to that of our Theorem 1.3.

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To clarify this, we will describe how it appears in coordinates. First, note that for the classical Lagrangian L(xi , z, pi)dx, the Euler-Lagrange equation E(xi , z, pi, pij ) = ( Dj Lpj − Lz )(xi , z, pi, pij ) = 0

deﬁnes a locus M (1) ⊂ J 2 (Rn, R), and the ﬁrst prolongation of the EulerLagrange system (J 1(Rn , R), EL) discussed previously is given by the restriction of the second-order contact Pfaﬃan system on J 2 (Rn, R) to this locus. Higher prolongations are deﬁned by setting M (k) = {EI = DI E = 0, |I| ≤ k − 1} ⊂ J k+1 (Rn, R) and restricting the (k + 1)st-order contact system I (k+1). We will consider generalized symmetries of (M (∞) , E (∞) ) which arise as restrictions of those generalized symmetries of (J ∞ (Rn , R), I) which are also tangent to M (∞) ⊂ J ∞ (Rn, R). This simpliﬁes matters insofar as we can understand generalized symmetries of I by their generating functions. The tangency condition is Lv (EI )|M (∞) = 0, |I| ≥ 0. (4.44) def This Lie derivative is just the action of a vector ﬁeld as a derivation on functions. Now, for a generalized symmetry v of the inﬁnite-order contact system, all of the conditions (4.44) follow from just the ﬁrst one, Lv (E)|M (∞) = 0. If we let v have generating function g = v θ ∈ R(J ∞ (Rn, R)), then we can see from (4.40, 4.41) that this condition on g is

∂E ∂pI DI g |I|≥0

= 0 on M (∞) .

(4.45)

We are again using p = z for convenience. For instance, E = pii − f(z) deﬁnes the Poisson equation ∆z = f(z), and the preceding condition is

2 Di g − f (z)g = 0 on M (∞) .

(4.46)

i

We now consider the Noether map for M (∞) ⊂ J ∞ (Rn, R). We write the Poincar´-Cartan form pulled back to M (∞) using coframes adapted to this ine ﬁnite prolongation, starting with dLpi = Dj (Lpi )dxj + Lpi z θ + Lpi pj θj ,

4.3. HIGHER-ORDER CONSERVATION LAWS and then the Poincar´-Cartan form on J ∞ (Rn , R) is e Π = d(L dx + θ ∧ Lpi dx(i))

179

= θ ∧ (−Di (Lpi ) + Lz )dx − θj ∧ Lpi pj dx(i) .

Restriction to M (∞) ⊂ J ∞ (Rn, R) kills the ﬁrst term, and we have Π = −Lpi pj θ ∧ θj ∧ dx(i). We then apply a vector ﬁeld vg with generating function g, and obtain vg Π = −gLpi pj θj ∧ dx(i) + (Dj g)Lpi pj θ ∧ dx(i).

This will be the “diﬀerentiated form” of a conservation law precisely if Lvg [Π] = 0, that is, if d(vg Π) ≡ 0 (mod I 3 Ωn+1(M ) + dI 2 Ωn(M )).

Concerning generalized symmetries of a PDE, note that in the condition (4.45) for g = g(xi, p, pi, . . . , pI ) ∈ Rk = C ∞ (J k (Rn , R)), the variables pI with |I| > k appear only polynomially upon taking the total derivatives DJ g. In other words, the condition on g is polynomial in the variables pI for |I| > k. Equating coeﬃcients of these polynomials gives a PDE system to be satisﬁed by a generalized symmetry of an Euler-Lagrange equation. With some eﬀort, one can analyze the situation for our Poisson equation ∆z = f(z) and ﬁnd the following. Proposition 4.6 If n ≥ 3, then a solution g = g(xi , p, pi, . . . , pI ) of order k to (4.46) is equal on M (∞) to a function that is linear in the variables pJ with |J| ≥ k − 2. If in addition f (z) = 0, so that the Poisson equation is nonlinear, then every solution’s restriction to M (∞) is the pullback of a function on M (0) ⊂ J 2 (Rn, R), which generates a classical symmetry of the equation. In other words, a non-linear Poisson equation in n ≥ 3 independent variables has no non-classical generalized symmetries, and consequently no higher-order conservation laws. Proof. Because the notation involved here becomes rather tedious, we will sketch the proof and leave it to the reader to verify the calculations. We previously hinted at the main idea: the condition (4.46) on a generating function g = g(xi , p, . . . , pI ), |I| = k, is polynomial in the highest order variables with coeﬃcients depending on partial derivatives of g. To isolate these terms, we ﬁlter the functions on M (∞) by letting Rl be the image of Rl under restriction to M (∞) ; in other words, Rl consists of functions which can be expressed as functions of xi, pJ , |J| ≤ l, after substituting the deﬁning relations of M (∞) , pJ ii = d|J | f . dxJ

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CHAPTER 4. ADDITIONAL TOPICS

To calculate in Rl we will need to deﬁne variables qJ to be the harmonic parts of pJ ; that is, qi qij qijk = = = = = pi , 1 pij − n δij pll

1 pij − n δij f(p), 1 pijk − n+2 (δij pkll + δjk pill + δki pjll )

pijk −

1 (δ p n+2 ij k

+ δjk pi + δki pj )f (p),

&c.

These, along with xi and p, give coordinates on M (∞) . In addition to working modulo various Rl to isolate terms with higher-order derivatives, we will also at times work modulo functions that are linear in the qI . In what follows, we use the following index conventions: p(l) = (p, pj , . . . , pJ ) (with |J| = l) denotes the derivative variables up to order l, and the multi-indices I, K, A, satisfy |I| = k, |K| = k − 1, |A| = k − 2. Now, starting with g = g(xj , p(k)) ∈ Rk , we note that 0= i 2 Di g − f g ∈ Rk+1;

that is, the possible order-(k + 2) term resulting from two diﬀerentiations of g already drops to order k when restricted to the equation manifold. We consider this expression modulo Rk 9 , and obtain a quadratic polynomial in qIj with coeﬃcients in Rk . We consider only the quadratic terms of this polynomial, which are 0≡ ∂2g qIiqI i . ∂pI ∂pI (4.47)

|I|,|I |=k 1≤i≤n

2

g To draw conclusions about ∂p∂ ∂p from this, we need the following fundamental I I lemma, in which the diﬀerence between the cases n = 2 and n ≥ 3 appears:

If Hl ⊂ Sym l (Rn )∗ denotes the space of degree-l homogeneous harmonic polynomials on Rn , n ≥ 3, then the O(n)-equivariant contraction map Hl+1 ⊗ Hm+1 → Hl ⊗ Hm given by XC z C ⊗ Y Q z Q → i XBi z B ⊗ YP i z P

is surjective; here, |C| − 1 = |B| = l, |Q| − 1 = |P | = m. We will apply this in situations where a given g BP ∈ Sym l (Rn) ⊗ Sym m (Rn) is known to annihilate all XBi YP i with XC , YQ harmonic, for then we have g BP

9 In this context, “modulo” refers to quotients of vector spaces by subspaces, not of rings by ideals, as in exterior algebra.

4.3. HIGHER-ORDER CONSERVATION LAWS

181

orthogonal to Hl ⊗ Hm ⊂ Sym l (Rn)∗ ⊗ Sym m (Rn )∗ . In particular, from (4.47) we have ∂2g qI qJ ≡ 0. ∂pI ∂pJ This means that the restriction of the function g to the hyperplanes pAii = is linear in the highest pI ; in other words, we can write g(xi , p(k)) = h(x, p(k−1)) + hI (x, p(k−1))pI for some hI ∈ Rk−1. We can further assume that all hAii = 0, where |A| = k−2. This completes the ﬁrst step. The second step is to simplify the functions hI (x, p(k−1)), substituting our new form of g into the condition (4.46). Again, the “highest” terms appear modulo Rk , and are 0≡2 ∂hI ∂hI qKiqIi + 2 qAiqIi ; ∂pK ∂pA d|A| f dxA

here we recall our index convention |A| = k −2, |K| = k −1, |I| = k. Both terms must vanish separately, and for the ﬁrst, our lemma on harmonic polynomials def ∂hI gives that [h]IK = ∂pK is orthogonal to harmonics; but then our normalization hypothesis hAii = 0 gives that [h]IK = 0. We conclude that g(xi , p(k)) = h(x, p(k−1)) + hI (x, p(k−2))pI , For the second term, our lemma gives similarly that

∂hI ∂pA

|I| = k. = 0, so have

g(xi , p(k)) = h(x, p(k−1)) + hI (x, p(k−3))pI . This completes the second step. For the third step, we again substitute the latest form of g into the condition (4.46), and now work modulo Rk−1. The only term non-linear in the pI is 0≡ ∂2h pKi pK i , ∂pK ∂pK

and as before, the lemma implies that h is linear in pK . Now we have g(xi , p(k)) = h(x, p(k−2)) + hK (x, p(k−2))pK + hI (x, p(k−3))pI .

∂hK ∂pA pAi pKi ,

Again working modulo Rk−1, the only term that is non-linear in (pK , pI ) is K so as before, we can assume ∂hA = 0 and write ∂p g(xi , p(k)) = h(x, p(k−2)) + hK (x, p(k−3))pK + hI (x, p(k−3))pI . The ﬁnal step is similar, and gives that h is linear in pA . This yields g(xi , p(k)) = h + hApA + hK pK + hI pI , (4.48)

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CHAPTER 4. ADDITIONAL TOPICS

where each of h, hA, hK , hI is a function of (xi , p(k−3)). This is the ﬁrst statement of the proposition. To derive the second statement, we use the form (4.48) in the condition (4.46) modulo Rk−2, in which the only term non-linear in (pK , pI ) is 0 ≡ ≡ hI (xi , p(k−3)) d|I|f dxI I i (k−3) h (x , p )f (p) i∈I pI\i pi.

In particular, if f (p) = 0, then we must have hI = 0. Therefore g ∈ Rk actually lies in Rk−1, and we can induct downward on k, eventually proving that g ∈ R1 , as desired. We mention two situations which contrast sharply with that of the non-linear Poisson equation in dimension n ≥ 3. First, in the case of a linear Poisson equation ∆z = f(z), f (z) = 0 (still in n ≥ 3 dimensions), one can extend the preceding argument to show that a generating function for a conservation law is linear in all of the derivative variables pI . In particular, the inﬁnite collection of conservation laws for the Laplace equation ∆z = 0 can be determined in this manner; it is interesting to see how all of these disappear upon the addition of a non-linear term to the equation. Second, in dimension n = 2, there are well-known non-linear Poisson equations ∆u = sinh u and ∆u = eu having inﬁnitely many higher-order conservation laws, but we will not discuss these.

4.3.3

In §1.4, we constructed Monge-Ampere systems on the contact manifold M 5 of oriented tangent planes to Euclidean space E3 , whose integral manifolds corresponded to linear Weingarten surfaces. We brieﬂy recall this setup for the case of surfaces with Gauss curvature K = −1. Our index ranges are now 1 ≤ a, b, c ≤ 3, 1 ≤ i, j, k ≤ 2. Let F → E3 be the Euclidean frame bundle, with its global coframing ω a , b a ωb = −ωa satisfying the structure equations a dωa = −ωb ∧ ωb , a a c dωb = −ωc ∧ ωb . 3 We set θ = ω3 , πi = ωi , and then θ ∈ Ω1(F) is the pullback of a global contact 3 form on M = G2 (T E ). The forms that are semibasic over M are generated by θ, ωi , πi ∈ Ω1 (F). We deﬁne the 2-forms on F

The K = −1 Surface System

Θ = dθ = −π1 ∧ ω1 − π2 ∧ ω2 , Ψ = π 1 ∧ π2 + ω 1 ∧ ω 2 ,

which are pullbacks of uniquely determined forms on M . On a transverse integral element E 2 ⊂ Tm M of the contact system I = {θ, Θ}, on which ω 1 ∧ω2 = 0,

4.3. HIGHER-ORDER CONSERVATION LAWS there are relations πi = hij ωj , In this case, π1 ∧ π2 = Kω1 ∧ ω2 , hij = hji.

183

where K = h11h22 − h12h21 is the Gauss curvature of any surface N 2 → E3 whose 1-jet graph in M 5 is tangent to E ⊂ Tm M . Therefore, transverse integral manifolds of the EDS E = {θ, Θ, Ψ} (4.49)

correspond locally to surfaces in E3 with constant Gauss curvature K = −1. The EDS (M, E) is an example of a hyperbolic Monge-Ampere system; this notion appeared in §2.1, where we used it to specify a branch of the equivalence problem for Poincar´-Cartan forms on contact 5-manifolds. The deﬁning e property of a hyperbolic Monge-Ampere system E = {θ, Θ, Ψ} is that modulo the algebraic ideal {θ}, E contains two distinct (modulo scaling) decomposable 2-forms; that is, one can ﬁnd two non-trivial linear combinations of the form λ1 Θ + µ 1 Ψ λ2 Θ + µ 2 Ψ = = α 1 ∧ β1 ,

α 2 ∧ β2 .

This exhibits two rank-2 Pfaﬃan systems Ii = {αi, βi }, called the characteristic systems of E, which are easily seen to be independent of choices (except for which one is I1 and which one is I2 ). The relationship between the geometry of the characteristic systems and that of the original hyperbolic Monge-Ampere system is very rich (see [BGH95]). Of particular interest are those hyperbolic systems whose characteristic systems each contain a non-trivial conservation law. We will show that this holds for the K = −1 system introduced above, but only after one prolongation. In other words, for the prolonged system E (1) , there is (1) also a notion of characteristic systems Ii which restrict to any integral surface (1) as the original Ii , and each of these Ii contains a non-trivial conservation law for E (1). Returning to the discussion of integral elements of E = {θ, Θ, Ψ}, note that for any integral element E ⊂ T(p,H) M , given by equations πi − hij ωj = 0, there is a unique frame (p, (e1 , e2, e1 × e2 )) ∈ F over (p, e1 ∧ e2 ) ∈ M for which the second fundamental form is normalized as h11 = a > 0,

1 h22 = − a ,

h12 = h21 = 0.

The tangent lines in E3 spanned by these e1 , e2 are the principal directions at p of any surface whose 1-jet graph is tangent to E; they deﬁne an orthonormal frame in which the second fundamental form is diagonal. The number a > 0 is

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CHAPTER 4. ADDITIONAL TOPICS

determined by the plane E ⊂ T M , so to study integral elements of (M, E), and in particular to calculate on its ﬁrst prolongation, we introduce F (1) = F × R∗, where R∗ has the coordinate a > 0. There is a projection F (1) → M (1) , mapping

1 (p, e, a) → (p, e1 ∧ e2 , {π1 − aω1 , π2 + a ω2 }⊥ ).

We deﬁne on F (1) the forms θ1 = π1 − aω1 , 1 θ2 = π 2 + a ω 2 , which are semibasic for F (1) → M (1). The ﬁrst prolongation of the system E on M is a Pfaﬃan system on M (1) , which pulls back to F (1) as E (1) = {θ, θ1 , θ2, dθ1, dθ2 }. We have structure equations dθ ≡ −θ1 ∧ ω1 − θ2 ∧ ω2 ≡ 0 2 1 dθ1 ≡ −da ∧ ω1 + 1+a ω2 ∧ ω2 a 2 1 1+a2 1 dθ2 ≡ a2 da ∧ ω + a ω2 ∧ ω1 (mod {θ, θ1 , θ2}),

and in particular, we have the decomposable linear combinations

1 1 −dθ1 − a dθ2 = (da − (1 + a2 )ω2 ) ∧ (ω1 + a ω2 ), 1 1 −dθ1 + a dθ2 = (da + (1 + a2 )ω2 ) ∧ (ω1 − a ω2 ).

(4.50)

The EDS E (1) is algebraically generated by θ, θ1 , θ2 , and these two decomposable 2-forms. The characteristic systems are by deﬁnition diﬀerentially generated by

1 1 I1 = {θ, θ1 , θ2 , da − (1 + a2 )ω2 , ω1 + a ω2 }, (1) 1 2 1 1 I2 = {θ, θ1 , θ2 , da + (1 + a )ω2 , ω − a ω2 }. (1)

(4.51)

Now, the “universal” second fundamental form can be factored as

1 1 1 II = a(ω1 )2 − a (ω2 )2 = a(ω1 + a ω2 )(ω1 − a ω2 ).

(4.52)

These linear factors, restricted any K = −1 surface, deﬁne the asymptotic curves of that surface, so by comparing (4.51) and (4.52) we ﬁnd that: On a K = −1 surface, the integral curves of the characteristic systems are the asymptotic curves. Now we look for Euclidean-invariant conservation laws in each Ii . Instead of using Noether’s theorem, we work directly. We start by setting

1 ϕ1 = f(a)(ω1 + a ω2 ) ∈ I1 , (1) (1)

4.3. HIGHER-ORDER CONSERVATION LAWS

185

and seek conditions on f(a) to have dϕ1 ∈ E (1). A short computation using the structure equations gives

1 1 dϕ1 ≡ (f (a)(1 + a2) − f(a)a)ω2 ∧ (ω1 + a ω2 )

(mod E (1)),

so the condition for ϕ1 to be a conserved 1-form is a f (a) = . f(a) 1 + a2 A solution is ϕ1 =

1 2

1 + a2 (ω1 + 1 ω2); a

1 the choice of multiplicative constant 2 will simplify later computations. A 1 similar computation, seeking an appropriate multiple of ω 1 − a ω2 , yields the conserved 1-form ϕ2 = 1 1 + a2 (ω1 − 1 ω2). 2 a

On any simply connected integral surface of the K = −1 system, there are coordinate functions s, t such that ϕ1 = ds, ϕ2 = dt.

If we work in these coordinates, and in particular use the non-orthonormal coframing (ϕ1 , ϕ2) then we can write ω1 ω2 I II = = = = 1 √ 1 + a2 a √ 1 + a2 (ϕ1 + ϕ2 ), (ϕ1 − ϕ2 ), 1 − a2 1 + a2 ϕ1 ϕ2 + (ϕ2 )2 , (4.53) (4.54) (4.55) (4.56)

(ω1 )2 + (ω2 )2 = (ϕ1 )2 + 2 4a 1 + a2 ϕ1 ϕ2 .

These expressions suggest that we deﬁne a = tan z, where a > 0 means that we can smoothly choose z = tan−1 a ∈ (0, π/2). Note that 2z is the angle measure between the asymptotic directions ϕ⊥ , ϕ⊥ , and 1 2 that ω1 = (cos z)(ϕ1 + ϕ2 ), ω2 = (sin z)(ϕ1 − ϕ2 ). (4.57)

The following is fundamental in the study of K = −1 surfaces.

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CHAPTER 4. ADDITIONAL TOPICS

Proposition 4.7 On an immersed surface in E3 with constant Gauss curvature K = −1, the associated function z, expressed in terms of asymptotic coordinates s, t, satisﬁes the sine-Gordon equation zst =

1 2

sin(2z).

(4.58)

One can prove this by a direct computation, but we will instead highlight certain general EDS constructions which relate the K = −1 diﬀerential system to a hyperbolic Monge-Ampere system associated to the sine-Gordon equation. One of these is the notion of an integrable extension of an exterior diﬀerential system, which we have not yet encountered. This is a device that handles a forseeable diﬃculty; namely, the sine-Gordon equation is expressed in terms of the variables s and t, but for the K = −1 system, these are primitives of a conservation law, deﬁned on integral manifolds of the system only up to addition of integration constants. One can think of an integrable extension as a device for appending the primitives of conserved 1-forms. More precisely, an integrable extention of π an EDS (M, E) is given by a submersion M → M , with a diﬀerential ideal E ∗ on M generated algebraically by π E and some 1-forms on M . In this case, the preimage in M of an integral manifold of E is foliated by integral manifolds of E . For example, if a 1-form ϕ ∈ Ω1 (M ) is a conservation law for E, then one can take M = M × R, and let E ⊂ Ω∗ (M ) be generated by E and ϕ = ϕ − ds, ˜ where s is a ﬁber coordinate on R. Then the preimage in M of any integral manifold of (M, E) is foliated by a 1-parameter family of integral manifolds of (M , E ), where the parameter corresponds to a choice of integration constant for ϕ.10 Proof. Because z is deﬁned only on M (1), it is clear that we will have to prolong once more to study zst . (Twice is unnecessary, because of the Monge-Ampere form of (4.58).) From (4.50), we see that integral elements with ϕ1 ∧ ϕ2 = 0 satisfy 1 1 dz − ω2 = 2pϕ1, dz + ω2 = 2qϕ2, for some p, q. These p, q can be taken as ﬁber coordinates on the second prolongation M (2) = M (1) × R2. Let θ3 = dz − pϕ1 − qϕ2 , 1 θ4 = ω2 + pϕ1 − qϕ2,

and then the prolonged diﬀerential system is

E (2) = {θ, θ1 , . . . , θ4 , dθ3, dθ4}. Integral manifolds for the original K = −1 system correspond to integral manifolds of E (2); in particular, on such an integral manifold f (2) : N → M (2) we have 0 = dθ3|N (2) = −dp ∧ ϕ1 − dq ∧ ϕ2

10 For

more information about integrable extensions, see §6 of [BG95b].

4.3. HIGHER-ORDER CONSERVATION LAWS and 0 = = = = dθ4|N (2) Kω1 ∧ ω2 + dp ∧ ϕ1 − dq ∧ ϕ2 (−1)(cos z)(ϕ1 + ϕ2) ∧ (sin z)(ϕ1 − ϕ2) + dp ∧ ϕ1 − dq ∧ ϕ2 sin(2z)ϕ1 ∧ ϕ2 + dp ∧ ϕ1 − dq ∧ ϕ2 .

187

Now we deﬁne the integrable extension M (2) = M (2) × R2, where R2 has coordinates s, t, and on M (2) we deﬁne the EDS E (2) to be generated by E (2), along with the 1-forms ϕ1 − ds, ϕ2 − dt. An integral manifold (2) f (2) : N → M (2) of E (2) gives a 2-parameter family of integral manifolds fs0 ,t0 : N → M (2) of E (2) . On any of these, the functions s, t will be local coordinates, and we will have 0 = dz − p ds − q dt, 0 = −dp ∧ ds − dq ∧ dt, 0 = sin(2z)ds ∧ dt − ds ∧ dp − dq ∧ dt. These three clearly imply that z(s, t) satisﬁes (4.58). Note that we can start from the other side, deﬁning the diﬀerential system for the sine-Gordon equation as EsG = {dz − p ds − q dt, −dp ∧ ds − dq ∧ dt, ds ∧ dp + dq ∧ dt − sin(2z)ds ∧ dt}, which is a Monge-Ampere system on the contact manifold J 1(R2 , R). One can form a “non-abelian” integrable extension P = J 1(R2 , R) × F of (J 1(R2 , R), EsG) by taking 1 ω − (cos z)(ds + dt), 2 ω − (sin z)(ds + dt), 3 ω , EP = EsG + ω2 + p ds − q dt, 1 3 ω − (sin z)(ds + dt), 1 3 ω2 + (cos z)(ds − dt).

.

This system is diﬀerentially closed, as one can see by using the structure equations for F and assuming that z satisﬁes the sine-Gordon equation. Though the

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CHAPTER 4. ADDITIONAL TOPICS

following diagram is complicated, it sums up the whole story: (M (2) , E (2) ) OOO o OOO ooo OOO ooo OOO oo woo ' (M (1) , E (1) ) (M (2), E (2)) OOO o OOO ooo OOO ooo oo OO' wooo (M (1) , E (1)) (M, E) The system (M (1) , E (1) ) was not introduced in the proof; it is the integrable extension of (M (1) , E (1)) formed by adjoining primitives s, t for the conserved 1forms ϕ1, ϕ2 , and its prolongation turns out to be (M (2) , E (2) ). In other words, starting on M (1) , one can ﬁrst prolong and then adjoin primitives, or vice versa. The main point of this diagram is: The identiﬁcation (M (2) , E (2) ) −→ (P, EP ) is an isomorphism of exterior diﬀerential systems. In other words, modulo prolongations and integrable extensions, the K = −1 system and the sine-Gordon system are equivalent. Note that while conservation laws are preserved under prolongation, there is an additional subtlety for integrable extensions. In this case, only those conservation laws for the sine-Gordon system that are invariant under s, t-translation give conservation laws for the K = −1 system. Conversely, only those conservation laws for the K = −1 system that are invariant under Euclidean motions (i.e., translations in F) give conservation laws for the sine-Gordon system. There is even a diﬃculty involving trivial conservation laws; namely, the non-trivial, Euclidean-invariant conservation laws ϕ1, ϕ2 for the K = −1 system induce the trivial conservation laws ds, dt for the sine-Gordon system. However, we do have two conservation laws for sine-Gordon, obtained via Noether’s theorem applied to s, t-translations and the Lagrangian

1 ΛsG = (pq − 2 (cos(2z) − 1))ds ∧ dt;

/ (P, EP ) (J 1 (R2 , R), EsG)

they are

1 ψ1 = 1 p2ds − 4 (cos(2z) − 1)dt, 2 1 2 1 ψ2 = 2 q dt + 4 (cos(2z) − 1)ds.

The corresponding conserved 1-forms on (M (2), E (2)) are

1 ϕ3 = 1 p2ϕ1 − 4 (cos(2z) − 1)ϕ2, 2 1 1 2 ϕ4 = 2 q ϕ2 + 4 (cos(2z) − 1)ϕ1 .

4.3. HIGHER-ORDER CONSERVATION LAWS

189

In the next section, we will introduce a B¨cklund transformation for the sinea Gordon equation, which can be used to generate an inﬁnite double-sequence ψ2k−1, ψ2k of conservation laws (see [AI79]). These in turn give an inﬁnite double-sequence ϕ2k+1 , ϕ2k+1 of independent conservation laws for K = −1, extending the two pairs that we already have. Although we will not discuss these, it is worth pointing out that the generalized symmetries on (M (∞) , E (∞)) to which they correspond under Noether’s theorem are not induced by symmetries at any ﬁnite prolongation. For this reason, they are called hidden symmetries. Finally, we mention the following non-existence result, which is similar to the result in Proposition 4.6 for higher-dimensional non-linear Poisson equations. Proposition 4.8 In dimension n ≥ 3, there are no second-order Euclideaninvariant conservation laws for the linear Weingarten system for hypersurfaces in En+1 with Gauss curvature K = −1. Proof. In contrast to our proof of the analogous statement for non-linear Poisson equations, we give here a direct argument not appealing to generating functions. We work on the product F (1) = F × {(a1, . . . , an) ∈ Rn : ai = −1},

where F is the Euclidean frame bundle for En+1 , and the other factor parameterizes eigenvalues of admissible second fundamental forms. We use the usual structure equations, but without the sum convention: dωi dθ dωij dπi = − = − = − = − ωij ∧ ωj + πi ∧ θ, πj ∧ ω j ωik ∧ ωkj + πi ∧ πj , πj ∧ ωji.

j

j

k

j

There is a Pfaﬃan system I on F (1) whose transverse n-dimensional integral manifolds correspond to K = −1 hypersurfaces; it is diﬀerentially generated by θ and the n 1-forms θ i = π i − a i ωi .

A conservation law for this system is an (n − 1)-form on F (1) whose exterior derivative vanishes on any integral manifold of I. A conservation law is invariant under Euclidean motions if its restriction has the form ϕ= fi (a1 , . . . , an)ω(i).

What we will show is that for n ≥ 3, such a form cannot be closed modulo I unless it equals 0.

190

CHAPTER 4. ADDITIONAL TOPICS First, we calculate using the structure equations that dϕ ≡ (dfi − fj ωij ) ∧ ω(i) (mod I). (4.59)

i

j

To proceed further, we want an expression for ωij ∧ ω(i), which we obtain ﬁrst by computing 0 ≡ dθi − j ≡

(mod I)

πk ∧ ωki − dai ∧ ωi + ai

k

ωik ∧ ωk

≡

−dai ∧ ωi +

k

(ai − ak )ωik ∧ ωk .

and then by multiplying the last result by ω(ij): 0 ≡ dai ∧ ω(j) + (ai − aj )ωij ∧ ω(i) Using this in (4.59), we obtain dfi + fi daj ∧ ω(i) aj − a i daj . ai − a j (mod I).

dϕ ≡

i

j

(mod I).

So ϕ is a conservation law only if for each i, dfi = fi j Keep in mind that we are requiring this equation to hold on the locus F (1) where Πai = −1. Wherever at least one fi (a1 , . . . , an) is non-zero, we must have However, when n ≥ 3, the summands in the expression are linearly independent 2-forms on F (1) . It is worth noting that when n = 2, this 2-form does vanish, and we can solve for f1 (a1 , a2), f2 (a1, a2) to obtain the conservation laws ϕ1 , ϕ2 discussed earlier. The same elementary method can be used to analyze second-order conservation laws for more general Weingarten equations; in this way, one can obtain a full classiﬁcation of those few Wiengarten equations possessing higherorder conservation laws. 0 = d j daj =− ai − a j

j

dai ∧ daj . (ai − aj )2

4.3. HIGHER-ORDER CONSERVATION LAWS

191

4.3.4

Two B¨cklund Transformations a

We have seen a relationship between the K = −1 surface system, and the sineGordon equation zxy =

1 2

sin(2z).

(4.60)

Namely, the half-angle measure between the asymptotic directions on a K = −1 surface, when expressed in asymptotic coordinates, satisﬁes the sine-Gordon equation. We have also interpreted this relationship in terms of important EDS constructions. In this section, we will explain how this relationship connects the B¨cklund transformations associated to each of these systems. a There are many deﬁnitions of B¨cklund transformation in the literature, and a instead of trying to give an all-encompassing deﬁnition, we will restrict attention to Monge-Ampere systems E = {θ, Θ, Ψ},

where θ is a contact form on a manifold (M 5 , I), and Θ, Ψ ∈ Ω2(M ) are linearly ¯ ¯ independent modulo {I}. Suppose that (M, E) and (M , E) are two MongeAmpere systems, with E = {θ, Θ, Ψ}, ¯ ¯ ¯ ¯ E = {θ, Θ, Ψ}.

¯ ¯ A B¨cklund transformation between (M, E) and (M , E), is a 6-dimensional suba ¯ such that in the diagram manifold B ⊂ M × M ~ ~ π ~~ ~ ~ ~~ BA AA π ¯ AA AA ¯ M, (4.61)

M

¯ • each projection B → M , B → M is a submersion; and • pulled back to B, we have ¯ ¯ ¯ {Ψ, Ψ} ≡ {Θ, Θ} (mod {θ, θ}). The second condition implies that the dimension of the space of 2-forms spanned ¯ ¯ ¯ by {Θ, Ψ, Θ, Ψ} modulo {θ, θ} is at most 2. Therefore, ¯ ¯ ¯ {Θ, Ψ} ≡ {Θ, Ψ} (mod {θ, θ}). This consequence is what we really want, but the original formulation has the ¯ extra beneﬁt of ruling out linear dependence between Θ and Θ, which would lead to a triviality in what follows. A B¨cklund transformation allows one to ﬁnd a family of integral manifolds a ¯ ¯ of (M , E) from one integral manifold N 2 → M of (M, E), as follows. On the 3-dimensional preimage π −1 (N ) ⊂ B, the restriction π ∗ E is algebraically gen¯ ¯ ¯ erated by θ alone, and is therefore an integrable Pfaﬃan system. Its integral

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CHAPTER 4. ADDITIONAL TOPICS

manifolds can therefore be found by ODE methods, and they foliate π −1(N ) into a 1-parameter family of surfaces which project by π to integral manifolds ¯ ¯ ¯ ¯ ¯ of (M , E). In each of the following two examples, (M, E) and (M , E) are equal, so one can generate from one known solution many others. Example 1: B¨cklund transformation for the sine-Gordon equation. a The primary example concerns the sine-Gordon equation (4.60). The wellknown coordinate phenomenon is that if two functions u(x, y), u(x, y) satisfy ¯ the ﬁrst-order PDE system ux − ux = λ sin(u + u), ¯ ¯ 1 uy + uy = λ sin(u − u), ¯ ¯ (4.62)

where λ = 0 is any constant,11 then each of u(x, y) and u(x, y) satisﬁes (4.60). ¯ Conversely, given a function u(x, y), the overdetermined system (4.62) for un¯ known u(x, y) is compatible, and can therefore be reduced to an ODE system, if and only if u(x, y) satisﬁes (4.60). This indicates that given one solution of ¯ the sine-Gordon equation, ODE methods give a family of additional solutions. We ﬁt this example into our deﬁnition of a B¨cklund transformation as a follows. Start with two copies of the sine-Gordon Monge-Ampere system, one on M = {(x, y, u, p, q)} generated by θ = du − p dx − q dy, Θ = dθ = −dp ∧ dx − dq ∧ dy, , E= Ψ = dx ∧ dp + dq ∧ dy − sin(2u)dx ∧ dy ¯ the other on M = {(¯, y, u, p, q)} generated by x ¯ ¯ ¯ ¯ ¯ u ¯ x ¯ y θ = d¯ − p d¯ − q d¯, ¯ ¯ ¯ E= Θ = dθ = −d¯ ∧ d¯ − d¯ ∧ d¯, p x q y . ¯ Ψ = d¯ ∧ d¯ + d¯ ∧ d¯ − sin(2¯)d¯ ∧ d¯ x p q y u x y

¯ One can verify that the submanifold B ⊂ M × M deﬁned by ¯ ¯ x = x, y = y, p − p = λ sin(u + u), ¯ ¯ ¯ q + q = λ sin(u − u), ¯ 1

satisﬁes the criteria for a B¨cklund transformation, and that the process of a solving the overdetermined system (4.62) for u(x, y) corresponds to integrating the Frobenius system as described previously. For example, the solution u(x, y) = 0 of sine-Gordon corresponds to the ¯ ¯ integral manifold N = {(x, y, 0, 0, 0)} ⊂ M , whose preimage in B ⊂ M × M has coordinates (x, y, u) and satisﬁes u = p = q = 0, p = λ sin(u), q = ¯ ¯ ¯

11 This

1 λ

sin(u).

λ will not correspond to the integration parameter in the B¨cklund transformation. a It plays a role only in the relation to the K = −1 system, to be discussed shortly.

4.3. HIGHER-ORDER CONSERVATION LAWS The system E is algebraically generated by the form θ|π−1 (N ) = du − λ sin(u)dx − ¯

1 λ

193

sin(u)dy.

The problem of ﬁnding u(x, y) on which this θ vanishes is the same as solving the overdetermined system (4.62) with u = 0. It is obtained by integrating ¯ du − λ dx − sin u which has the implicit solution

1 − ln(csc u + cot u) − λx − λ y = c, 1 λ

dy = 0,

where c is the integration constant. This can be solved for u to obtain u(x, y) = 2 tan−1 (eλx+ λ y+c ). One can verify that this is indeed a solution to the sine-Gordon equation. In principle, we could rename this as u, and repeat the process to obtain more ¯ solutions. Example 2: B¨cklund transformation for the K = −1 system. a

1

¯ Suppose that f, f : N → E3 are two immersions of a surface into Euclidean ¯ space. We say that there is a pseudospherical line congruence between f, f if for each p ∈ N : ¯ 1. the line through f(p) and f (p) in E3 is tangent to each surface at these ¯ points (we assume f(p) = f (p)); ¯ 2. the distance r = ||f(p) − f (p)|| is constant; 3. the angle τ between the normals ν(p) and ν (p) is constant. ¯

¯ This relationship between f, f will play a role analogous to that of the system (4.62). We prove the following theorem of Bianchi. ¯ Theorem 4.3 If there is a pseudospherical line congruence between f, f : N → 3 ¯ has constant negative Gauss curvature E , then each of f and f K=− sin2 (τ ) . r2

¯ It is also true that given one surface f , there locally exists a surface f sharing ¯ ¯ a pseudospherical line congruence with f if and only if f has constant negative Gauss curvature. We will partly verify this claim, after proving Bianchi’s theorem.

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CHAPTER 4. ADDITIONAL TOPICS

¯ Proof. Choose Euclidean frame ﬁelds F, F : N → F which are adapted to the pair of surfaces in the sense that e1 = e1 , ¯ (4.63)

made possible by condition 1 above. Also, as usual, we let e3 , e3 be unit normals ¯ ¯ to f, f , respectively, which must then satisfy e2 = (cos τ )e2 + (sin τ )e3 , ¯ e3 = (− sin τ )e2 + (cos τ )e3 , ¯ with τ constant by condition 3. Now condition 2 says that ¯ f (p) = f(p) + re1 (p) for ﬁxed r. We can use the structure equations df = ei · ωi , j dei = ej · ωi ,

(4.64)

(4.65)

¯ e ¯ and similar for df, d¯i, to obtain relations among the pullbacks by F and F of the canonical forms on F. Namely, ¯ df = e1ω1 + ((cos τ )e2 + (sin τ )e3 )¯ 2 , ¯ ω and also so that ¯ ω1 = ω1 , 2 (cos τ )¯ 2 = ω2 + rω1 , ω 2 3 (sin τ )¯ = rω1 . ω (4.66)

2 3 ¯ df = d(f + re1 ) = e1 ω1 + e2 (ω2 + rω1 ) + e3 (rω1 ),

2 3 Note that ω2 +rω1 = (r cot τ )ω1 , and this gives a necessary condition on f alone to share a pseudospherical line congruence. Similar calculations using e1 = e1 ¯ yield 2 3 ω1 = (cos τ )ω1 + (sin τ )ω1 , ¯2 3 2 3 ω1 = −(sin τ )ω1 + (cos τ )ω1 , ¯

and diﬀerentiating the remaining relations (4.64) gives

3 ¯3 ω2 = ω2 ,

¯ giving complete expressions for F ∗ ω in terms of F ∗ ω. Note in particular that ω1 = ¯3 sin τ 2 ω . r

Now we can consider the curvature, expanding both sides of the deﬁnition ¯¯ d¯ 1 = −K ω1 ∧ ω2. ω2 ¯ (4.67)

4.3. HIGHER-ORDER CONSERVATION LAWS First, d¯ 1 ω2 = = = −¯ 3 ∧ ω1 ω2 ¯ 3 sin τ 2 2 −ω3 ∧ ω r sin τ 1 h21 ω ∧ ω2 , r

195

where h21 is part of the second fundamental form of F : N → M , deﬁned by 3 3 ωi = hij ωj . Note in particular that if h12 = 0, then ω1 is a multiple of ω1 , so 1 2 by (4.66), ω ∧ ω = 0, a contradiction; we can now assume that h12 = h21 = 0. ¯ ¯ On the right-hand side of (4.67), ¯¯ −K ω 1 ∧ ω 2 ¯ r ω3 sin τ 1 r ¯ = −Kh12 ω1 ∧ ω2 . sin τ ¯ = −Kω1 ∧

Equating these expressions, we have sin τ ¯ K =− 2 r as claimed. Now suppose given a surface f : N → E3 with constant negative Gauss ¯ curvature K = −1. We are interested in ﬁnding f which shares with f a pseudospherical line congruence. We start with local coordinates (s, t) on N whose coordinate lines ds = 0, dt = 0 deﬁne the asymptotic curves of f. It will be convenient to instead have orthogonal coordinate lines, so we deﬁne the coordinates x = s + t, y = s − t, for which dx = 0 and dy = 0 deﬁne the principal curves of f. We have seen in (4.55, 4.56) that the ﬁrst and second fundamental forms are given by I II = ds2 + 2 cos(2z)ds dt + dt2 = cos2 z dx2 + sin2 z dy2 , = 2 sin(2z)ds dt = sin(z) cos(z)(dx2 − dy2 ), (4.68) (4.69) (4.70) (4.71)

2

where 2z is half of the angle measure between the asymptotic directions and satisﬁes the sine-Gordon equation. One orthonormal coframing is given by (cos(z)dx, sin(z)dy); we consider an orthonormal coframing diﬀering from this one by rotation by some α: ω1 ω2 = = cos α sin α − sin α cos α cos(α − z) − sin(α − z) cos(z)dx sin(z)dy cos(α + z) − sin(α + z) ds dt . (4.72) (4.73)

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CHAPTER 4. ADDITIONAL TOPICS

The idea here is that we are looking for a function α on N for which this coframing could be part of that induced by a pseudospherical line congruence. The main compatibility condition, derived from (4.66), is

2 3 ω2 + rω1 = r(cot τ )ω1 .

(4.74)

2 We can compute the Levi-Civita connection form ω1 using the structure equai i j tions dω = −ωj ∧ ω , and ﬁnd 2 ω1 = (αs + zs )ds + (αt − zt )dt.

Similarly, we can compute from (4.70) and (4.73) the coeﬃcients of the second fundamental form with respect to the coframe (ω 1 , ω2 ), and ﬁnd

3 ω1 = sin(α − z)ds − sin(α + z)dt, 3 ω2 = cos(α − z)ds − cos(α + z)dt.

Substituting these into the compatibility condition (4.74), we obtain an equation of 1-forms whose ds, dt coeﬃcients are αs + z s αt − z t = = (csc τ + cot τ ) sin(α − z), 1 sin(α + z). csc τ + cot τ

We compare this to (4.62), and conclude that the local existence of a solution α is equivalent to having z satisfy the sine-Gordon equation. Note that the role played by λ in (4.62) is similar to that played by the angle τ in the pseudospherical line congruence. We conclude by exhibiting a surprising use of the B¨cklund transformation for the K = −1 system. This starts with an integral a manifold in M 5 of E (see (4.49)) that is not transverse as a Legendre submanifold, in the sense of being a 1-jet lift of an immersed surface in E3 . Instead, regarding the contact manifold M as the unit sphere bundle over E3 , N → M consists of the unit normal bundle of the line {(0, 0, w) : w ∈ R} ⊂ E3 . This Legendre surface is topologically a cylinder. To study its geometry, we will work in the circle bundle F → M . Its preimage there is parameterized by e1 = (sin u cos v, − sin u sin v, cos u), (u, v, w) → (0, 0, w), e2 = (cos u cos v, − cos u sin v, − sin u), , (4.75) e3 = (sin v, cos v, 0)

where (u, v, w) ∈ S 1 ×S 1 ×R. It is easily veriﬁed that this is an integral manifold for the pullback of E by F → M . We will apply the B¨cklund transformation a to this degenerate integral manifold, and obtain a non-trivial surface in E3 with Gauss curvature K = −1. ¯ We will take the B¨cklund transformation to be the submanifold B ⊂ F × F a ¯ (F is another copy of F) deﬁned by (4.63, 4.64, 4.65); this is a lift of the original picture (4.61) from M to F. We ﬁx the constants of the line congruence to be τ = π , 2

r = 1.

4.3. HIGHER-ORDER CONSERVATION LAWS

197

As a consequence, if our B¨cklund transformation gives a transverse Legendre a submanifold, then Theorem 4.3 states that the corresponding surface in E3 will have Gauss curvature K = −1. ¯ Now, the deﬁnition of B ⊂ F × F provides a unique lift π −1 (N ) of our ¯ degenerate integral manifold (4.75) to B. The “other” K = −1 system E should −1 ¯ to be algebraically generated by the 1-form restrict to π (N ) ⊂ B ⊂ F × F ¯ θ, and then π−1 (N ) will be foliated into surfaces which project into integral ¯ ¯ ¯ manifolds of E. So we compute θ = ω3 : ω3 ¯ = = = d¯, e3 x ¯ dx + de1, −e2 sin u dw − du.

Indeed, this 1-form is integrable, and its integral manifolds are of the form u = 2 tan−1 (exp(w + c)), (4.76)

where c is an integration constant. We will consider the integral manifold corresponding to c = 0. The Euclidean surface that we are trying to construct is now parameterized by x(u, v, w), constrained by (4.76). We obtain ¯ x(u, v, w) = ¯ = = = x(u, v, w) + e1 (u, v, w) (0, 0, w) + (sin u cos v, − sin u sin v, cos u)

2ew 1+e2w 2e cos v, − 1+e2w sin v, w + w 1−e2w 1+e2w

(sech w cos v, −sech w sin v, w − tanh w) .

This surface in E3 is the pseudosphere, the most familiar surface of constant negative Gauss curvature; we introduced both it and the “framed line” in §1.4, as examples of smooth but non-transverse Legendre submanifolds of the unit sphere bundle M → E3 . In principle, we could iterate this B¨cklund transformation, a obtaining arbitrarily many examples of K = −1 surfaces.

198

CHAPTER 4. ADDITIONAL TOPICS

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Index admissible lifting, 156–161, 165 aﬃne frame bundle, 58–65 fundamental forms, 59–65, 73 hypersurface, 37, 57–65, 73, 74 normal line, 60–63 transformation, x, 58–65 area functional, vii, 15, 24–35 asymptotic curve, 184, 190 B¨cklund transformation, xi, 23, 188, a 190–197 basic form, xii, 57 Betounes form, xi, 158–164 Bianchi identity, 91–92 Bianchi, L., 193 Blaschke, W., 61 calibration, 145–149, 156 Cartan lemma, 24, 59, 60, 94, 108 ´ Cartan, E., ix, xi, 37 Cauchy problem, 118–120 Cauchy-Schwarz inequality, 129 characteristic cohomology, 4 characteristic system, 183–185 Christodoulou, D., 118 coframe bundle, 38 conformal bundle, 104 curvature, 82 frame bundle, 83, 85 group, 40, 79, 97, 101, 111, 139 inner-product, 80, 83, 136 Laplacian, 79, 80, 93–97, 102, 120, 138 structure, x, 79–110, 133, 136– 140, 149 202 ﬂat model, 79–85, 91, 96–100, 102, 115–118 conformal energy, 127 connection, 42, 52 Cartan, 85, 91, 137 Levi-Civita, 22, 77, 87, 95, 141, 142 conservation law, ix, 14–21, 79 “almost”, 34–35, 125 for cmc hypersurfaces, 32–34 for harmonic maps, 166–168 for K = −1 surfaces, 183–190 for minimal hypersurfaces, 27– 32 for wave equations, 118–127 n+3 for z = Cz n−1 , 123–127 n+2 for ∆u = Cu n−2 , 110–118 “hidden”, 31 higher-order, 176–190 proper, 15, 18–19 trivial, 15, 116, 188 contact form, 2, 7, 15, 16, 98, 102, 103, 124 ideal, 3, 4, 10 line bundle, 4, 10, 16, 52, 123, 125 manifold, ix, 1–8, 10, 39, 77, 83, 97 Cotten tensor, 92 covariant derivative, 90, 92, 94, 95, 109, 138–144, 165 Darboux theorem, 50 decomposable form, 41, 53, 183 density line bundle, 92–97, 120, 124, 126, 138, 149

INDEX dilation, 34, 84, 116, 121, 123 divergence equivalence, 155, 156 energy, vii, 122–123, 127–129 of a map, 164, 165 equivalence of G-structures, 42 of conformal structures, 85–92 of Lagrangians, 3 of Monge-Ampere systems, 39– 52 of Poincar´-Cartan forms, 65– e 78, 102–110 equivalence method, ix, 37–38, 41– 53, 65–74, 79, 85–88, 102– 110, 134 Euclidean frame bundle, 21, 26, 28, 182 motion, 21–34, 78 space, vii, 21–35 Euler-Lagrange equation, vii, viii, 7, 10, 72, 99, 151, 156 system, ix, 1, 7–15, 40, 51, 133– 149 exterior diﬀerential system, xii ﬁeld, 147–150 ﬁrst variation, viii, 7, 133–134 Frobenius theorem, 55 functional, vii–ix, 3–7 Gauss curvature, 13, 41, 182–197 Gauss map, 23 generating function, 16–17 Grassi, M., 155n, 163n Grassmannian, 2, 13, 56, 142, 151 Green’s theorem, viii, 127–130 G-structure, 37–38, 41–50, 85–88, 102 H¨lder’s inequality, 130 o harmonic function, 29 map, 164–168 harmonic map, xi Hodge-Lepage decomposition, 4 holomorphic curve, 31–32

203

ideal algebraic, 2n, 154 diﬀerential, 2, 9, 10, 158 independence condition, 2, 20 integrable extension, xi, 185–188 integral manifold, xii, 2, 4, 15 transverse, 12 inverse problem, 1, 7, 10–14, 24, 39, 51 inversion, 84, 117, 121, 125 Jacobi operator, ix, 145, 149–150 jets, viin, 2, 99–100, 103 John, F., 129 K3 surface, 13 Killing vector ﬁelds, 166 Koszul complex, 144 Lagrangian, vii–ix, 3–7, 155–164 Lagrangian potential, 57–58, 73, 145 Laplace equation, 139 Legendre submanifold, 2–3, 7–11, 133– 145, 158 non-transverse, 23, 27, 196 transverse, 3, 9, 142 Legendre variation, 7–9, 134–149 Levine, H., 128n Lorentz conformal group, 119, 121, 126 frame, 81, 83, 98 metric, 78, 119 space, 80 Lorentz boost, 121 Maurer-Cartan equation, 58, 82 form, 80–82, 85, 100, 167 mean curvature, vii, 13, 24 constant, 24, 27, 141 prescribed, x, xi, 38, 74, 77, 105n, 133, 140–145 minimal surface, ix, 15, 24–35, 78, 145 Minkowski space, 20, 118–127

204 compactiﬁed, 119 moment conditions, 31–34 Monge-Ampere system, 10–14, 99 hyperbolic, ix, 40–52, 182 moving frames, 59 multi-contact geometry, 151–158 Noether’s theorem, ix, 1, 7, 10, 14– 21, 28, 29, 79, 114, 115, 163, 184 non-degenerate 1-form, 1 aﬃne hypersurface, 59–60 functional, 9, 10, 15 null vector, 80 positive, 80, 97 Pfaﬀ theorem, 2, 154 Pfaﬃan system, xii Plateau problem, 31 Pohoˇaev’s theorem, 117 z Poincar´ lemma, 11, 50, 55 e Poincar´-Cartan form, ix, 1, 6–11, e 133–147, 151–164 almost-classical, 52–56, 65–67 conformally invariant, 97–118 Euclidean-invariant, 25–26 neo-classical, x, 37, 52–59, 65– 67, 74–78, 98 deﬁnite, 67–74, 102–110, 135 Poisson equation, x, 11–13, 79 conformally invariant, 110–118 non-linear, 96–97, 102–110, 118, 188 primitive, 4, 9n, 10–12, 39–40 principal direction, 183 prolongation of a G-structure, x, 42, 85–88 of a linear Lie algebra, 87 of an EDS, 183 pseudo-connection, 42–52, 85–90, 103 pseudosphere, 23 pseudospherical line congruence, 192– 195 Ricci curvature, 95

INDEX Riemann curvature tensor, 22, 78, 89, 105, 137, 142 Riemannian frame bundle, 22, 77, 95, 141– 144, 164 Laplacian, 96, 144 manifold, 22, 74, 87 metric, 77–78, 80, 95, 140 rotation, 29, 34, 84, 116 Rumin, M., 9n scalar curvature, 95, 106 second fundamental form, 24, 26, 141, 183 of a map, 166 second variation, ix, x, 78, 133–140 for prescribed mean curvature, 140–145 semibasic form, xii, 38, 43, 57 simple foliation, 56, 78n, 97, 104 sine-Gordon equation, xi, 185–188, 190–195 space-like hypersurface, 78 Spencer cohomology, 44, 87 stationary, vii–ix, 7–10, 134–149 stereographic projection, 79, 119 Stokes’ theorem, 147–148 Strauss, W., 121n stress-energy tensor, 166 strong local minimum, 145–149 summation convention, 68 symbol, 10, 162 symmetric form, 159, 165 symmetry, ix, 14–21, 78–85, 102, 111– 113, 136 generalized, 188 hidden, 188 symplectic linear algebra, 4–6, 10, 11, 17– 18, 50, 164 transformation, 39–40 tautological 1-form, 38, 42–50, 85 time-orientation, 80 torsion absorption of, 44–45, 85

INDEX intrinsic, of a G-structure, 44, 46–47, 87 normalization of, 46, 85, 87–88 of a pseudo-connection, 43–52, 85–90, 103 total derivative, 10 transformation classical, vii, 101, 154 contact, viii, 16–17, 26–27, 56– 57, 151–155 gauge, vii, 154 point, viii, 26, 57, 151–155 translation, 27, 33, 84, 115 wave equation, 20 conformally invariant, 118–131 homogeneous, 39, 51 non-linear, 79, 118–131 wave operator, 118 conformal, 120 weak local minimum, 145–149 Weierstrass excess function, 148–149 Weierstrass representation, 32 Weingarten equation, ix, 13, 26–27, 182 Weyl tensor, 92, 106

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...Mostly Harmless Econometrics: An Empiricist’ Companion s Joshua D. Angrist Massachusetts Institute of Technology Jörn-Ste¤en Pischke The London School of Economics March 2008 ii Contents Preface Acknowledgments Organization of this Book xi xiii xv I Introduction 1 3 9 10 12 16 1 Questions about Questions 2 The Experimental Ideal 2.1 2.2 2.3 The Selection Problem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Random Assignment Solves the Selection Problem . . . . . . . . . . . . . . . . . . . . . . . . Regression Analysis of Experiments . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . II The Core 19 21 22 23 26 30 36 38 38 44 47 51 51 3 Making Regression Make Sense 3.1 Regression Fundamentals . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.1.1 3.1.2 3.1.3 3.1.4 3.2 Economic Relationships and the Conditional Expectation Function . . . . . . . . . . . Linear Regression and the CEF . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Asymptotic OLS Inference . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Saturated Models, Main E¤ects, and Other Regression Talk . . . . . . . . . . . . . . . Regression and Causality . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.2.1 3.2.2 3.2.3 The Conditional Independence Assumption . . . . . . . . . . . . . . . . . . . . . . . . The Omitted Variables Bias Formula . ....

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