THE L2 GEOMETRY OF THE SYMPLECTOMORPHISM GROUP A Dissertation Submitted to the Graduate School of the University of Notre Dame in Partial Fulfillment of the Requirements for the Degree of Doctor of Philosophy by James Benn, Gerard Misio lek,Director Graduate Program in Mathematics Notre Dame, Indiana April 2015 c Copyright by James Benn 2015 All Rights Reserved THE L2 GEOMETRY OF THE SYMPLECTOMORPHISM GROUP Abstract by James Benn In this thesis we study the geometry of the group of Symplectic diffeomor- phisms of a closed Symplectic manifold M, equipped with the L2 weak Riemannian metric. It is known that the group of Symplectic diffeomorphisms is geodesically complete with respect to this L2 metric and admits an exponential mapping which is defined on the whole tangent space. Our primary objective is to describe the structure of the set of singularities of associated weak Riemannian exponential mapping, which are known as conjugate points. We construct examples of con- jugate points on the Symplectomorphism group and solve the Jacobi equation explicitly along geodesics consisting of isometries of M. Using the functional cal- culus and spectral theory, we show that every such geodesic contains conjugate points, all of which have even multiplicity. A macroscopic view of conjugate points is then given by showing that the exponential mapping of the L2 metric is a non- linear Fredholm map of index zero, from which we deduce that conjugate points constitute a set of first Baire category in the Symplectic diffeomorphism group. Finally, using the Fredholm properties of the exponential mapping, we give a new characterization of conjugate points along stationary geodesics in terms of the linearized geodesic equation and coadjoint orbits. CONTENTS ACKNOWLEDGMENTS . iv CHAPTER 1: INTRODUCTION AND OVERVIEW . .1 CHAPTER 2: PRELIMINARIES . .8 2.1 Diffeomorphism Groups . .8 2.1.1 Manifolds of Mappings . .8 2.1.2 Groups of Diffeomorphisms . 11 2.1.3 Exponential Mappings . 12 2.2 Weak Riemannian Structure on the Diffeomorphism Group . 14 s 2.3 The Symplectic Diffeomorphism Group D! ............ 19 s 2.3.1 The Manifold Structure of D! ................ 19 2.3.2 Hodge Theory for Manifolds . 24 s 2.3.3 Weak Riemannian Structure for D!(M).......... 33 2.4 L2 Geodesic Completeness . 36 2.5 Curvature . 42 CHAPTER 3: CONJUGATE POINTS IN THE SYMPLECTOMORPHISM GROUP . 49 3.1 Introduction . 49 3.2 Adjoint and Coadjoint Operators . 51 3.3 The Second Fundamental Tensor and Curvature . 62 s n 3.4 Conjugate Points on D!(CP ).................... 71 3.5 Geodesics in the Isometry Subgroup . 77 3.6 The Jacobi Equation along Geodesics of Isometries . 82 CHAPTER 4: FREDHOLM PROPERTIES OF THE L2 EXPONENTIAL s MAP ON D! ................................ 92 4.1 Introduction . 92 4.2 The Jacobi Equation . 94 4.3 Proof of Fredholmness in Hσ ..................... 98 4.4 Proof of Fredholmness in Hs ..................... 108 ii s n CHAPTER 5: CONJUGATE POINTS ON DHam(M ) IN DIMENSIONS n =2AND4 ................................ 114 5.1 Introduction . 114 5.2 A Conservation Law . 118 5.3 A Characterization of Conjugate Points along Stationary Geodesics 120 5.4 Proof of Proposition 5.3.1 . 122 APPENDIX A: SOBOLEV SPACES . 130 A.1 Sobolev Spaces of Bounded Domains in Rn ............. 130 A.2 Sobolev Spaces on Compact Manifolds . 131 A.3 Sobolev Spaces on Vector Bundles . 132 APPENDIX B: SYMPLECTIC VECTOR SPCAES AND MANIFOLDS . 135 B.1 Symplectic Manifolds . 135 B.2 Symplectomorphisms . 137 APPENDIX C: FREDHOLM OPERATORS . 140 C.1 Compact Operators . 140 C.2 Fredholm Operators . 142 APPENDIX D: SPECTRAL INTEGRALS . 146 D.1 Spectral Measures . 146 D.2 Stone's Theorem . 148 BIBLIOGRAPHY . 151 iii ACKNOWLEDGMENTS A very special thanks goes out to my friend and advisor, Gerard Misio lek,for introducing me to the subject and for his endless ideas. Thankyou to Alex Hi- monas for his ongoing support and to Steve Preston for his interest and useful suggestions. My family, my friends; I thank all of you deeply. iv CHAPTER 1 INTRODUCTION AND OVERVIEW Let M be a closed Symplectic manifold with Symplectic form ! and Rie- mannian metric g. We assume that ! and g are compatible, in the sense that there exists an almost complex structure J : TM ! TM satisfying J 2 = −I, g(Jv; Jw) = g(v; w), and g(v; Jw) = !(v; w), for any vector fields v; w on M (see s appendix A). Let D! (M) denote the group of all diffeomorphisms of Sobolev class s dim M s H preserving the Symplectic form ! on M. If s > 2 + 1 then D! becomes an infinite dimensional Hilbert manifold whose tangent spaces at a point η consists of Hs sections X of the pull-back bundle η∗TM for which the corresponding vector −1 field v = X ◦ η on M satisfies Lv! = 0, where L is the usual Lie derivative. Using right-translations, the L2 inner product on vector fields, Z s (u; v)L2 = g (u; v) dµ, u; v 2 TeD!; (1.1) M defines a right-invariant metric on the group. This thesis is concerned with the 2 s s L geometry of the group D!, and it's finite codimensional subgroup DHam - the group of Hamiltonian diffeomorphisms. Diffeomorphism groups can be realized as the configuration spaces of a number of equations in mathematical physics, which provides a strong motivation to study their geometry. Perhaps the most famous example is the Euler equations of hy- 1 drodynamics, where Arnold, [A], noticed that a curve η(t) in the group of smooth s 2 volume preserving diffeomorphisms (Dµ) is a geodesic of the L metric (1.1) if and only if the vector field v, defined by @tη = v ◦ η, solves the Euler equations of hydrodynamics. The L2 metric is simply the kinetic energy of the fluid, and the geodesic equation is a manifestation of Newton's second law F = ma. s 2 Analogously, a curve η(t) in D!(M) is a geodesic of the L metric starting from the identity in the direction vo if and only if the time dependent vector field v =η _ ◦ η−1 on M solves the Symplectic Euler equations ! @tv + Pe (rvv) = 0 (1.2) Lv! = 0 v(0) = vo; ! where Pe is the orthogonal projection onto the space of Symplectic vector fields. The subgroup of Hamiltonian diffeomorphisms plays a role in plasma dynamics analogous to the role played by the volume preserving diffeomorphism group in incompressible hydrodynamics, see Arnold and Khesin [A-Kh], Holm and Tronci [H-T], Morrison [Mo], Marsden and Weinstein, [M-W], for details. Chapter 2 contains a review of the manifold structure of mapping spaces and diffeomorphism groups. We briefly describe the shortcomings of the group expo- nential map on diffeomorphism groups and motivate the endowment of a weak Riemannian structure (the L2 metric (1.1)) on these manifolds. Section 2.3 fo- cuses our attention on the Symplectomorphism and Hamiltonian subgroups (and submanifolds!) of the diffeomorphism group. Here we recall the Hodge decompo- sition of forms and the fundamental results of Ebin and Marsden, [E-M], which 2 prove the existence of a smooth right-invariant connection and exponential map- ping associated to the weak L2 metric. We define the weak Riemannian curvature tensors on the Symplectomorphism group and indicate that they are trilinear op- erators bounded in the strong Sobolev Hs topology, as shown in Misiolek [M1]. Finally, in section 2.4 we review Ebin's [Eb] and Khesin's [Kh] proof that solutions to the Symplectic Euler equations (1.2) exist globally in time for any Symplectic s 2 manifold, so that the group D!(M) is L geodesically complete. Arnold ([A]) computed sectional curvatures of diffeomorphism groups and found that they were mostly negative, although in some small regions they were s positive. He asked if there are conjugate points on Dµ and called for a description s of them. Much progress in understanding conjugate points on Dµ has been made since the work of Misiolek. In [M1], some simple examples of conjugate points in the Volumorphism group were constructed, answering Arnold's first question in the affirmative. More examples were later provided by Misiolek [M2], Preston [P2], and Shnirelman [Sh2]. In contrast with finite dimensional geometry, two types of conjugate points can occur in infinite dimensions. Grossman [Gro] gave the first examples of the two types of conjugacies that may occur: on a sphere in Hilbert space the differential of the exponential map may have infinite dimensional kernel corresponding to an infinite-dimensional family of geodesics joining two antipodal points. In addition, on an infinite dimensional ellipsoid the exponential map differential fails to be surjective, even though it is injective, in certain directions. s A natural question to ask is whether conjugate points exist on D!. This is the contents of chapter 3: Some simple examples of conjugate points are constructed on the Symplectomorphism group of the complex projective plane. In particular, 3 s n n Theorem. (3.4.3) Conjugate points exist on D! (CP ), for s > 2 + 1 and n ≥ 2. s We then show that geodesics which lie in the isometry subgroup of D! always carry conjugate points, all of which have even multiplicity. 2 Theorem. (3.6.2, 3.6.4) Let η(t) = exp(tvo) be a geodesic of the L metric (1.1) generated by a Killing vector field vo. Let J(t) be a Jacobi field along η(t), with 0 initial conditions J(0) = 0, J (0) = wo. Then tKv0 −tKv e − I J(t) = Dη(t) · e 0 w0; Kv0 s where Kvo (·) is a compact, skew self-adjoint operator on TeD!, and we have the −tK R itλ etKv0 −I R eitλ−1 spectral representations e vo = e dE(λ) and = dE(λ) which R Kv0 R iλ s are linear operators on TeD!.