The L2 Geometry of the Symplectomorphism Group

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

James Benn,

Gerard Misio lek,Director

Abstract by James Benn

In this thesis we study the geometry of the group of Symplectic diffeomorphisms 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 conjugate points on the Symplectomorphism group and solve the Jacobi equation explicitly along geodesics consisting of isometries of M. Using the functional calculus 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 nonlinear 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 Dω ...... 19 s 2.3.1 The Manifold Structure of Dω ...... 19 2.3.2 Hodge Theory for Manifolds ...... 24 s 2.3.3 Weak Riemannian Structure for Dω(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 Dω(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 Dω ...... 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 ω and Rie- mannian metric g. We assume that ω 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) = ω(v, w), for any vector ﬁelds v, w on M (see

s appendix A). Let Dω (M) denote the group of all diﬀeomorphisms of Sobolev class

s dim M s H preserving the Symplectic form ω on M. If s > 2 + 1 then Dω 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 Lvω = 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 ∈ TeDω, (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 Dω, 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 Dω(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

ω ∂tv + Pe (∇vv) = 0 (1.2)

Lvω = 0

v(0) = vo,

ω 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 exponential 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 decomposition 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 mapping associated to the weak L2 metric. We deﬁne the weak Riemannian curvature tensors on the Symplectomorphism group and indicate that they are trilinear operators 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 Dω(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 Dω. 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 Dω (CP ), for s > 2 + 1 and n ≥ 2.

s We then show that geodesics which lie in the isometry subgroup of Dω 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 TeDω, 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 TeDω. Consequently, the multiplicity of each conjugate point along η(t) is even.

s Conjugate points on Dµ(M), M a closed surface, were studied extensively by Ebin, Misiolek and Preston in [E-M-P], where it was shown that the exponential map of the L2 metric is a non-linear Fredholm map of index zero. A corollary of this is that the two types of conjugacies mentioned above coincide. Moreover, conjugate points are isolated and of ﬁnite multiplicity. The conclusion is that the exponential map behaves like that of a ﬁnite dimensional manifold. When M is

s three dimensional, the singularities of the exponential mapping on Dµ typically behave pathologically and the exponential map is no longer a Fredholm map, cf. [E-M-P], Misiolek-Preston [M-P], [P2], [Sh2]. In chapter 4 we will show

Theorem. (4.1.1) Let M be a closed Symplectic manifold of dimension n = 2m

2 s and s > m + 1. Then the exponential map of the L metric on Dω(M) is a nonlinear Fredholm map of index zero.

4 This result provides a distinction between Symplectic diﬀeomorphisms and Volume preserving diﬀeomorphisms, when equipped with the L2 metric. It holds

s for any closed Symplectic manifold of dimension 2n, but fails for the group Dµ of manifolds of dimension 3 and higher, cf. [E-M-P], [P2]. The relationship between Fredholmness of the L2 exponential map and known classiﬁcations (e.g. C0 closure, Gromov’s non-squeezing Theorem) of Symplectic diﬀeomorphisms is, at this point, unclear. In chapter 5 we give a new geometric characterization of conjugate points along a stationary geodesic, and relate their existence to the right-invariance of the L2 metric.

Theorem. (5.3.2) Let M be a two or four dimensional Symplectic manifold and

2 s dim M η(t) a stationary geodesic of the L metric on Dω, s > 2 + 1, with initial ∗ velocity vo. Then η(t ) is conjugate to the identity if and only if there exists a

⊥ s v ∈ (ker Kvo ) ⊂ TeDω such that

∗ ∗ S(t )v = Adη−1(t∗)v,

∗ 2 where Adη(t) is the formal L adjoint of the push forward of vector ﬁelds operation,

s Kvo (·) a compact skew self-adjoint operator on TeDω, and S(t) is the solution operator of the linearized Symplectic Euler equations.

That is, conjugate points occur when a solution to the linearized Symplectic

s Euler equations (with initial value v), which can be thought of as a curve in TeDω, intersects the coadjoint orbit of its initial value v which can also be thought of as

s a curve in TeDω. Moreover, we are able to express solutions of the Jacobi equation in terms of solutions to the linearized Symplectic Euler equation and coadjoint

5 orbits. Namely,

Theorem. (5.3.3) Let M be a two or four dimensional Symplectic manifold and

2 s dim M η(t) be a stationary geodesic of the L metric on Dω, s > 2 +1. Then the Jacobi 0 ⊥ ﬁeld J(t) = u(t) ◦ η(t) with initial conditions J(0) = 0, J (0) = wo ∈ (ker Kvo ) is given by X 1 u(t) = (g (t) − a (t)) v , λ i i i i i

0 where {vi}i∈N is a complete orthonormal set of eigenvectors of Kvo spanning TeDω, P Kvo vi = λivi, S(t)wo = i gi(t)vi solves the linearized Symplectic Euler equations ∗ P and Adη−1(t)wo = ai(t)vi.

In particular, the growth of Jacobi fields (measured in some norm) is determined by how much solutions of the linearized Symplectic Euler equations differ from coadjoint orbits (measured in the same norm). It is interesting to understand what information conjugate points carry about the qualitative behavior of flows. For example, the inviscid Burgers’ equation

∂tv + ∇vv = 0

v(0) = vo describes the motion of a collection of particles moving without any internal forces. It is also the geodesic equation of the L2 metric on the full diffeomorphism group. It has been shown by Khesin and Misiolek, [Kh-M], that the Burgers’ equation has solutions in which particles begin colliding with one-another, forming shock- waves, in finite time. That is, geodesics (i.e. particle trajectories) cease to be diffeomorphisms (or reach the boundary of the group) in finite time. The first conjugate point along a geodesic generated by such a solution signals the onset of

6 shock-waves in the material space. On the other hand, solutions of the 2D Euler- equations of hydrodynamics (and the Symplectic Euler equations (1.2)) exist for all time and yet some of the corresponding geodesics contain conjugate points.

7 CHAPTER 2

PRELIMINARIES

2.1 Diﬀeomorphism Groups

2.1.1 Manifolds of Mappings

The basic idea of giving a manifold structure to mapping spaces was first laid down by Eells [E1] in 1958 where he constructed a smooth manifold out of the set of continuous maps between two manifolds. Constructing a manifold from Ck- diffeomorphisms of a compact manifold without boundary was done independently around 1966 by Abraham, Eells and Leslie, [E2], [L]. The Sobolev Hs case was done by Ebin and Marsden [E-M] where they gave a manifold structure to the Hs diffeomorphism group, the Volume-preserving diffeomorphism subgroup and the Symplectic diffeomorphism subgroup of a compact manifold with, or without, boundary. The construction is as follows. Let M and N be two compact manifolds each endowed with a Riemannian metric, g and h, and suppose N is without boundary. For an integer s, a map f : M → N is of Sobolev class Hs (write f ∈ Hs(M,N)) if for any point p ∈ M and chart around p,(Up, ϕ), and any chart (V, ψ) around f(p) the composite map ψ◦f◦ϕ−1 : ϕ(U) → Rn is in Hs(ϕ(U), Rn). If the sobolev index

m s 0 s satisﬁes s > 2 then by the Sobolev embedding Lemma, H (M,N) ⊂ C (M,N), and the above notion is well-deﬁned and independent of the charts chosen. We

8 refer the reader to appendix A for a review of Sobolev spaces. In order to deﬁne charts on Hs(M,N) we need to determine the correct mod- eling space for Hs(M,N). Just as in ﬁnite dimensions, we use the tangent space as the model space for a manifold and we shall proceed similarly here. With this in mind, we shall look for a good description of the tangent space at a point f ∈ Hs(M,N). Consider a curve c :(−, ) → Hs(M,N) such that c(0) = f. For a point p ∈ M, the map t 7→ c(t)(p) is a curve in N. Now c(0)(p) = f(p) and so the

d derivative of this curve at 0 is dt c(t)(p)|t=0 and is an element of Tf(p)N. Therefore, d the map p 7→ dt c(t)(p)|t=0 is a map from M to TN and such that the canonical projection πN : TN → N covers f. Making the identiﬁcation

d d c(t)| (p) = c(t)(p)| , dt t=0 dt t=0 the tangent space at a point f ∈ Hs(M,N) is

s s Tf H (M,N) = {X : M → TN : X ∈ H (M,TN), πN ◦ X = f} .

Here, Hs(M,TN) is the space of all sections from M to TN which have L2 derivatives up to order s. Deﬁne the inner product

Z X k k (V,W )s = g ∇ V, ∇ W dµ |k|≤s M where ∇k means the k − th order covariant derivative given by the Riemannian metric on N. With this inner product, Hs(M,TN) is a Hilbert space and all

9 m sections are continuous by the Sobolev Lemma (since s > 2 ) and the topology of Hs(M,TN) is stronger than that of uniform convergence. In order to construct an f-centered chart for Hs(M,N) we use the Riemannian exponential map of N. Since N is closed, it is geodesically complete and for each

N x ∈ N, the exponential map exp x : TxN → N is deﬁned on the whole of TxN.

N N Consequently, exp x can be extended to a map exp : TN → N, where for

N N vx ∈ TxN, exp (vx) = exp x(vx). Since f(M) is compact, there is a number

λf > 0 such that any point of N whose distance from f(x) is less than λf can be joined by a unique geodesic arc of length less than λf . That is, for any point p of

N whose distance from f(x) is less than λf there is an X(x) ∈ Tf(x)N which lies

N in the disk of radius λf centered at 0, such that expf(x) X(x) = p and the map

N x 7→ expf(x) X(x) is a map from M to N. Consequently, the map

s s Ψ: Tf H (M,N) → H (M,N)

X 7→ expN X

s gives a bijective correspondence between the disc of radius λf in Tf H (M,N)

s centered at 0 and the disc of radius λf in H (M,N) centered at f. Since exp : TN → N is a local diﬀeomorphism, the transition functions are compositions of smooth maps and hence Hs(M,N) has a smooth manifold structure. Moreover, compactness of M and N is used to show that the topology deﬁned on Hs(M,N) is independent of the metric.

10 2.1.2 Groups of Diﬀeomorphisms

The diﬀeomorphism groups have a very rich and complicated structure which is still not very well understood. Let M be a compact manifold without boundary

n s 1 1 and assume that s > 2 + 1 so that the H topology is stronger than C . Let C D be the group of C1 diﬀeomorphisms of M, to itself, and let Ds(M) = Hs(M,M)∩ C1D. According to Theorem 1.7 of Hirsch’s Diﬀerential Topology, [Hi], C1D is open in C1(M,M), so that Ds(M) is open in Hs(M,M) and hence inherits its manifold structure. Ds(M) is also a topological group with composition as the group operation. Right multiplication is smooth:

s s Rη : D → D

ξ 7→ ξ ◦ η.

s ˙ Indeed, let t 7→ ξ(t) be a curve in D with ξ(0) = ξ, ξ(0) = X, then dξRη(X) =

d d dt |t=0(Rη(ξ(t))) = dt |t=0(ξ(t) ◦ η) = X ◦ η which is another right translation. However, Left multiplication

s s Lη : D → D

ξ 7→ η ◦ ξ

is only continuous and its tangent map is given by dξLη(X) = Dηξ · X. The manifold Ds(M) is not precisely a Lie group but has some similarities. If we were to work with D∞(M) instead of Ds(M) then we would have a genuine Lie group; however, we would no longer have a Banach manifold in which important theorems like the inverse function theorem hold, see [H].

11 If G is a Lie group and e ∈ G the identity element then the Lie algebra of G is

s s s identified with TeG. Similarly, TeD (M) = H (TM), which are H vector fields on M, serves as the Lie Algebra of Ds(M). Since right multiplication is smooth we are able to talk about right-invariant vector fields on Ds. Given any vector

s field v on M, we define a right-invariant vector field vη on D (M) by the formula

vη = v ◦ η.

Since these are ﬁelds of class C1 we are able to deﬁne the Lie bracket. The Lie bracket [uη, vη] is calculated as

[uη, vη](η) = dvη(uη) − duη(vη) = (dv ◦ η)(u ◦ η) − (du ◦ η)(v ◦ η)

= (dv · u − du · v) ◦ η = dRη [u, v] .

s s−1 s However, since u and v are both in H , the bracket is only H and hence TeD is not closed under the bracket operation.

2.1.3 Exponential Mappings

Let M be a closed, orientable manifold of dimension n and let D∞ be the

∞ group of smooth diﬀeomorphisms of M with Lie Algebra TeD (M) of smooth

∞ vector fields on M. For a vector field v ∈ TeD (M), it’s flow η(t): M → M is defined and is called a one-parameter subgroup of D∞(M). Therefore, the group exponential mapping

G ∞ ∞ exp : TeD → D

v 7→ η1

12 is defined. Here, η1 is the value of the one-parameter subgroup ηt = exp (tv) corresponding to t = 1. However, the group exponential mapping has a number of shortcomings. It is not even a homeomorphism in a neighborhood of the identity. There exist diffeomorphisms arbitrarily close to the identity which are not embeddable in a flow ([Ko]). This was shown for even the simplest case of M = S1. The group exponential map is in fact much worse! In the work of Grabowski ([G1]) it was shown that for the group of compactly supported diffeomorphisms of a Ck manifold (k = 0, 1, 2, ...) there exist arcwise-connected, non-trivial free subgroups of diffeomorphisms which embed in no flow. It is shown in [E-M] that, just like a Lie group, the group of Sobolev Hs diffeomorphisms of a closed, orientable manifold M admits an exponential mapping which associates to every tangent vector at the identity a one parameter subgroup of Ds(M). Recall that such a tangent vector is an Hs vector field on M and the one parameter subgroup is the flow generated by the vector field.

n Theorem 2.1.1. Let M be a compact manifold without boundary, s > 2 + 2 and Ds(M) the group of Hs diﬀeomorphisms.

s 1. If V is an H vector ﬁeld on M, its ﬂow ηt is a one-parameter subgroup of Ds.

1 2. The curve t 7→ ηt is of class C .

s s 1 3. The mapping exp : TeD → D , V 7→ η1 is continuous (but not C ).

We refer the reader to [E-M] for the proof of this Theorem. Note that the exponential mapping is not C1 because it does not cover any neighborhood of the identity. Since smooth diﬀeomorphisms embed densely in the set of Hs diﬀeomorphisms the

13 above results regarding local injectivity and surjectivity of the exponential mapping apply to the exponential mapping of the Hs diffeomorphisms: there exist diffeomorphisms arbitrarily close to the identity which embed in no flow. More- over, there exist arcwise-connected, non-trivial free subgroup of diffeomorphisms which embed in no flow. The solution to this problem is to use an exponential map associated to a weaker metric defined on Ds(M). It is not automatic that a weak metric admit an exponential mapping but this turns out to be the case on Ds(M).

2.2 Weak Riemannian Structure on the Diﬀeomorphism Group

Let M be a closed, orientable manifold of dimension n, endowed with a Rie- mannian metric g and volume form dµ deﬁned by g. The Riemannian metric

s s on M deﬁnes a weak Riemannian structure on D (M): Let η ∈ D (M) and uη,

s vη ∈ TηD (M), then,

Z −1 ∗ (uη, vη)L2 = g(u, v)(η ) dµ. (2.1) M

s This is a symmetric bilinear form defined on each tangent space TηD (M) whose norm is an L2-norm. The Riemannian structure is said to be weak because it generates the L2 topology on Ds(M) rather than the stronger Hs topology. Geodesics on a Hilbert manifold N locally minimize the L2-energy functional R t 2 0 kη˙(s)kN ds. The equations defining geodesics are then a second order system of differential equations η¨(t) = F (η, η˙) with F a vector field on TTN ∼= TN. According to the fundamental existence

14 and uniqueness theorem of ODE’s in Banach space, if F is smooth in both it’s arguments η andη ˙ then there exists a unique solution η(t) deﬁned on an open interval around 0 ∈ R. It would then follow that around every point x ∈ N there exists a neighbourhood U of x and a number ε > 0 so that for each p ∈ U and each tangent vector v ∈ TpN with length less than ε there is a unique geodesic

η :(δ1, δ2) → N satisfying the initial conditions

η(0) = p η˙(0) = v.

Let v ∈ TpN be a tangent vector at a point p ∈ N and suppose there exists a geodesic η(t) : [0, 1] → N satisfying the initial conditions η(0) = p, η˙(0) = v. The point η(1) ∈ N will be denoted by

N expp (v) and called the exponential of the tangent vector v. The geodesic η(t) can then be described by

N η(t) = expp (tv)

N and expp is called the exponential map.

Theorem 2.2.1. ([E-M] Theorem 9.1) Let M be compact, without boundary,

s and endowed with a Riemannian metric g and in every tangent space TηD , s >

dim M 2 + 1, let the inner product be deﬁned by (2.1). Then

1. (2.1) deﬁnes a weak Riemannian structure on Ds(M)

15 2. (2.1) has a unique torsion-free aﬃne connection ∇¯ associated to it; that is, for smooth vector ﬁelds u, v and w on Ds(M), we have

¯ ¯ u (v, w)L2 = ∇uv, w L2 + v, ∇uw

¯ ¯ ∇uv − ∇vu = [u, v] .

3. Let expM : TM → M be the exponential mapping corresponding to the connection ∇ on M. Then expDs : T Ds → Ds deﬁned by expDs (v) = expM ◦v is the exponential mapping of the Riemannian connection ∇¯ on Ds. The map expDs is deﬁned only on a neighborhood of the zero-section of T Ds, and is a C∞-mapping onto a neighborhood of e ∈ Ds.

The connection ∇¯ is completely determined by the connection ∇ on M. For any

s s v ∈ TeD , the geodesic emanating from e ∈ D (M) in the direction v is given by

Ds Ds M t 7→ expe (tv). For any x ∈ M, t → expe (tv)(x) = expx (tv(x)) is the geodesic

Ds starting at x in the direction v(x) in M. Therefore, t 7→ expe (tv) represents the totality of geodesics on M in the direction of the vector ﬁeld v. Conversely, geodesics on M combine to form geodesics on Ds(M). The L2 inner product just deﬁned is neither left- nor right-invariant: Let

−1 −1 u = dRη uη and v = dRη vη so that u and v are vector ﬁelds on M. Then, using the change of variables formula for integrals, we have

Z −1 ∗ (uη, vη)L2 = g(u, v)(η ) dµ M where (η−1)∗dµ is the pull-back of the volume form by the diﬀeomorphism η−1. From this we see that the inner product is right-invariant if and only if η (and

16 hence its inverse) is a volume-preserving diﬀeomorphism.

s If uη and vη are right-invariant vector ﬁelds on D (M) then we have the follow-

s ing formula for the covariant derivative in terms of of the ﬁelds u, v ∈ TeD (M):

¯ ∇uη vη = (∇uv) ◦ η

See [E-M] for details. Although the above formula is not the most general formula for the covariant derivative (it only holds for right-invariant vector fields), it is enough for our purposes. We will, however, need a general formula for the differentiation of a time-dependent vector field along a curve η :(−, ) → Ds(M). Defining a vector field v(t) on M by the relationη ˙(t) = v(t) ◦ η(t), the covariant derivative of uη(t) = u(t) ◦ η(t) along η(t) (where u(t) is a time-dependent vector field on M) is given by ∂u ∇¯ u = ◦ η(t) + (∇ u(t)) ◦ η(t) η˙ η ∂t v(t)

This formula follows from the above covariant derivative formula, cf. [E-M] for details. Using the formula for the covariant derivative of right-invariant vector ﬁelds and the formula for covariant derivative of vector ﬁelds along curves, we are able

s ¯ to write down the geodesic equation in D (M): ∇η˙ η˙ = 0. Letting v(t) be deﬁned byη ˙(t) = v(t) ◦ η(t) we obtain

∂v + ∇ v = 0 ∂t v

There is no explicit dependence on η and so we can think of the geodesic equation as a partial diﬀerential equation in the space of vector ﬁelds on M.

17 Remark 2.2.2. The geodesic equation on Ds(M) is the Burgers’ equation, which describes the motion of a cloud of particles moving without any internal forces. It is not always true that a weak metric yields geodesics (and hence an exponential map). For if M has non-empty boundary, the weak metric may yield geodesics which could try to cross the boundary of M:

Consider the unit disk in R2, described by polar coordinates as

D2 = {(r, θ): r ∈ [0, 1] , θ ∈ [0, 2π]}

  1 0   with metric g =  . The christoﬀel symbols are given by 0 r2

r r θ θ Γrθ = Γrr = Γθθ = Γrr = 0

r θ Γθθ = −r Γθr = r.   1   r θ Let v(r, θ) =   so that in components v = 1 and v = 0, and v points in the 0 direction of the boundary of D2. Then, in coordinates

X ∂vk ∇ v = vi + Γk vivi ∂ v ∂xi ij k i

X k i i = Γijv v ∂k i

k r r k r θ k θ θ = Γrrv v + 2Γrθv v + Γθθv v ∂k

= 0 so that v deﬁnes a stationary solution to the Burgers’ equation on the disk D2.

18 Since v is constant and points in the direction of the boundary of D2, particles will cross the boundary of D2 in ﬁnite time. Particles initially on the boundary of D2 leave the disk instantaneously.

s 2.3 The Symplectic Diﬀeomorphism Group Dω

s 2.3.1 The Manifold Structure of Dω

Let M be a closed Symplectic manifold of dimension 2n with Symplectic form ω. Endow M with a Riemannian metric g which is compatible with the Symplectic form, i.e. there exists an almost complex structure J : TM → TM satisfying J 2 = −I, g(Jv, Jw) = g(v, w), and g(v, Jw) = ω(v, w), for any vector ﬁelds v, w on M. In this case, the Riemannian volume form coincides with ωn. Let

s s ∗ Dω(M) = {η ∈ D (M): η ω = ω} be the subgroup of diﬀeomorphisms of Sobolev class Hs preserving the Symplectic form ω.

s Let η :(−, ) → Dω(M) be a curve with η(0) = e, the identity element, and velocityη ˙(t) = v(t) ◦ η(t). Then η(t)∗ω = ω for all t. Diﬀerentiating this identity with respect to t we obtain

∗ η(t) (Lvω) = 0

where Lvω is the Lie derivative given by Lvω = ivdω + divω = divω. Since η(t) is a diﬀeomorphism for each t the above implies

divω = 0

19 s Therefore, the Lie algebra of Dω(M) can be identiﬁed with

s s TeDω = {v ∈ H (TM): divω = 0} .

Theorem 2.3.1. [E-M] Let M be a closed, orientable manifold of dimension n

n and let ω be a Symplectic form. Then for s > 2 + 1

s s ∗ Dω(M) = {η ∈ D (M): η ω = ω}

s s is a smooth, closed submanifold of D (M). Moreover, Dω(M) is a subgroup of Ds(M) and is therefore a topological group whose group operations have the

s s same smoothness as those on D (M). The Lie algebra TeDω consists of (locally Hamiltonian) vector ﬁelds on M.

The main ideas in the proof are as follows, see [E-M] for more details. If ω is the Symplectic form on M then consider its Hs cohomology class

s+1 ∗ [ω]s = ω + dα : α ∈ H (T M) ,

Deﬁne a map

s Ψ: D (M) −→ [ω]s

η 7−→ η∗(ω).

If we can show that the map Ψ is a submersion onto [ω]s then the pre-image of ω under Ψ is a submanifold. We do this in three steps:

s 1. Show that Ψ actually maps D (M) into [ω]s

2. Compute the derivative DΨ at the identity

20 3. Show that DΨ is surjective.

1. We write Ψ in the following way: For a curve η(t) in Ds(M) with η(0) = e and vector ﬁeld v deﬁned byη ˙ = v ◦ η

Z t d Ψ(η(t)) = η(t)∗(ω) = ω + η(t)∗ωdt 0 dt

Z t ∗ = ω + d η(t) ivωdt ∈ [ω]s 0

2. To compute the derivative of Ψ at e consider a curve η(t) in Ds(M) with η(0) = e and vector ﬁeld v deﬁned byη ˙ = v ◦ η. Then

d DΨ(X) = | η(t)∗ω = L ω = di ω dt t=0 v v

s+1 ∗ 3. The tangent space to [ω]s is naturally identiﬁed with {dα : α ∈ H (T M)} and since ω is a Symplectic form (and hence non-degenerate), the map v 7→

s s+1 ∗ ivω is an isomorphism. It follows that DΨ: TeD → {dα : α ∈ H (T M)} is surjective and that Ψ is a submersion. In later chapters we shall formulate our results on the subspace (of ﬁnite codi-

s mension) of globally Hamiltonian vector fields TeDHam. The set of globally Hamil- tonian vector fields consists of those vector fields v on M such that ivω = dF for some Hs+1 function F on M, or, equivalently, v = J∇F . From now on we will

s denote elements of TeDHam by vF = J∇F to indicate their Hamiltonian function. Let Z s+1 s+1 H0 (M) = {F : M → R : F ∈ H (M), F dµ = 0}. M

s s+1 To each element in TeDHam there corresponds a unique element of H0 . Also, each

s+1 s element of H0 uniquely determines an element of TeDHam. So we have an iso-

21 s s+1 morphism between the sets TeDHam and H0 (M) and we denote this isomorphism by

s+1 s T : H0 (M) → TeDHam (2.2)

F 7→ J∇F

The Lie bracket of any two globally Hamiltonian vector fields is again a globally Hamiltonian vector field. To the space of globally Hamiltonian vector fields there corresponds the “Lie subgroup” of Hamiltonian diffeomorphisms denoted by

s DHam(M), see [E-M] or, alternatively the work of Ratiu and Schmid [R-S].

Theorem 2.3.2. ([E-M] Theorem 8.4) Let M be a closed, orientable manifold of dimension 2n and let ω be a Symplectic form. Then, for s > n + 1

s s ∗ DHam(M) = {η ∈ D (M): η ω = ω}

s s is a smooth, closed submanifold of D (M). Moreover, DHam(M) is a subgroup of Ds(M) and therefore is a topological group whose group operations have the

s s same smoothness as those on D (M). The Lie algebra TeDHam consists of globally Hamiltonian vector ﬁelds on M.

Denote by Adη the pushfowrd operation by a Symplectic diﬀeomorphism η:

−1 Adηv = Dη · v ◦ η .

Then AdηvF = vF ◦η−1 . Indeed,

−1 AdηJ∇F = (Dη · J∇F ) ◦ η

22 −1 = J DηT ∇F ◦ η−1 = J Dη−1T ∇F ◦ η−1

= J∇ F ◦ η−1 , where in the second equality we used that Dη is a Symplectic matrix: DηJDη> = J.

Deﬁne the Poisson bracket of two functions F,G : M → R by

{F,G} = ω(vF , vG). (2.3)

If η is a Symplectomorphism, then

{F,G} ◦ η = ω(vF , vG) ◦ η

ω(vF ◦η◦η−1 , vG◦η◦η−1 ) ◦ η

= ω(AdηvF ◦η, AdηvG◦η) ◦ η

∗ = (η ω)(vF ◦η, vG◦η)

= ω(vF ◦η, vG◦η)

= {F ◦ η, G ◦ η}.

That is, Symplectomorphisms preserve the Poisson bracket. It follows that the Poisson bracket satisﬁes the Jacobi identity:

{F, {G, H}} + {G, {H,F }} + {H, {F,G}} = 0. (2.4)

23 s Indeed, let η(t) be a geodesic in Dω. Then, diﬀerentiating both sides of

{F,G} ◦ η(t) = {F ◦ η(t),G ◦ η(t)} yields the Jacobi identity.

2.3.2 Hodge Theory for Manifolds

Let Λk be the vector bundle over M whose ﬁber at a point x ∈ M consists of k-

∗ dim M k linear skew-symmetric maps from Tx M to R. For each x, ⊕k=0 Λx forms a graded algebra with the wedge-product. Thus Hs(Λk) is the space of Hs diﬀerential k- forms. Recall the exterior derivative

d : Hs(Λk) → Hs−1(Λk+1) satisfying d2 = 0.

It drops one degree of differentiability because d differentiates once, i.e. is a first order differential operator. The Riemannian metric g(·, ·) on M gives rise to an

∗ inner product gx h·, ·i on Tx M for each x ∈ M, and then to an inner product on

k ∗ Λ Tx M, via

X hα1 ∧ ... ∧ αk, β1 ∧ ... ∧ βki = (sgnπ) gx α1, βπ(1) · ... · gx αk, βπ(k) , π

24 where π ranges over the set of permutations of {1, ...k}. Consequently, there is an L2 inner product on k-forms (i.e. sections of Λk) given by

Z (α, β) = hα, βi ωn. (2.5) M

On one forms, the inner product (2.5) is the same as the L2 metric (2.1). There is a ﬁrst order diﬀerential operator

δ : Hs(Λk) → Hs−1(Λk+1) which is the formal adjoint of d:

(dα, β) = (α, δβ) , α ∈ Λk(M), β ∈ Λk+1(M).

On functions, δ is the zero functional. We shall also make use of the linear operator ?, called the Hodge star, which assigns to each k-form on M a (dim M − k)-form and satisﬁes

?? = (−1)k(dim M−k).

The Hodge star is uniquely speciﬁed by the relation

α ∧ ?β = hα, βi ωn for any two Hs k-forms α and β and

δ = (−1)dim M(k+1)+1 ? d ? .

25 The δ operator corresponds to the classical divergence operator div. Deﬁne a

[ ∗ map g : TM → T M which assigns to each vector ﬁeld X the one form iX g, where g is the metric on M. This is an isomorphism from the tangent bundle to the cotangent bundle of M. The inverse is deﬁned as g] : T ∗M → TM and corresponds to contracting with the inverse components of the metric tensor. Let

[ X be a vector ﬁeld on M with corresponding 1-form g (X). Let LX µ be the Lie derivative of the volume form µ on M. Then, by deﬁnition

divX · µ = LX µ and the divergence measures how much the volume form µ changes in the direction of the ﬂow of X. Cartan’s formula for the Lie derivative gives

divX · µ = iX dµ + diX µ = diX µ,

[ since dµ = 0. Using the Hodge star we have that iX µ = ?g (X) and ?µ = 1 so that divX = ?divX · µ = ?d ? g[(X).

Since k = 1 and δ = (−1)dim M(k+1)+1 ? d? it follows that

divX = −δg[(X).

The Hodge Laplacian on k-forms,

4 : Hs(Λk) → Hs−2(Λk)

26 is defined by −4 = dδ + δd, and is a self-adjoint (in the L2 metric), second order differential operator which commutes with ?. The Laplacian is a well-defined operator acting on forms. In the case of functions

1 −1 4 : H0 (M) → H0 (M).

−4F = δdF = −div∇F

∞ Let F ∈ C0 (M), be a smooth function on M with zero mean. Then

(−4F,F )L2 = kdF kL2 .

Using the Poincare inequality, whose proof can be found in Evans [Ev],

kF kL2 ≤ kdF kL2 we are able to deﬁne an inner product which is equivalent to the H1 inner product and involves only the derivatives of functions but not the functions themselves:

2 2 kdF k 2 ≈ kF k 1 . L0 H0

Consequently,

2 (−4F,F ) 2 ≥ C kF k 1 (2.6) L0 H0 so that

k4F k −1 ≥ C kF k 1 . (2.7) H0 H0

27 1 −1 Theorem 2.3.3. The Hodge Laplacian is an isomorphism from H0 (M) to H0 (M).

Proof. The estimate (2.7) implies that 4 is injective with closed range. Indeed,

If 4Fn is a Cauchy sequence in the range of 4, then the inequality

1 kF − F k ≤ k4F − 4F k n m H1 C n m H−1

implies Fn is also a Cauchy sequence. Hence, if F = lim Fn then 4x = lim 4Fn is also in the range of 4 and hence the range of 4 is closed in H−1. Since the range of 4 is closed, if 4 were not surjective then there must exist

1 a function G ∈ H0 (M) which is orthogonal to the range:

(4F,G)L2 = 0 for any F . Setting F = G we see that G = 0 by (2.6).

Therefore, we have a uniquely determined inverse

−1 −1 1 4 : H0 → H0 .

If we consider the restriction of 4−1 to L2 then 4−1 is also self-adjoint in the L2

−1 2 1 metric. By Rellich’s embedding Lemma, 4 : L0(M) → H0 (M) is a compact operator.

k Theorem 2.3.4. Let k be a non-negative integer and suppose G ∈ H0 (M) and

1 F ∈ H0 (M) is a solution of 4F = G.

28 Then F ∈ Hk+2 and

kF kHk+2 ≤ C kGkHk , the constant C depending only on M and k. Consequently, the Laplacian

k+2 k 4 : H0 (M) → H0 (M) is an isomorphism.

Proof. The proof is by induction on k, see [Ev] chapter 6.

For any k-form α, 4α = 0 if and only if dα = 0 and δα = 0 and such a form is said to be harmonic. Let H denote the set of Harmonic forms and Hs Λk the space of sections of k-forms which are of Sobolev class Hs. The main theorem of this section is the Hodge Decomposition which says that the equation 4α = β has a unique solution α on M if and only if the k-form β is orthogonal to the space of Harmonic forms. This is analogous to the treatment of functions above, where we removed the Harmonic functions by considering only those Sobolev functions with mean zero. It follows from general elliptic regularity theory that the space of Harmonic forms is ﬁnite dimensional. One may consult Warner [Wa] for an elementary discussion. We shall write the Hodge decomposition in a form that is most convenient for our purposes, which is to identify the tangent space at the identity of the Sym- plectomorphism group and its orthogonal complement in the space of all vector ﬁelds.

Theorem 2.3.5. (Morrey [M], Theorem 7.4.2) For 0 ≤ s ≤ ∞, let ζ ∈ Hs(Λk).

29 Then there is an α ∈ Hs+1(Λk−1), a β ∈ Hs+1(Λk+1) and γ ∈ C∞(Λk) such that

ζ = dα + δβ + γ with 4γ = 0. In particular, we have the following decomposition of k-forms which are orthogonal in the L2 metric

Hs(Λk) = dHs+1(Λk−1) ⊕ δHs+1(Λk+1) ⊕ H, so that dα, δβ and γ are uniquely determined. The projection maps onto the ﬁrst, second, and third summands are continuous; that is, each subspace is closed. Moreover, the space of harmonic forms H is ﬁnite dimensional.

For k = 1 the Hodge decomposition yields

Hs(T ∗M) = dHs+1(T ∗M) ⊕ δHs+1(T ∗M) ⊕ H.

Define a map ω[ : TM → T ∗M which assigns to each vector field X the one form iX ω, where ω is the Symplectic form on M. This is an isomorphism from the tangent bundle to the cotangent bundle of M. The inverse is defined as ω] : T ∗M → TM and corresponds to contracting with the inverse components of the metric tensor. According to the Hodge decomposition, any Hs vector field X on M decomposes as

X = ω] (dδα + δdβ + γ) where α and β are Hs+2 1-forms and γ is a harmonic form. More generally, the

30 Lie algebra of Ds(M) decomposes as

s ] s+2 ∗ s+2 ∗ TeD = ω dδH (T M) ⊕ δdH (T M) ⊕ H . (2.8)

s Recall that the Lie algebra TeDω is given by locally Hamiltonian vector ﬁelds

[ (vector ﬁelds X which satisfy diX ω = dω (X) = 0). Therefore,

s s ] s ∗ TeD = TeDω ⊕ ω (δdH (T M))

s s and since the action of Dω on D by composition on the right is an isometry of (2.1) we obtain an L2-orthogonal splitting in each tangent space

s s ] s ∗ TηD = TηDω ⊕ ω (δdH (T M)) ◦ η. (2.9)

Deﬁne a projection

ω s s Pe : TeD → TeDω

X = ω](dδα + δdβ + γ) 7−→ ω](dδα + h) which sends a vector ﬁeld X on M to its locally Hamiltonian part v, where α, β and γ are as above. By the Hodge decomposition this is an orthogonal projection

s s for the weak metric (2.1). For vη ∈ TηD , η ∈ Dω

ω ω −1 Pη (vη) = Pe vη ◦ η ◦ η which makes P ω right-invariant and is correct since the metric (2.1) restricted to

s Dω is right-invariant.

ω ω Lemma 2.3.6. The projections Pe and Qe , given by orthogonal projection onto

31 the ﬁrst and second summands of (2.9), respectively, are given by

ω −1 Pe = −J∇4 divJ

ω ] −1 [ Qe = −ω δ4 dω .

s [ −1 [ Proof. For any v ∈ TeDω we have ω (v) = dδβ + σ = d4 δω (v) + σ for some

s+2 ∗ ω s s β ∈ H (T M) and σ ∈ H. Hence the orthogonal projection Pe : TeD → TeDω can be written as

ω ] ] ] −1 [ H v 7→ Pe (v) = ω dδβ + ω (σ) = ω d4 δω (v) + Pe (v)

H where Pe denotes the projection onto the ﬁnite-dimensional space of harmonic forms which we henceforth neglect. Using the compatible structure J we may further write

ω ] [ ] −1 [ ] [ −1 −1 −1 Pe (v) = ω g g d4 δg g ω (v) = J ∇4 divJ(v) = −J∇4 divJ(v).

ω ω Qe is computed as I − Pe .

ω ∞ The projection Pe is a C bundle map (see [E-M] Appendix A). An alternative reason for this can be seen by deﬁning another metric on Ds by

s s 2 2 (u, v)s = (u, v)L2 + 4 u, 4 v L2 where 4 is the Hodge Laplacian. This metric is smooth and by regularity properties of the Laplacian can be shown to generate the Hs topology on Ds. The Hodge decomposition is orthogonal with respect to this stronger metric and it

32 follows that the projection P ω is smooth. A proof along these lines may be found in [E]. We also obtain an L2 orthogonal splitting for the space of Hamiltonian vector ﬁelds:

s s ] s+2 ∗ TηD = TηDHam ⊕ ω δdH (T M) ◦ η ⊕ H ◦ η (2.10)

Deﬁne the corresponding orthogonal projection

H s s Pe : TeD → TeDHam which is also seen to be right invariant. Observe that

P ω = P H + P H where P H is the orthogonal projection onto the ﬁnite dimensional space of Har- monic vector ﬁelds.

s 2.3.3 Weak Riemannian Structure for Dω(M)

The smoothness of the projection P ω yields a smooth right-invariant connection and exponential mapping on the Symplectomorphism group. Since P ω and P H are smooth P H is smooth aswell and we obtain a smooth right-invariant connection and exponential mapping on the Hamiltonian diﬀeomorphism group.

Theorem 2.3.7. ([E-M], Theorem 8.5 (i)(ii)) Let M be a closed Symplectic man-

s ifold with Symplectic form ω. Then the metric (2.1) deﬁned on Dω(M) is a smooth

s s Dω-right-invariant weak Riemannian structure. On Dω, (2.1) induces the aﬃne connection ∇ω = P ω ◦ ∇¯ ,

33 where ∇¯ is the connection of Theorem 2.2.1, and the exponential mapping expω. Moreover, ∇ω and expω are invariant under right-translations.

s s The metric (2.1) restricted to DHam(M) is a smooth Dω-right-invariant weak

s Riemannian structure. On DHam(M), (2.1) induces the aﬃne connection

∇H = P H ◦ ∇¯

and the exponential mapping expH. Moreover, ∇H and expH are invariant under right-translations.

s Following Smolentsev, [Smo], deﬁne another inner product on TeDHam by

Z hvF , vH i = F Hdµ. (2.11) M

s+1 For F,H ∈ H0 (M) we have

h−4F,Hi = hdF, dHiL2 = (vF , vH )L2 so that the weak L2 metric (2.1) on vector ﬁelds is related to the inner product (2.11) by

(vF , vH )L2 = hv−4F , vH i . (2.12)

Deﬁne a map

4 s s−2 T : TeDHam → TeDHam (2.13)

vF 7→ v4F which is an isomorphism by Theorems 2.3.3 and 2.3.4.

s s ∗ Let Dµ = {η ∈ D : η µ = µ} denote the volume-preserving diﬀeomorphism

34 group of a closed Riemannian manifold M with volume form µ. Just like the

s s Symplectomorphism group, Dµ is a smooth submanifold of D whose tangent space at the identity consists of divergence free vector ﬁelds on M. According to the Hodge decomposition (Theorem 2.3.5),

s s ] s+2 2 TeDµ = {v ∈ TeD : divv = 0} = g δH (Λ ) ⊕ H .

If M is also a Symplectic manifold, with a Symplectic form ω which is compatible

n s s with the Riemannian metric, then µ = ω and Dω is a smooth submanifold of Dµ. Moreover, the tangent space admits a decomposition

s s s ⊥ TeDµ = TeDω ⊕ (TeDω) .

s [ s+2 ∗ For any v ∈ TeD we have g (v) = dδα + δdβ + σ for some α, β ∈ H (T M) and σ ∈ H. Then δg[(v) = δdδα = −4δα.

Since the Laplacian is an isomorphism,

4−1δg[(v) = −δα,

µ s s and hence the orthogonal projection Qe : TeD → TeDµ can be written as

ω ] ] −1 [ Qe (v) = g dδα = −g d4 δg (v)

= ∇4−1divv.

µ ω The projection Pe is computed as I − Qe .

35 In complete analogy to Theorems 2.3.1 and 2.3.7 we have

Theorem 2.3.8. Let M be a closed Riemannian manifold with volume form µ.

dim M s s Then, for s > 2 + 1, the group Dµ(M) is a smooth submanifold of D whose tangent space at the identity is given by

s s TeDµ = {v ∈ TeD : divv = 0} .

s s The metric (2.1) deﬁned on Dµ(M) is a smooth Dµ-right-invariant weak Rieman-

s nian structure. On Dµ, (2.1) induces the aﬃne connection

µ µ ¯ ∇ = Pe ◦ ∇,

¯ µ where ∇ is the connection of Theorem 2.2.1, Pe the smooth orthogonal projec-

s µ µ µ tion onto TeDµ, and the exponential mapping exp . Moreover, ∇ and exp are invariant under right-translations.

2.4 L2 Geodesic Completeness

ω We shall use the orthogonal projection Pe of Lemma 2.3.6 to write down the

s equations describing geodesics in the submanifold Dω. Recall that a geodesic on a manifold is a curve η(t) such that the accelerationη ¨(t) is identically zero. In the case of Ds(M) and a geodesic η(t) with ﬂow equation

η˙(t) = v(t) ◦ η(t) (2.14) we have

0 =η ¨(t) = ∂tη˙(t) = ∂t (v(t) ◦ η(t))

36 ¯ = (∂tv(t)) ◦ η(t) + ∇η˙ (v(t) ◦ η(t)) = (∂tv + ∇vv) ◦ η(t) and since η(t) is a curve of diﬀeomorphisms it follows that

∂tv + ∇vv = 0. (2.15)

Remark 2.4.1. Equation (2.15) is the well-known Burgers’ equation, describing the motion of a cloud of particles moving without any internal forces.

If N is a submanifold of a manifold M then geodesics on N correspond to curves η(t) such that the projection of the accelerationη ¨(t) onto TN is identically

s zero. With this we are able to write down the geodesic equation on Dω:

ω Pe (¨η(t)) = 0

ω ∂tv + Pe (∇vv) = 0. (2.16)

Equation (2.16) is known as the Symplectic Euler equation and who’s study is relatively new. By Lemma 2.3.6

ω ] −1 [ Qe (u) = ω δ4 dω (u) (2.17)

ω ω and since Pe = I − Qe equation (2.16) may be rewritten as

] −1 [ ∂tv + ∇vv = ω δ4 dω (∇vv) . (2.18)

s Therefore, if η(t) is a geodesic in Dω, withη ˙(t) = v(t)◦η(t), it satisﬁes the equation

37 ] −1 [ −1 η¨(t) = ω δ4 dω ∇η˙◦η−1 η˙ ◦ η = F (η, η˙). (2.19)

Theorem 2.3.7 guarantees the existence of a smooth right invariant exponential

s map, deﬁned on an open set U ⊂ TeDω around 0, so that geodesics are unique and are deﬁned on short time-intervals:

s Theorem 2.4.2. For any vo ∈ TeDω there exists t1 and t2 and a unique curve

s η :(t1, t2) → Dω such that η˙ is an integral curve of (2.19).

From the theory of ODE’s the extendability of a solution varies proportionally to the inverse of the norm of the vector field. If the C1 norm of the velocity vector field is globally bounded then solutions may be extended indefinitely. This is the basic idea behind showing that solutions of the Symplectic Euler equations (1.2) exist globally in time. Ebin’s global existence Theorem has been generalized by Khesin in [Kh], where ω need not be compatible in with g. However, it is assumed that the volume form defined by the metric g coincides with the volume form defined by ωn.

Theorem 2.4.3. [Eb] Let M be a compact Riemannian manifold without boundary, endowed with a Symplectic form and compatible Riemannian metric g. Then

s dim M geodesics in Dω(M), s > 2 + 1, exist globally in time.

We now outline the main idea in showing that the C1 norm of v remains bounded in time.

Definition 2.4.4. [Kh] Define the Symplectic vorticity for a symplectic vector field v to be the 2n form defined by ξ := dα ∈ Ω2n(M), where 2n = dimM, v and α are related by α = g[(v) ∧ ωn−1.

38 Lemma 2.4.5. The Symplectic vorticity, ξ = dg[(v) ∧ ωn−1, is given explicitly as

ξ(t) = 4F (t)ωn where 4 is the Laplacian and v = ω](dF (t) + h(t)).

Proof. Using the almost complex structure J we have

g[ω] = −g[J 2ω] = −g[g]ω[g]ω[ω] = −ω[g] and therefore g[(v) = −ω[g]dF − ω[g](h) so that

[ n−1 n n g (v) ∧ ω = −i∇F ω − ig]hω .

Since g](h) is harmonic and ω is closed

[ n−1 n n n n dg (v) ∧ ω = −di∇F ω = −L∇F ω = −(div∇F )ω = 4F ω .

Lemma 2.4.6. The Symplectic Euler equation (2.18) has the vorticity formulation

∂t4F (t) + {4F (t),F (t)} = 0 (2.20) or

∂t(4F (t) ◦ η(t)) = 0, (2.21) where {·, ·} is the Poisson bracket and η(t) is the solution of η˙(t) = v(t) ◦ η(t),

39 and v(t) = vF (t) + h(t) solves (2.18).

[ [ 2 Proof. Recall that g (∇vv) = Lvg (v) + d |v| , where Lv is the Lie derivative and |v|2 = g(v, v). Applying dg[ to both sides of (2.18)

[ [ [ ] −1 [ ∂tdg (v) + Lvdg (u) = dg ω δ4 dω (∇uu) .

n−1 n−1 Wedging both sides with ω , and observing the Lvω = 0 we get

[ ] −1 [ n−1 ∂tξ(t) + Lvξ(t) = dg ω δ4 dω (∇uu) ∧ ω .

To simplify the right hand side we use properties of the almost complex structure J as in Lemma 2.4.5. Namely, g[ω] = −ω[g] so that

[ ] −1 [ n−1 [ ] −1 [ n−1 n ] −1 [ dg ω δ4 dω (∇uu) ∧ ω = −dω g δ4 dω ∇vv ∧ ω = −dig δ4 dω ∇vvω .

] −1 [ [ n Now we observe that g δ4 dω ∇vv is divergence free (divu = −δg u) and ω is the volume form on M. Hence

n n ] −1 [ ] −1 [ −dig δ4 dω ∇vvω = Lg δ4 dω ∇vvω = 0 and the vorticity formulation of the Symplectic Euler equations is

∂tξ(t) + Lvξ(t) = 0.

By Lemma 2.4.5 this equation is given explicitly as

n n ∂t4F ω + Lv4F ω = 0 (2.22)

40 Since ω is time-independent and v is divergence free, equation (2.22) becomes

∂t4F + Lv4F = 0. (2.23)

Composing both sides of equation (2.23) with η(t) we obtain

∂t(4F (t) ◦ η(t)) = 0

The solution of (2.21) is clearly given by

4F (t) ◦ η(t) = 4F (0). (2.24)

Therefore the vorticity and the vorticity function are both invariant under the flow. This shows that the C0 norm and all Lp norms of ξ(t) are constant in time. In particular, since ξ = d g[(v) ∧ ωn−1 we are able to obtain global estimates on the derivatives of v. We also mention a special type of geodesic called a stationary geodesic. A geodesic is said to be stationary if its velocity vector v defined byη ˙(t) = v ◦ η(t) is independent of time. That is, v is a time independent solution of the Symplectic Euler equations (2.16) and satisfies both

ω ω Pe ∇vv = 0 Qe ∇vv = ∇vv.

If v = vFo is a globally Hamiltonian vector ﬁeld then the Hamiltonian function Fo

41 solves the vorticity equation (2.21):

∂t4Fo + {4Fo,Fo} = 0, which reduces to

{4Fo,Fo} = 0. (2.25)

Conversely, any function F satisfying (2.25) generates a time-independent solution

s to the Symplectic Euler equations (1.2) and a stationary geodesic η(t) in DHam.

If Fo generates a stationary geodesic η(t) then

∗ η(t) Fo = Fo ◦ η(t) = Fo.

Indeed,

∗ ∂tη(t) Fo = g(∇Fo, vFo ) ◦ η(t)

= ω (vFo , vFo ) ◦ η(t)

= 0.

2.5 Curvature

In view of Theorem 2.3.7 we are able to deﬁne the weak Riemann curvature

s s tensor of Dω and DHam:

ω s s s s Re : TeDω × TeDω × TeDω → TηDω

ω ω ω ω ω ω Re (u, v)w = ∇u ∇v w − ∇v ∇u w − ∇[u,v]w

42 H s s s s Re : TeDω × TeDω × TeDω → TeDω

H H H H H H Re (u, v)w = ∇u ∇v w − ∇v ∇u w − ∇[u,v]w

s s where u, v, w ∈ TeDω (u, v, w ∈ TeDHam resp.). Since the connection is right

s s invariant we deﬁne the curvature tensor in every tangent space TηDω (TηDHam) by right translation:

Rω(u , v )w = (∇ω∇ωw) − (∇ω∇ωw) − ∇ω w . η η η η u v η v u η [u,v] η

Proposition 2.5.1. [M1] Let M be a compact Symplectic manifold. The curvature

ω s H s tensor R of Dω and R of DHam are trilinear operators, invariant with respect

s s s to right multiplication by Dω (resp. DHam), which are continuous in the H (s >

dim M 2 + 1) topology.

s For any v and w ∈ TeDω, we deﬁne the sectional curvature of the plane σ spanned by v and w by

ω ω (Re (v, w)w, v)L2 . Ke (v, w) = (2.26) q 2 2 kvkL2 kwkL2 − (v, w)L2

ω We are only interested in the sign of Ke (v, w) so we will ignore the normalizing

ω factor and compute only with (Re (v, w)w, v)L2 . The following Lemma is from [P4].

s Lemma 2.5.2. Let v and w ∈ TeDω. Then the sectional curvature of the plane spanned by v and w is given by

Z Z ω n ω ω n Ke (v, w) = g(Awv, v) ω + g(Pe ∇vw, Pe ∇vw) ω , (2.27) M M

43 where A is a linear operator depending on w.

Proof. Using the deﬁnition of the curvature

ω ω ω ω ω ω Re (u, v)w = ∇u ∇v w − ∇v ∇u w − ∇[u,v]w, we compute

Z Z ω ω ω n ω ω n (Re (v, w)w, v)L2 = g(∇v ∇ww, v) ω − g(∇w∇v w, v) ω M M

Z ω n − g(∇[v,w]w, v) ω M Z Z ω n ω n = g(∇v∇ww, v) ω − g(∇w∇v w, v) ω M M Z n − g(∇[v,w]w, v) ω M Z Z ω n ω n = g(∇v∇ww − ∇vQe ∇ww, v) ω − g(∇w∇vw − ∇wQe ∇vw, v) ω M M Z n − g(∇[v,w]w, v) ω M Z Z Z n ω n ω n = g(R(v, w)w, v) ω − g(∇vQe ∇ww, v) ω + g(∇wQe ∇vw, v) ω M M M Z Z Z n ω ω n ω ω n = g(R(v, w)w, v) ω + g(Qe ∇ww, Qe ∇vv) ω − g(Qe ∇vw, Qe ∇wv) ω M M M Z Z Z n ω ω n ω ω n = g(R(v, w)w, v) ω + g(Qe ∇ww, Qe ∇vv) ω − g(Qe ∇vw, Qe ∇vw) ω M M M

Note that the ﬁrst two terms involve integrals of quantities which are bilinear in v. This will be useful for their point wise analysis in later chapters. The last term

44 ω ω is non-positive and we write it as ∇vw = Pe ∇vw + Qe ∇vw:

Z Z ω n ω n (Re (v, w)w, v)L2 = g(R(v, w)w, v) ω − g(∇vQe ∇ww, v) ω M M

Z Z n ω ω n − g(∇vw, ∇vw) ω + g(Pe ∇vw, Pe ∇vw) ω M M and the ﬁrst three terms are integrals of quantities bilinear in v. Thus, we write

Z Z ω n ω ω n (Re (v, w)w, v)L2 = g(Auv, v) ω + g(Pe ∇vw, Pe ∇vw) ω . M M

Here we will assume that w generates a time-independent solution to the Sym-

ω plectic Euler equations (2.16): Qe ∇ww = ∇ww. The operator Aw may be written in index notation as

i X i k l j i k i l Aj = Rjklw w + ∇j w ∇jw − gkl∇jw ∇ w , k,l

j i where Rijkl are the components of the curvature tensor on M, ∇ww = w ∇jw =

j ∂wi i k l i l ∂wl l m w ∂xj + Γklw w and ∇ w = ∂xi + Γmiw , where we have used the Einstein summation convention. The following Lemmas are from [P4] and will be of use in chapter 3.

Lemma 2.5.3. Let w be a stationary solution to the Symplectic Euler equations

(2.16) and suppose that there exists a point p ∈ M such that Lwg(p) 6= 0. Then the trace of the operator Aw, at p, is strictly negative.

45 Proof. The trace of the operator Aw may be computed:

i i k l j i mi k l Ai = Riklw w + ∇i w ∇jw − gklg ∇iw ∇mw

i k l j i k i = Riklw w + ∇i w ∇jw + ∇iw ∇ wk

i k l j i j i j i = Riklw w + ∇i w ∇jw − w ∇i∇jw − ∇iw ∇ wj

i k l j i j i i k j i = Riklw w + ∇i w ∇jw − w ∇j∇iw + Rijkw − ∇iw ∇ wj

j i j i = ∇i w ∇jw − ∇iw ∇ wj

ik jl ik jl = −g g ∇iwj∇lwk − g g ∇iwj∇kwl.

To simplify this computation further set Sij = ∇iwj + ∇jwi which corresponds to the components of the deformation tensor Lwg. Also, let νij = ∇iwj − ∇jwi. Then

i ik jl Ai = −g g (∇iwj) Skl

1 = − gikgjl (S + ν ) S 2 ij ij kl 1 = − gikgjlS S 2 ij kl since

ik jl jl ik ik jl g g νijSkl = g g νjiSlk = −g g νijSkl and is therefore zero.

At any point p ∈ M we may choose normal coordinates so that gij(p) = δij. Then, at p n n 1 X X Ai = − S2 . i 2 ij i=1 j=1

46 In particular, if Lvg(p) 6= 0 then the trace of A is strictly negative.

Lemma 2.5.4. If A and B are two linear transformations of Rn, with A symmet-

n 1 ric and TrB = 0, then there is a w ∈ R such that kwk = 1, g(w, Aw) ≤ n TrA and g(w, Bw) = 0, where g is an inner product on Rn

Proof. Let S be the symmetric part of B; then clearly g(w, Sw) = g(w, Bw) and the trace of S is equal to the trace of B. Thus we may assume B is symmetric. P Choose an orthonormal basis {vi} such that B = diag{λ1, . . . λn} with i λi = 0. Consider the set of 2n vectors deﬁned by

1 1 J = √ ( v + ··· + v ): 2 = 1, ∀i = √ : ∈ {−1, 1}n . n 1 1 n n i n

P 2 Then, for each w ∈ J, we have g(w, Bw) = i λii = 0. We compute the sum of g(w, Aw) over all w ∈ J:

n n X 1 X X 1 X X g(w, Aw) = a = a . n ij i j n ij i j w∈J ∈{−1,1}n i,j=1 i,j=1 ∈{−1,1}n

P n−2 The sum ∈{−1,1}n ij is 2 ((−1)(−1) + (−1)(1) + (1)(−1) + (1)(1)) = 0 if i 6= j and is equal to 2n−1 ((−1)2 + (1)2) = 2n if i = j. So

X 2n g(w, Aw) = Trace(A) n w∈J

1 and therefore the average value of g(w, Aw) over J is n Trace(A). So at least one element of J must have 1 g(w, Aw) ≤ Trace(A). n

47 CHAPTER 3

CONJUGATE POINTS IN THE SYMPLECTOMORPHISM GROUP

3.1 Introduction

Since the Symplectomorphism group is L2 geodesically complete, the exponen-

ω s tial mapping expe is deﬁned on the whole tangent space TeDω. The exponential mapping is the data to solution map of the Symplectic Euler equations (1.2). A

ω natural question to ask is does the exponential map expe have singularities; that

s is, a point w ∈ TeDω at which the linear operator

D expe(w) fails to be an isomorphism.

s Deﬁnition 3.1.1. Let η(t) be a geodesic in Dω with η(0) = e and η˙(0) = vo. The point η(t∗) is said to be conjugate to η(0), t∗ ∈ (0, t], if the linear operator

∗ s s D expe(t vo): TeDω → Tη(t∗)Dω

∗ is not an isomorphism. If dim ker D expe(t vo) = k , k is called the multiplicity of the conjugate point.

In contrast with ﬁnite dimensions, a linear operator between Hilbert spaces with empty kernel need not be an isomorphism. Therefore, η(t∗) may be a conju-

48 ∗ gate point even if dim ker D expe(t vo) = 0. Therefore two types of singularities of the exponential map can occur in inﬁnite dimensions. Following Grosman ([Gro]), we have

Deﬁnition 3.1.2. Let η(t) be a geodesic in a Hilbert manifold. A point η(t0) is monoconjugate to η(0) if D expη(0) (t0η˙(0)) fails to be injective. A point η(t0) is epiconjugate to η(0) if D expη(0) (t0η˙(0)) fails to be surjective.

ω s Let η(t) = expe (tvo) be a geodesic in Dω withη ˙(t) = v(t) ◦ η(t). Deﬁne v(s, t) = tvo(s), where vo(s) is a variation of the initial condition vo depending on s. Then, η(s, t) = expe(tvo(s)) is a variation of geodesics, with

∂v(s, t) | = tw , ∂s s=0 o

s for some wo ∈ TeDω, and

∂η(s, t) ∂ u(t) ◦ η(t) = | = | exp (tv (s)) = D exp (tv )tw . ∂s s=0 ∂s s=0 e o e o o

The vector field u(t) is called the variation field, or Jacobi field, of the variation η(s, t) of geodesics.

s The purpose of this chapter is to construct examples of conjugate points on Dω and describe their distribution along a certain class of geodesics. To ﬁnd examples of conjugate points, the above observation shows that it is enough to construct a variation of geodesics such that the variation ﬁeld is 0 at t = 0 and 0 at some later time t∗. The point η(t∗) will then be monoconjugate to e.

s Section 3.2 we collect the properties of some useful operators deﬁned on TeDω. It is a classical result of Riemannian geometry that manifolds of non-positive curvature do not contain conjugate points; in 3.3 we compute sectional curvatures

49 s in Dω to assist in locating conjugate points along geodesics. Explicit examples of conjugate points are constructed in 3.4 and a complete characterization of conjugate points along geodesics generated by Killing vector ﬁelds is given in section 3.5 and 3.6.

3.2 Adjoint and Coadjoint Operators

In this section we will deﬁne and compute the adjoint and coadjoint operators

s on TeDω which will be used throughout the rest of this text. Every Lie group has two distinguished representations: the adjoint and coadjoint representations. Any element g in a group G deﬁnes an automorphism cg of the group G by conjugation:

−1 cg : h 7→ ghg .

The diﬀerential of cg at the identity e ∈ G maps the Lie algebra g to itself and therefore deﬁnes an automorphism of g called the group adjoint operator

Adg : g → g.

Although the diffeomorphism group Ds is not exactly a Lie group we may still define the group adjoint operator. Indeed, let h(t) be a curve of diffeomorphisms

d s s with h(0) = e and dt |t=0h(t) = X ∈ TeD . Then, for any η ∈ D

d d c X = | c h(t) = | R −1 L h(t) η∗ dt t=0 η dt t=0 η η

−1 = dLηdRη−1 X = Dη · X ◦ η

50 which is the usual pushfoward operation on vector ﬁelds. In the last inequality we have used that dLη = Dη - the Jacobian matrix of η - and dRη is again a right translation by η. Consequently

s s Adη : TeD → TeD (3.1)

X 7→ Dη · X ◦ η−1.

The diﬀerential of Adg at the group identity g = e deﬁnes the algebra adjoint representation

aduv : g → g

d ad v = | Ad v u dt t=0 η(t) where η(t) is a curve on the group G issued from the identity η(0) = e with

d velocity dt |t=0η(t) = u. s The algebra adjoint representation on TeD is given by the negative of the usual Lie bracket of vector ﬁelds. Indeed, let η(t) be a curve in Ds with η(0) = e

d and velocity dt |t=0η(t) = u. Also, let h(s) be a curve of diffeomorphisms with d s h(0) = e and ds |s=0h(s) = v ∈ TeD . The diffeomorphisms η(t) and h(s) can be expressed to first order in s and t (in local coordinates) as

η(t): x 7→ x + tu(x) + o(t), t → 0

h(s): x 7→ x + sv(x) + o(s), s → 0.

Then η−1(t): x 7→ x − tu(x) + o(t), so that

h(s)η−1(t): x 7→ x − tu(x) + o(t) + sv (x − tu(x) + o(t)) + o(s).

51 As t → 0, we recognize the term sv (x − tu(x) + o(t)) as the directional derivative

∂v s v(x) − tu(x) ∂x , so we write

∂v h(s)η−1(t): x 7→ x − tu(x) + s v(x) − tu(x) + o(s) + o(t). ∂x

Now ∂v η(t)h(s)η−1(t) = x − tu(x) + s v(x) − tu(x) + ∂x

∂v +tu x − tu(x) + s v(x) − tu(x) + o(t) + o(s) + o(s) + o(t) ∂x

∂v = x − tu(x) + s v(x) − tu(x) + ∂x

∂v +tu x + sv(x) − tu(x) − stu(x) + o(s) + o(t) ∂x

∂v As s → 0, along with t → 0, we recognize the term tu x + sv(x) − tu(x) − stu(x) ∂x ∂u as the directional derivative tu(x) + stv(x) ∂x so that

−1 cη(t)h(s) = η(t)h(s)η (s)

∂u ∂v = x + s v(x) + t v(x) − u(x) + o(t) + o(s). ∂x ∂x

Since d d ad v = | | c h(s) u dt t=0 ds s=0 η(t) it follows that ∂u ∂v ad v = v(x) − u(x) = −[u, v]. (3.2) u ∂x ∂x

The group adjoint operator gives a representation of the group on its algebra. The orbits of the group G in its algebra are called the adjoint orbits of G. The

52 algebra adjoint is given by the rate at which the group adjoint leaves the identity, the velocity of an adjoint orbit. In the next Proposition we will describe the group Adjoint and algebra Adjoint operators on the Symplectomorphism and Hamiltonian diﬀeomorphism groups.

Proposition 3.2.1. Suppose G is the Symplectomorphism group or the Hamil- tonian diﬀeomorphism group. Then the Lie group adjoint and the Lie algebra adjoint are given by the following formulas:

s s • If G is the Symplectomorphism group Dω(M) with Lie algebra TeDω, then

s s the group adjoint Adη : TeDω → TeDω is given by

] −1 ] −1∗ n−1 Adηv = ω d δα ◦ η + g η ? h ∧ ω , (3.3)

where ? is the Hodge star operator, while the Lie algebra adjoint adu :

s s TeDω → TeDω is given by

aduv = − [u, v] = J∇ω(u, v) (3.4)

• If G is the Hamiltonian diffeomorphism group with Lie algebra of Hamilto- nian vector fields of the form v = J∇F , F an Hs+1 function on M, then the adjoints above can be written in the simplified form

−1 Adηv = J∇(F ◦ η ) (3.5)

The Lie algebra adjoint can be written in the form

aduv = J∇ {F,G} (3.6)

53 where u = J∇G and {F,G} is the Poisson bracket deﬁned by (2.3)

s s Proof. Let G be the Symplectomorphism group Dω(M) with Lie algebra TeDω of locally Hamiltonian vector ﬁelds of the form v = ω](dδα + h) = J∇ (δα) + ω] (h). Then, using formula (3.1)

] −1 ] Adη J∇ (δα) + ω (h) = (Dη · J∇ (δα)) ◦ η + Adηω (h)

T −1 −1 ] −1T −1 ] = J Dη ∇ (δα) ◦ η + Adηω (h) = J Dη ∇ (δα) ◦ η + Adηω (h)

−1 ] = J∇ δα ◦ η + Adηω (h) , where in the second equality we used that Dη is a Symplectic matrix (cf. Appendix

] A). To compute Adηω (h), let β be any one-form. Then

Z Z ] ] β η∗ω (h) dµ = β η∗ω (h) ◦ ηdµ M M

Z Z = η∗β, g[ω] (h) dµ = − η∗β, ω[g] (h) dµ M M since g[ω] = −g[J 2ω] = −ω[g].

Z Z ∗ ∗ = − η β, ig]hω dµ = − η β ∧ ? ig]hω dµ M M

Z Z = − η∗β ∧ h ∧ ωn−1dµ = − β ∧ ? η−1 ? h ∧ ωn−1 dµ M M

Since this holds for any 1-form we obtain (3.3). The algebra adjoint (3.4) follows from

[ [ ω (advw) = −i[v,w]ω = diviwω = ω J∇ω(v, w).

The result for the Hamiltonian diﬀeomorphism group follows from the result

54 on the Symplectomorphism group.

Recall that for any Hilbert space H with inner product (·, ·) there exists a unique operator T ∗ on H such that

(T f, g) = (f, T ∗g) for any f and g in H.

2 s s The formal L adjoint of the operator Adη : TeDω → TeDω is called the group

∗ s s coadjoint operator Adη : TeDω → TeDω, and is deﬁned by

Ad∗u, v = (u, Ad v) ∀v ∈ T Ds (3.7) η L2 η L2 e ω

∗ s s and the Lie algebra adjoint adu : TeDω → TeDω is deﬁned so that

∗ s (aduv, w)L2 = (v, aduw)L2 ∀w ∈ TeDω (3.8)

From here on when we refer to the coadjoint operators we mean the operators deﬁned by (3.7) and (3.8).

Proposition 3.2.2. Suppose G is the Symplectomorphism group or the Hamilto- nian diﬀeomorphism group of a closed Symplectic manifold M. The the Lie group coadjoint and Lie algebra coadjoint operators are given by the following formulas

s s • If G = Dω(M) with Lie algebra TeDω, then the group coadjoint action is given by

∗ −1 ] ∗ n−1 Adη = J∇4 (4δα ◦ η) − g ? η h ∧ ω (3.9)

55 The Lie algebra coadjoint action is given by

∗ ω aduv = Pe (div(Jv)Ju) (3.10)

• If G is the Hamiltonian diﬀeomorphism group then the group coadjoint action is given by

∗ −1 AdηvF = J∇4 (4F ◦ η) . (3.11)

The Lie algebra coadjoint action is given by

ad∗ v = P ω (4G · ∇F ) (3.12) vG F e

∗ ad v = v −1 (3.13) vG F 4 {4F,G}

with {, } the Poisson bracket deﬁned by (2.3).

Proof. Let G be the Symplectomorphism group so that g is the set of vector ﬁelds v = J∇F + h for a function F on M and a harmonic vector ﬁeld h. For the Lie algebra coadjoint

∗ (aduv, w)L2 = (v, aduw)L2 Z Z Z = hv, J∇ω(u, w)i dµ = − ω(u, w)div(Jv)dµ = hdiv(Jv)Ju, wi dµ M M M

= (div(Jv)Ju, w)L2

s ∗ ω Since this holds for any w ∈ TeDω we conclude that aduv = P (div(Jv)Ju), proving (3.10).

Suppose G is the Hamiltonian diﬀeomorphism group. Letting vF and vG ∈

56 s TeDHam

Ad∗v , v = (v , Ad v ) = J∇F,J∇ G ◦ η−1 η F G L2 F η G L2 L2

Z Z = ∇F, ∇ G ◦ η−1 dµ = − (4F ) G ◦ η−1 dµ M M Z Z = − (4F ◦ η) Gdµ = − 44−1 (4F ◦ η) Gdµ M M Z = J∇4−1 (4F ◦ η) ,J∇G , M which gives (3.11)

s For formula (3.12) let vH be any vector in TeDHam. Using (3.6) and (2.12)

ad∗ v , v = (v , ad v ) vG F H L2 F vG H L2

= − v4F , v{G,H} = − {G, H} · 4F dµ M Z Z = ω (vH , vG) · 4F dµ = g(vH , ∇G) · 4F dµ M M

= (4F · ∇G, vH )L2 consequently ad∗ v = P (4F · ∇G) vG F e which yields (3.11). Using the formula for the projection P contained in Lemma 2.3.6 we have ad∗ v = −J∇4−1divJ (4F · ∇G) vG F

= −J∇4−1g(∇4F,J∇G)

−1 = −J∇4 ω(vG, v4F )

57 = v4−1{4F,G} which gives (3.13). Formula (3.9) follows from the coadjoint formula on the Hamiltonian diﬀeo- morphism group and

Ad∗ω] (h) , ω] (k) = ω] (h) , Ad ω] (k) . η L2 η L2

s For any v ∈ TeDω deﬁne an operator

s s Kv : TeDω → TeDω

∗ u 7→ Kv(u) := aduv. (3.14)

We will now record some properties of Kv which will be useful in later sections and chapters.

2 Lemma 3.2.3. The operator Kv is skew self-adjoint in the L metric.

s 0 Proof. Let u ∈ TeDω and w ∈ TeDω. Then

∗ ∗ (Kv u, w)L2 = (u, Kvw)L2 = (u, adwv)L2

= (adwu, v)L2 = − (aduw, v)L2

∗ = − (aduv, w)L2

= − (Kvu, w)L2 .

58 0 Since this holds for any w ∈ TeDω it follows that Kv is skew self-adjoint.

Lemma 3.2.4. Suppose v(t) solves the Symplectic Euler equations (1.2). Then the operator Kv(t) satisﬁes

∗ −1 Kv(t) = Adη−1(t)Kvo Adη (t).

∗ Proof. Since KvX = advX = advX = 0 for any harmonic vector ﬁeld X, it suﬃces

s to prove the Lemma for the subspace TeDHam. Let v = vF (t) solve the Symplectic

s Euler equations (1.2) and vG ∈ TeDHam. In view of formula (3.13), (3.6) and (3.10)

−1 KvF (t) = v4 {4F (t),·}

−1 −1 = v4 {4Fo◦η (t),·}

−1 −1 = v4 {4Fo,·◦η(t)}◦η (t)

∗ −1 −1 = Adη (t)KvFo Adη (t), where in the second equality we have used (2.24).

The next two Lemmas show that the operator Kv is a compact operator for

s any v ∈ TeDω. Since the space of Hamiltonian vector ﬁelds is of ﬁnite codimension

s s s+1 in TeDω, it suﬃces to prove the Kv is compact on TeDHam. Let H = ∩s≥0H0 (M) be the subspace consisting of smooth functions on M with zero mean and observe

s+1 s+1 that H is dense in H0 (M). Given F ∈ H0 (M), let Fk be a sequence of smooth

s+1 functions in H approximating F in the H norm. Consequently, vFk is a smooth

s s sequence of H vector ﬁelds approximating vF in the H norm.

σ For σ ≥ 0, let TeDω denote the closure of the space of locally Hamiltonian

59 vector ﬁelds in the Hσ norm. By the Hodge decomposition this is a closed subspace

σ dim M in the space of all H vector ﬁelds ([M]). For σ > 2 + 1 this coincides with σ s the actual tangent space to Dω. However, for smaller σ the group DHam is not necessarily a smooth manifold.

dim M Lemma 3.2.5. Let s > 2 + 1, s ≥ σ + 1 and let F and {Fk}k∈N be as above. σ Then, Kv → Kv in the H norm. Fk F

σ Proof. Let vH ∈ TeDHam. We estimate

KvF vH − KvF vH = kPe (4F · ∇H − 4Fk · ∇H)kHσ k Hσ

. k(4F − 4Fk) · ∇HkHσ

. k4F − 4FkkHσ · kvH kHσ

. kF − FkkHσ+2 · kvH kHσ

. kF − FkkHs+1 · kvH kHσ and the Lemma follows.

dim M s Lemma 3.2.6. Let s > 2 + 1, s ≥ σ + 1. For any vector ﬁeld vF ∈ TeDHam σ the operator KvF deﬁned in Proposition 4.2.2 is compact on TeDHam.

s Proof. By Lemma 3.2.5 we can approximate vF in the H norm by a sequence of

σ smooth vector ﬁelds vF such that Kv → Kv in the H operator norm. Since k Fk F a limit of compact operators is compact it suﬃces to show that KvF is compact when vF is smooth.

By Proposition 4.2.2 and Lemma 3.2.2 (3.12), the operator KvF may be written

60 as

KvF (vH ) = P (4F · ∇H) = P (H · ∇4F )

s since the projection of a gradient ﬁeld vanishes. Then, for any vH ∈ TeDHam,

kKvF (vH )kHσ+1 = kP (H · ∇4F )kHσ+1

. kH · ∇4F kHσ+1 . kHkHσ+1

. kvH kHσ

σ σ+1 Therefore the map vH 7→ KvF (vH ), as a map from H vector ﬁelds to H vector ﬁelds, is compact by the Rellich embedding Theorem (Appendix A Theorem A.2.2).

3.3 The Second Fundamental Tensor and Curvature

Our goal in this section is to understand the sectional curvature of the group

s Dω and use these heuristics to locate examples of conjugate points. The Lemmas of section 2.5 will be of use here. First we will describe the second fundamental tensor of the Symplectomorphism group inside the Volume preserving diﬀeomorphism group. Let M be a closed Symplectic manifold with compatible Riemannian metric

s g. It is useful to consider the group Dω as a subgroup of the group of Volume-

s preserving Diﬀeomorphism group, Dµ.

61 s s The Levi-Civita connection on Dµ = Dωn at the identity is given by

µ µ ¯ ∇ = Pe (∇), (3.15)

µ s ¯ where P is the orthogonal projection onto TeDµ, and ∇ the Levi Civita connection

s of Theorem 2.2.1, cf. Theorem 2.3.8. The Levi-Civita connection on Dω at the identity is given by ω ω ¯ ∇ = Pe (∇). (3.16)

s s ] s+1 2 ∗ For each η ∈ Dω, the tangent space TηDµ = g (δH (Λ T M) ⊕ H) admits an orthogonal splitting

s s s ⊥ TηDµ = TηDω ⊕ (TηDω)

s ⊥ s s s ⊥ s where (TηDω) is the normal space to Dω at η, in Dµ.(TηDω) consists of H sections X of the pull-back bundle η∗TM for which the corresponding vector ﬁeld

−1 s ⊥ u = X ◦η on M is not locally Hamiltonian but is divergence free. Let (T Dω) =

s ⊥ s ⊥ s s s s ∪η∈Dω(M) (TηDω) and π :(T Dω) → Dω where π(TηDω) = η, ∀η ∈ Dω.

s s s Any divergence free vector ﬁeld Z : Dω → T Dµ along Dω decomposes uniquely

> ⊥ > s s ⊥ s as Z = Z + Z , where Z is an H vector ﬁeld tangent to Dω and Z is an H

s normal ﬁeld to Dω.

µ s s ˜ s Deﬁne ∇ to be the restriction of the Levi-Civita connection to T Dµ|Dω , π, Dω . Then ˜ µ ˜ ω ∇X Y = ∇X Y + α(X,Y ) (3.17) where ⊥ ˜ α(X,Y ) = ∇X Y (3.18)

62 s s s for all H vector ﬁelds X and Y on Dω. For η ∈ Dω,

s s s ⊥ αη : TηDω × TηDω −→ (TηDω(M))

⊥ ˜ αη(X,Y ) = ∇X Y , η as deﬁned in (3.18), is the 2nd Fundamental Tensor.

s We now proceed to compute the 2nd Fundamental tensor of Dω as it sits inside

s s s Dµ. This will be given by the diﬀerence of the covariant derivatives on Dµ and Dω. However, we intuitively expect that the 2nd Fundamental Form is given by the

¯ s ω s ⊥ projection of ∇X Y onto TeDµ, followed by the projection Qe onto (TηDω) . This

s is because the projections are orthogonal so that by ﬁrst projecting onto Dµ we

ω remove any gradient parts of the covariant derivative. Then using Qe we remove any locally Hamiltonian parts of the covariant derivative so that what we are left

s s with is an element of the complement of TeDω in TeDµ.

Proposition 3.3.1. Let M be a closed Symplectic manifold. The second funda-

s s mental form of Dω(M) as a submanifold of Dµ is given by

ω µ ¯ α(X,Y ) = Q P (∇X Y )

¯ s µ ω Proof. Let u = ∇vw, for v, w ∈ TeDω. Then Pe (u) = Pe (u) + α(, v, w). Recall that

ω ] −1 [ Qe = ω δ4 dω from Lemma 2.3.6 and

µ ] −1 Qe = g d4 div.

63 We compute α(v, w) = P µ(u) − P ω(u)

= Qω(u) − Qµ(u)

= −ω]δ4−1dω[(u) − g]d4−1div(u)

= −ω]δ4−1dω[(u) − ω]ω[g]d4−1div(u)

= −ω] δ4−1dω[(u) + ω[g]d4−1div(u)

] −1 [ = −ω δ4 dω (u) + i∇4−1div(u)ω

= −ω] δ4−1dω[(u) + δ (4−1div(u))ω

1 where we have used the formula δ (fα) = fδα + i∇f α, for any C function f and any k-form α, and noting that δω = 0. Continuing,

α(v, w) = −ω]δ 4−1dω[(u) + 4−1div(u) ω

= −ω]δ4−1 dω[(u) + 4 4−1div(u) ω

= −ω]4−1δ dω[(u) − dδ 4−1div(u) · ω − δd 4−1div(u) ω

since δ commutes with the inverse Laplacian Therefore, using δ (fα) = fδα+i∇f α once more,

α(v, w) = −ω]δ4−1 dω[(u) − dδ 4−1div(u) ω

] −1 [ = −ω δ4 dω (u) − di∇4−1div(Z)ω

= −ω]δ4−1 dω[ u − ∇4−1div(u)

64 = −ω]δ4−1dω[ (I − Qµ)(u)

= QωP µ(u).

The next result is from [M1] and [P4] and gives a general criterion for a geodesic

s in Dµ to have non-negative curvature in all plane sections along it which contain its tangent vector.

s Theorem 3.3.2. [M1][P4] If η :(−, ) → Dµ is a geodesic, with a possibly time- dependent velocity ﬁeld v, then the sectional curvature Kµ(v, w) is non-negative for every divergence free vector ﬁeld if and only if η(t) is an isometry of M for all t.

By Lemma 2.4.2 the sectional curvature of the plane spanned by v and w ∈

s TeDω is given by

Z Z ω n ω ω n Ke (v, w) = g(Awv, v) ω + g(Pe (∇vw),Pe (∇vw)) ω M M

s s for a linear operator Aw, which depends on w. Since Dω(M) ⊂ Dµ(M) we expect Theorem 3.3.2 to hold on the Symplectomorphism group:

s Theorem 3.3.3. If η :(−, ) → Dω is a geodesic, with a possibly time-dependent velocity ﬁeld v, then the sectional curvature Kω(v, w) is non-negative for every locally Hamiltonian vector ﬁeld if and only if η(t) is an isometry of M for all t.

The Theorem will follow from the next two Lemmas

Lemma 3.3.4. Let v be a Killing vector ﬁeld on a compact, Riemmanian manifold M endowed with a Symplectic form which is compatible with the metric. Then v

65 s generates a stationary geodesic η in Dω(M) such that the curvatures of the two- dimensional planes spanned by η˙ and any vector ﬁeld along η are non-negative. In fact,

ω ω 2 (R (v, w)w, v)L2 = kP (∇wv)kL2

s ω s for any w ∈ TeDµ(M), where R is the curvature tensor of Dω(M) at the identity.

s ∞ Proof. Let v ∈ TeDω be a Killing vector ﬁeld. Let w be any other C vector ﬁeld on M. We compute

g(∇vv, w) = vg(v, w) − g(v, ∇vw)

= (Lvg)(v, w) + g(v, Lvw) − g(v, ∇vw)

= g(v, ∇vw) − g(v, ∇wv) − g(v, ∇vw)

1 = −g(v, ∇ v) = − w · g(v, v) w 2 1 g(− ∇ |v|2 , w) 2 and since this holds for any w

1 ∇ v = − ∇ |v|2 . v 2

s 2 Since a gradient vector ﬁeld is orthogonal to TeDω in the L metric by the Hodge decomposition, Theorem 2.3.5, we have

1 P ω(∇ v) = P ω(− ∇ |v|2) = 0 e v e 2

66 so that v generates a stationary solution to the Symplectic Euler equations (1.2).

s We use the Gauss equations; for any u, v, w, y ∈ TeDω,

ω µ (Re (u, v)w, y)L2 = (Re (u, v)w, y)L2 + (α(v, w), α(u, y))L2

− (α(u, w), α(v, y))L2 , where α is the second fundamental tensor computed in Proposition 3.3.1.

s Let v be a Killing ﬁeld and w ∈ TeDω. Then

ω µ (Re (v, w)w, v)L2 = (Re (v, w)w, v)L2 + (α(w, w), α(v, v))L2

− (α(u, w), α(v, y))L2

µ = (Re (v, w)w, v)L2 − (α(v, w), α(v, w))L2

µ 2 ω µ 2 = kPe (∇wv)kL2 − kQe Pe (∇wv)kL2

ω 2 = kPe (∇wv)kL2 ≥ 0

ω µ since α(v, v) = Qe Pe (∇vv) = 0.

The following Lemma is from [P4].

Lemma 3.3.5. Let M be a compact, Riemannian manifold, with or without boundary, endowed with a Symplectic form compatible with the metric. If v is any locally

Hamiltonian vector ﬁeld, tangent to the boundary, such that (Lvg)(p) 6= 0 at some point p ∈ M, then there is a locally Hamiltonian vector ﬁeld z, with support in a

ω neighborhood of p, such that (R (u, v)v, u)L2 < 0.

Proof. By Lemma 2.5.3, TraceAv(p) < 0 and by Lemma 2.5.4 we can ﬁnd a

67 vector field u ∈ TpM such that g(u, u) = 1, g(u, Avu) < 0, and g(u, ∇uv) = 0. p Let w = ∇uv ∈ TpM, and a = g(w, w). We first define normal coordinates

(x1, y1, x2, y2, ..., xn, yn) in a neighborhood Ω of the point p such that gij|p = δij,

∂x1 (p) = up and ∂y1 (p) = wp and the Symplectic form ω takes its standard form

Pn i i i=1 dx ∧ dy , by the Darboux Theorem. Let be a small positive number. Let ψ : [0, ∞) → [0, ∞) be a C∞ function positive on [0, 1) and zero elsewhere. Let ξ : M → [0, ∞) be deﬁned in coordinates by

x 2 y 2 x 2 y 2 ξ(x , y , . . . , x , y ) = ψ 1 + 1 + ··· + n + n 1 1 n n 2

on Ω and 0 elsewhere on M. Let Ω0 be the inverse image of (0, ∞) under ξ.Ω0 is open. The vector ﬁeld z is deﬁned by

1 z = √ 2 (− (∂ ξ) ∂ + (∂ ξ) ∂ ) det g y1 x1 x1 y1

clearly satisﬁes diwω = 0 on Ω0 and is zero outside of Ω0.

2 In view of our normal coordinates we have gij = δij + O(|x| ) so that around p we have det g = 1 + O(2). Thus we can write z as

2ψ0(ρ2) 1 z = − y ∂ + x ∂ , 1 + O(2) 2 1 x1 1 y1

2 x1 2 y1 2 xn 2 yn 2 1 where ρ = + 2 + ··· + + . The term − 2 y1 is o(1) inside Ω0 and the term x1 is O() in Ω0. So to ﬁrst order in we can write

1 z = −2ψ0(ρ2) y ∂ + O(). 2 1 x1

68 Then we compute ∇zv to lowest order in :

1 ∇ v = −2ψ0(ρ2) y ∂ X + O() z 2 1 x1

1 = −2ψ0(ρ2) y ∇ v + O() 2 1 u 1 == −2ψ0(ρ2) y w + O() 2 1 a = −2ψ0(ρ2) y ∂ + O(). 2 1 y1

We now construct a function which gives, to ﬁrst order in , an approximation of

−1 ω 4 divJ (∇Y X), used for the projection Pe (∇Y X). Let

2 0 2 σ(x1, y1, . . . , xn, yn) = a ψ (ρ ).

Then

y ∇σ = 2a2ψ0(ρ2) x ∂ + 1 ∂ + ... + x ∂ + y ∂ + O(2) 1 x1 2 y1 n xn n yn

0 2 = 2aψ (ρ )y1∂y1 + O().

Therefore we have

ω ω Pe (∇zv) = Pe (∇zv + ∇σ) = Pe(O()) and so Z Z ω ω n n g(Pe (∇zv),Pe (∇zv)) ω = g(O(),O()) ω M Ω0

2 n+3 = O( )VolΩ0 = O( ).

69 On the other hand, the term (Av, z)L2 in

Z ω ω ω n K (z, v) = (Avz, z)L2 + g(Pe (∇zv),Pe (∇zv)) ω , M is of lower order. Since

a a z = −2ψ0(ρ2) y ∂ + O() = −2ψ0(ρ2) y u + O(), 2 1 y1 2 1 we have a A z = −2ψ0(ρ2) y A u + O(). v 2 1 v

Therefore, Z n (Avz, z)L2 = g(Avz, z) ω M Z 2 2 0 2 2 a y1 n = 4ψ (ρ ) 4 g(Avu, u) ω + O(). Ω0

Since g(Avw, w) was chosen to be strictly negative, we ﬁnd that by taking suﬃciently small we have Kω(z, v) < 0.

s n 3.4 Conjugate Points on Dω(CP )

The results of the previous section show that positive curvature is mostly concentrated around the set of Killing vector ﬁelds on M. For this reason we look for examples of conjugate points along geodesics which are generated by Killing vector ﬁelds. A manifold M is a Kahler manifold if it can be endowed with a compatible Symplectic form and Riemannian metric such that ω, or equivalently the almost complex structure J, is parallel with respect to the Levi-Civita connection of g.

70 That is, M is Kahler if ∇J = ∇ω = 0.

s Let M be a Kahler manifold with compatible triple (J, g, ω) and let Dω(M) be the group of diﬀeomorphisms of M preserving the Kahler form, ω. The form ω is

s also a Symplectic form and is harmonic with respect to the metric. Thus Dω(M) is the group of Symplectomorphisms of M.

Remark 3.4.1. The isometry subgroup is wholly contained in the group of Sym- plectomorphisms. For an isometry η and a harmonic form α, η∗α is also harmonic. If G is any Lie group then the action of G on H∗(M, R) by α 7→ η∗α is trivial. That is, if α is a representative of a cohomology class [α] then η∗α ∈ [α]. By the Hodge isomorphism, each cohomology class has a unique harmonic representative and consequently harmonic forms are invariant under the action of the isometry group of M. Since the Symplectic form ω is harmonic every isometry is contained in the group of Symplectomorphisms of M.

Denote by Iso(M) the isometry group of M. For more details on the Isometry groups of Riemannian manifolds we refer the reader to Poor, [Po].

s Proposition 3.4.2. Let v ∈ TeDω be a Killing vector ﬁeld. Then v generates a stationary solution to the Symplectic Euler equations (1.2) and the corresponding

s geodesic in Dω consists of isometries for all t.

Proof. That v generates a stationary solution to the Symplectic Euler equations (2.16) follows from Lemma 3.3.4.

s If η(t) is the unique geodesic in Dω with initial velocity v, then

d η(t)∗g = η(t)∗L g = 0 dt v so that η(t) consists of isometries for each t.

71 Complex projective space CP n is a complex manifold that can be described by n + 1 coordinates (zo, z1, ..., zn) where coordinates which diﬀer by an over all rescaling are identiﬁed:

(zo, z1, ..., zn) ≡ (λzo, λz1, ..., λzn); λ ∈ C \{0}.

The complex projective space CPn is a Kahler manifold with the Fubini-Study metric, which is given in components as

2 ¯ (1 + |z| )δi¯j − z¯izj hi¯j = h(∂i, ∂j) = (1 + |z|2)2

n 2 2 2 where z = (z1, ..., zn) is a point in CP , |z| = z1 + ... + zn. The isometry group of CPn is given by PU(n + 1), the projective unitary group. PU(n + 1) is given by the quotient of the unitary group, U(n + 1), by it’s center, U(1), embedded as scalars. Thus, in terms of matrices, U(n + 1) consists of complex n + 1 × n + 1 matrices whose center consists of elements of the form eiθI. Elements of PU(n+1) correspond to equivalence classes of unitary matrices, where two matrices A and B are equivalent if A = eiθI × B and we write A ≡ B. Composition is given by the usual matrix multiplication

s n Theorem 3.4.3. Conjugate points exist on Dω(CP ) for all n ≥ 2.

Proof. If n is even, consider the following 2-parameter variation of isometries

γ(s, t) = A(s)B(t)A−1(s)

72 where   i    0   A (s) 0      A(s) =  A0(s)       ..   0 .    A0(s)

  B0(t)    0   B (t) 0     .  B(t) =  ..       0   0 B (t)    i

    i cos s sin s i cos t sin t 0   0   with A (s) =   and B (s) =   are 2 × 2 block sin s i cos s sin t i cos t matrices. Observe that γ(s, 0) = iI ≡ I for all s. We shall show that for each

s n s, γ(s, t) is a family of geodesics in Dω(CP ) and compute its variation field, d ds |s=0γ(s, t), along γ(0, t). We first find that d γ(s, t) ◦ γ−1(s, t) = dt

73   0 −i cos s sin s      −i cos s 0 0 C (s)   1       − sin s 0 0       C2(s) 0 0 C1(s)      =  0 0       ..   C2(s) 0 0 .       0 0     . .   .. ..      with

    − sin s cos s −i sin2 s sin s cos s −i cos2 s     C1(s) =   C2(s) =   −i cos2 s sin s cos s −i sin2 s − sin s cos s where γ−1(s, t) = (A(s)B(t)A−1(s))−1 = A(s)B−1(t)A−1(s).

† The matrices C1 and C2 are both skew hermitian matrices. Moreover, C1 =

−C2, where † denotes the conjugate transpose. Therefore, the matrix associated

d −1 with the vector ﬁeld V (s, t) = dt γ(s, t) ◦ γ (s, t) is a skew-hermitian matrix and lies in the Lie algebra to PU(n + 1). Thus V (s, t) is a Killing vector ﬁeld so that V generates a stationary solution to the Symplectic Euler equations (2.16) by

s n Proposition 3.4.2. Hence γ(s, t) satisﬁes the geodesic equation (2.19) in Dω(CP ).

74 The variation ﬁeld of this family of geodesics is given by

  0 D1(t)      −D1(t) 0 D1(t)     .   −D (t) 0 ..   1  d   J(t) = | γ(s, t) =  .. ..  , s=0  . . D1(t)  ds    ..   D1(t) 0 .     . .   .. .. D (t)   2    −D2(t) 0

      0 0 − sin t 0 − sin t 0       0 =   D1 =   D2 =   0 0 0 sin t 0 i(1 − cos t) which clearly vanishes at t = 0 and t = 2π. Therefore, the point γ(2π) = B(2π) is conjugate to the identity γ(0) = e. The proof for odd n is exactly the same except we take

γ(s, t) = A(s)B(t)A−1(s) with   i      A (s)   1       A1(s)  A(s) =    ..   .       A (s)   1    i

75   B1(s)      B (s)   1     ..   .  B(t) =      B1(s)       i 0      0 i as our two-parameter variation of isometries, where A1 and B1 are as in the n even case.

Remark 3.4.4. In the particular case when n = 2, the geodesic

  i cos t sin t 0     γ(t) =  sin t i cos t 0      0 0 i has a point, conjugate to the identity, at t = 2π.

3.5 Geodesics in the Isometry Subgroup

s Having established the existence of conjugate points in the group Dω we now look for a a more general description of them. In this section we prove that every geodesic of the L2 metric, which is generated by a Killing vector ﬁeld, has conjugate points. In section 3.6 we solve the Jacobi equation along a geodesic generated by a Killing vector ﬁeld and show that all conjugate points along such a geodesic have even multiplicity.

2 s Lemma 3.5.1. The restriction of the L metric to Iso(M) ⊂ Dω(M) is biinvariant. Consequently, for a killing ﬁeld v, which generates a stationary geodesic

76 s η(t) in Dω, ∗ Adη(t)w = Adη−1(t)w (3.19)

∗ advw = −advw (3.20)

s for any vector ﬁeld w ∈ TeDω.

s 2 Proof. Since Dω(M) is a group and right translation is an isometry of the L metric (2.1) it suﬃces to do the necessary computations at the identity. We will show that the restriction of (2.1) to Iso(M) is invariant under the adjoint action restricted to Iso(M). Let ηt be a geodesic of the metric (2.1) in Iso(M) with initial

s velocity vo and u, w ∈ TeDω(M). Then, using the formula for the Adjoint action

Z −1 −1 (Adηt u, Adηt w)L2 = g Dηt · u ◦ ηt , Dηt · w ◦ ηt dµ M

Z ∗ = (ηt g)(u, w) dµ = (u, w)L2 M since ηt ∈ Iso(M). To prove the formula (3.20) we diﬀerentiate both sides of the identity

(Adηt u, Adηt w)L2 = (u, w)L2 in t and set t = 0 with u and w as above. We obtain

(advu, w)L2 + (u, advw)L2 = 0 and the formula is proved.

s Corollary 3.5.2. 1. Let v be a Killing ﬁeld and u ∈ TeDω. Then the covariant

77 derivative of the L2-metric (2.1) reduces to

1 ∇ωu = − ad∗ v. v 2 u

1 ∇ωv = −ad u + ad∗ v u v 2 u

If u is also a Killing ﬁeld then

1 ∇ωu = − ad u v 2 v

2. For Killing vector ﬁelds u, v, w ∈ TeIso(M)

1 Rω (u, v) w = [w, [u, v]] (3.21) 4

Proof. This follows directly from the bi-invariance of the L2 metric when restricted

s to Iso(M). However, to see this another way let v be a Killing ﬁeld and u ∈ TeDω. We have 1 P ω(∇ u) = P ω (ad u + ad∗u + ad∗ v) e v 2 e v v u

∗ By Lemma 3.5.1, advu = −advu so that

1 P ω(∇ u) = P ω (ad u − ad u + ad∗ v) e v 2 e v v u

1 = P ω(ad∗ v). 2 e u

We also have 1 P ω(∇ v) = P ω (ad∗u + ad∗ v − ad u) e u 2 e v u v 1 = P ω(−ad u + ad∗ v − ad u) 2 e v u v

78 1 = −ad u + ad∗ v v 2 u

The expression for the curvature tensor follows from the expression for the covariant derivative.

Recall that for a general finite dimensional Riemannian manifold the Ricci tensor is defined as the contraction, with respect to the metric, of the full Riemann curvature tensor. That is, the Ricci tensor Ric(u, v) evaluated at (u, v) is defined as the linear map

ω Trace(w 7→ Re (w, u)v)

In infinite dimensions, the Ricci tensor is not necessarily a well defined object: the trace may not converge. However, when considering the restriction of the L2 metric to the subgroup Iso(M), which is finite dimensional, the curvature tensor behaves as in finite dimensions and the Ricci tensor is well defined when restricted to this subgroup. Using the expression for the curvature tensor (3.21) we can write

1 1 Rω(w, u)v = [v, [w, u]] = − [v, [u, w]] e 4 4

1 = − ad ◦ ad w 4 v u so that the Ricci tensor may be realized as the trace of the map

Rv,u : TeIso(M) → TeIso(M) (3.22)

1 w 7→ − ad ◦ ad w 4 v u

Theorem 3.5.3. Every geodesic of the L2 metric (2.1) which is generated by a Killing ﬁeld and which is of length greater than πr (for some positive constant r)

79 has conjugate points.

Proof. The restriction of the L2 metric to Iso(M) reduces to ﬁnite dimensional calculations. Iso(M) has a matrix representation where the Lie derivative reduces to the usual Lie bracket of matrices. The operator adv is skew self-adjoint and we can choose an orthonormal basis for Iso(M) so that the matrix for adv has the form   0 a1      −a1 0      adv =  0 a   2     ..   −a2 0 .   . .  .. ..

s For any ﬁxed Killing ﬁeld v ∈ TeDω there must exist at least one vector u 6= v, also tangent to Iso(M), that does not commute with v. For otherwise v must lie in the center of the Lie subalgebra to Iso(M) but the center of any matrix group consists only of scalar multiples of the identity matrix which does not belong to the Lie subalgebra. Consequently at least one of the ai’s is non-zero and the matrix of

1 the operator Rv,v = − 4 adv ◦ adv is given by

 2  a1    2   a1  1    2  Rv,v =  a  4  2     2   a2    ...

80 Let v be a Killing vector ﬁeld. The Ricci tensor (3.22) is given by

dim Iso(M) 1 X Trace(R ) = a2, v,v 2 i i=1 which is strictly greater than zero since not all ai’s are zero by the above construction. The statement now follows directly from the classical theorem of Bonnet and Myers, (Theorem 1.26, Cheegar and Ebin [C-E]).

3.6 The Jacobi Equation along Geodesics of Isometries

We now solve the Jacobi equation along a geodesic of isometries explicitly and use this solution to describe conjugate points explicitly. This will be done using spectral integrals and spectral representations of self-adjoint operators.

Let B(R) be the Borel σ-algebra of R and H a Hilbert space. A spectral measure on B(R) is a mapping E of B(R) into the orthogonal projections on H such that E(R) = I and E is countably additive. Using E we can deﬁne operator valued integrals Z I(f) = f(λ) dE(λ) R for a general measurable function f : R → C ∪ {∞}. According to the spectral Theorem for (skew) self-adjoint operators on Hilbert space, to each self-adjoint operator A is associated a unique spectral measure EA such that

Z A = λ dEA(λ). R

LetS ≡ S(R, B(R),EA) be the set of all EA-a.e. ﬁnite Borel functions f : R →

81 C∪{∞}. For a function f ∈ S, we write f(A) for the spectral integral I(f). Then

Z f(A) = f(λ) dEA(λ),

so that we may deﬁne functions of an operator A. For the operator Kvo deﬁned by

eitλ−1 (3.14), which is skew self-adjoint by Lemma 3.2.3, and the function f(λ) = iλ we write tK itλ e v0 − I Z e − 1 f(Kvo ) = = dE(λ) Kv0 R iλ We refer the reader to Appendix D for properties of spectral integrals and the functional calculus.

s Let vo be any vector in TeDω and let η be the geodesic of (2.1) starting from

ω the identity with initial velocity vo. The Jacobi equation along expe (tvo) is given by

ω ω ω ∇η˙ ∇η˙ J + R (J, η˙)η ˙ = 0 (3.23) with initial conditions

0 J(0) = 0,J (0) = wo. (3.24)

The vector field v(t) defined by the flow equations

η˙(t) = v(t) ◦ η(t) (3.25) solves the Symplectic Euler equations

ω ∂tv(t) + P (∇v(t)v(t)) = 0. (3.26)

Let η(s, t) a family of geodesics depending on a parameter s, with η(0, t) = η(t) =

82 ω s expe (tvo), and v(s, t) denote the corresponding family of curves in TeDω. Deﬁne

∂v(s, t) w(t) = | ∂s s=0 and let u(t) be deﬁned by the equation

∂η(s, t) | = y(t) ◦ η(t). ∂s s=0

Diﬀerentiating equations (3.25) and (3.26) with respect to s and setting s = 0 we obtain

ω ∂tw + Pe (∇vw + ∇wv) = 0 (3.27)

∂ty + [v, y] = w (3.28)

Plugging equation (3.28) into (3.27) yields the usual Jacobi equation (3.23). Using the metric (2.1) and the algebra adjoint operator one can compute that

ω ∗ ∗ P ∇v(t)w(t) + ∇w(t)v(t) = adv(t)w(t) + adw(t)v(t) (3.29) so that equations (3.28) and (3.27) become

∗ ∗ ∂tw(t) + adv(t)w(t) + adw(t)v(t) = 0 (3.30)

∂ty(t) − adv(t)y(t) = w(t). (3.31)

83 Using the deﬁnition of the operator Kv in (3.14) we have

∗ ∂t + adv(t) + Kv(t) ∂t − adv(t) y(t) = 0. (3.32)

Lemma 3.6.1. Equation (3.32) is equivalent to

∗ (∂t + adv)(∂t − adv + Kv) y = 0 (3.33)

Proof. Expanding equations (3.32) and (3.33) and setting their diﬀerence equal to zero gives the equation

∗ ∂tKv(t) = adv(t)Kv(t) + Kv(t)adv(t). (3.34)

That is, if we knew equation (3.34) holds then equations (3.32) and (3.33) would be equal. Recall Lemma 3.2.4 which states that

∗ −1 −1 Kv(t) = Adη (t)KvFo Adη (t).

Diﬀerentiating both sides of this identity yields (3.34) so that equations (3.32) and (3.33) are equivalent.

Using Lemma 3.6.1 we rewrite (3.32) as a system of equations

∗ (∂t + adv) w = 0

(∂t − adv + Kv) y = w.

∗ We can rewrite the operators adv and adv in terms of the push-forward Adη and

84 ∗ its adjoint Adη, d Ad −1 = −Ad −1 ad (3.35) dt η η v

d ∗ ∗ ∗ Ad −1 = −ad Ad −1 (3.36) dt η v η

Using these equations, the factored, right-translated Jacobi equation can be written as the pair of equations

∗ d ∗ Ad −1 Ad w(t) = 0 (3.37) η(t) dt η(t)

d Ad Ad −1 y(t) + K (y(t)) = w(t) (3.38) η(t) dt η(t) v(t)

∗ The solution of (3.38) is obviously w(t) = Adη(t)−1 wo, and from this we rewrite (3.38) as

d ∗ Ad −1 y(t) + Ad −1 K (y(t)) = Ad −1 Ad −1 w . (3.39) dt η(t) η(t) v(t) η(t) η(t) o

2 Theorem 3.6.2. Let η(t) = exp(tvo) be a geodesic of the L metric (2.1) generated by a Killing vector ﬁeld vo. Let J(t) be a Jacobi ﬁeld along η(t), with initial

0 conditions J(0) = 0, J (0) = wo. Then

−tK I − e v0 J(t) = Dη(t) · w0. Kv0

0 s Proof. We will solve the Jacobi equation in TeDω. Let vo be any vector in TeDω and let η be the geodesic of (2.1) starting from the identity with initial velocity vo. By Theorem 2.4.3, the Cauchy problem for the Symplectic Euler equations

85 s (1.2) is globally well posed and it follows that the corresponding geodesic in Dω can be extended indeﬁnitely. From equation (3.39), the Jacobi equation along η is equivalent to

d ∗ Ad −1 y(t) + Ad −1 K (y(t)) = Ad −1 Ad −1 w , (3.40) dt η(t) η(t) v(t) η(t) η(t) o where J(t) = y(t) ◦ η solves the Jacobi equation (3.23) with initial conditions (3.24). By Lemma 3.2.4

∗ −1 Kv(t) = Adη−1(t)Kv0 Adη (t) (3.41)

Using equation (3.41) and letting u(t) = Adη(t)−1 Y (t), equation (3.40) becomes

∗ ∗ −1 −1 ∂tu(t) = Adη(t) Adη(t)−1 wo − Adη(t) Adη(t)−1 Kv0 u(t) (3.42)

ω Now suppose that η(t) = expe (tv0) is a geodesic generated by a Killing ﬁeld ∗ vo. Then Adη(t)−1 = Adη(t), by Lemma 3.5.1, and equation (3.42) reduces to

∂tu(t) = w0 − Kv0 u(t).

Since Kvo is skew self-adjoint by Lemma 3.2.3, the solution to the Homogeneous equation is given by Stone’s Theorem , cf. Theorem E.3.3,

−tK u(t) = e v0 u(0), (3.43)

86 where e−tKvo has the spectral representation

Z e−tKvo = e−itλ dE(λ) R

and E is the unique spectral measure associated to the operator Kvo . Therefore, the solution to the inhomogeneous equation is given by Duhamel’s principle:

Z t Z t −tKv −(t−s)Kv −(t−s)Kv u(t) = e 0 u(0) + e 0 w0 ds = e 0 w0 ds 0 0

since u(0) = Adη−1(0)y(0) = y(0) = 0. Explicitly, from (3.43)

Z t Z −tKv isλ −tKv u(t) = e 0 e dE(λ) ds w0 = e 0 U(t)wo, 0 R where Z t Z isλ U(t) = e dE(λ) ds w0 0 R

0 For any x ∈ TeDω, we have

Z t Z Z t Z isλ isλ e d (E(λ)x, x)L2 ds = e dEx(λ) < ∞, 0 R 0 R

0 where Ex(λ) = (E(λ)x, x)L2 is a real, finite scalar measure. For any y ∈ TeDω, there is a complex measure Ex,y on B(R): Ex,y(λ) = (E(λ)x, y)L2 which is a linear combination of four positive finite measures, each of which yield a finite integral as above, see the polarization formula (D.1) in Appendix D. We deduce that the measure space (R, B(R),Ex,y) is finite and

Z 1 Z Z 1 Z isλ isλ e d (E(λ)x, y)L2 ds = e dEx,y(λ) < ∞. 0 R 0 R

87 Making use of the Fubini-Tonelli Theorem we have

Z 1 Z Z Z 1 isλ isλ (S(t)wo, x)L2 = e d (E(λ)wo, x)L2 ds = e ds d (E(λ)wo, x)L2 0 R R 0

Z eiλ − 1 Z eiλ − 1 = d (E(λ)wo, x)L2 = dE(λ)wo, x . R iλ R iλ L2

0 Since this relation holds for any x ∈ TeDω we deduce that

iλ tK Z e − 1 e v0 − I S(t)wo = dE(λ)wo = wo. R iλ Kv0

Consequently, the Jacobi ﬁelds along η(t) are given explicitly as

−tK I − e v0 J(t) = Dη(t) · wo. Kv0

Deﬁnition 3.6.3. Let A be a bounded linear operator on H. The resolvent set, denoted by ρ(A), is deﬁned by

ρ(A) = {λ ∈ R :(A − λI) is bijective from H to H} .

The spectrum, denoted by σ(A), is the complement of the resolvent set, i.e. σ(A) =

R \ ρ(A). A real number λ is said to be an eigenvalue of A if

ker (A − λI) 6= {0} with corresponding eigenspace ker (A − λI).

Corollary 3.6.4. Let η(t) be a geodesic of the L2 metric generated by a Killing

88 ∗ ∗ 2πik ﬁeld vo on M. Then η(t ), t > 0, is conjugate to the identity if and only if t∗ is an eigenvalue of Kvo for some non-zero integer k. Consequently, the multiplicity of every conjugate point along η(t) is even.

Proof. Let η(t) be a geodesic of the L2 metric, generated by a Killing ﬁeld v on M. By the above construction, Jacobi ﬁelds along η(t) have the form

etKvo − I −tKvo J(t) = Dη(t) · e wo Kvo where

0 J(0) = 0 J (0) = wo.

A point η(t∗) is conjugate to the identity if and only if there exists a Jacobi ﬁeld

J(t), along η(t), such that J(t∗) = 0. Since both Dη(t) and e−tKvo are invertible linear operators, conjugate points are determined by the operator

∗ ∗ et Kvo − I Z eit λ − 1 = dE(λ). Kvo R iλ

∗ The spectrum of the operator et Kvo −I is determined by the essential range of Kvo ∗ ∗ the function f(λ) = eit λ−1 , [Sch]. In particular, 0 is an eigenvalue of et Kvo −I if iλ Kvo ∗ n eit λ−1 o and only if E λ ∈ R : iλ = 0 6= 0. From the Taylor series of the function ∗ eit λ−1 f(λ) = iλ we see that f(0) = 1. Therefore,

it∗λ e − 1 ∗ λ ∈ : = 0 = λ ∈ \{0} : eit λ = 1 . R iλ R

∗ This shows that 0 is an eigenvalue of et Kvo −I if and only if 1 is an eigenvalue of Kvo ∗ et Kvo . Since the function f(λ) = etλ is an analytic function, the semigroup etKvo is an analytic semigroup. Applying the spectral mapping theorem (see Appendix

89 ∗ ∗ D) to the operator et Kvo we have that 1 is an eigenvalue of et Kvo if and only if

2πik t∗ is an eigenvalue of Kvo for some non-zero integer k. We have the following chain of equalities

it∗λ e − 1 ∗ ∗ E λ ∈ : = 0 = E λ ∈ \{0} : eit λ = 1 = σ et Kvo R iλ R

∗ 2πik = et σ(Kvo ) = E : k ∈ \{0} t∗ Z

∗ 2πik which shows that η(t ) is conjugate to the identity if and only if t∗ is an eigenvalue of Kvo for some non-zero integer k. By Lemma 3.2.6, the operator Kvo is compact and skew self-adjoint in the L2 metric and therefore has a complete set

0 of orthonormal eigenvectors spanning TeDω, see Theorem C.1.4. Since complex eigenvalues always occur in conjugate pairs, whose associated eigenvectors are orthonormal, the multiplicity of every conjugate point along η(t) is even.

90 CHAPTER 4

2 s FREDHOLM PROPERTIES OF THE L EXPONENTIAL MAP ON Dω

4.1 Introduction

Having established the existence of conjugate points along geodesics in the Symplectomorphism group we now proceed to describe their distribution. After proving the L2 geodesic completeness of the Symplectomorphism group, Ebin asked if the exponential map is a Fredholm map. A bounded linear operator T between Banach spaces is said to be F redholm if it has closed range, finite dimensional Kernel and finite dimensional co-kernel. T is said to be semi-Fredholm if it has closed range and at least one of the other two conditions holds. The index of a semi-Fredholm operator is defined as

ind T = dim ker T − dim co ker T ∈ Z

Semi-Fredholm operators form an open set in the space of all bounded linear operators and the index is a continuous function on this set into Z ∪ {±∞}. Kato treats unbounded semi Fredholm operators in [K3], [K2] and shows that the index is well deﬁned for a semi-Fredholm operator. The deﬁnition of a Fredholm map is due to Smale, [S]: A smooth map f : M → N between Banach manifolds is called a Fredholm map if its Frechet derivative

91 df(p) is a Fredholm operator for each p. If the domain of f is connected then the index of the operator df(p) is independent of p and by deﬁnition is the index of f. In this chapter we answer Ebin’s question in the aﬃrmative:

Theorem 4.1.1. Let M be a closed Riemannian manifold of dimension n = 2m endowed with a ﬁxed symplectic form ω compatible with the Riemannian metric

2 s g, and let s > m + 1. Then the exponential map of the L metric on Dω(M) is a nonlinear Fredholm map of index zero.

A corollary of Theorem 4.1.1 is that the set of singularities of the exponential

ω s map expe is of first Baire category in TeDω - which follows from the Sard-Smale Theorem on Fredholm maps, cf. [Sm]. Moreover, conjugate points are isolated and of finite multiplicity so that the exponential map behaves like that of a finite dimensional manifold. To prove Theorem 4.1.1 we show that the differential of the exponential map,

D expe is a Fredholm operator of index zero. In the next section we show that

ω D expe is the solution operator to the Jacobi equation. Using a convenient de-

ω composition of the Jacobi equation we are able to write D expe as the sum of an

ω invertible operator and a compact operator, from which it follows that D expe is a Fredholm operator of index zero. This will be carried out in 4.3. Unfortunately,

σ the decomposition is only deﬁned on TeD , for s ≥ σ + 1, but are able to compen-

ω s sate for this in section 4.4 where we prove that D expe is semi-Fredholm on TeDω so that we can deﬁne it’s index. The index of a Fredholm operator is a continuous

s function and for any v ∈ TeDω, D expe(tv) is the identity operator at t = 0 so

ω that D expe is of index zero and therefore Fredholm for all t. We remark that Theorem 4.1.1 provides a distinction between Symplectic diffeomorphisms and Volume preserving diﬀeomorphisms, when equipped with the

92 L2 metric. Theorem 4.1.1 holds for any closed Symplectic manifold of dimension 2n, while Theorem 4.1.1 fails for the Volume preserving diffeomorphism group of manifolds of dimension 3 and higher, cf. [E-M-P], [P2]. The relationship between Theorem 4.1.1 and known classifications (e.g. C0 closure, Gromov’s non-squeezing Theorem) of Symplectic diffeomorphisms is, at this point, unclear. In section 4.2 we define relevant objects for the proof of Theorem 4.1.1 and collect their useful properties. Section 4.3 begins an analysis of the Jacobi equation and its solution operator, with the proof of Theorem 4.1.1 finally given in 4.4.

4.2 The Jacobi Equation

s Let vo be any vector in TeDω and let η be the geodesic of (2.1) starting from

ω the identity with initial velocity vo. The Jacobi equation along expe (tvo) is given by

ω ω ω ∇η˙ ∇η˙ J + R (J, η˙)η ˙ = 0 (4.1) with initial conditions

0 J(0) = 0,J (0) = wo. (4.2)

The fact that Rω is a bounded multi-linear operator implies the Jacobi ﬁelds exist, are unique and are deﬁned for as long as η is, cf. [M1]:

Proposition 4.2.1. [M1] Let M be a compact Symplectic manifold. The curvature

ω s H s tensor R of Dω and R of DHam are trilinear operators, invariant with respect

s s s to right multiplication by Dω (resp. DHam), which are continuous in the H (s >

dim M 2 + 1) topology.

Proof. The right-invariance of Rω follows from the right-invariance of the connec-

93 ω s tion ∇ . Let u, v ∈ TeDω and deﬁne

s (u, v)s = (u, v)L2 + (u, 4 v)L2 ,

s where 4 is the Hodge Laplacian. Extending (u, v)s to T Dω by right-invariance

s yields a smooth right -invariant Riemannian metric for Dω. The topology induced

s by this metric is equivalent to the underlying topology of Dω.

s ∞ dim M Let u, v, w ∈ TeDω and z ∈ C (TM) with dizω = 0. Since s > 2 + 1 and ω 2 s ω ω ¯ using the fact that Pe is an L orthogonal projection onto TeDω and ∇ = Pe ◦∇ is a weak connection, we have

ω (R (u, v)w, z)L2

ω ω ω = − (Pe ∇vw, ∇uw)L2 + (Pe ∇uw, ∇vz)L2 − Pe ∇[u,v]w, z L2

≤ C kukHs kvkHs kwkHs kzkH1 .

From the tensorial character of Rω we also obtain

ω s (R (u, v)w, 4 z)L2 ≤ C kukHs kvkHs kwkHs kzkHs , so that

ω ω ∞ kRe (u, v)wks = sup {(Re (u, v)w, z)s : z ∈ C (TM), dizω = 0, kzks < 1}

≤ C kukHs kvkHs kwkHs where the constant C depends on M. The Proposition now follows from the right-invariance of Rω.

94 The relationship between the Jacobi equation (4.1) and the exponential map is

ω as follows. Let η(t) = expe (tvo) be as above. Deﬁne v(s, t) = tvo(s), where vo(s) is a variation of the initial condition vo depending on s. Then, η(s, t) = expe(tvo(s)),

∂v(s, t) | = tw , ∂s s=0 o

s for some wo ∈ TeDω, and

∂η(s, t) ∂ u(t) ◦ η(t) = | = | exp (tv (s)) = D exp (tv )tw . ∂s s=0 ∂s s=0 e o e o o

Combining this observation with the above, we see that the diﬀerential of the exponential map yields solutions of the Jacobi equation (4.1) satisfying the initial conditions (4.2). If J is the Jacobi ﬁeld along η with initial conditions (4.2), then

Φ(t)wo := D expe(tvo)two = J(t) (4.3)

s s deﬁnes a family Φ(t) of bounded linear operators from TeDω to Tη(t)Dω. From section 3.6, equation (3.42) the Jacobi equation can be written as

d ∗ Ad −1 y(t) + Ad −1 K (y(t)) = Ad −1 Ad −1 w , (4.4) dt η(t) η(t) v(t) η(t) η(t) o where J(t) = y(t) ◦ η(t) is the Jacobi ﬁeld along η(t) with initial conditions (4.2).

σ For σ ≥ 0, let TeDω denote the closure of the space of locally Hamiltonian vector ﬁelds in the Hσ norm. By the Hodge decomposition, Theorem 2.3.5, this

σ dim M is a closed subspace in the space of all H vector ﬁelds ([M]). For σ > 2 + 1 σ s this coincides with the actual tangent space to Dω. However, for smaller σ, DHam

95 is not necessarily a smooth manifold. We have the following convenient decomposition of the solution operator (4.3) to the Jacobi equation (4.1). The decomposition loses one derivative (the equations are only deﬁned on Hσ if the initial velocity deﬁning the geodesic is in Hσ+1), nevertheless we will be able to compensate for this later.

s dim M Proposition 4.2.2. If η is a geodesic in Dω(M), with s > 2 + 1, s ≥ σ + 1 then the map Φ(t) deﬁned in (4.3) extends to a continuous linear operator from

σ σ TeDω to Tη(t)Dω. In addition, we have the formula

Φ(t) = Dη(t) (Ω(t) − Γ(t)) (4.5)

where Z t ∗ Ω(t) = Adη(τ)−1 Adη(τ)−1 dτ (4.6) 0 is a self-adjoint operator in the L2 metric and

Z t Γ(t) = Adη(τ)−1 Kv(τ)dRη−1(τ)Φ(τ) dτ, (4.7) 0 where v solves the Symplectic Euler equations (1.2).

Proof. Consider equation (4.4), which we write as

d ∗ Ad −1 u(t) = Ad −1 Ad −1 w − Ad −1 K u(t). dt η(t) η(t) η(t) o η(t) v(t)

By deﬁnition, u(t) ◦ η(t) = J(t) = Φ(t)wo so that u(t) = dRη−1(t)Φ(t)wo. Writing this into (4.4) and integrating both sides in t yields

Z t Z t ∗ Adη(t)−1 u(t) = Adη(s)−1 Adη(s)−1 wo ds − Adη(s)−1 Kv(s)dRη−1(s)Φ(s)wo ds. 0 0

96 Using the deﬁnition of Adη−1(t) = dLη−1(t)dRη(t) we arrive at

Φ(t)wo = u(t) ◦ η(t) = Dη(t) (Ω(t) − Γ(t)) wo.

s To see that Ω(t) is self-adjoint let v, w ∈ TeDω:

∗ (Ω(t) v, w)L2 = (v, Ω(t)w)L2

Z t ∗ = v, Adη(s)−1 Adη(s)−1 w ds 0 L2 Z t ∗ = v, Ad −1 Ad −1 w ds η(s) η(s) L2 0 Z t ∗ = Ad −1 Ad −1 v, w ds η(s) η(s) L2 0

= (Ω(t)v, w)L2 .

As this holds for any w the statement follows.

4.3 Proof of Fredholmness in Hσ

We are now in a position to prove Theorem 4.1.1 by showing that the operator Ω(t) deﬁned in (4.6) is invertible and the operator Γ(t) deﬁned by (4.7) is compact.

s s As observed in chapter 1, TeDHam is of ﬁnite codimension in TeDω and thus it

s suffices to prove that Φ(t) of Proposition 4.2.2 is Fredholm on TeDHam. We first prove invertibility of the operator Ω(t) in a series of lemmas. Invertibility is first

2 0 established in the L topology; if Ω(t) has empty kernel in TeDHam it will certainly

σ 0 have empty kernel in TeDHam ⊂ TeDHam, for all σ > 0. We have a very useful criteria for determining when an operator is invertible: an operator T is invertible on H if and only if it is bounded below and has dense range. An operator T on a

97 Hilbert space H is bounded below if there exists an ε > 0 such that kT xk ≥ ε kxk for x ∈ H. Indeed, if T is invertible then the range of T is H which is dense. Moreover,

1 −1 1 kT xk ≥ T T x = kxk kT −1k kT −1k for x ∈ H and therefore T is bounded below. Conversely, if T is bounded below there exists an ε > 0 such that kT xk ≥ ε kxk for x ∈ H. If T xn is a Cauchy sequence in the range of T , then the inequality

1 kx − x k ≤ kT x − T x k n m ε n m

implies xn is also a Cauchy sequence. Hence, if x = lim xn then T x = lim T xn is also in the range of T and hence the range of T is closed in H. If we assume the range is dense then the range of T is equal to H. Since T is bounded below we know that T is one-to-one and the inverse transformation is well-deﬁned. Moreover, if y = T x, then

−1 1 1 T y = kxk ≤ kT xk = kyk ε ε and hence T −1 is bounded. Invertibility of Ω(t) in the Hσ topology then follows from the estimate

kwokHσ . kΩ(t)wokHσ + kwokHσ−1 , (4.8)

s for any wo ∈ TeDω, which implies that Ω(t) has closed range and ﬁnite dimensional kernel. To see that such an estimate implies closed range let Ω(t)xn → y in Hσ. We need an x such that Ω(t)x = y. Let L = ker Ω(t). If the distance between xn and L remains bounded take zn = xnmodL, kznkHσ ≤ 2a < ∞;

98 σ−1 Then Ω(t)zn = Ω(t)xn → y. Passing to a subsequence, znk → w in H . Then, applying the estimate (4.8) to zn − zm, implies that the sequence zn is Cauchy so zn → z and Ω(t)z = y. If the distance between xn and L tends to inﬁnity we can assume that the distance is greater than two for all n. Pick un = xnmodL such that dist(x ,L) ≤ ku k ≤ dist(x ,L) + 1 and set w = un . Now 1 ≤ dist(w ,L) and n n n n kunk 2 n σ−1 since kwnk = 1 we can choose a subsequence converging in H (by the Rellich

Lemma). Also, Ω(t)wn → 0 so that the estimate (4.8) applied to wm − wnimplies

σ 1 wn is a Cauchy sequence. Thus wn → w ∈ H with 2 ≤ dist(wn,L) and Ω(t)w = 0, which is a contradiction. Thus this latter case is impossible and the claim is

σ proved. To prove ﬁnite dimensionality of the kernel, let wn be an H sequence on the unit sphere in the kernel of Ω(t) so that

kwnkHσ . kwnkHσ−1 .

σ σ−1 Since H embeds compactly in H , wn contains a subsequence converging in

σ−1 the H topology. But the estimate implies wn must contain a subsequence converging in the Hσ topology aswell, which implies that the unit sphere in the kernel of Ω(t) is locally compact. Consequently, the kernel is ﬁnite dimensional. With this we are ready to prove invertibility of the operator Ω(t) in the L2 topology.

dim M Lemma 4.3.1. For s > 2 +1 the operator Ω(t) deﬁned by (4.6) is an invertible 0 linear operator on TeDHam with

kΩ(t)vH kL2 ≥ C(t) kvH kL2 , (4.9)

99 0 for any vH ∈ TeDHam, and

Z t 1 C(t) = −1 dτ. 0 kDη(τ)kL∞ kDη (τ)kL∞

Proof. We compute

Z t ∗ (vH , Ω(t)vH ) = (vH , Adη−1 Adη−1 vH ) dτ 0

Z t Z t ∗ 2 1 2 = Ad −1 v dτ ≥ dτ kv k η H L2 ∗ 2 H L2 0 0 Ad −1 η (t) L2 Z t 1 2 ≥ dτ kvH k 2 , T 2 L 0 kDη(t)Dη(t) kL∞ where

∗ 2 2 T Ad −1 = Ad −1 ≤ Dη(τ) Dη(τ) . η (τ) L2 η (τ) L2 L∞

By the Schwarz inequality,

C(t) kvH kL2 ≤ kΩ(t)vH kL2 so that Ω(t) has empty kernel and closed range. Since Ω(t) is self-adjoint by Proposition 4.2.2, and has closed range we have RanΩ(t) = (ker Ω(t))⊥ = {0}⊥ =

0 0 TeDHam and therefore Ω(t) is invertible on TeDHam.

Let O be a coordinate patch on M and H ∈ Hσ(O). Recall that a coordinate patch consists of an open set O and a tuple (x1, x2, . . . , xn) ∈ Rn, where each xi : O → R is an invertible map which assigns to each point p ∈ O coordinates

100 1 n (xp, . . . , xp ). The Sobolev topology can be deﬁned locally by

X α kH ◦ ηkHσ(O) = k∂ H(η)kL2(O) , |α|≤σ

Pn α α1 αn where α = (α1, . . . αn) is a multi-index with |α| = j=1 αj and D = ∂x1 ··· ∂xn . We compute (in coordinates)

X α kΩ(t)vH kHσ = kD Ω(t)vH kL2

X α X α ≥ kΩ(t)D vH kL2 − k[D , Ω(t)] vH kL2

X α ≥ C(t) kvH kHσ − k[D , Ω(t)] vH kL2 , where we have used Lemma 4.3.1 in the ﬁrst inequality. To prove the estimate (4.8) we must show that

X α k[D , Ω(t)] vH kL2 . kvH kHσ−1 .

This is the contents of the next Lemma. The proof of (4.8) will be completed in Lemma 4.3.3.

Lemma 4.3.2. Suppose M is a compact Symplectic manifold of dimension 2n,

s without boundary, and O a coordinate patch in M. Suppose η ∈ Dω,with s ≥

dim M σ + 1 and s > 2 + 1. Then, for any multi-index α with |α| ≤ σ and any w ∈ Hσ(TM), we have the estimate

∂α,P ω w ≤ C kwk (4.10) η L2(O) Hσ−1(O) for some constant C.

101 Proof. Let w ∈ Hσ. Then, according to the Hodge decomposition, Theorem 2.3.5, we can write w = v + Jg]δα

σ+1 s σ+1 for some H 2-form α, v = J∇F ∈ TeDHam for some H function F and J = ω]g[. We prove the estimate (4.10) by induction on σ.

ω ω Consider ∂xi ,Pη (where Pη = dRη ◦ Pe ◦ dRη−1 ):

ω ω −1 ∂xi ,Pη (w ◦ η) = ∂xi (v ◦ η) − Pe (∂xi (w ◦ η) ◦ η ) ◦ η.

We will prove

ω [∂ i ,P ]w ◦ η ≤ b kwk . (4.11) x η L2 i L2

j ∂ηj −1 k lm With Ni = ∂xi ◦ η , Jl the components of the almost complex structure, g the components of the inverse metric, (δα)m the components of the 1-form δα,

k X k −1 X X j ∂w ∂ i (w ◦ η) ◦ η ∂ = N ∂ x k i ∂xj k k k j

k k lm X X ∂v ∂J ∂g ∂(δα)m = N j + N j l glm(δα) + N jJ k (δα) + N jJ kglm ∂ . i ∂xj i ∂xj m i l ∂xj m i l ∂xj k k j

∂(δα)m ∂α Now ∂xj = (δ ∂xj )m (since δ = − ? d?, with ? the Hodge star operator) and - j ∂α j ∂α j ∂α j ∂α j ∂α ∂α N ?d? j = −?N d? j = −?d?(N j )+? (dN ) ∧ ? j = δ(N j )+i j j i ∂x i ∂x i ∂x i ∂x i ∂x ∇Ni ∂x

(which follows from the formula δ(fγ) = fδγ + i∇f γ for any function f and any k-form γ). Consequently

X k −1 ∂xi (w ◦ η) ◦ η ∂k = k

102 k k lm X X j ∂v j ∂Jl lm j k ∂g k lm ∂α Ni + Ni g (δα)m + Ni Jl (δα)m + Jl g (i∇N j )m ∂k ∂xj ∂xj ∂xj i ∂xj k j ∂α +Jg]δ(N j ). i ∂xj

ω −1 2 Since Pe = J∇4 divJ and J = −I, the last term projects to zero. Thus the

k −1 expression ∂xi (w ◦ η) ◦ η ∂k contains only ﬁrst derivatives of α and v so we have essentially gained a derivative. Therefore

k ω X X ω j ∂v ∂ i ,P (w ◦ η) = Q (N ∂ ) ◦ η x η e i ∂xj k k j

k lm ω j ∂Jl lm j k ∂g k lm ∂α −Pe (Ni g (δα)m + Ni Jl (δα)m + Jl g (i∇N j )m)∂k ◦ η. ∂xj ∂xj i ∂xj

We now estimate the L2 norm of the term

k lm ω j ∂Jl lm j k ∂g k lm ∂α Pe (Ni g (δα)m + Ni Jl (δα)m + Jl g (i∇N j )m)∂k ◦ η ∂xj ∂xj i ∂xj

. Composition on the right by η corresponds to a change of variables so we estimate

k lm ω j ∂Jl lm j k ∂g k lm ∂α Pe (N g (δα)m + N Jl (δα)m + Jl g (i j )m)∂k ◦ η i j i j ∇Ni j ∂x ∂x ∂x L2

k lm j ∂Jl lm j k ∂g k lm ∂α ≤ (N g (δα)m + N Jl (δα)m + Jl g (i j )m)∂k i j i j ∇Ni j ∂x ∂x ∂x L2

k lm j ∂Jl lm j k ∂g k lm ∂α ≤ (N g (δα)m + N Jl (δα)m + Jl g (i j )m i j i j ∇Ni j ∂x L2 ∂x L2 ∂x L2

k lm ∂Jl lm k ∂g lm ∂α g (δα)m + Jl (δα)m + g (i j )m . j j ∇Ni j ∂x L2 ∂x L2 ∂x L2

lm lm ∂α g (δα)m 2 + g,j (δα)m 2 + (i j )m . L L ∇Ni j ∂x L2

103

∂α . k(δα)mkL2 + k(δα)mkL2 + ( j )m ∂x L2

. kwkL2

To complete the proof it suffices to bound the first term by the L2 norm of w. For any vector field w

ω ] −1 [ Qe (w) = ω δ4 dω (w), by Lemma 2.3.6. Using the Leibniz rule and the fact that dω[(v) = 0, the expres-

ω j k sion Qe (Ni v,j∂k) ◦ η only involves ﬁrst derivatives of v. Hence

ω −1 [ kQe (v)kL2 ≤ 4 dω (v) H1 ≤ C kvkL2 ≤ C kwkL2 and we obtain (4.11). Inductively, the estimate (4.11) implies that for any multi- index α, with no more than σ terms, we will have (4.10).

dim M Lemma 4.3.3. For s > 2 + 1, s ≥ σ + 1 the operator Ω(t) deﬁned by (4.6) is σ an invertible linear operator on TeDHam with

C(t) kvH kHσ ≤ kΩ(t)vH kHσ + C kvH kHσ−1 , (4.12)

σ for some constant C and any vH ∈ TeDHam, and

Z t 1 C(t) = −1 dτ, 0 kDη(τ)kL∞ kDη (τ)kL∞

104 Proof. We have

X α kΩ(t)vH kHσ = k∂ Ω(t)vH kL2 0≤|α|≤σ

X α X α ≥ kΩ(t)∂ vH kL2 − k[∂ , Ω(t)] vH kL2 0≤|α|≤σ 0≤|α|≤σ

X α ≥ C(t) kvH kHσ − k[∂ , Ω(t)] vH kL2 , 0≤|α|≤σ where we have used Lemma 4.3.1 in the last inequality. With the deﬁnition of Ω(t) we write

Z t −1 ω −1 T Ω(t) = Dη dRηPe dRη−1 (Dη ) dτ 0

α σ−1 and use this to show that k[∂ , Ω(t)] vH kL2 is bounded above by the H norm of vH . It is enough to show that, for each τ,

X ∂α, Dη−1P ω(Dη−1)T v ≤ C kv k , (4.13) η H L2 H Hσ−1

ω ω where dRηPe dRη−1 = Pη . Now

X X ∂α, Dη−1P ω(Dη−1)T v ≤ ∂α, Dη−1 P ω(Dη−1)Tv η H L2 η H L2

X + Dη−1 ∂α,P ω (Dη−1)Tv η H L2 X + Dη−1P ω ∂α, (Dη−1)T v . η H L2

Both Dη−1 and (Dη−1)T are matrices of Hσ functions. The commutator of a σ- order diﬀerentiation operator and a multiplication operator is an operator of order σ − 1, by the Leibniz rule. Therefore, the ﬁrst and third terms above are bounded above by kvH kHσ−1 . Applying Lemma 4.3.2 to the second term we obtain (4.13)

105 and therefore (4.12). The estimate (4.12) shows that Ω(t) has empty kernel (since Ω(t) has empty

2 σ σ kernel in L ) and closed range on TeDHam. Choose any vH ∈ TeDHam. Since Ω(t) is

0 0 invertible on TeDHam we can ﬁnd a vF ∈ TeDHam such that Ω(t)vF = vH . But now

σ the estimate (4.12) shows that vF is in H as well and Ω(t) is therefore invertible

σ on TeDHam.

To prove that the operator Γ(t) is compact we will need the results from chapter 3 which show that the operator Kv(t) is compact. For a geodesic η(t) with

η˙(t) = v(t) ◦ η(t) and initial velocity vo we have that

∗ −1 Kv(t) = Adη−1(t)Kvo Adη (t),

s by Lemma 3.2.4. Lemma 3.2.6 states that for any vo ∈ TeDω the operator Kvo is

σ compact on TeDω. Since the set of compact operators is a closed, two-sided Ideal ∗ in the space of bounded linear operators, and the operators Adη−1(t), Adη−1(t) are

σ bounded it follows that Kv(t) is compact on TeDω.

dim M Lemma 4.3.4. Suppose M is a closed Symplectic manifold and let s > 2 + 1, s s ≥ σ + 1. Let vo ∈ TeDHam and η(t) = expe(tvo). Then, the operator Γ(t) deﬁned

σ by (4.7) is compact on TeDHam.

−1 Proof. Since the operators Φ(t), dRη and Adη are all continuous and Kv(t) is compact, the composition appearing under the integral in (4.7) is compact. Thus the integral, as a limit of Riemann sums of compact operators, is compact:

Z t Γ(t) = Adη(τ)−1 Kv(τ)dRη−1(τ)Φ(τ) dτ 0

106 n X ∗ = lim Adη(t∗)−1 Kv(t∗)dRη−1(t∗)Φ(ti )(ti − ti−1) , n→∞ i i i i=1

∗ where ti−1 ≤ ti ≤ ti and P = {[0, t1], [t1, t2],..., [tn−1, tn = t]} is a partition of the interval [0, t].

4.4 Proof of Fredholmness in Hs

0 Proof of Theorem 4.1.1 Since Ω(t) is invertible and Γ(t) is compact on TeDHam,

0 the sum Ω(t) − Γ(t) is Fredholm on TeDHam. Since Dη(t) is continuous and invert-

0 0 ible on TeDHam, the operator Φ(t) = Dη [Ω(t) − Γ(t)] is Fredholm on TeDHam (see

Theorem C 3.3, Appendix C) and hence d expe(tvF ) is Fredholm aswell. In order to prove the result in Hs we need to approximate η by a smootherη ˜ since if η is in Hs the decomposition (4.5) only works in Hs−1 due to the presence of derivatives of η. We prove that Φ(t) has closed range and ﬁnite dimensional kernel in Hs. Then Φ(t) is semi-Fredholm and we can compute its index.

ω s Let us calculate D expe (0): For v ∈ TeDHam

d D expω(0)v = | expω(tv) e dt t=0 e

d = | η(t) dt t=0

= v(t) ◦ η(t)|t=0

= v,

ω where η(t) = expe (tv) is a geodesic with initial velocity v. From the relationship (4.3) we see that Φ(0) is the identity with index zero so we can conclude that Φ(t) is Fredholm of index zero for all time.

107 Finite dimensionality of the kernel in Hs follows from ﬁnite dimensionality of the kernel in L2 since the former is a subset of the latter. In order to prove closed range it will suﬃce to establish an estimate of the form

A ku0kHs ≤ kΦ(t)u0kHs + kκ(t)u0kHs (4.14) for some positive constant A and a compact operator κ(t) on Hs.

s ∞ The exponential mapping on T Dω restricts to the same map on T Dω . Any

∞ s s vo ∈ T Dω is also in T Dω so we can construct its geodesic in Dω. But for any

s˜ s˜ s˜ > s, vo ∈ T Dω so we get a geodesicη ˜ in Dω. By the uniqueness of geodesics

∞ we must have η =η ˜ for anys ˜ > s and hence η ∈ Dω , cf. [E-M], [Eb]. Choose a

∞ s C vector ﬁeldv ˜F0 close to vF0 in the H topology, in a sense to speciﬁed shortly.

Then the geodesicη ˜(t) deﬁned byv ˜F0 is smooth and is deﬁned for all time. From the decomposition (4.5) we have

h i Φ(˜ t) = Dη˜ Ω(˜ t) − Γ(˜ t) with

Z t ˜ ∗ Ω(t) = Adη˜(τ)−1 Adη˜(τ)−1 dτ 0 Z t ˜ −1 −1 ˜ Γ(t) = Adη˜(τ) Kv˜F (τ)dRη˜ (τ)Φ(τ) dτ 0

0 which is valid on TeDHam as well. Since solutions to the geodesic and Jacobi equations depend continuously on the initial conditions, Φ˜ is close to Φ in the Hs operator norm andη ˜ is close to η

108 in the Hs norm. Consequently,

˜ ˜ kΦ(t)u0k ≥ Φ(t)u0 − Φ(t) − Φ(t) ku0kHs Hs Hs

˜ ˜ ˜ ≥ Dη˜(t) · Ω(t)u0 − Dη˜(t) · Γ(t)u0 − Φ(t) − Φ(t) ku0kHs Hs Hs Hs ˜ C(t) ˜ ˜ ≥ ku0k s − Dη˜(t) · Γ(t)u0 − Φ(t) − Φ(t) ku0k s H s s H kDη˜kL∞ H H where we have used the estimate (4.12). Therefore, with κ(t) = Dη˜(t) · Γ(˜ t) we obtain the estimate

A ku0kHs ≤ kΦ(t)u0kHs + kκ(t)u0kHs with ! C(t) C(t) C˜(t) A = − − − Φ(t) − Φ(˜ t) . s kDηkL∞ kDηkL∞ kDη˜kL∞ H

The number C(t) of Lemma 4.3.1 depends only on the C1 norm of η, as does

s ˜ kDηkL∞ . So if we choosev ˜F0 close enough to vF0 in the H norm then C is close

˜ to C, kDη˜kL∞ is close to kDηkL∞ . Also, Φ(t) − Φ(t) is close to zero so that Hs A can be made positive and the estimate (4.14) is satisﬁed. Thus Φ(t), and hence

s D expe(tvF0 ), has closed range in the H topology. This completes the proof of Theorem 4.1.1.

Corollary 4.4.1. Monoconjugate and Epiconjugate points coincide.

s ∗ Proof. Let η(t) be a geodesic in Dω withη ˙(0) = vo and suppose that η(t ) is

∗ epiconjugate to e. Since the range of D expe(t v) is closed we have an orthogonal

109 splitting of the tangent space

s ∗ ∗ Tη(t∗)Dω = RanD expe(t v) ⊕ Co ker D expe(t v).

∗ By assumption, the cokernel of D expe(t v) is non-empty. Since the index of

∗ ∗ D expe(t v) is zero it follows that the kernel of D expe(t v) is non-empty aswell. Thus η(t∗) is also monoconjugate to the identity. That monoconjugate implies epiconjugate is shown in the same way.

s Let η : [0, a] → Dω(M) be a geodesic. Denote by V(0, a) = V the vector space

s formed by H Symplectic vector ﬁelds yη along η which vanish at the end points of η, i.e. y(0) ◦ η(0) = y(a) ◦ η(a) = 0. The index form of η is the quadratic form

Ia deﬁned on V by

Z a I (u , w ) = u0 , v0 + (Rω(u , η˙)η, ˙ v ) dt, a η η η η L2 e η η L2 0

where uη, wη ∈ V. In general, given a symmetric bilinear form B over a space V, we define the index of B as the maximal dimension of all subspaces of V on which the quadratic form is negative definite. The nullity of B is defined to be the dimension of the subspace of V formed by elements v ∈ V such that B(v, w) = 0 for all w ∈ V; such a subspace is called the nullspace of B. We say that B is degenerate if its nullity is strictly positive.

An element uη ∈ V belongs to the null space of Ia if and only if uη is a Jacobi

ﬁeld along η. Therefore, Ia is degenerate if and only if η(0) and η(a) are conjugate to one another. In this case the nullity of Ia is equal to the multiplicity of η(a) as a conjugate point. Since the exponential map is a non-linear Fredholm map of

110 index zero, the multiplicty of any conjugate point is ﬁnite and hence the nullity of Ia is ﬁnite dimensional.

s Since η([0, a]) is compact and each point of Dω is contained in a normal neighbourhood, we can choose a subdivision

0 = to < t1 < ... < tk−1 < tk = a

of [0, a] such that each η|[tj−1,tj ] is contained in a normal neighbourhood. Thus, each η|[tj−1,tj ] is a minimal geodesic and doesn’t contain conjugate points. Let

− − V (0, a) = V be the subspace of V formed from the felds vη such that vη|(tj−1,tj ), j = 1, ..., k, is a Jacobi ﬁeld. The space V− has ﬁnite dimension because the

+ nullity of Ia has ﬁnite dimension. Let V be the subspace of V consisting of

− vector ﬁelds wη such that wη(t1) = wη(t2) = ... = wη(tk−1) = 0. . The spaces V

+ − and V are orthogonal with respect to Ia: Given vη ∈ V, let uη ∈ V be given by

uη(tj) = vη(tj). Since η|[tj−1,tj ] does not contain conjugate points, such a uη exists

+ + − + and is unique. Hence vη − uη ∈ V and, therefore, V = V ⊕ V . Also, if wη ∈ V

− and uη ∈ V , we have

k X Z tj I (u , w ) = u00, w + (Rω(u , η˙)η, ˙ w ) dt = 0 a η η η η L2 e η η L2 j=1 tj−1

since wη vanishes at the end points and uη|[tj−1,tj ], j = 1, ..., k, is a Jacobi field. We will now show that the index form is positive definite on V+ so that the space on which the index form is negative definite is contained in a finite dimensional

subspace .Since η|[tj−1,tj ], j = 1, ..., k, are minimizing geodesics and do not contain conjugate points, each η|[tj−1,tj ] has less energy than any other path between its

+ endpoints. Therefore, if wη ∈ V then Ia(wη, wη) ≥ 0. It remains to show that

111 + Ia(wη, wη) > 0 if wη ∈ V \{0}. On the contrary, suppose that Ia(wη, wη) = 0

− − with wη 6= 0. If uη ∈ V then Ia(wη, uη) = 0. If uη ∈ V , then we have

2 0 ≤ Ia(wη + cuη, wη + cuη) = 2cIa(wη, uη) + c Ia(uη, uη), valid for all c ∈ R. This inequality says that there exist real numbers A ≥ 0 and B such that Ac2 + 2Bc ≥ 0 for all c ∈ R. This is only possible if B = 0, i.e.

Ia(wη, uη) = 0. This shows that wη lies in the null space of Ia and is therefore a

Jacobi ﬁeld. But wη(tj) = 0, j = 1, ..., k − 1, and each η|[tj−1,tj ] do not contain conjugate points. Thus wη = 0.

Corollary 4.4.2. Any ﬁnite geodesic segment contains ﬁnitely many isolated conjugate points.

Proof. Let tk ∈ [0, a] be a sequence such that η(tk) is conjugate to η(0) = 0 and let Jk(t) be the corresponding Jacobi field vanishing at t = 0 and t = tk. For each k define a vector field

   Jk(t) t ∈ [0, tk] Yk(t) = .   0 t ∈ [tk, a]

− Then each Yk ∈ V and Ia(Yk,Yk) = 0. Thus, Yk ∈ V and there are only ﬁnitely

0 many linearly independent ﬁelds Jk(t) and consequently only ﬁnitely many tks.

Corollary 4.4.3. The set of points conjugate to the identity along any geodesic (emanating from the identity) is of ﬁrst Baire Category.

Proof. This is a direct consequence of expe being a nonlinear Fredholm map of index zero and Smale’s inﬁnite dimensional version of Sard’s Theorem ([Sm]).

112 CHAPTER 5

s n CONJUGATE POINTS ON DHam(M ) IN DIMENSIONS n = 2 AND 4

5.1 Introduction

In this chapter we focus our attention on the ﬁnite codimensional subgroup of Hamiltonian diﬀeomorphisms. Recall that the linearized Symplectic Euler equations were given by

∗ ∂t + adv(t) + Kv(t) vG(t) = 0

vG(0) = vGo cf. section 3.6, equation (3.30). Denote by S(t) the solution operator to this

equation deﬁned by vG(t) = S(t)vGo . In this chapter we will show that a solution to the linearized Symplectic Euler equation S(t)vGo = vG(t), satisﬁes

∗ vG(t) = Adη−1(t)vGo − KvF (t)vH (t) (5.1)

−1 where KvF (t) is the operator deﬁned by (3.14), vF (t) =η ˙(t) ◦ η (t) and vH (t) is the right translated Jacobi ﬁeld J(t) = vH(t) ◦ η(t) along η(t) with J(0) = 0 and

0 ∗ J (0) = vGo . A point η(t ) is then conjugate to the identity if and only if

∗ ∗ vG(t ) = Adη−1(t∗)vGo (5.2)

113 Recall that we had two factorizations of the Jacobi equation (cf. section 4.2). The right factored Jacobi equation

∂ + ad∗ + K ∂ − ad v (t) = 0 (5.3) t vF (t) vF (t) t vF (t) H or in system form ∂ + ad∗ + K z = 0 (5.4) t vF vF

(∂t − advF ) vH = z. (5.5)

And the left factored Jacobi equation

∗ (∂t + adv)(∂t − adv + Kv) vH = 0 (5.6) and in system form ∂ + ad∗ z˜ = 0 (5.7) t vF

(∂t − advF + KvF ) vH =z. ˜ (5.8)

Condition (5.2) tells us that conjugate points occur when a solution z(t) = vG(t) to the linearized Symplectic Euler equation, which can be thought of as a curve in

s ∗ TeDHam, intersects a solutionz ˜(t) = Adη−1(t)vGo to the left-factored equation (5.7)

∗ s at a time t > 0, which is also a curve in TeDHam. The fact that KvFo is compact tell us that the solution operator to the linearized Symplectic Euler equations diﬀers from the coadjoint operator by a compact operator. We also solve the Jacobi equation (4.1) explicitly in terms of solutions to the linearized Symplectic Euler equations and coadjoint orbits. This shows that the

114 stability of geodesics (i.e. the growth of Jacobi ﬁelds in some norm) in the Sym- plectomorphism group depends on how far a solution to the linearized Symplectic Euler equations is from being a coadjoint orbit (measured in the same norm). We will need the following deﬁnition

Deﬁnition 5.1.1. [A-Kh] A function H on a compact Symplectic manifold M does not admit extra symmetries if an arbitrary function G satisfying {H,G} = 0 is constant on connected components of the level sets of H (i.e., {H,G} = 0 implies that the diﬀerential dG is proportional to dH ; or dG = Ψ(H)dH for some function of H, Ψ.

On a two dimensional Symplectic manifold no functions admit extra symmetries. Indeed, consider the ﬂat 2-Torus T2 with standard Symplectic form ω = dx ∧ dy, and let F , G be two functions on T2 satisfying {F,G} = 0. Then,

∂F ∂G ∂F ∂G 0 = {F,G} = − ∂x ∂y ∂y ∂x

∂ ∂F ∂2F ∂ ∂F ∂2F = G − G − G + G ∂y ∂x ∂y∂x ∂x ∂y ∂x∂y

∂ ∂F ∂ ∂F = G − G ∂y ∂x ∂x ∂y    − ∂F = div G  ∂y    ∂F  ∂x

= div (GJ∇F )

= ?LGJ∇F ω

= ?d Gω[J∇F

115 = ? dG ∧ ω[J∇F + Gdω[J∇F

= ? dG ∧ ω[J∇F , where in the third to last equality we have used Cartan’s magic formula for the Lie derivative, and in the last equality we used that J∇F is Hamiltonian. In particular, dG ∧ dF = 0.

Now, the wedge product of two one forms is non-zero if and only if the forms are linearly independent. Indeed, suppose dG is proportional to dF ; then the wege product is zero since it contains repeated forms. If dG and dF are linearly independent then ∇F and ∇G are linearly independent and the matrix A = [∇F ∇G] has non-zero determinant. That is

0 6= det A = dF ∧ dG.

Consequently, if {F,G} = 0 then dG is proportional to dF . Conjecturally, a generic function on a compact Symplectic manifold of any dimension does not admit extra symmetries. Markus and Meyer have shown that it is true for dim M = 4, in [M-Me]. In the next section we will prove the identity (5.1) which holds for any geodesic of the L2 metric and any Symplectic manifold M. From section 5.3 onwards, we assume that M is either 2 or 4 dimensional and η(t) is a stationary geodesic. A

−1 stationary geodesic η is a geodesic whose velocity vFo =η ˙ ◦ η is independent of time and satisﬁes the Symplectic Euler equations (1.2). In particular, the

116 Hamiltonian function Fo solves the vorticity equation (2.21):

∂t4Fo + {4Fo,Fo} = 0, which reduces to

{4Fo,Fo} = 0. (5.9)

The solution of the vorticity equation (2.21) is given by

−1 4F (t) = 4Fo ◦ η (t), (5.10) and reduces to

−1 4Fo = 4Fo ◦ η (t)

when Fo deﬁnes a stationary solution to the Symplectic Euler equations (1.2). Conversely, any function F satisfying (5.9) generates a time-independent solution

s to the Symplectic Euler equations (1.2) and a stationary geodesic η(t) in DHam. In section 5.3 we prove that if η(t) is a stationary geodesic then η(t∗) is conju-

∗ ∗ gate to the identity if and only if S(t )vGo = Adη−1(t)vGo . In order to do this we will need a lemma, whose proof will be postponed until section 5.4.

5.2 A Conservation Law

2 s Let η(t) = expe(tvFo ) be any geodesic of the L metric in DHam with initial ve-

s locity vFo ∈ TeDHam. As in section 4.3 we linearize the Symplectic Euler equations (1.2) and ﬂow equations (2.14) to obtain

∂ v (t) + ad∗ v (t) + K v (t) = 0 (5.11) t G vF (t) G vF (t) G

117 ∂tvH (t) − advF (t)vH (t) = vG(t) (5.12) where

vG(t) = ∂s|s=0vF (s, t) and vH (t) ◦ η(t) = ∂s|s=0η(s, t) and (s, t) ∈ (−δ, δ) × R, δ > 0. Recall that plugging equation (5.11) into equation (5.12) yields the usual Jacobi equation (4.1), cf. section 3.6.

2 s Lemma 5.2.1. Let η(t) be any geodesic of the L metric in DHam, with velocity

−1 ﬁeld vF (t) =η ˙(t) ◦ η (t). Then the solution of (5.11) satisﬁes

∗ vG(t) = Adη−1(t)vGo − KvF (t) vH (t) (5.13)

s Proof. Let η(s, t) be a smooth variation of geodesics in DHam, and vF (s, t) the corresponding variation of solutions to the Symplectic Euler equations (1.2). For each s and t, the variation F (s, t) satisﬁes (5.10):

4F (s, t) ◦ η(s, t) = 4F (s, 0).

Diﬀerentiating both sides of this equation in s and setting s = 0 yields

4G(t) ◦ η(t) − {H(t), 4F (t)} ◦ η(t) = 4Go, where {·, ·} is the Poisson bracket deﬁned by (2.3). Composing both sides of this equation, ﬁrst with η−1(t) and then applying 4−1 we obtain

−1 −1 G(t) = −4 {4F (t),H(t)} + 4 Rη−1(t)4Go,

118 where G(t) = ∂s|s=0F (s, t). Applying J∇ to both sides of this equation yields the Hamiltonian vector ﬁelds:

v (t) = v −1 − v −1 G 4 Rη−1(t)4Go 4 {4F (t),H(t)}

∗ and using the formulas for Adη−1(t) and KvF (t) in Proposition 3.2.2 we recover

(5.13), where vH (t) ◦ η(t) solves the Jacobi equation (4.1) and vGo is the initial condition of (4.2).

5.3 A Characterization of Conjugate Points along Stationary Geodesics

Let η(t) be a stationary geodesic with initial velocity vFo . The operator KvFo 0 is compact and skew self-adjoint on TeDHam, by Lemmas 3.2.3 and 3.2.6. Thus, by the spectral theorem for compact (skew) self-adjoint operators (cf. Theorem

0 C.2.6) there exists an orthonormal basis of TeDHam consisting of eigenvectors of

KvF (t) , {vψi } and 0 ⊥ TeDHam = ker KvFo ⊕ ker KvFo .

⊥ Let {vψi } denote the basis for ker KvFo and vψj the basis for ker KvFo . Given s 0 ker a vector ﬁeld w ∈ TeDHam ⊂ TeDHam we will write w for the part of w in ker KvFo ⊥ ⊥ and w for the part of w in ker KvFo .

Proposition 5.3.1. Suppose η(t) is a stationary geodesic of the L2 metric in

s DHam(M), generated by the vector vFo , and J(t) = vH (t) ◦ η(t) the Jacobi ﬁeld

0 along η(t) with initial conditions J(0) = 0, J (0) = vGo . Then,

ker ker 1. If vGo 6= 0 then vH (t) is never zero for t > 0. ker ker 2. If vGo = 0 then vH (t) = 0 for all t.

The proof Proposition 5.3.1 will be postponed until the next section, and relies

119 critically on the Fredholm results of Chapter 4.

2 s Theorem 5.3.2. Suppose η(t) is a stationary geodesic of the L metric in DHam(M),

∗ ∗ generated by the vector vFo . Then η(t ), t > 0, is conjugate to the identity if and ⊥ only if there exists a vGo ∈ ker KvFo such that

∗ ∗ vG(t ) = Adη−1(t∗)vGo . (5.14)

Proof. Suppose η(t∗) is a point conjugate to the identity along η(t). By Theorem 4.1.1 the exponential map is a non-linear Fredholm map of index zero so that monoconjugate and epiconjugate points coincide. Therefore, there exists a Jacobi

∗ ﬁeld along η(t) vanishing at t > 0. Let J(t) = vH (t) ◦ η(t), with initial conditions

0 ∗ J(0) = 0, J (0) = vGo , be the Jacobi ﬁeld along η(t) with J(t ) = 0. By Propo-

ker sition 5.3.1, vGo = 0, for otherwise J(t) 6= 0 for all t. By Lemma 5.2.1 it follows ⊥ that (5.14) holds with vGo ∈ ker KvFo . ⊥ Conversely, suppose that (5.14) holds with vGo ∈ ker KvFo . Let J(t) = 0 vH (t) ◦ η(t) be the Jacobi ﬁeld along η(t) with initial conditions J(0) = 0, J (0) =

ker vGo . Then vH (t) = 0 for all t by Lemma 5.3.1. Let vG(t) be a solution of the linearized Symplectic Euler equations (3.30) with initial conditions vGo . Expand

∗ −1 vG(t), Adη (t)vGo and vH(t) in the orthonormal basis {vψi } of KvFo :

X vH (t) = hi(t)vψi i

X vG(t) = gi(t)vψi i

∗ X Adη−1(t)vGo = ai(t)vψi . i

120 By Lemma 5.2.1 we have

gi(t) = ai(t) − λihi(t)

∗ ∗ for each i and by (5.14) gi(t ) = ai(t ) for each i. Therefore

∗ hi(t ) = 0

∗ ∗ for each i. Thus vH (t ) = 0 and η(t ) is monoconjugate, and also epiconjugate, to the identity.

Corollary 5.3.3. Let η(t) be a stationary geodesic. Then the Jacobi ﬁeld J(t) =

0 ⊥ vH (t) ◦ η(t) with initial conditions J(0) = 0, J (0) = vGo ∈ ker KvFo is given by X 1 v (t) = (g (t) − a (t)) v , H λ i i ψi i i P where KvFo vψi = λivψi , S(t)vGo = i gi(t)vψi solves the linearized Symplectic ∗ P Euler equations (3.30) and Adη−1(t)vGo = ai(t)vψi .

5.4 Proof of Proposition 5.3.1

The proof of Proposition 5.3.1 will be broken up into a series of lemmas, which rely critically on the Fredholm results of Chapter 4.

For a stationary geodesic with velocityη ˙(t) = vFo ◦ η(t), equation (5.12) can be written as

∂tvH (t) − advFo vH (t) = vG(t)

Since η(t) is a stationary geodesic, the operator Adη−1(t) is a C0 semigroup with

−1 inﬁnitesimal generator −advFo , cf. Appendix D. Therefore −Adη (t)advFo =

−1 −1 −advFo Adη (t). Multiplying both sides of this equation by Adη (t) and using

121 the product rule we obtain

Adη−1(t)vG(t) = Adη−1(t)∂tvH (t) − advAdη−1(t)vH (t) = ∂t Adη−1(t)vH (t) .

Integrating both sides in t yields

Z t vH (t) = Adη(t) Adη−1(s)vG(s) ds. 0

ker ⊥ Write vG (t) for the part of vG(t) in ker KvFo and vG(t) for the part of vG(t) in ⊥ ker KvFo . Then

Z t Z t ker ⊥ vH (t) = Adη(t) Adη−1(s)vG (s) ds + Adη(t) Adη−1(s)vG(s) ds. (5.15) 0 0

To prove Lemma 5.3.1 we will show the following chain of equalities

Z t ker ker vH (t) = Adη(t) Adη−1(s)vG (s) ds (5.16) 0

Z t ker ∗ −1 = Adη(t) Adη (s)Adη−1(s)vGo ds (5.17) 0

Z t ∗ ker −1 −1 = Adη(t) Adη (s)Adη (s)vGo ds, (5.18) 0 with each equality established in a separate lemma. Recall that the operator

Z t ∗ Ω(t) = Adη−1(s)Adη−1(s) ds 0

is the invertible part of D expe(tvFo ) since any Fredholm operator of index zero is

122 the sum of an invertible operator and a compact operator, compare with Lemma

ker ker ker 4.3.3. Therefore, if vGo 6= 0 then vH(t) 6= 0 for all t and if vGo = 0 then vH(t) = 0 for all t. This proves (i) and (ii) of Proposition 5.3.1. We now prove the ﬁrst equality (5.16).

Lemma 5.4.1. For all t we have

Adη(t) ker KvFo ⊆ ker KvFo and ⊥ ⊥ Adη(t) ker KvFo ⊆ ker KvFo .

In particular, each eigenvector vψj of KvFo which lies in ker KvFo is invariant under the adjoint action.

Proof. By Proposition 3.2.2, KvFo vψj = 0 if and only if

−1 4 {4Fo, ψj} = 0.

Since 4 is an isomorphism, cf. Theorems 2.3.3 and 2.3.4, this condition is equivalent to

{4Fo, ψj} = 0. (5.19)

As the manifold M is 2 or 4 dimensional there are no functions on M with extra symmetries, cf. Deﬁnition 5.1.1, and therefore condition (5.19) is equivalent to

dψj = Ψ(4Fo)d4Fo.

123 But

4Fo = Φ(Fo)

for some function Φ, since vFo generates a stationary geodesic. Therefore

0 dψj = Ψ ◦ Φ(Fo) · Φ (Fo)dFo

by the chain rule. As a result, for each vψj ∈ ker KvFo we have

∗ 0 ∗ Adη(t)vψj = Ψ ◦ Φ(η Fo) · Φ (η Fo)Adη(t)vFo

0 −1 = Ψ ◦ Φ(Fo) · Φ (Fo)vFo◦η (t)

0 = Ψ ◦ Φ(Fo) · Φ (Fo)vFo

vψj ,

∗ since η Fo = Fo ◦ η(t) = Fo for all t (cf. section 3.2) and in the second equality

we have used (3.5) of Proposition 3.2.1. This also shows that Adη(t) ker KvFo ⊆ ⊥ ⊥ ker KvFo , and therefore Adη(t) ker KvFo ⊆ ker KvFo .

We now prove equality (5.17).

Lemma 5.4.2. Let vG(t) be a solution of (5.11) with initial condition vG(0) = vGo . Then

ker ∗ ker vG (t) = Adη−1(t)vGo for all t.

Proof. Let vG(t) be a solution of (5.11) with initial condition vGo . Write vG(t) = P ∗ P i gi(t)vψi and Adη−1(t)vGo = i ai(t)vψi , using the orthonormal basis {vψi , λi}

124 of eigenvectors provided by KvFo . By Lemma 5.2.1

X X X gi(t)vψi = ai(t)vψi − λihi(t)vψi (5.20) i i

where J(t) = vH(t) ◦η(t) is the Jacobi ﬁeld along η(t) with initial conditions J(0) =

0 ker P ∗ ker P 0, J (0) = vGo . Write vG(t) = j gj(t)vψj and Adη−1(t)vGo = j aj(t)vψj . Then by (5.20)

gi(t) = ai(t) − λihi(t) and therefore

gj(t) = aj(t) for all t and each j. Therefore

ker ∗ ker vG(t) = Adη−1(t)vGo

Using Lemma 5.4.2 we compute

Z t ker ker vH(t) = Adη(t) Adη−1(s)vG(s) ds 0

Z t ∗ ker −1 = Adη(t) Adη (s) Adη−1(s)vGo ds 0

Z t ker ∗ ker −1 = Adη(t) Adη (s)Adη−1(s)vGo ds = Adη(t) (Ω(t)vGo ) . (5.21) 0

In the third inequality we have used Lemma 5.4.1: since ker KvFo is invariant ∗ −1 under Adη(t), applying Adη(t) to the part of Adη (s)vGo in ker KvFo is equivalent

125 ∗ −1 to applying Adη(t) to Adη (t)vGo and then orthogonally projecting onto ker KvFo . This gives the second equality (5.17) . The next Lemma will allow us to write

Z t ker Z t ∗ ∗ ker −1 −1 −1 −1 Adη (s)Adη (s)vGo ds = Adη (s)Adη (s)vGo ds. 0 0

Lemma 5.4.3. We have

⊥ ⊥ Ω(t) ker KvFo ⊆ ker KvFo and

Ω(t) ker KvFo ⊆ ker KvFo ,

R t ∗ −1 where Ω(t)vGo = 0 Adη (s)Adη−1(s)vGo ds.

⊥ P Proof. Suppose vGo ∈ ker KvFo and write vGo = gi(0)vψi . Using the formulas ∗ −1 −1 for Adη (t), Adη (t) and KvFo found in Propositions 3.2.1 and 3.2.2 and using the fact that each vψi is an eigenvector of KvFo with non-zero eigenvalue we write

∗ X Ad −1 Ad −1 v = g (0)v −1 6= 0. η (s) η (s) Go i Rη−1 4 Rη4ψi i

By Proposition 3.2.2,

−1 λivψi = KvFo vψi = v4 {4Fo,ψi}.

Since each Hamiltonian vector ﬁeld is uniquely determined by its Hamiltonian function, cf. (2.2), we must have

−1 λiψi = 4 {4Fo, ψi}.

126 Since the Laplacian is an isomorphism and λi 6= 0:

1 4ψi = {4Fo, ψi}, λi and therefore 1 v −1 = v −1 Rη−1 4 Rη4ψi Rη−1 4 Rη{4Fo,ψi} λi 1 = v −1 Rη−1 4 {4Fo,Rηψi} λi 1 −1 = AdηKvFo Adη vψi , λi where in the second equality we have used that Rη(t) preserves the Poisson bracket

(see section 2.3) and vFo is a stationary solution. ⊥ −1 −1 By Lemma 5.4.1 Adη vψi ∈ ker KvFo which implies that KvFo Adη vψi ∈ ⊥ ⊥ −1 ker KvFo and thus AdηKvFo Adη vψi ∈ ker KvFo . Therefore

∗ X 1 ⊥ Ad −1 Ad −1 v = g (0) Ad K Ad −1 v ∈ ker K . η (s) η (s) Go i λ η vFo η ψi vFo i i

∗ −1 P Writing Adη (s)Adη−1(s)vGo = i fi(t)vψi , in the orthonormal basis {vψi }, we conclude that

Z t Z t ∗ X ⊥ −1 −1 Ω(t)vGo = Adη (s)Adη (s)vGo ds = fi(s) ds vψi ∈ ker KvFo . 0 i 0

s ker Let vGo ∈ TeDHam with vGo 6= 0. By Lemma 5.4.3 and (5.21) we have

ker ker vH(t) = Adη(t) (Ω(t)vGo )

127 ker ⊥ ker = Adη(t) Ω(t)vGo + Ω(t)vGo

kerker = Adη(t) Ω(t)vGo

ker = Adη(t)Ω(t)vGo and this completes the chain of equalities in (5.18).

Remark 5.4.4. In Chapter 3 we saw that all solutions to the equation (5.11) were constant. That is, every Jacobi ﬁeld along a geodesic of isometries has constant velocity. Moreover any Jacobi ﬁeld whose initial velocity was an eigenvector of

∗ KvFo vanished for some t > 0. One may apply the above results to constant speed Jacobi ﬁelds along general stationary geodesics. It is possible to prove that if

∗ ⊥ there exists an eigenvector of Kv which lies in ker ad + Kv ∩ ker Kv Fo vFo Fo Fo then η(t) has conjugate points. In particular, the only constant speed Jacobi ﬁelds along η(t) which vanish at some time t∗ > 0 are those ﬁelds whose velocity is an

eigenvector of KvFo . If η(t) is a stationary geodesic generated by an eigenfunction of the Laplacian, i.e. those functions on M satisfying

−4F = λF λ > 0, then the constant speed Jacobi ﬁelds along η(t) have initial velocities given by

v = v −1 Go (λI−4) Ψ(Fo).

−1 Moreover, if there exists a function Ψ such that (λI + (−4)) Ψ(Fo) = vψi ∈/

ker KvFo is an eigenfunction of KvFo then η(t) has conjugate points.

128 APPENDIX A

SOBOLEV SPACES

A.1 Sobolev Spaces of Bounded Domains in Rn

Our main reference for this appendix is Taylor, [T]. Let Ω ⊂ Rn be an open bounded set with smooth boundary ∂Ω, and let Ω¯ denote the closure of Ω. Deﬁne

C∞(Ω, Rn) to be the set of smooth functions from Ω to Rn that can be extended n ¯ ∞ n to a smooth function on some open subset of R containing Ω. Let C0 (Ω, R ) = {f ∈ C∞(Ω, Rn) : The support of f is contained in a compact subset of Ω}.

An m multi index is a set of m ordered integers. If k = (k1, . . . km) is an m

Pm ∞ n k multi index then set |k| = i=1 ki. If f ∈ C (Ω, R ) deﬁne D f to be

∂|k| Dkf = k1 km ∂x1 . . . ∂xm and D0f = f. For f ∈ C∞(Ω, Rn), deﬁne

Z X k 2 kfkHs = D f(x) dx. Ω 0≤|k|≤s

s n s n ∞ n Now H (Ω, R ) (resp. H0 (Ω, R ))is deﬁned to be the completion of C (Ω, R )

∞ n s n (resp. C0 (Ω, R )) in the norm k·kHs . The space H (Ω, R ) is a Hilbert space

129 with the inner product

Z X k k (f, g)Hs = D f(x) · D g(x) dx. Ω 0≤|k|≤s

A.2 Sobolev Spaces on Compact Manifolds

Letg be the Riemannian metric on M and ∇ the corresponding Levi-Civita connection. For an integer k and a smooth function f : M → R, denote by ∇kf the kth order covariant derivative of f and ∇kf the norm of ∇kf deﬁned in a local chart by

k k k i1j1 i j k k ∇ f = g(∇ f, ∇ f) = g ··· g k k ∇ f ∇ f . i1···ik j1···jk

∂f For k = 1, (∇f)i = while ∂xi

2 2 ∂ f k ∂f ∇ f ij = − Γij . ∂xi∂xj ∂xk

For f ∈ C∞(M) deﬁne

Z 2 X k k kfkHs = g(∇ f, ∇ f) dµ, 0≤|k|≤s M

X k 2 = ∇ f L2 . (A.1) 0≤|k|≤s

The Sobolev space Hs(M) is deﬁned as the completion of C∞(M) in the norm (A.1). For any integer s, the Sobolev space Hs(M) is a Hilbert space with the

130 inner product Z X k k (f, g)Hs = g(∇ f, ∇ g) dµ. 0≤|k|≤s M

If M is compact with two Riemannian metrics g andg ˜ then the norms induced by each of these metrics are equivalent so that the deﬁnition of Hs(M) does not depend on the metric.

Theorem A.2.1. Let M be a compact manifold with boundary and suppose that

dim M s > 2 + k. Then, 1. Hs(M) embeds continuously into Ck(M). 2. Hs(M) is a ring under point wise multiplication.

We also have the important Rellich Lemma

Theorem A.2.2. For any integers s ≥ 0 and t > s, the space Ht(M) embeds compactly into Hs(M).

The proofs of these theorems may be found in the comprehensive work of Hebey, [He].

A.3 Sobolev Spaces on Vector Bundles

Let M be a compact, n-dimensional Riemannian manifold without boundary, and let E be a vector bundle over M with canonical projection π : E → M. For each point x ∈ M, the ﬁber π−1(x) over x is isomorphic to Rm and there is an −1 ∼ n open cover {Ui} of M such that each Ui is a chart on M and π (Ui) = Ui × R for each i. Such a covering is called a trivialization.

A section of E is a map h : M → E such that π ◦ h = IdM . For an integer s ≥ 0, we deﬁne Hs(E) to be the space of sections of E whose derivatives up to

131 order s are in L2. This makes sense in view of the trivialization because a section can be thought of locally as a map from Rn to Rm. When E is the tangent bundle TM of M we deﬁne Hs(TM) to be the completion of C∞(TM) in the norm deﬁned by (A.1). The Sobolev space Hs(TM) is a ring under point-wise multiplication and embeds continuously in Ck(E) for

dim M s > 2 + k. As before, compactness of M is used to show that the deﬁnition of Hs(TM) is independent of the metric. It is convenient to work in coordinates on M and for this we shall use the trivialization. For a section X : M → TM of the tangent bundle of M and a chart Ui ⊂ M we write

l X = X ∂xl

in local coordinates in Ui. The Sobolev norm is then deﬁned by

X X X k l kXk s = D X 2 , H L (Ui) i l 0≤|k|≤s where 1 Z 2 2 kgk 2 = |g(x)| dx . L (Ui) Ui More generally, we can use the covariant derivative on M to give a coordinate free description of this norm and inner product:

Z X k k kXkHs = g(∇ X, ∇ X) dµ 0≤|k|≤s M

Z X k k (X,Y )Hs = g(∇ X, ∇ Y ) dµ, 0≤|k|≤s M where X and Y are sections of TM.

132 The Riemannian metric g(·, ·) on M gives rise to an inner product gx h·, ·i on

∗ p ∗ Tx M for each x ∈ M, and then to an inner product on Λ Tx M, via

X hα1 ∧ ... ∧ αk, β1 ∧ ... ∧ βki = (sgnπ) gx α1, βπ(1) · ... · gx αk, βπ(k) , π where π ranges over the set of permutations of {1, ...k}. Consequently, there is an L2 inner product on k-forms (i.e. sections of Λk) given by

Z (α, β) = hα, βi dµ. (A.2) M

In local coordinates (that is, in a chart Ui), a p-form α can be written as

dim M X i1 ip α = fi1···ip dx ∧ · · · ∧ dx ,

i1,...ip=1

ij ∗ where fi1...ip is a collection of functions and each dx is basis element of T M. The Sobolev norm is then deﬁned analogously to the Sobolev norm of vector ﬁelds:

dim M X X X k kαk s = D fi1···ip 2 . H L (Uj ) j i1···ip=1 0≤|k|≤s

We also deﬁne the Sobolev inner product as in (A.1) without coordinates:

Z X k k (α, β)Hs = ∇ α, ∇ β dµ 0≤|k|≤s M where

k k X k k k k ∇ α, ∇ β = (sgnπ) gx ∇ α1, ∇ βπ(1) · ... · gx ∇ αp, ∇ βπ(p) . π

133 APPENDIX B

SYMPLECTIC VECTOR SPCAES AND MANIFOLDS

B.1 Symplectic Manifolds

Our main reference for this section is McDuﬀ and Salamon, [M-S]. A Sym- plectic structure on a smooth manifold M is a non-degenerate, closed two form

2 ω ∈ Ω (M). That is, each tangent space (TxM, ωx) is a Symplectic vector space. The manifold is necessarily of even dimensional and the n-fold wedge product

ω ∧ ... ∧ ω deﬁnes an volume form on M. The sphere S2 is a Symplectic manifold with Symplectic form

ωx(u, v) = hx, u × vi ,

2 for u, v ∈ TxS . Where h·, ·i is the euclidean inner product induced from the am- bient space and × is the cross product of vectors. The non-degeneracy condition is purely algebraic and means that there is an isomorphism between the tangent and cotangent bundles of M:

ω] : TM → T ∗M

u 7→ iuω = ω(x, ·).

134 The closedness of ω is a geometric condition.

Deﬁnition B.1.1. An almost complex structure on a manifold M is a smooth ﬁeld of complex structures on the tangent spaces:

x 7→ Jx : TxM → TxM

2 Jx = −I.

The pair (M,J) is called an almost complex manifold.

Let (M, ω) be a Symplectic manifold. An almost complex structure J on M is called compatible if the assignment

x 7→ gx : TxM × TxM → R

gx(u, v) = ωx(u, Jv) is a Riemannian metric on M. The triple (ω, g, J) is called a compatible triple when g(·, ·) = ω(·,J·).

Proposition B.1.2. Let (M, ω) be a Symplectic manifold and g a Riemannian metric on M. Then there exists a canonical almost complex structure J on M which is compatible.

135 B.2 Symplectomorphisms

A Symplectomorphism of a Symplectic manifold (M, ω) is a diﬀeomorphism η which preserves the Symplectic form

η∗ω = ω.

The group of Symplectomorphisms of M will be denoted Dω(M). The non-degeneracy of ω gives us a one-to-one correspondence between vector ﬁelds and one-forms via the map ω]. A vector ﬁeld X on M is called Symplectic

] if the one-form iX ω = ω (X) is closed.

Proposition B.2.1. Let M be a closed manifold. If t 7→ η(t) is a smooth family of diﬀeomorphisms generated by a family of vector ﬁelds X(t) via

∂tη(t) = X(t) ◦ η(t), η(0) = Id,

then η(t) ∈ Dω(M) for every t if and only if X(t) is Symplectic for every t. Moreover, if X, Y are Symplectic then [X,Y ], the Lie bracket of vector ﬁelds, is Symplectic.

Proof. Then ﬁrst statement follows from

∗ ∗ ∗ ∂tη(t) ω = η(t) iX(t)dω + diX(t)ω = η(t) diX(t)ω, where we have used Cartan’s formula for the Lie derivative and that ω is closed. Now let X and Y be Symplectic vector ﬁelds with corresponding ﬂows η(t) and γ(t). Then

∗ [X,Y ] = LY X = ∂t|t=0γ(t) X

136 so that

i[X,Y ]ω = ∂t|t=0iγ(t)∗X ω = LY (iX ω) = diY iX ω = dω(X,Y ).

For any smooth function H : M → R the vector ﬁeld XH : M → TM deﬁned by the identity

iXH ω = dH is called the Hamiltonian vector ﬁeld associated to the Hamiltonian function H.

If M is closed, the vector ﬁeld XH generated a smooth 1-parameter group of diﬀeomorphisms ηH (t) satisfying

∂tηH (t) = XH ◦ ηH (t), ηH (0) = Id.

This is called the Hamiltonian ﬂow associated to H. The identity

dH(XH ) = (iXH ω)(XH ) = ω(XH ,XH ) = 0

show that the vector ﬁeld XH is tangent to the level sets H = const. of H. A smooth function F ∈ C∞(M) is constant along the orbits of the ﬂow of H if and only if the Poisson bracket

{F,H} = ω(XF ,XH ) = dF (XH ) vanishes. Because ω is closed, The Poisson bracket deﬁnes a Lie algebra structure on the space of smooth functions on M.

137 Proposition B.2.2. Let (M, ω) be a Symplectic manifold.

(i) Wherever deﬁned, the Hamiltonian ﬂow ηH (t) is a Symplectomorphism, which is tangent to the level surfaces of H.

(ii) For every Hamiltonian function H : M → R and every Symplectomor-

∗ phism η ∈ Dω(M) we have XH◦η = η XH .

(iii) The Lie bracket of two Hamiltonian vector ﬁelds XF and XG is the Hamil- tonian vector ﬁeld [XF ,XG] = X{F,G}.

Proof. Statement (i) was proved in A.4.1. Statement (ii) follows from the identity

∗ ∗ ∗ iXH◦η ω = d(H ◦ η) = η dH = η iXH ω = iη XH ω.

To prove statement (iii), we have

∗ [XF ,XG] = −∂t|t=0ηF (t) XG = −∂t|t=0XG◦ηF (t).

Hence

i[XF ,XG]ω = −∂t|t=0d (G ◦ ηF (t)) = −d∂t|t=0G ◦ ηF (t) = d {F,G}

The shows that the Hamiltonian vector fields form a Lie subalgebra of Sym- plectic vector fields. The map X 7→ XH is a surjective Lie algebra homomorphism from the Lie algebra of smooth functions on M with the Poisson bracket to the Lie algebra of Hamiltonian vector fields. The kernel of this homormorphism consists of constant functions.

138 APPENDIX C

FREDHOLM OPERATORS

C.1 Compact Operators

The reader is referred to Schechter, [Sc], for details on this appendix. Let X and Y be normed vector spaces. A linear operator K from X to Y is called compact if the domain of K is X and for every sequence {xn} ⊂ X such that kxnkX ≤ C, the sequence {Kxn} has a subsequence which converges in Y . The set of all compact operators from X to Y is denoted K(X,Y ).

A compact operator is clearly a bounded linear operator. The sum of two compact operators is compact and the product of a compact operator and a scalar is compact. Hence K(X,Y ) is a subspace of the space of bounded linear operators from X to Y : L(X,Y ).

Theorem C.1.1. Let X be a normed vector space and Y a Banach space. Then, K(X,Y ) is a closed, two-sided ideal in L(X,Y ) If K ∈ K(X,Y ) then K∗ ∈ K(Y ∗,X∗).

We make frequent use of the following Theorem.

Theorem C.1.2. Let A be a bounded linear operator on a Hilbert space H and suppose the estimate

kxkH . kAxkH + kKxkH (C.1)

139 is satisﬁed for any x ∈ H, and some compact operator K ∈ K(H). Then A has closed range and ﬁnite dimensional kernel.

Proof. Let Axn → y in H. We need an x such that Ax = y. Let L = ker A. If the distance between xn and L remains bounded take zn = xnmodL, kznkHσ ≤ 2a < ∞; Then Az = Ax → y. Passing to a subsequence, z0 = Kz → w in n n nk nk

H and then, applying the estimate (C.1) to znk − zmk , implies that the sequence zn is Cauchy so zn → z and Az = y. If the distance between xn and L tends to inﬁnity we can assume that the distance is greater than two for all n. Pick u = x modL such that dist(x ,L) ≤ ku k ≤ dist(x ,L) + 1 and set w = un . n n n n n n kunk 1 Now 2 ≤ dist(wn,L) and since kwnk = 1 we can choose a subsequence converging σ−1 in H (by the Rellich Lemma). Also, T wn → 0 so that the estimate (C.1)

σ applied to wm − wnimplies wn is a Cauchy sequence. Thus wn → w ∈ H with

1 2 ≤ dist(wn,L) and Ω(t)w = 0, which is a contradiction. Thus this latter case is impossible and the claim is proved. To prove ﬁnite dimensionality of the kernel,

σ let wn be an H sequence on the unit sphere in the kernel of Ω(t) so that

kwnkHσ . kKwnkH .

Now this estimate implies wn must contain a convergent subsequence which implies that the unit sphere in the kernel of A is locally compact. Consequently, the kernel is ﬁnite dimensional.

We now record some further useful properties on the spectrum of compact operators

Deﬁnition C.1.3. Let A be a bounded linear operator on H. The resolvent set,

140 denoted by ρ(A), is deﬁned by

ρ(A) = {λ ∈ R :(A − λI) is bijective from H to H} .

The spectrum, denoted by σ(A), is the complement of the resolvent set, i.e. σ(A) =

R \ ρ(A). A real number λ is said to be an eigenvalue of A if

ker (A − λI) 6= {0} with corresponding eigenspace ker (A − λI).

Theorem C.1.4. Let H be a separable Hilbert space and K ∈ K(H) a compact operator. Then we have: 1. 0 ∈ σ(K) 2. σ(K) \{0} consists entirely on non-zero eigenvalues of K. 3. One of the following holds: (a) σ(K) = {0}, (b) σ(K) \{0} is a ﬁnite set, (c) σ(K) \{0} is a sequence converging to 0. 4. If K is also self-adjoint then there exists an orthonormal basis of H consisting of eigenvectors of K.

C.2 Fredholm Operators

Let X and Y be Banach spaces. An operator A ∈ L(X,Y ) is said to be a Fredholm operator from X to Y if

1. dim ker A is ﬁnite.

2. R(A) is closed in Y

3. dim co ker A is ﬁnite.

141 The set of Fredholm operators will be denoted by Φ(X,Y ). If Y = X and K ∈ K(X) then by Theorem A.2.4 I − K is a Fredholm operator. The index of a Fredholm operator is deﬁned by

i(A) = dim ker A − dim co ker A.

For a compact operator K ∈ K(X) the index of the operator I − K is zero, by Theorem A.2.4. The next Theorem gives a very useful criterion for an operator to be Fredholm

Theorem C.2.1. Let A ∈ L(X,Y ) and assume that there are operators A1,

A2 ∈ L(X,Y ), K1 ∈ K(X), K2 ∈ K(Y ) such that

1. A1A = I − K1 on X

2. A2A = I − K2 on Y . Then A ∈ Φ(X,Y ).

The index of a Fredholm operator is also deﬁned for products of Fredholm operators:

Theorem C.2.2. If A ∈ Φ(X,Y ) and B ∈ Φ(Y,Z), then BA ∈ ⊕(X,Z) and

i(BA) = i(A) + i(B).

In view of Theorems C.2.3 and C.3.1, if K ∈ K(X), then A = I − K is a Fredholm operator. If we add any compact operator to A, it remains a Fredholm operator. This is also true if we add a compact operator to any Fredholm operator.

Theorem C.2.3. If A ∈ Φ(X,Y ) and K ∈ K(X,Y ), then A + K ∈ Φ(X,Y ) and

i(A + K) = i(A).

142 In particular, it follows that the index of any Fredholm operator of the form A = I + K is zero. More generally, if A is invertible then B = A + K, for some compact operator K, is a Fredholm operator of index zero. A Fredholm operator has index zero if and only if it is the sum of an invertible operator and a compact operator. One may relax the above restrictions somewhat, and consider a slightly more general operator. An operator A is semi-Fredholm if the range of A is closed and at least one of the other two conditions hold. Unbounded semi-Fredholm operators were treated by Kato in [K2] and [K3] where it is shown that the index of a semi-Fredholm operator is well-deﬁned. Denote the set of semi-Fredholm operators between two Banach spaces X and

Y , whose kernel is ﬁnite dimensional, by Φ+(X,Y ).

Theorem C.2.4. Suppose A is a bounded linear operator from X to Y . Then

A ∈ Φ+(X,Y ) if and only if there is a semi-norm |·| compact relative to the norm of X such that kxk ≤ C kAxk + |x| , x ∈ X.

Observe that this Theorem is analogous to Theorem C.2.4 and the estimate characterizes semi-Fredholm operators with ﬁnite dimensional kernel. If we denote by A∗, the adjoint of a semi-Fredholm operator A, then A is semi-Fredholm with

∗ ∗ ∗ ﬁnite dimensional cokernel if A ∈ Φ+(Y ,X ). Deﬁne the set Φ−(X,Y ) to be

∗ ∗ ∗ those bounded linear operators A with A ∈ Φ+(Y ,X ). We now collect some properties of semi-Fredholm operators which are analogous to the Fredholm case:

Proposition C.2.5. 1. An operator A is Fredholm if and only if A ∈ Φ+(X,Y )

∗ ∗ ∗ and A ∈ Φ+(Y ,X ).

2. If A ∈ Φ+(X,Y ) and B ∈ Φ+(Y,Z) then BA ∈ Φ+(X,Z).

143 3. If A ∈ Φ+(X,Y ) and K ∈ K(X,Y ), then A + K ∈ Φ+(X,Y ).

4. If A ∈ Φ−(X,Y ) and K ∈ K(X,Y ), then A + K ∈ Φ−(X,Y ).

144 APPENDIX D

SPECTRAL INTEGRALS

D.1 Spectral Measures

Let B(R) be the Borel σ-algebra of subsets of R, and let H be a Hilbert space with inner product h·, ·i.

Deﬁnition D.1.1. A spectral measure on B(R) is a mapping E of B(R) into the orthogonal projections on H such that

i) E(R) = I ∞ P∞ ii) E is countably additive, that is, E (∪n=1Mn) = n=1 E(Mn) for any sequence (Mn)n∈N of pairwise disjoint sets from B(R) whose union is also in B(R).

Remark D.1.2. Let E be a spectral measure on the σ-algebra B(R). Each vector x ∈ H gives rise to a scalar positive measure Ex on B(R) by

2 Ex(M) := kE(M)xk = hE(M)x, xi ,M ∈ U.

2 2 The measure Ex is ﬁnite, since Ex(R) = kE(R)xk = kxk . Let x, y ∈ H, then there is a complex measure Ex,y on B(R) given by Ex,y(M) = hE(M)x, yi.

The complex measure is a linear combination of four positive, ﬁnite measures Ez, z ∈ H, given by the polarization formula

1 E = (E − E + iE − iE ) . (D.1) x,y 4 x+y x−y x+iy x−iy

145 Using E,one may deﬁne spectral integrals of the form

Z Z I(f) = f(t) dE(t) = f dE Ω Ω of E-a.e. ﬁnite U-measurable functions f :Ω → C ∪ {∞} with respect to the spectral measure E. Roughly speaking, the idea of our construction is to deﬁne

Z I(χM ) ≡ χM (t) dE(t) := E(M) Ω

for characteristic functions χM of sets M ∈ U and to extend this deﬁnition by linearity and by taking limits to general measurable functions. We refer the reader to Schmudgen, [Sch], for details of this construction.

Theorem D.1.3. Let A be a bounded self-adjoint operator on a Hilbert space H.

Let J = [a, b] be a compact interval on R such that σ(A) ⊆ J . Then there exists a unique spectral measure E on the Borel σ-algebra B(J ) such that

Z A = λ dE(λ). J

If F is another spectral measure on B( ) such that A = R λ dF (λ), then we have R R E(M ∩ J ) = F (M) for all M ∈ B(R).

LetS ≡ S(R, B(R),EA) is the set of all EA-a.e. ﬁnite Borel functions f : R → C∪{∞}. For a function f ∈ S, we write f(A) for the spectral integral I(f). Then

Z f(A) = f(λ) dEA(λ)

146 is a normal operator on H with dense domain

Z 2 D(f(A)) = x ∈ H : |f(λ)| d hEA(λ)x, xi < ∞

Proposition D.1.4. Let f, g ∈ S, α, β ∈ C, x, y ∈ D(f(A)), and B ∈ B(H). Then we have: R (i) hf(A)x, yi = f(λ) d hEA(λ)x, yi . 2 R (ii) kf(A)xk = f(λ) d hEA(λ)x, xi

∞ (iii) f(A) is bounded if and only if f ∈ L (R,EA). In this case kf(A)xk = kfk∞ . ¯ ∗ (iv) f(A) = f(A) . In particular, f(A) is self-adjoint if f is real EA-a.e. on

R. (v) (αf + βg)(A) = αf(A) + βg(A). (vi) (fg)(A) = f(A)g(A)

P n P n (vii) p(A) = n αnA for any polynomial p(t) = n αnt ∈ C[t].

(viii) χM (A) = EA(M) for M ∈ B(R).

−1 1 (ix) If f(t) 6= 0 EA-a.e. on R, then f(A) is invertible, and f(A) = f (A).

(x) If f(t) ≥ 0 EA-a.e. on R then f(A) ≥ 0.

D.2 Stone’s Theorem

For a complete account of strongly continuous semigroups, the reader is referred to Engel and Nagel, [E-N]. A family T (t) of bounded linear operators on a Banach space X is called a strongly continuous semigroup if it satisﬁes

T (0) = I

147 T (t + s) = T (t)T (s)

lim T (t)x = x t→0 for all x ∈ X. Each strongly continuous semigroup has an inﬁnitesimal generator given by a, possibly unbounded, operator A satisfying

∂tT (t) = AT (t) = T (t)A.

A strongly continuous semigroup is said to be analytic if the map t 7→ T (t) P is analytic in a sector of C. For a comprehensive treatment of semigroups we refer the reader to Engel and Nagel, [E-N].

Theorem D.2.1. (Spectral mapping Theorem) If T (t) is an analytic semigroup on a Banach space then σ(T (t)) \{0} = etσ(A), with σ denoting the spectrum.

Let A be a self-adjoint operator with spectral measure EA. By the functional calculus of self-adjoint operators the operator eitA is deﬁned by

Z itA itλ e := e dE(λ), t ∈ R. R

We have the following characterization of semigroups satisfying T (t)T ∗(t) = I, which are called unitary semigroups.

Theorem D.2.2. (Stone’s Theorem) If U is a strongly continuous one-parameter unitary group on H, then there is a unique self-adjoint operator A on H such that

U(t) = eitA for t ∈ R. The operator iA is called the inﬁnitesimal generator of the

148 unitary group U.

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