Symplectic Homology

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MSc Mathematics

Master Thesis

Symplectic Homology

Author: Supervisor:

Robbert Evers

dr. O. Fabert

Examination date:

September 28, 2016

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Abstract

In this thesis, we explain C. Viterbo’s construction of symplectic homology and its connection to Floer homology, starting at the basics of Hamiltonian dynamics. Moreover, we discuss how the esteemed Weinstein conjecture can be shown for hypersurfaces in R2n using symplectic homology and explain how this gives a scheme for proving some other interesting cases of the Weinstein conjecture as well.

Title: Symplectic Homology

Author: Robbert Evers, robbert evers@hotmail.com, 10217037 Supervisor: dr. O. Fabert

Second Examiner: dr. R.R.J. Bocklandt Examination date: September 28, 2016 Korteweg-de Vries Institute for Mathematics University of Amsterdam

Science Park 105-107, 1098 XG Amsterdam http://kdvi.uva.nl

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Introduction

Like many subjects in mathematics, the roots of symplectic homology and the Weinstein conjecture lie in the study of classical mechanics, in which one studies the motion of bodies when influenced by a certain system of forces. As a very fundamental subject all throughout history, classical mechanics has been studied intensively. An important development in the mathematical study of classical mechanics was the reformulation known as Hamiltonian mechanics. This reformulation by Irish mathematician William R. Hamilton in 1833, improved the mathematical understanding of mechanical systems. Another reformulation of classical mechanics is given by Lagrangian mechanics, on which Hamiltonian mechanics is originally based. One advantage of studying a system in the formulation of Hamilton is that it can easily be generalized to abstract symplectic manifolds.

A big part of the study of mechanical systems is the search for its periodic orbits. Besides being mathematically interesting, questions about such periodic orbits naturally appear in many problems arising from classical mechanics. Examples include orbits of particles, but also of satellites, planets and other astronomical objects. In the language of symplectic geometry this corresponds to searching for periodic solutions of a first order differential equation. If one considers a time-independent Hamiltonian, it is not hard to see that any periodic Hamiltonian orbit is completely contained in an energy hypersurface, a result which we will discuss in Proposition 1.14. For this reason, the study of Hamiltonian dynamics in symplectic manifolds focuses on periodic orbits on energy hypersurfaces.

Among many others, Paul H. Rabinowitz and Alan Weinstein have been studying such periodic orbits. In 1978, Rabinowitz [23] showed the existence of periodic orbits on star shaped hypersurfaces in R2n. Inspired by this result, Weinstein soon after conjectured that such an existence result would always hold on energy hypersurfaces in a symplectic manifold that satisfy certain topological conditions. This conjecture has lead to many developments in symplectic geometry, and showed important connections with analysis and topology.

Although having been heavily studied, the conjecture of Weinstein remains a math-ematically challenging and still unsolved problem in its full generality. It has however been shown to hold in several interesting cases. A classical result is the proof by C. Viterbo in 1987 [29], who showed that the Weinstein conjecture holds for hypersurfaces in R2n. Another example is the 1993 article by H. Hofer [14] who proved that in di-mension three, overtwisted manifolds carry a contractible Reeb orbit. Furthermore, he showed that the same holds true for manifolds of dimension three that are equivalent to the sphere S3 or have non-trivial second homotopy class. In 2007, C. Taubes [28] was even able to show that the Weinstein conjecture holds for all three dimensional

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manifolds, by making use of Seiberg-Witten Floer theory.

Originally, symplectic homology was introduced by A. Floer and H. Hofer in 1992 [11]. Even though it was presented for different purposes, in 1999 C. Viterbo [30] adapted this idea to prove existence results on periodic orbits and more specifically, the Weinstein conjecture. In this thesis we will be studying Viterbo’s construction of symplectic ho-mology and its application on the Weinstein conjecture. Moreover, we will discuss some more results that have been found.

In Chapter 1 we will discuss some background theory in Hamiltonian dynamics, and explain the fundamental idea of Floer homology on which the construction of symplectic homology is based. Although plenty can be discussed about this, the purpose of this explanation is rather to illustrate the concepts that will be used later on, than to give a detailed exposition of the theory.

Chapter 2 gives the statements we will eventually attempt to prove using symplectic homology. However, the Weinstein conjecture in the standard symplectic space can also be shown by a very accessible argument using symplectic capacity. In this chapter we will discuss this proof comprehensively.

Throughout Chapter 3 we will give a detailed construction of symplectic homology, also including the reasoning that is adapted from Floer homology. The purpose of this is to give a complete overview of the many methods used. In particular, it becomes clear how concepts from analysis and topology have deep connections with the geometric principles.

Having built the necessary machinery, in Chapter 4 we will finally be able to prove the Weinstein conjecture in the standard symplectic space by applying symplectic ho-mology. After the expansive discussion from the previous chapters, this should be a comprehensible argumentation.

Ultimately, we will discuss in Chapter 5 what a blueprint for a proof of the Weinstein conjecture by means of symplectic homology could look like. Moreover, we will give an overview of some interesting developments that resulted from this.

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Acknowledgments

First of all, I would like to express my gratitude to Oliver Fabert for the helpful discus-sions and enthusiastic supervising during the research period. On top of that, I got to enjoy his lectures through the whole of my masters education. Even though it was not my research field of choice, during this period he has stirred up an interest in symplectic geometry in me, ultimately leading to the subject of this thesis.

I am also grateful to Alan, Berrie and Lucy, for keeping me motivated and because they were happy to read and comment on some sections of the thesis. Lastly, I would like to thank my family and my scouting group for their motivating words and for giving me the space I needed.

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1 Hamiltonian dynamics

In classical mechanics, one often studies the dynamics of a mechanical system. Such a system models the motion of bodies through space and can be mathematically described by a C1-path q : [0, 1] → Rn. Problems in classical mechanics can be described by a C2-function L : R × Rn× Rn→ R called the Lagrangian

Definition 1.1. . The Lagrangian is defined by the property that q minimizes the action integral

IL(p) =

Z 1

0

L(t, p(t), ˙p(t)) dt as a function over the C1-path space between two fixed points.

One can check that if a path q : [0, 1] → Rn minimizes the action integral, then it satisfies the Euler-Lagrange equations

d dt ∂L ∂ ˙qk = ∂L ∂qk k ∈ {1, . . . , n}. (EL)

Another way to look at this, is to define a new variable pk:= ∂L/∂ ˙qkfor k ∈ {1, . . . , n}

and consider the Hamiltonian function H : R × Rn× Rn→ R corresponding to L defined by H(t, q, p) := n X k=1 pkq˙k− L(t, q, ˙q).

It is well-defined if the Legendre condition is fulfilled, meaning that the Lagrangian function has det_{∂ ˙}_q∂2L

k∂ ˙q`

6= 0 for each k, ` ∈ {1, . . . , n}. This way the Euler-Lagrange equations are equivalent to Hamilton’s equations

˙ qk= ∂H ∂pk , ˙pk= − ∂H ∂qk k ∈ {1, . . . , n}.

More precisely, a C1-path q : [0, 1] → Rn solves (EL) if and only if the path (q, p) : [0, 1] → R2n solves Hamilton’s equations, where p = ∂L/∂ ˙q. Substituting the definition of the Hamiltonian function in the action integral, we can express this completely in terms of H and we find that the (Hamiltonian) action integral AH of a path x = (q, p)

is A_H(x) = Z 1 0 n X k=1 pkq˙k− H(x) dt. (1.1)

A reason to consider Hamiltonian’s equations over the Euler-Lagrange equations, is that the former is a family of first order differential equation, while the latter actually

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is a family of second order differential equations and therefore probably more difficult. Moreover, a first order differential equation corresponds to the flow of a vector field, which is something we can easily extend to a more abstract setting like on an arbitrary manifold.

Throughout this thesis, we will be studying a symplectic manifold.1

Definition 1.2. If M is a smooth, 2n-dimensional manifold and ω ∈ Ω2(M ) is a closed volume form, which means that it satisfies dω = 0 and ωn 6= 0, then the pair (M, ω) is called a symplectic manifold . The 2-form ω is called the symplectic form.

Remark. The symplectic form is always bilinear and skew-symmetric since it is a differ-ential form. Furthermore, it is non-degenerate which means that at any point x ∈ M , the only tangent vector X ∈ TxM such that ω(X, Y ) = 0 for any other Y ∈ TxM is

X = 0.

Example 1.3. A symplectic manifold is by definition always even-dimensional, therefore we will sometimes write elements x ∈ M as x = (q, p) where q and p denote n-vectors. As an example of a symplectic manifold, one can consider the complex space Cn which is of course even dimensional. Clearly, the notation x = (q, p) corresponds to using the identification Cn ' R2n _{given by q + ip ∼ (q, p). With this we can define the}

corresponding symplectic form on R2n to be ω0 := Pnk=1dqk ∧ dpk. The symplectic

manifold (R2n, ω0) is called the standard symplectic manifold and ω0 is known as the

standard symplectic form. Actually, a symplectic manifold can be seen as the “complex space in manifold theory”. It always carries an almost complex structure, denoted by J . For example, on (R2n, ω0) one has the standard complex structure

J0 = 0 1n×n −1n×n 0 .

We will give some more details on almost complex structures below.

Remark. Sometimes we also assume that the symplectic form is exact , meaning that there exists a 1-form λ ∈ Ω1(M ) such that ω = dλ. The standard symplectic form from Example 1.3 is exact too, with

λ0= 1 2 n X k=1 (qkdpk− pkdqk). (1.2) Indeed, we have dλ0= 1 2 X (dq ∧ dp + q ∧ ddp − dp ∧ dq − p ∧ ddq) = 1 2 X (dq ∧ dp − dp ∧ dq) = 1 2 X 2dq ∧ dp =Xdq ∧ dp = ω0.

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Here the sums run over k = 1, . . . , n and we shortened pk and qk by just writing p

and q respectively. This is a notation we will use often when we discuss the standard symplectic space.

We will now discuss some terminology used in the studies of Hamiltonian dynamical systems on symplectic manifolds.

Definition 1.4. Given a Hamiltonian function H : M → R on a symplectic manifold (M, ω), its Hamiltonian vector field is defined as the vector field XH that satisfies dH =

iXHω.

One can check that this is well defined, that is, the vector field XH is unique.

Definition 1.5. The Hamiltonian flow is the flow φH_t : M → M defined by XH, so

d dtφ

H

t = XH ◦ φHt and φH0 = 1.

Given an initial point x(0) in M , the Hamiltonian flow defines a Hamiltonian orbit x : R → M with initial value x(0) via x(t) := φHt (x(0)).

Example 1.6. On the standard symplectic manifold (R2n, ω0) the Hamiltonian vector

field is given by XH(q, p) = (∂pH(q, p), −∂qH(q, p)). Indeed,

iXHω0 =

X

(∂pHdp − (−∂qH)dq) = dH.

A fixed point x = (q, p) of the time one flow φH₁ corresponds to a solution of Hamilton’s equations ( ˙ q = ∂H_∂p ˙ p = −∂H_∂q.

We would like to extend these equations to an arbitrary symplectic manifold. Note that they are equivalent to ˙x = J0∇H(x), where ∇H = (∂iH)i=1,...,2n is the the metric

gradient of H on R2n, and J0is the standard complex structure on R2n as stated before.

Note that if we write x = (q, p) then J0(x) = (p, −q) =

n

X

k=1

(pk∂qk− qk∂pk) . (1.3)

Similarly, one can define an almost complex structure on an arbitrary manifold M as a map J : T M → T M such that J2 _{= −1. Using the symplectic form and the complex}

structure, one can define a metric.

Definition 1.7. The metric h·, ·i on (M, ω) is defined by ω = −hJ ·, ·i.

If (ω, J, h·, ·i) is a triple that satisfies this definition, then we say that this is a symplectic structure on M . It is sometimes referred to as a compatible structure. Actually, any two entries in such a triple determine the third uniquely.

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Example 1.8. On the standard symplectic manifold (R2n, ω0) we have determined ω

and J , so the standard metric is already uniquely defined. Indeed, note that for x = (q, p) and y = (q0, p0) we have hx, yi₀ = ω0(J0(x), y) = X (dq(J0(x))dp(y) − dq(y)dp(J0(x))) =X pp0− q0(−q) =X(pp0+ qq0)

which agrees with the usual Euclidean structure on R2n.

Definition 1.9. The metric gradient is the vector ∇H that is uniquely determined by the equation dH = h∇H, ·i.

Using the definition of the Hamiltonian vector field, it follows that h∇H, ·i = dH = i_X_Hω = h−J XH, ·i,

which means that

XH = J ∇H. (1.4)

Since the Hamiltonian vector field is defined on arbitrary symplectic manifolds, we can thus extend Hamilton’s equations as

˙

x(t) = XH(x(t)). (HE)

This is the equation we will mainly be studying throughout this thesis.

The following properties of a Hamiltonian dynamical system are often useful. Proposition 1.10. Let H : M → R be a Hamiltonian function. Then

(i) H ◦ φH t = H,

(ii) (φH_t )∗ω = ω.

Both of these properties can be easily shown by calculating the differential with respect to the t-coordinate, which vanishes in both cases, and then use the fact that at time t = 0 the flow behaves like the identity.

Remark. Property (ii) means by definition that the Hamiltonian flow φH_t is a symplec-tomorphism, i.e., it leaves the symplectic form invariant.

Until now we have only considered a Hamiltonian system in which time had no influ-ence on the dynamics. In other words, we have only discussed an autonomous mechanical system. A system where time does influence the behaviour can also be described by a function.

Definition 1.11. A time-dependent or non-autonomous Hamiltonian is a function H : S1×M → R and the corresponding Hamiltonian vector field is defined as the Hamiltonian vector field of H(t, ·).

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The definitions as stated above can now be applied to the Hamiltonian vector field of such a non-autonomous Hamiltonian too. While property (ii) of the previous proposition remains true in this situation, as the symplectic form remains independent of the time variable, property (i) does not hold for Hamiltonians that depend on time.

Definition 1.12. Let H : M → R be a Hamiltonian function. When a 1-periodic solution x of Hamilton’s equations (HE) satisfies

det 1 − dφH1 (x(0)) 6= 0

we say that it is non-degenerate .

Definition 1.13. A submanifold of codimension 1 in M is called a hypersurface. In our situation, these are all submanifolds of dimension 2n − 1. An example of a hypersurface is an energy hypersurface or level set of a Hamiltonian H, which is a set

Σc= {x ∈ M : H(x) = c}

for c ∈ R.

Note that by property (i) of the previous proposition we have H(φH_t (Σc)) = H(Σc) = {c}.

Proposition 1.14. Let H : M → R and H0 : M → R be Hamiltonian functions. (i) The Hamiltonian flow always stays at the same level set, i.e., φH_t (Σc) ⊂ Σc.

(ii) If H and H0 define the same energy hypersurface Σ, then there exists a non-zero scaling function ρ : Σ → R \{0} such that

XH = ρXH0 on Σ.

We will now show that an orientable compact hypersurface always corresponds to a level set of some Hamiltonian in a neighborhood of it.

Lemma 1.15. Let Σ be a compact hypersurface in a symplectic manifold (M, ω) carrying some nowhere vanishing vector field. Then there exists a neighborhood U of Σ and a Hamiltonian function H : U → R such that Σ is the level set H−1(0).

Proof. We know that we can find a vector field X which is nowhere vanishing on Σ, and without loss of generality we may assume that it is normal, so everywhere perpendicular to T Σ. Let φX denote the flow of X. Then the map

Φ : Σ × (−ε, ε) → R2n (x, t) 7→ φX_t (x)

is smooth, and for ε > 0 small enough it is a diffeomorphism onto its image. Note that

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by definition of a flow. Since we also know that Φ is a diffeomorphism and Σ is compact, the set U := Φ(Σ × (−ε, ε)) is an open neighborhood of Σ. The projection

π : Σ × (−ε, ε) → (−ε, ε) now results in a smooth Hamiltonian

H := π ◦ Φ−1: U → (−ε, ε) ⊂ R . Lastly, by (1.5) we have

H−1(0) = Φ(Σ, 0) = Σ as desired.

We are interested in finding periodic solutions to Hamilton’s equations (HE). Note that by Proposition 1.14 (i) we know that if x(0) ∈ Σc, then the whole solution x maps

in Σc. As pointed out before, the 1-periodic solutions are in one-to-one correspondence

with the fixed points of the flow φH₁ . Therefore, we will be searching for 1-periodic orbits on M . In order to describe these without having to specify the flow map, one can consider the notion of closed characteristics. This is closely related to the following. Definition 1.16. Given a hypersurface Σ ⊂ M , its characteristic line bundle is defined at x ∈ M as

LΣ(x) = {ξ ∈ TxΣ | ωx(ξ, η) = 0 for all η ∈ TxΣ}.

Remark. (i) Note that if we consider a level set Σc of some Hamiltonian H, then

TxΣc= {ξ ∈ TxM | dH(x)ξ = 0}. Therefore,

ω(XH(x), η) = dH(x)η = 0

whenever η ∈ TxΣc. In particular, this means that XH(x) ∈ LΣc(x).

(ii) If x is a 1-periodic solution to Hamilton’s equations (HE) on M , so x : S1 → M satisfies ˙x = XH(x), then its image is a parametrization of the circle S1. Writing

U for the image, we know that the direction of the tangent space to U is given by the Hamiltonian vector field, so we have TxU = XH(x) ∈ LΣ(x) by the above.

Also, U ⊂ Σ by Proposition 1.14 (i). Since TxU ∈ LΣ(x) is true for any x ∈ M ,

we know that T U ⊂ LΣ and restricting this to U we see that T U = LΣ|U.

This example motivates the following definition.

Definition 1.17. A closed characteristic of a hypersurface Σ is some embedded circle U such that

T U = LΣ|U.

Remark. Note that this definition allows us to think of periodic orbits on a hypersurface without having to specify a Hamiltonian.

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1.1 Reeb dynamics

Again, consider a symplectic manifold (M, ω) together with a hypersurface Σ ⊂ M . By definition Σ itself is a manifold of dimension 2n − 1. We will consider the symplectic structure under the inclusion map j : Σ ,→ M . Note that j∗ω is still a closed form, as

d(j∗ω) = j∗dω = 0.

Nevertheless, since Σ is obviously not even dimensional, we cannot really say that j∗ω is a symplectic form on Σ. This does however give us an idea of what an odd dimensional variant of symplectic geometry should behave like. More formally, this is a concept known as contact geometry.2

Definition 1.18. A contact manifold is a pair (Σ, α) where Σ is a smooth 2n − 1-dimensional manifold and α ∈ Ω1(Σ) is a smooth 1-form such that

α ∧ (dα)n−16= 0 on Σ, (1.6)

so it is a volume form. A 1-form with this property is called a contact form.

Since a 1-form is locally linear, its kernel consists of a hyperplane on each fibre. There-fore, we can associate to a 1-form a hyperplane distribution. If a hyperplane distribution ξ corresponds to a 1-form α that satisfies (1.6), then we say that ξ is maximally non-integrable.

Remark. It may seem a bit strange to assign (1.6) as a property of ξ instead of α. However, if α satisfies (1.6) and defines the hyperplane distribution ξ, then any other 1-form that defines ξ will be of the form f α where f : Σ → R is a smooth, non-vanishing function. Furthermore,

(f α) ∧ (d(f α))n−1= (f α) ∧ (f dα + df ∧ α)n−1 = (f α) ∧ (f dα)n−1

= fn(α ∧ (dα)n−1) 6= 0 (1.7)

where the second equality is an application of α ∧ α = 0 on all of the terms in the expansion of (f dα + df ∧ α)n−1, except for (f dα)n−1. We conclude that if ξ is given as the kernel of some 1-form that satisfies (1.6), then any other 1-form that defines ξ satisfies it too. Therefore, it is a property of the hyperplane distribution ξ rather than it is of the specific choice of α.

In the specific case that Σ is a compact hypersurface, A. Weinstein [32] introduced a different, very useful, definition. He made use of what is called a Liouville vector field. Definition 1.19. A Liouville vector field on a symplectic manifold (M, ω) is a vector field Y on M with LYω = ω. A vector field is said to be transverse to Σ in a point x ∈ Σ

when it is not in TxΣ. We say that a compact, oriented hypersurface Σ ⊂ (M, ω) is of

contact type if there exists a Liouville vector field, defined on a neighborhood of Σ, that is everywhere transverse to Σ.

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Example 1.20. Consider the real unit sphere

S2n−1 = {x ∈ R2n | |x| = 1} ⊂ R2n.

The symplectic form on R2nwas given by ω0 =P dp∧dq. Since lines through 0 intersect

the circle centered at the origin perpendicularly, we choose a vector field Y (x) = c · x for some scalar c ∈ R which is definitely everywhere transverse to S2n−1. When we set c = 1₂ we find that iY (x)ω0 = 1 2ω0(x, ·) = 1 2 X (qdp − pdq) = λ0

where λ0 is the 1-form such that dλ0 = ω0 which we encountered in (1.2). Now from

Cartan’s formula we conclude that

L_{Y (x)}ω0 = diY (x)ω0− iY (x)dω0= diY (x)ω0 = ω0

which means that Y (x) = 1₂x is a Liouville vector field on the unit sphere S2n−1, and hence the sphere is of contact type in R2n. Note that this argument actually works for any star-shaped hypersurface in R2n that encloses the origin, since the radial projection of such a star-shaped hypersurface to the sphere is a diffeomorphism.

Figure 1.1:The Liouville vector field Y (x) = 1 2x for S

2n−1_.

Remark. Define on M the 1-form λ = iYω.

(i) As LYω = ω and dω = 0 we have

dλ = d(iYω) = LYω − iYdω = ω. (1.8)

So the symplectic form is exact. For this reason a symplectic manifold admitting a global Liouville vector field is always an exact symplectic manifold. Furthermore, this also means that for a contact type hypersurface the symplectic form can be considered locally exact.

(ii) In the situations above, since iYλ = ω(Y, Y ) = 0 and dλ = ω we also have

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Lemma 1.21. Let Σ be a smooth 2n − 1 dimensional manifold. Then there exists a symplectic manifold (M, ω) such that Σ ⊂ (M, ω) is a hypersurface of contact type if and only if Σ is a contact manifold.

Proof. If Σ is of contact type with Liouville vector field Y , then we set α := iYω since

Cartan’s formula shows that diYω = LYω = ω and therefore

d iYω ∧ (diYω)n−1 = d iYω ∧ ωn−1 = (diYω ∧ ωn−1) + 0 = ωn6= 0.

In particular, we can conclude that iYω ∧ (diYω)n−1 6= 0 too and hence α = iYω restricts

to a contact form on Σ.

In order to understand the other direction of the statement, assume that Σ is a contact manifold with contact form α, so α ∧ (dα)n−1 is a volume form on Σ. We want to find a symplectic manifold in which Σ is a codimension one submanifold, so a hypersurface. Therefore we consider the 2n-dimensional manifold that is a tubular neighborhood of Σ, thus for ε > 0 this is (−ε, ε) × Σ. The contact 1-form α can be extended to (−ε, ε) × Σ by (s, x) 7→ esα(x). This way, we make sure that (Σ, α) corresponds to the hypersurface {0} × Σ where e0_{α = α. The symplectic structure on (−ε, ε) × Σ will be defined as}

d(esα), which clearly satisfies d(d(esα)) = 0 since d2 = 0. By (1.7) we know that esα ∧ (d(esα))n−1 is also a volume form on Σ, and therefore

esds ∧ α ∧ (d(esα))n−1

is a volume form on (−ε, ε) × Σ too. Since

d(esα) = d(es) ∧ α + esdα = es(ds ∧ α + dα)

we see that

(d(esα))n= es(ds ∧ α + dα) ∧ (d(esα))n−16= 0.

We conclude that d(esα) is a symplectic form on (−ε, ε)×Σ and hence Σ is a hypersurface in ((−ε, ε) × Σ, d(esα)). It remains to show that there exists a Liouville vector field on Σ that is transverse to it. By construction, the s-direction is transverse to Σ. Therefore we set Y = _∂s∂ and we will show that LY(d(esα)) = d(esα). Using Cartan’s formula we

find that

L_Y(d(esα)) = diYd(esα) = diY(es(ds ∧ α + dα)) = d(esiY(ds ∧ α + dα)).

By construction α is independent of s, meaning that

iY(ds ∧ α + dα) = (iYds)α − ds ∧ iYα + iYdα = ds(∂/∂s)α + 0 = α

and therefore

L_Y(d(esα)) = d(esiY(ds ∧ α + dα)) = d(esα).

This shows that Y is a Liouville vector field on Σ. So it is a contact type hypersurface in ((−ε, ε) × Σ, d(esα)) as desired.

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Definition 1.22. The symplectic manifold ((−ε, ε) × Σ, d(esα)) from the construction we used in the second part of the proof above is known as the symplectization of the contact manifold (Σ, α).

Example 1.23. (i) Consider the real unit sphere from Example 1.20. There we found the Liouville vector field Y (x) = 1₂x on S2n−1. We also calculated that iYω0 = λ0.

From the proof above we know that this must be a contact form for S2n−1_.

(ii) On (R2n, ω0) one can equivalently define a hypersurface Σ to be of contact type by

requiring the existence of a 1-form α on Σ such that dα = ω0|Σ and α(ξ) 6= 0 for

any non-zero ξ ∈ LΣ. The first property is obvious in the example of (i), while the

second follows from non-degeneracy of ω0. Namely, for any 0 6= ξ ∈ LΣ we have

α(ξ) = iYω0(ξ) = ω0(ξ, Y ) 6= 0.

Using the contact form, we can define a unit vector field along Σ.

Definition 1.24. Let (Σ, α) be a contact manifold. The Reeb vector field of α is the unique vector field Rα such that

(i) iRαdα = 0 and

(ii) iRαα = 1.

The corresponding flow is known as the Reeb flow and an orbit of the Reeb flow is called a Reeb orbit.

Note that the first condition determines the direction of the vector field, where the second normalizes it. Also note that the Reeb vector field is certainly dependent on the choice of a contact 1-form α. For example, by multiplying α with a non-vanishing function the normalization condition can easily be changed, forcing the Reeb vector field to be rescaled too. Moreover, the same example could similarly conflict with the first condition.

Example 1.25. We continue with the example of the unit sphere S2n−1. In Example 1.23 we concluded that λ0 = 1₂P q dp − p dq is the contact 1-form and Y (x) = 1₂x the

Liouville vector field. Since dλ0= ω0 we have

iJ0Ydλ0= ω0(J0Y, ·) = −hJ

2

0Y, ·i0 = hY, ·i0

which is always zero since the Liouville vector field is everywhere transverse to Σ. So the Reeb vector field points in the direction perpendicular to the direction of the Liouville vector field. In order to obtain the second condition too, we might need to rescale the vectors. Note that

iJ0Yλ0= iJ0YiYω0 = ω0(Y, J0Y ) = −hJ0Y, J0Y i0 = −J

2

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For any x ∈ S2n−1 we have hY (x), Y (x)i0 = 1 2x, 1 2x 0 = 1 4|x| 2 ₌ 1 4.

So in order to fulfill the second condition, we need to rescale the vector field by 4 and we conclude that Rλ0 = 4J0Y = 2J0 is the Reeb vector field on S

2n−1_.

Since J0 corresponds to multiplication with i if we identify R2n ' Cn, the flow of the

Reeb vector field can be described by φR_tλ0(x) = e2itx. Indeed, φR₀λ0(x) = x and d

dtφ

R_λ0

t (x) = 2ie2itx = 2iφ R_λ0

t (x) = (Rλ0◦ φ

R_λ0

t )(x).

Therefore, the Reeb orbits are given by x(t) = φR_tλ0(x(0)). In particular, we see that for ` ∈ N we have

x(t) = e2itx(0) = e2i(t+`π)x(0) = x(t + `π) meaning that the action of the Reeb orbits is `π.

Later on, we will mainly consider a special type of almost complex structure when we are studying the symplectization of a contact manifold (Σ, α).

Definition 1.26. A time-independent almost complex structure J on the symplectiza-tion ((−ε, ε) × Σ, d(esα)) of (Σ, α) is called of contact type if it makes the Reeb vector field transverse to the Liouville vector field: Rα= J Y .

1.2 Floer theory

In this section we will describe the idea of Floer theory very concisely. More details can be found in the lecture notes of D. Salamon, see [24]. Moreover, we will encounter some of these arguments when we construct symplectic homology in section 3.

Let (M, ω) be a closed, symplectic manifold and H : S1× M → R a (time-dependent) Hamiltonian function. In Floer thoery, one aims to prove the Arnold conjecture. Basi-cally, it provides information about the number of 1-periodic Hamiltonian orbits depend-ing on the topological structure of the manifold. Denote by P (H) te set of 1-periodic Hamiltonian orbits of a Hamiltonian H, and denote the i-th homology group of M by Hi(M ). The Arnold conjecture states the following.

Conjecture 1.27. If (M, ω) is a closed, symplectic manifold and H : S1× M → R a time-dependent Hamiltonian such that all its 1-periodic orbits are non-degenerate, then #P (H) ≥P2n

i=0dim Hi(M ).

1.2.1 Morse homology

In the case of an autonomous Hamiltonian system, we have XH(x) = 0 if and only

if dH(x) = 0 by definition of XH. So every critical point of H corresponds to a

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dH(x) = 0}, we have #Crit(H) ≤ #P (H). Using Morse theory we will explain why

#Crit(H) ≥ P2n

i=0dim Hi(M ) if H is a Morse function. Morse theory itself is actually

a construction that works on arbitary closed manifolds, so for now we do not assume M to be symplectic. In particular, this means that we are free in choosing a Riemannian metric.

Definition 1.28. Let H : M → R be a smooth function. A critical point x of H is called non-degenerate when the Hessian of H is non-degenerate as a bilinear map on TxM × TxM . If all critical points of H are non-degenerate, then H is called a Morse

function. The Morse index of a non-degenerate critical point x is the dimension of a maximal subspace of TxM on which the Hessian is negative definite, and we denote it

by ind(x).

Let H : M → R be a Morse function. The moduli space of negative gradient trajecto-ries from x− to x+_{, where x}± _{are critical points of H, is defined as}

M(x–, x+) := {u : R → M | ∂su + ∇H(u) = 0, lim

s→±∞u(s) = x ±

}/ R .

Note that ∇H depends on the Riemannian metric. If the metric h·, ·i is chosen such that the moduli space is a smooth manifold of dimension dim M(x−, x+_{) = ind(x}−_{) −}

ind(x+) − 1, then the pair (H, h·, ·i) is called a Morse-Smale pair .

Remark. We have ind(x−) = dim Wu(x−), where Wu(x−) denotes the unstable man-ifold, and ind(x+) = 2n − dim Ws(x+), where Ws(x+) denotes the stable manifold. If these intersect transversally, then we can write the moduli space as M(x−, x+) = (Wu(x−) ∪ Ws(x+)) / R and hence dim M(x−, x+) = ind(x−) − ind(x+) − 1 as desired. Given a Morse-Smale pair, we can define its i-th Morse homology HMi(H, h·, ·i) as

the homology with chain groups

CMi(H) := Z2hx ∈ Crit(H) | ind(x) = ii

and boundary operator defined by

∂i : CMi(H) → CMi−1(H) x−7→ X x+_∈Crit(H) ind(x+)=i−1 #M(x− , x+) mod 2 · x+.

So the homology groups are given by

HMi(H, h·, ·i) :=

ker ∂i

im ∂i−1

.

By definition, the boundary operator counts the number of negative gradient trajectories from x− to x+ modulo 2. Of course, this is only well-defined if there are only finitely many such trajectories. In order to see this, note that we take the sum over values x+ whose index satisfies ind(x−) − ind(x+) = 1, which means that dim M(x−, x+) = 0

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since we are considering a Morse-Smale pair, so it is a discrete set. One can also show that is is compact, so that #M(x−, x+) < ∞ indeed holds and the boundary operator is well-defined. The fact that it actually is a boundary operator, so ∂2= 0, follows from the property that if ind(x−) − ind(x+) = 2, then the moduli space can be compactified to a 1-dimensional manifold with boundary given by

∂M(x−, x+) = [

x0_∈Crit(H)

ind(x0)=ind(x±)∓1

M(x−, x0) × M(x0, x+),

meaning that the limit of a sequence of trajectories can at most “break” at one point x0. In particular, when applying the boundary operator twice we obtain

∂i−1(∂i(x−)) = X ind(x0)=i−1 #M(x−_{, x}0₎ _{mod 2 · ∂}i−1_(x0₎ = X ind(x0)=i−1 #M(x− , x0) mod 2 · X ind(x+)=i−2 #M(x0_{, x}+₎ _{mod 2 · x}+ = X ind(x+)=i−2 #∂M(x−_{, x}+₎ _{mod 2 · x}+_{= 0.}

The important property of the Morse homology, is that it actually is isomorphic to the usual homology. That is, HMi(H, h·, ·i) ' Hi(M ). We can conclude that

2n X i=0 dim Hi(M ) = 2n X i=0 dim HMi(H, h·, ·i) ≤ 2n X i=0 dim CMi(H) = #Crit(H).

The inequality we applied here is known as the (weak) Morse inequality, see (the proof of) Corollary 1.13 in [24]. Combining this result with our findings in the beginning of this section, the Arnold conjecture follows when considering a time-independent Hamiltonian system.

1.2.2 Floer homology

Using the idea of the construction in the autonomous case, we would like to prove the Arnold Conjecture for time-dependent Hamiltonians H : S1 × M → R. To this end, we will define a functional whose critical points correspond to 1-periodic orbits (like the action integral we saw in (1.1)) and make a construction that is similar to the one we did in Morse theory applied to this functional. Instead of looking at arbitary orbits, we will consider the space of smooth loops on M and try to find the ones that correspond to a Hamiltonian orbit. So define the loop space LM := C∞(S1, M ) as the space of smooth maps x : S1 → M . The reason why this is useful, is that the loop space is actually a manifold itself. It is an infinite dimensional Frechet manifold with tangent spaces given by TxLM = C∞(S1, x∗T M ).

On this manifold, consider the action 1-form αH ∈ Ω1(LM ) defined as

αH(x)ξ :=

Z

S1

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Note that αH(x) vanishes exactly on the 1-periodic Hamiltonian orbits, since ˙x(t) −

XH(x(t)) = 0 for any such orbit and ω is a non-degenerate 2-form. It is a closed 1-form

on LM , but not exact. We can however expand LM to the universal cover gLM by attaching to each loop x ∈ LM a disc, i.e., it consists of tuples (x, [u]) where [u] is a homotopy class of discs whose boundary is x. We can obviously lift αH to gLM by setting

αH(x, [u]) := αH(x) for each (x, [u]) ∈ gLM . We will abbreviate the elements of gLM by

˜ x.

Loop x Disc u

Figure 1.2: One obtains ˜x by attaching a disc u to the loop x.

Definition 1.29. On the universal cover of LM , define the symplectic action functional AH : gLM → R by AH(˜x) := Z D2 u∗ω − Z S1 H(x(t)) dt where D2 is the unit disc, and S1 the unit sphere.

If ω = dλ is exact, then Stokes’ Theorem shows that Z D2 u∗ω = Z D2 u∗(dλ) = Z D2 d(u∗λ) = Z S1 u∗λ

in correspondence with the symplectic action functional as we defined it at the start, see (1.1). The reason for defining the universal cover of the loop space and a corresponding action functional, is that the action 1-form actually is exact on gLM . Indeed, one can explicitly calculate that αH = dAH. In particular, since αH vanishes on the 1-periodic

orbits, these correspond to critical points of the action functional.

Due to this correspondence between 1-periodic orbits of Hamilton’s equations and critical points of the action functional, we will focus on the (negative) gradient flow lines of AH. In particular, we need the universal cover gLM to possess a metric. To this

purpose, consider the L2_{-metric on g}_{LM , which we define by}

hξ0, ξ1iL2 :=

Z

S1

hξ0(t), ξ1(t)i dt for ξ0, ξ1∈ TxLMg

where h·, ·i is the standard metric on M .

Now the L2-gradient of AH is by definition the vector ∇L2A_H such that for any

˜

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using the action 1-form, dAH(˜x)(ξ) = αH(x)(ξ) = Z S1 ω( ˙x(t) − XH(x(t)), ξ(t)) dt = Z S1 −hJ (x(t)) ˙x(t) − J (x(t))XH(x(t)), ξ(t)i dt = Z S1 h−J (x(t)) ˙x(t) − ∇H(x(t)), ξ(t)i dt. We conclude that the L2-gradient of AH is given by

∇_L2A_H(˜x)(t) = −J (x(t)) ˙x(t) − ∇H(x(t)).

Hence, the equation for a gradient flow line u : R ×S1 → LM , which is ∂su = ∇L2A_H(u),

becomes

∂su + J (u)∂tu + ∇H(u) = 0.

This is called the Floer equation and we denote it by ∂J,H(u) = 0. It might be noteworthy

that this is actually just a Cauchy-Riemann equation, perturbed by the ∇H-term. We will use this later on, when we apply ideas from complex analysis to it.

By using the Floer equation, we can try to find gradient flow lines for the action functional directly on M instead of LM . The reason for doing this, is that the loop space is an infinite dimensional manifold, which makes solving such equations a lot harder. However, the Floer equation is not necessarily well-defined on the L2-space since it involves derivatives. One solution to this problem might be to look at a Sobolev space instead, and thus use weak derivatives. Another solution is the idea that Floer came up with. His observation was that one could just forget about the theoretical background with gradient flows where the equation originated and instead just study the solutions by looking at a moduli space.

Definition 1.30. For ˜x−, ˜x+ ∈ gLM , the moduli space of (unparameterized) gradient trajectories from ˜x− to ˜x+ is defined as

M(˜x–, ˜x+) :=    u : R ×S1→ M | ∂J,H(u) = 0, lims→±∞u(s, ·) = ˜x± [u−] = [u+# u]    / R .

Here [u+#u] is the disc obtained by glueing the disc u+ to the cylinder u, see Figure 1.3. Moreover, we factor out the R-action (r, u) 7→ u(r + ·, ·) ∈ M(x–, x+), which just means that the gradient trajectories will be unparameterized.

Consider the set B(˜x−, ˜x+) := u : R ×S1→ M | lims→±∞u(s, ·) = ˜x ± [u−] = [u+_{# u]} .

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Disc u+

u

Figure 1.3:One obtains [u+_{#u] by attaching a disc u}+ _{to the cylinder u.}

One could look at the moduli space as the kernel of ∂J,H as an operator on B(˜x−, ˜x+),

divided out to R. We will be looking into this in more detail in section 3.4 when we define symplectic homology. Although the setting is a little different, the argumentation is completely analogous.

It is a more extensive argumentation to show that, for a generic choice of Hamiltonian H and almost-complex structure J , the moduli space is actually compact and that the dimension equals the difference of the Conley-Zehnder indices minus 1 after dividing out the R-action. We will show these properties when we discuss symplectic homology in section 3. Now, we see that if the difference of the Conley-Zehnder indices actually equals 1, then the moduli space is a 0-dimensional submanifold of B(˜x−, ˜x+). In particular, it then is a finite set.

Using this we will build a homology called the Floer homology based off the construc-tion of Morse homology. So we define the Floer homology chain groups over a field K as

CFi(H, J ) := Khx ∈ P (H) : cz(x) = i mod N i

where N is the minimal Chern number. If K = Z2 then the boundary operator can be

defined equally as we did it in Morse homology, namely ∂F,i : CFi(H, J ) → CFi−1(H, J )

x− 7→ X

x+_{∈P (H)}

cz(x+)=i−1

#M(x−, x+) · x+.

This is well-defined because by construction the difference of the Conley-Zehnder indices is 1 and we already concluded that then the moduli space is always a finite set. The proof that (∂F)2= 0 is actually completely analogous to the Morse homology case again. Now

the Floer homology HFi(H, J ) over Z2 is defined as the homology of the chain complex

(CFi(H, J ), ∂F). So

HFi(H, J ) :=

ker ∂F,i

im ∂F,i−1

.

In the case where K is some other field, one can still define a similar boundary operator by choosing an appropriate orientation for the moduli spaces.

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For now, the last thing that is in our interest regarding Floer homology is to know how the Arnold conjecture follows for a time-dependent Hamiltonian too. This is a consequence of the following important result.

Theorem 1.31. For any generic choice of pairs (H, J ) and (H0, J0), there is a linear isomorphism Φ : HFi(H, J ) → HFi(H0, J0) of the Floer homology groups over a field K.

In particular, given any dependent Hamiltonian H we can consider a time-indepent Hamiltonian H0 whose Floer homology group is isomorphic to the Floer homol-ogy group of H. Since we applied the same construction as we did in Morse homolhomol-ogy, the Floer homology group of H0 is actually isomorphic to its Morse homology group. Since we already showed that the Arnold conjecture holds for this case, the time-dependent case boils down to be somewhat of a consequence of this!

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2 The Weinstein Conjecture

The objective of this section is to give a broad overview of the proof of the Weinstein Conjecture for the standard symplectic space using symplectic homology. This result was first shown by Claude Viterbo [29] in 1987. The original conjecture was stated by Alan Weinstein [32] as follows:

Conjecture 2.1. (Weinstein, 1979) Any compact hypersurface Σ ⊂ (M, ω) of

con-tact type for which H1(Σ; R) = 0 carries a closed characteristic.

The Arnold conjecture and the Weinstein conjecture are in a sense dual, at least in a finite dimensional setting. Basically, the Arnold conjecture seeks periodic orbits with a fixed period, but puts no restrictions on the hypersurface it can be found on. On the other hand, the Weinstein conjecture seeks periodic orbits without restrictions on the period, but considers a fixed hypersurface. Note that a hypersurface can be defined as the level set of some Hamiltonian, but the choice of Hamiltonian does not affect the result of the Weinstein conjecture, as long as it defines the correct hypersurface. In particular, one can rescale these in such a way that an orbit of a certain period will be a 1-periodic orbit in the rescaled hypersurface. Therefore, one might be able to use the methods used for the Arnold conjecture to find such periodic orbits in the Weinstein conjecture.

In case one considers the standard symplectic space (R2n, ω0) it will appear that

requiring H1(Σ; R) to be zero is unnecessary. Therefore, we aim to prove the following. Theorem 2.2. Any compact hypersurface of contact type in (R2n, ω0) carries a closed

characteristic.

More precisely, the proof will show that periodic orbits exist for a specific vector field, namely the Reeb vector field. Therefore, we will prove Theorem 2.2 by showing that any compact hypersurface of contact type in (R2n, ω0) admits a periodic Reeb orbit. There

are various ways to prove this, and each proof has its own benefit. We will first consider a proof using capacities on R2n, which is quite a well-known argument. For a second proof we shall utilize the theory of symplectic homology. This is interesting because symplectic homology is a more abstract concept which can be applied in more general situations than the capacities.

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2.1 Proof of the Weinstein conjecture using capacities

In this section, we will cover the argument as shown by Helmut Hofer and Eduard Zehnder in [16]. Let U ⊂ R2n open. A Hamiltonian H : U → R will be called a special Hamiltonian on U if

1. there is a compact subset K ⊂ U such that H|_{U \K} ≡ m(H) is constant,

2. there is a open subset O ⊂ U such that H|O≡ 0,

3. 0 ≤ H(x) ≤ m(H) for all x ∈ U

and such a special Hamiltonian will be admissible if furthermore

4. XH has no fast periodic orbit, i.e. each non-constant orbit has a period that is

greater than 1.

Here m(H) is also called the oscillation of H. The Hofer-Zehnder capacity of an open set U ⊂ R2n is defined as chz(U ) := sup{m(H) : H is admissible}. It is a symplectic

capacity, in the sense that it is

1. invariant, which means that c_hz(φ(U )) = c_hz(U ) for any symplectomorphism φ,

2. monotone, meaning that chz(V ) ≤ chz(U ) when V ⊂ U ,

3. conform, meaning that c_hz(λU ) = λ2_c

hz(U ) for λ ∈ R, and

4. normalizing, so c_hz(B2n(0, 1)) = π = chz(Z2n(1))).

Note that if c_hz(U ) < ∞, then for any special Hamiltonian H we can find some T ∈ N such that m(T H) = T · m(H) > c_hz(U ). Looking at the definition of the capacity we find that T H cannot be an admissible Hamiltonian. Since it is clearly still a special Hamiltonian, we conclude that it must have a fast periodic orbit. So T H is a Hamiltonian that has a periodic orbit of period at most 1. In particular, H must have a periodic orbit whose period is at most T . We have thus showed the following.

Lemma 2.3. If U ⊂ R2n has finite capacity c_hz(U ) < ∞, then every special Hamiltonian has a non-constant periodic orbit.

We can strengthen this result as follows.

Proposition 2.4. (Hofer-Zehnder) Consider a Hamiltonian function H : R2n→ R

and let c ∈ R be a real number such that Σ := H−1(c) is a regular hypersurface. If there is an open neighborhood Σ ⊂ U ⊂ R2n such that c_hz(U ) < ∞, then there exists a sequence {c_j}_j∈N of energy values converging to c such that each hypersurface Σj := H−1(cj)

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Proof. In order to prove this, we will consider a particular family of special Hamiltonians whose hypersurfaces carry some periodic orbit. Then we will show that we can manip-ulate this orbit to find a periodic orbit on a hypersurface for our original Hamiltonian.

Without loss of generality, we will assume that c = 0 (otherwise one can consider H − c). For j ∈ N choose εj > 0 sufficiently small and such that εj ↓ 0 when j → ∞.

Define a smooth function fj : R → R by

fj(s) :=

1 if |s| ≥ εj

0 if |s| ≤ εj

2

in such a way that fj is strictly decreasing on (−εj, −ε₂j) and strictly increasing on

(εj 2, εj). See Figure 2.1. 0 εj −εj εj 2 −εj 2 1 fj

Figure 2.1: We consider smooth functions fj of a special form.

Now we define a family of Hamiltonian functions Hj := fj◦ H. By the definition of fj

it is clear that these are examples of special Hamiltonian functions on U as we defined them above with K = H−1([−εj, εj]) and O = H−1((−ε₂j,ε₂j)).

Now we apply Lemma 2.3 to obtain a non-constant periodic orbit xj of the special

Hamiltonian Hj. So ˙xj = XHj(xj). We can relate the Hamiltonian vector fields of H

and Hj by XHj = (f 0 j◦ H)XH because ω0(XHj, ·) = dHj = d(fj ◦ H) = (f 0 j◦ H)dH = ω0((fj0 ◦ H)XH, ·).

Using this we see that d dtH(xj(t)) = dH(xj(t)) ◦ ˙xj(t) = dH(xj(t)) ◦ XHj(xj(t)) = dH(xj(t)) ◦ [fj0(H(xj(t))) · XH(xj(t))] = f_j0(H(xj(t))) · ω0(XH(xj(t)), XH(xj(t))) | {z } =0 = 0.

So H(xj(t)) is constant in t. Now set H(xj(t)) = cj. Since xj itself was a non-constant

orbit, we have XHj(xj(t)) 6= 0 and XH(xj(t)) 6= 0. So by the relation XHj = (f

0

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we find that f_j0(cj) = fj0(H(xj(t))) 6= 0 which means that ε₂j < |cj| < εj by definition of

fj (as its derivative should be zero otherwise). Writing α := fj0(cj) 6= 0 we let

yj(t) := xj

t α

.

Then yj(αt) = xj(αt_α) = xj(t) so yj is a non-constant periodic orbit too whose period is

α times the period of xj. Furthermore,

˙ yj(t) = 1 αx˙j t α = 1 αXHj xj t α = 1 α(f 0 j◦ H) xj t α XH xj t α = 1 αf 0 (cj) · XH(yj(t)) = 1 αα · XH(yj(t)) = XH(yj(t)).

Thus yj(t) is a periodic solution for our original Hamiltonian H too. So we find a

sequence {cj}j∈N such that each hypersurface Σj carries a periodic orbit. Since εj ↓ 0 as

j → ∞ and |cj| < εj, we have cj → 0 as j → ∞. Thus {cj}j∈N is the desired sequence

of energy values.

Proposition 2.4 tells us that if a regular hypersurface Σ has a neighborhood of finite capacity, then we can find periodic orbits on hypersurfaces arbitrarily close to Σ. One also says that on Σ periodic orbits almost exist. In order to prove the Weinstein con-jecture, we will now assume that Σ is a compact contact type hypersurface in R2n and show that the periodic orbits we’ve found can be used to find a periodic orbit on Σ itself. More specifically, we’ll flow solutions on neighboring hypersurfaces to a solution on Σ.

Let ψtbe the flow corresponding to the Liouville vector field Y on Σ. Since LYω0= ω0

it is clear that

d dt(ψt)

∗_ω

0= (ψt)∗LYω0= (ψt)∗ω0.

Thus (ψt)∗ω0 = cetω0 for some constant c. Using the definition of a flow, we have ψ0 = 1

which means that c = 1 and therefore

(ψt)∗ω0= etω0. (2.1)

Because Σ ⊂ R2n is compact there is some r > 0 such that the ball B2n(0, r) contains

Σ. Since

c_hz(B2n(0, r)) = chz(rB2n(0, 1)) = r

2_{π < ∞}

we can apply Proposition 2.4 to Σ in order to find a family {Σj}j∈N of regular

hyper-surfaces with energies cj each carrying a periodic orbit. Let y be such a periodic orbit

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Claim 2.5. In this situation x := ψ−cj(y) is a closed characteristic for Σ itself.

Proof. For any ξ ∈ TxΣ we need to show that ω( ˙x, ξ) = 0 so that ˙x ∈ LΣ(x). Since y is

a periodic orbit on Σj we already know that ˙y ∈ LΣj(y). We have ˙x = d(ψ−cj)y( ˙y) and

writing ξ = d(ψ−cj ◦ ψcj)xξ = d(ψ−cj)ψ_cj(x)◦ d(ψcj)x ξ = d(ψ−cj)y d(ψcj)xξ we find that ω( ˙x, ξ) = ω d(ψ−cj)y( ˙y) , d(ψ−cj)y d(ψcj)xξ = (ψ−cj) ∗ yω0 y, d(ψ˙ cj)xξ = e−cj _{· ω} 0 y, d(ψ˙ cj)xξ = 0.

The latter equality is an immediate consequence of that fact that we have ˙y ∈ LΣj(y)

and d(ψcj)xξ ∈ TyΣj.

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3 Symplectic homology

We will now proceed with another proof of the Weinstein conjecture using symplectic ho-mology. Basically, symplectic homology is a way to learn about the topological structure of a space using the knowledge we have about Hamiltonian functions. On closed mani-folds, this is precisely what one does in Floer homology. We will find that for a suitable family of Hamiltonian functions, we can still use the arguments from the construction of Floer homology even when the manifold is not closed.1

3.1 Admissible Hamiltonians

Consider the following example. Whenever Σ is a compact hypersurface in (R2n, ω0),

its compactness ensures that it encloses a bounded area W . We will take W closed (so Σ ⊂ W is its boundary). This way we obtain a compact, symplectic manifold as a submanifold of (R2n, ω0). If Σ is of contact type, we will say that W is a Liouville

domain. In more generality, we can consider arbitrary symplectic manifolds.

W

Σ

Figure 3.1:A compact hypersurface in R2n gives rise to a compact submanifold.

Definition 3.1. A compact, symplectic manifold (M, ω) with boundary is a Liouville domain if ∂M is of contact type and its Liouville vector field points outwards.

Example 3.2. An easy example of a Liouville domain is the unit disc D2n. In Example 1.20 we already saw that its boundary S2n−1 _{is of contact type and the Liouville vector}

field Y (x) = 1₂x does clearly point outward (see Figure 1.1).

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Let (M, ω) be a Liouville domain. Similar to the symplectization of ∂M , the Liouville vector field Y gives us a sense of direction on a neighborhood (−ε, ε) × ∂M . Extending this direction we find an oriented set R ×∂M , whose orientation is given by the Liouville vector field. The R-coordinate is called the radial coordinate and the direction given by the Liouville vector field is called the radial direction. We can turn R ×∂M into a symplectic manifold by extending the symplectic form on ∂M in the radial direction by the same technique we applied in the symplectization. To see this, note that the contact 1-form α on ∂M is given by α = λ|∂M, where λ is the 1-form iYω so that we have dλ = ω

on ∂M by (1.8) (the remark following Example 1.20). Denote by s the radial coordinate and let ψt be the flow of the Liouville vector field Y again. By definition this means

that Y = _∂s∂.

∂M

Y

s = 0

Figure 3.2:Following the radial direction. The map

Ψ : (−ε, 0] × ∂M ,→ M (s, x) 7→ ψs(x)

is an embedding such that Ψ(0, x) = x for all x ∈ ∂M . Moreover, for (s, x) ∈ (−ε, 0]×∂M it satisfies

Ψ∗λ(s, x) = λ(ψs(x))

= (ψs)∗λ(x)

= esλ(x) = esα(x).

Here the third equality follows from an equivalent calculation as we have done in (2.1) and using that the pull-back and exterior derivative commute. This shows that

Ψ∗ω = d(Ψ∗λ) = d(esα).

Thus d(esα) extends the symplectic form to R ×∂M , as then the flow map Ψ becomes a symplectomorphism. In the proof of Lemma 1.21 we showed that this does indeed define a symplectic form on the symplectization. Remark that the 1-form esα does indeed equal α = λ on ∂M as s = 0 there.

In conclusion, a Liouville domain (M, ω) can be extended to an open, symplectic manifold ( cM ,ω) by attaching a cylindrical end to the boundary following the radial_b

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direction. We will write the radial coordinate as r := es in order to keep the notation more concise. So

c

M := M ∪ ([1, ∞) × ∂M ).

See Figure 3.3. The symplectic form is defined by ω|_bM = ω andω|b [0,∞)×∂M = d(rα).

Definition 3.3. If (M, ω) is a Liouville domain, then the open, symplectic manifold ( cM ,ω) is called the completion of M ._b

1. ∂M M Cylindrical end 2. W Σ Radial directions

Figure 3.3: (1) Attaching a cylindrical end. (2) Viewed in R2n, here c_{W = R}2n.

Let Σ be a compact hypersurface of contact type (e.g. ∂M ) with Reeb vector field Rα. Assume that we are interested in finding a k-periodic orbit of Rα for some k ∈ R+.

Then this corresponds to a 1-periodic solution for the vector field k · Rα. Using a Floer

homology type argument we may be able to find such 1-periodic orbits, thus it is in our interest to consider a Hamiltonian whose vector field is a rescaled version of the Reeb vector field. To this purpose we consider a set of admissible Hamiltonians.

Definition 3.4. A Hamiltonian H : cM → R is called admissible if (i) H is a C2-small Morse function on M ,

(ii) on [1, ∞) × ∂M the Hamiltonian depends only on the radial coordinate, so there is some function h : [1, ∞) → R such that H(r, p) = h(r) for all (r, p) ∈ [1, ∞) × ∂M , (iii) this function h is convex,

(iv) ∃R ∈ R≥1 such that on [R, ∞) × ∂M it holds that H(r, p) = h(r) is linear in r.

Note that on [1, ∞) × ∂M an admissible Hamiltonian H depends only on the radial coordinate, which means that every point in [1, ∞) corresponds to a level set for H. Since h is linear, we have h0 ≡ k for some k ∈ R and hence ∇H(r, p) = h0(r) · Y = k · Y

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M

H

1 R Radial direction Linear

C2-small on M

Figure 3.4: An admissible Hamiltonian.

points in the direction of the Liouville vector field Y . This slope k of h0 is called the slope at infinity .

We assume that we have a complex structure J that is of contact type, which means that it satisfies J Y = Rα as in Definition 1.26. Using equality (1.4) from section 1, we

know that the Hamiltonian vector field XH is given by J ∇H. In particular, we see that

on [R, ∞) we have

XH = J ∇H = k · J Y = k · Rα. (3.1)

So, as we explained above, on this part a 1-periodic orbit for XH corresponds to a

k-periodic solution for Rα. As a Hamiltonian function is assumed to be smooth, in the

interval [1, R) the slope has to increase smoothly and therefore admit all values in [0, k). If we want to find all the orbits we need to be able to detect any period, so we need to let k tend to infinity. From now on we denote by Hk an admissible Hamiltonian whose

slope at infinity is k ∈ R.

Radial direction

Figure 3.5: Increasing the slope.

If we choose an increasing divergent sequence {kn}n∈N such that for any n ∈ N there

exists no kn-periodic orbit for the Reeb vector field, then this corresponds to a sequence

of admissible Hamiltonians Hkn where the slopes at infinity increase. It is possible to

choose such a sequence since the set of such possible periods of Reeb orbits is countable for a generic choice of contact 1-form. The reason to assume that the slopes kn are not

equal to the period of a Reeb orbit, is to make sure that there won’t occur 1-periodic orbits in the infinite cylindrical end. Moreover, when we have such a sequence then the

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1-periodic orbits of Hkn all correspond to Reeb orbits of different periods. In particular,

since the sequence diverges, we find all the different periodic Reeb orbits when we take a limit.

We will mimic the arguments from Floer homology in order to find 1-periodic orbits for XHk. Recall from section 1.2.2 that we defined the Floer homology chain groups by

CFi(Hk, J ) = Z2hx ∈ P (Hk) : cz(x) = ii

where cz is the Conley-Zehnder index. The boundary operator was given by ∂F : CFi(Hk, J ) → CFi−1(Hk, J )

x−7→ X

x+∈P (Hk)

cz(x+)=i−1 mod N

# M(x–, x+) ·x+

where M(x–, x+) is the moduli space corresponding to the Floer equation for the Hamil-tonian Hkand N is the minimal Chern number. The Floer homology groups HFi(Hk, J )

have been defined as the homology of the chain complex (CFi(Hk, J ), ∂F). Of course,

we need to check if this is still well-defined in the new situation.

The admissible Hamiltonians Hk form a directed set, by setting Hk1 Hk2 whenever

Hk1 ≤ Hk2 in the cylindrical end, or equivalently if it holds that k1 ≤ k2. Therefore we

can take a direct limit and we define the i-th symplectic homology group by SHi(M ) := lim_−→HFi(Hk, J ).

We will explain this construction in more detail in section 3.4.

Even though we defined the boundary operator the same way as we did in Floer homology, it is not clear that the corresponding moduli spaces are finite again. In Floer homology this is the case because the moduli spaces are compact and discrete, but since

c

M is by definition unbounded this might not be true in our situation. In order to define the boundary operator we thus need to make sure that the moduli spaces are compact and 0-dimensional again.

3.2 Moduli space for an admissible Hamiltonian

Let (M, ω) be a Liouville domain and assume that the symplectic form is exact, so there is a λ ∈ Ω1(M ) such that ω = dλ. Let J be a contact type complex structure on M and denote by h·, ·i = ω(·, J ·) the induced Riemannian metric. Let cM be its completion, and consider an admissible Hamiltonian H on cM .

Analogous to the Floer theory situation, instead of looking at the space of solutions to Hamilton’s equations and trying to find a periodic one, we will look at the space of loops and search for loops that correspond to a solution. So, we write L cM := C∞(S1, cM ) for the loop space again. On the loop space we define the symplectic action functional

AH(x) = Z S1 x∗λ − Z S1 H(x(t)) dt x ∈ L cM .

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Note that, unlike before, we can actually define the action functional on the loop space immediately, and do not need to consider the universal cover, since we have assumed that the symplectic form is exact. Now we have

dAH(x)(ξ) = Z S1 x∗(dλ)(ξ) − Z S1 dH(ξ) dt = Z S1 x∗ω(∂t)(ξ) dt − Z S1 ω(XH(x(t)), ξ(t)) dt = Z S1 ω( ˙x(t), ξ(t)) − ω(XH(x(t)), ξ(t)) dt = Z S1 ω( ˙x(t) − XH(x(t)), ξ(t)) dt.

Thus we obtain dAH = αH again, and therefore the critical points of AH correspond to

periodic solutions for Hamilton’s equation (HE).

Remark. Consider a 1-periodic Hamiltonian orbit x on a hypersurface {r}×∂M ⊂ cM for some r > R. Since H is assumed to be admissible, by definition we have λ = erα there, H(x) = h(er_{) is linear with slope h}0_(er_{) and it only depends on the radial coordinate r.}

Therefore A_H(x) = Z S1 x∗λ − Z S1 H(x(t)) dt = er Z S1 x∗α − h(er) = er Z S1 α( ˙x) − h(er) = er Z S1 α(XH) − h(er) = erh0(er) Z S1 α(Rα) − h(er) = erh0(er) − h(er),

where the last equality follows from the definition of the Reeb vector field since we have iRαα = 1.

In order to be able to find gradient flow lines, we again use the L2-gradient on the loop space and perform an analogous calculation to find that the L2-gradient of the action functional is given by ∇L2A_H(x)(t) = −J (x(t)) ˙x(t) − ∇H(x(t)) again. In particular,

the gradient flow lines correspond to solutions of the Floer equation

∂J,H(u) = ∂su + J (u)∂tu + ∇H(u) = 0. (FE)

Hence the moduli space of gradient trajectories on cM can be defined by M(x–, x+) := u : R ×S1 → cM smooth | ∂J,H(u) = 0, lims→±∞u(s, ·) = x± .

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Now consider the space B(x−, x+) :=

u : R ×S1 → cM a smooth map | lim

s→±∞u(s, ·) = x ±

of smooth x−, x+-paths. The moduli space obviously is a subspace of B(x−, x+), and locally at u ∈ M(x–, x+), its tangent space is given by C₀∞(R ×S1, u∗T cM ). In particular, we can interpret

M(x–, x+) = ∂_J,H−1(0) ⊂ B(x−, x+).

In order to determine the dimension of the moduli space, we will linearize the Floer operator ∂J,H. That is, for a fixed trajectory u : R ×S1 → M in B(x−, x+) we will

construct a linear operator Du: C0∞(u∗T cM ) → C∞(u∗T cM ) such that

TuM(x–, x+) = ker(Du) ⊂ C0∞(u ∗

T cM ) = TuB(x−, x+).

To this end we will make use of the Levi-Civita connection on T cM → cM , which gives rise to a connection ∇ on u∗T M → R ×S1 _{for u ∈ B(x}−_{, x}+_).2

Proposition 3.5. The linearization of ∂J,H at u ∈ B(x−, x+) is

Duξ = ∇sξ − ∇ξ∇L2A_H(u)

for ξ ∈ C₀∞(u∗T cM ).

Proof. We consider functions ur∈ B(x−, x+) such that _drd|r=0 ur= ξ, namely

ur := expu(rξ).

Also, let Φu(ξ) : TuM → Tc _exp

u(ξ)M be the parallel transport along uc r for the Levi-Civita

connection. Then the linearization is given by Duξ = d dr r=0 Φu(rξ)−1◦ ∂J,H(ur) = d dr r=0 Φu(rξ)−1(∂sur+ J (ur)∂tur+ ∇H(ur)) = ∇sξ + d dr r=0 Φu(rξ)−1(J (ur)∂tur) | {z } (T ) +∇ξ∇H(u).

For the middle term we calculate (T ) = J (u) · d dr r=0 Φu(rξ)−1(∂tur) + d dr r=0 Φu(rξ)−1(J (ur)) · ∂tu = J (u)∇tξ + (∇ξJ )(u) · ∂tu. 2

A great exposition of connections and the Levi-Civita connection is given in chapters 11 and 12 in the lecture notes by E. Lerman [18].

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Therefore, we can conclude that

Duξ = ∇sξ + J (u)∇tξ + (∇ξJ )(u) · ∂tu + ∇ξ∇H(u)

= ∇sξ + ∇ξ(J (u) · ∂tu) + ∇ξ∇H(u)

= ∇sξ − ∇ξ∇L2A_H(u)

as desired.

In order to simplify upcoming calculations we would like to identify Du with some

linear operator on the standard symplectic space. A way to do this is to use a unitary trivialization.

Definition 3.6. Let (M, ω) be a smooth manifold and consider a smooth vector bundle E → M that carries a compatible triple (ω, J, h·, ·i). A unitary trivialization of E is a smooth mapping

M × R2n→ E

(p, x) 7→ Ψ(p)x such that

Ψ∗J = J0, Ψ∗ω = ω0 and Ψ∗h·, ·i = h·, ·i0.

Given u ∈ B(x−, x+) we consider a unitary trivialization of u∗T cM , which we denote by Ψu : R ×S1× R2n→ u∗T cM , that identifies the compatible triple (ω, J, h·, ·i) with the

standard symplectic structure (ω0, J0, h·, ·i0). We apply this unitary trivialization to Du.

Proposition 3.7. Under the unitary trivialization, the linearized operator D := Ψ−1_u ◦ Du◦ Ψu is of the form

D : C₀∞(R ×S1, R2n) → C₀∞(R ×S1, R2n) (3.2)

ξ 7→ ∂sξ + J0∂tξ + Sξ

where S : R ×S1 → R2n×2n is given by

S = Ψ−1_u (∇sΨu+ J (u)∇tΨu+ ∇ΨuJ (u)∂tu − ∇Ψu∇H(u)) .

Remark. If we let s → ±∞, then we obtain S±: S1→ R2n×2nwhich are two S1-families of matrices corresponding to x±∈ P (H).

We shall now show that D is a Fredholm operator after an extension to a completed Sobolev space W1,p(R ×S1, R2n).

3.2.1 Fredholm theory

Before proving that D is a Fredholm operator, let us revisit some basics of the theory.3

3

The following is based on the lecture notes by D. Salamond [24] and handwritten lecture notes by O. Fabert.

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Definition 3.8. Let X and Y be two Banach spaces and T : X → Y a bounded linear operator. The cokernel of T is the quotient Y / im(T ) and is denoted by coker(T ).

(A) The operator T is called a Fredholm operator when (i) im(T ) ⊂ Y is closed,

(ii) dim ker(T ) < ∞ and (iii) dim coker(T ) < ∞.

(B) If only (i) and (ii) hold, then T is called a semi-Fredholm operator .

(C) If U ⊂ X is open and f : U → Y is of class Ck for some k ∈ N, then f is called Fredholm whenever df (x) : X → Y is Fredholm for any x ∈ U .

Remark. Definition (C) tells us that it suffices to show that the linearization of ∂J,H is

Fredholm. Therefore, we will show that the map D from (3.2) is a Fredholm operator. Definition 3.9. A linear operator K : X → Y between Banach spaces X and Y is called compact if K(BX) ⊂ Y is relatively compact as a subset of Y . Here BX denotes

the closed unit ball in X.

Remark. The image of a bounded sequence under a compact operator always has a convergent subsequence, as it can be rescaled to a sequence contained in the unit ball which has a compact image under compact operators.

Proposition 3.10. Let X, Y and Z be Banach spaces. Assume that we are given a bounded linear operator T : X → Y between Banach spaces. If there exists a compact linear operator K : X → Z and there is a c ≥ 0 such that for every x ∈ X we have

kxkX ≤ c (kT xkY + kKxkZ) , (3.3)

then T : X → Y is semi-Fredholm.

Proof. We will start this proof by showing what consequence the assumed bound (3.3) has.

Claim 3.11. Consider a sequence {xj} ⊂ X. If both {T xj} and {Kxj} converge, then

{x_j} is a convergent sequence.

Proof. Let ε > 0. Because of the convergence we can find an N ∈ N such that if n, m ≥ N then kT xn− T xmkY < _2cε and kKxn− KxmkZ < _2cε. From (3.3) it now

follows that kxn− xmkX < ε, so {xj} is a Cauchy sequence. Since X is a Banach space

this means that it converges.

In order to prove that T is semi-Fredholm, we check properties (i) and (ii) from the definition, starting with (ii). To prove this, we show that the set BX∩ker(T ) is compact,

as ker(T ) must be finite dimensional then.

To this end, let {xj} ⊂ BX ∩ ker(T ) be a sequence. Then {T xj} ≡ {0} is obviously

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we find a subsequence {xj0} such that {Kx_j0} is convergent. By the claim {x_j0} itself

must be convergent, which means that BX ∩ ker(T ) is compact as desired. This proves

(ii).

To prove (i) we will assume that ker(T ) = {0}, which doesn’t violate the generality.4 Now consider a point y ∈ im(T ). Then there is a sequence {xj} in X so that {T xj}

converges to y. We’ll show that there exists an x ∈ X such that y = T x meaning that y ∈ im(T ), so im(T ) is closed.

In order to show the contrary, assume that {xj} is unbounded. So there exists some

subsequence {xj0} with kx_j0k_X → ∞. Since this also assures that kx_j0k_X 6= 0 we can

consider the normalized sequence {ξj0} defined by

ξj0 := xj 0

kxj0k_X.

Clearly we have kξj0k_X = 1, so in particular it is a bounded sequence which means that

there exists a subsequence {ξj00} such that {Kξ_j00} converges. Since kx_j0k_X → ∞ while

T xj0 → y we also have T ξ_j0 → 0. By the claim this means that {ξ_j00} itself must converge

too, say to a limit ξ ∈ X. Then ξ satisfies kξkX = 1 and kT ξkY = 0 which contradicts

the assumption that ker(T ) = {0}. We conclude that {xj} must be a bounded sequence.

Again using the compactness of K, we can find a new subsequence {xj0} for which

{Kx_j0} converges. By the claim we conclude that this subsequence itself converges to

some limit x ∈ X that satisfies T x = limj0T x_j0 = y as we wanted to show. Hence (i)

holds and thus T is a semi-Fredholm operator.

We shall now consider the operator D from (3.2). In order to apply the theory above we need D to be an operator on a Banach space, which it now is not. Therefore, we need to complete the space C₀∞(R ×S1, R2n). For 1 < p < ∞ the Lp-norm

kξkp =

Z

R ×S1

|ξ|p 1/p

of measurable functions ξ : R ×S1→ R2n _{defines the complete L}p_-space

Lp(R ×S1, R2n) = {ξ | kξkp < ∞}/ ∼

where ξ ∼ η when kξ − ηkp= 0. However, our operator D is not well-defined on Lp since

it involves first-order derivatives which are not guaranteed to exist for a function from Lp. One can solve this problem by considering a Sobolev space instead. This means that we will be taking weak derivatives now, even though we will keep the notation the same.5 For 1 < p < ∞, the operator

kξk_1,p = Z R ×S1 |ξ|p_{+ |∂} sξ|p+ |∂tξ|p 1/p

4_{This is because we could otherwise consider the quotient space X}

0 = X/ ker(T ) with quotient map

π : X → X/ ker(T ). Indeed, then im(T ◦ π−1) = T (X + ker(T )) = im(T ) and ker(T ◦ π−1) = {0}.

Symplectic Homology

MSc Mathematics

Master Thesis