Equivariant symplectic homology and multiple closed Reeb orbits

Equivariant symplectic homology and multiple closed Reeb orbits

Abstract.

We study the existence of multiple closed Reeb orbits on some contact manifolds by means of -equivariant symplectic homology and the index iteration formula. We prove that a certain class of contact manifolds which admit displaceable exact contact embeddings, a certain class of prequantization bundles, and Brieskorn spheres have multiple closed Reeb orbits.

Key words and phrases:
-equivariant symplectic homology, Index iteration formula, closed Reeb orbit

1. Introduction

After Weinstein’s famous conjecture [Wei79], the existence problem of a closed Reeb orbit has been extensively studied. It is natural to study the multiplicity of (simple) closed Reeb orbits in contact manifolds that are known to have one. This has been addressed in [HWZ98, HWZ03, GHHM12] for tight 3-spheres and in [HT09, CGH12] for general contact 3-manifolds. To the authors knowledge, there are few multiplicity results for general higher dimensional contact manifolds but there are a number of theorems [EL80, BLM85, EH87, LZ02, WHL07, Wan11] for pinched or convex hypersurfaces in .

In the present paper we study the multiplicity problem of closed Reeb orbits for nondegenerate contact manifolds which admit displaceable exact contact embeddings, prequantization bundles, and Brieskorn spheres. Our approach is based on -equivariant symplectic homology and the index iteration formula. Although we only treat those three cases, we expect that our method can apply for other contact manifolds for which the formulas of -equivariant symplectic homology (or contact homology) are nice in a sense that will be explained below.

An embedding of a contact manifold into a symplectic manifold is called an exact contact embedding if is bounding, for some 1-form , and there exists a contact form on such that and is exact. Throughout this paper we identify with and every contact manifold is assumed to be closed. We also tacitly assume that every manifold is connected. Here by bounding we mean that separates into two connected components of which one is relatively compact. We denote by the relatively compact domain. This embedding is said to be displaceable if there exists a function such that the associated Hamiltonian diffeomorphism displaces from itself, i.e. . A symplectic manifold is called convex at infinity if there exists an exhaustion of by compact sets with smooth boundaries such that , are contact forms.

The Reeb vector filed on is characterized by and . We recall that a closed Reeb orbit is nondegenerate if the linearized Poincaré return map associated to the orbit has no eigenvalue equal to 1. A contact from on is called nondegenerate if every closed Reeb orbit is nondegenerate.

Theorem A. Suppose that a closed contact manifold of dimension admits a displaceable exact contact embedding into which is convex at infinity and satisfies . Assume that at least one of the following conditions is satisfied:

  • for some

  • for all even degree

where is the relatively compact domain bounded by . Then there are at least two closed Reeb orbits contractible in for any nondegenerate contact form on such that is exact.

One may ask if there are more than two closed Reeb orbits when both conditions (i) and (ii) are fulfilled. This question does not seem to be easily answered in general. However the Conley-Zehnder index of closed Reeb orbits on 3-dimensional contact manifolds is special enough to answer this question and the precise statement is given below.

The following contact manifolds meet the condition (ii) in the theorem.

  • is a rational homology sphere;

  • is a -injective fillable 5-manifold;

  • is a Weinstein fillable -manifold;

  • is a subcritical Weinstein fillable -manifold.

It is worth pointing out that due to [FSvK12, Lemma 3.4]

if is displaceable in and in particular implies . Moreover this equation implies .

Question.

Does every (nondegenerate) subcritical Weinstein fillable closed contact manifold has two closed Reeb orbits? More generally, does every closed contact manifold admitting a displaceable exact contact embeddings possess two closed Reeb orbits?

We expect that the above question will be answered positively. There is no particular reason for conditions (i) and (ii) in Theorem A to be essential. We include some examples in the appendix which do not meet such conditions but have two closed Reeb orbits.

As we mentioned above, it turns out that every 3-dimensional closed contact manifold has two closed Reeb orbits [CGH12]. Moreover if a nondegenerate closed contact 3-manifold is not a lens space there are at least three closed Reeb orbits [HT09]. In the following we show that if a contact manifold in Theorem A is of dimension 3, we have at least closed Reeb orbits where denotes the third Betti number.

Corollary A. Suppose that a 3-dimensional closed contact manifold admits an exact contact embedding into which is convex at infinity and satisfies . If displaceable in , then for any nondegenerate contact form such that is exact,

Moreover, -many simple closed Reeb orbits are of Conley-Zehnder index 2. In particular, if is subcritical Weinstein,

Remark 1.1.

A closed Reeb orbit of Conley-Zehnder index 3 in Corollary 4.4 and -many simple closed Reeb orbits of Conley-Zehnder index 2 in the above corollary have the following nice property. There exist gradient flow lines of the symplectic action functional which connect such closed Reeb orbits with Morse critical points in . These gradient flow lines can be used to obtain finite energy planes. This will be discussed in the forthcoming paper [FK14]. For instance for subcritical Weinstein fillable contact 3-manifolds, using such finite energy planes, we are able to prove that if any closed Reeb orbit is linked with such , then the linking number is always positive.

As a matter of fact, the proof of Theorem A heavily relies on the facts that the positive part of -equivariant symplectic homology is periodic, i.e. for not small and that the positive part of -equivariant symplectic homology vanishes for low degrees (condition (ii) in Theorem A guarantees this). In other words, we can find more than one closed Reeb orbits if (the positive part of) the -equivariant symplectic homology of a fillable contact manifold is nice in such a sense. A certain class of prequantization bundles and Brieskorn spheres which are treated below have nice -equivariant symplectic homologies and thus, for them, we are able to find more than one closed Reeb orbit.

Let be a symplectic manifold with an integral symplectic form , i.e. . For each , there exists a corresponding prequantization bundle over with . Due to [BW58], such a prequantization bundle is a contact manifold with a connection 1-form . The following theorem proves the existence of two closed Reeb orbits for a certain class of prequantization bundles which naturally arise from the Donaldson’s construction, see Remark 4.6.

Theorem B. Let be a prequantization bundle over a simply connected closed integral symplectic manifold with and for some . Suppose that is primitive in and for some and that admits an exact contact embedding into with which is -injective. Then for a nondegenerate contact form , there are two closed Reeb orbits contractible in and hence in .

More generally, -orbibundles over symplectic orbifolds provide more examples of contact manifolds. In particular Brieskorn spheres, one of the simplest examples, are of our interest since the positive part of the equivariant symplectic homology (contact homology) was already computed in [Ust99]. For , we define

which is singular when . Then a 1-form on is a contact form. We call a Brieskorn manifold. When is odd and mod 8 and , is diffeomorphic to and called a Brieskorn sphere. As mentioned, Brieskorn manifolds are generalized example of prequantization bundles. Indeed, all Reeb flows of are periodic and thus Brieskorn manifolds can be interpreted as principal circle bundles over symplectic orbifolds. Furthermore a Brieskorn manifold is Weinstein fillable and in fact a filling symplectic manifold is with . We refer to [Gei08, Section 7.1] for detailed explanation about Brieskorn manifolds.

Theorem C. Brieskorn spheres with nondegenerate contact forms have two closed Reeb orbits.

Except 3-dimensional displaceable case, we only can find two closed Reeb orbits but we do not think that this lower bound is optimal. For instance, it is interesting to ask:

Question.

Can one find more than two closed Reeb orbits on Brieskorn spheres with nonperiodic contact forms?

Acknowledgments

I would like to thank Otto van Koert for fruitful discussion. His comments and suggestions led to Theorems B and C of the present paper. I also thank my advisor Urs Frauenfelder for consistent help. Many thanks to Peter Albers and Universität Münster for their warm hospitality. Finally, I thank the referee for careful reading of the manuscript and the valuable comments. This work is supported by the NRF grant (Nr. 2010-0007669) and by the SFB 878-Groups, Geometry, and Actions.

2. -equivariant symplectic homology

2.1. Borel type construction


-equivariant symplectic homology theory was first introduced in [Vit99]. Recently -equivariant symplectic homology theory was rigorously studied and written up in [BO09b, BO12b]. In the present paper following [Vit99, BO09a, BO09b] we use the Borel type construction of -equivariant (Morse-Bott) symplectic homology and refer to [BO12b] for other constructions, their equivalences, and applications.

Let be a contact manifold which admits an exact contact embedding into a symplectic manifold that is convex at infinity. Consider a nondegenerate contact from on such that is exact. Upto adding a compactly supported exact 1-form to we can assume that , see [CF09, page 253]. We denote by the bounded region of . A neighborhood of in can be trivialized by the Liouville flow as . The symplectic completion of is defined by

We denote by a primitive 1-form of which is on and on .

We choose an almost complex structure on which is compatible with and preserves the contact hyperplane field . We extend this on so that is invariant under the -action and and . Such a will be called admissible. Here is the Reeb vector field associated to and we denote by the flow of . The Hamiltonian vector field associated to a Hamiltonian function is defined by .

Since we have assumed that is nondegenerate, periods of closed Reeb orbits on form a discrete subset in . We define a family of -invariant admissible Hamiltonians , to have the following properties:

  • for ;

  • is Morse-Bott and invariant under the -action on given by for , ;

  • On , and is a -small Morse function;

  • On , for some strictly increasing function satisfying on for some ;

  • for some on .

With a family of admissible Hamiltonians , we define a family of action functionals , where denotes the space of contractible loops in , by

We note that this action functional is -invariant with respect to the following torus action on ,

That is , and thus the critical points set is -invariant as well. Here if and only if

There are two types of critical points of :

  • where and where is a critical point of the Morse function ;

  • where and where lying on levels , is a solution of

    (2.1)

    Here is the projection to the second factor.

The second type solutions correspond to closed Reeb orbits with period . They are transversally nondegenerate (see [BO09a, Lemma 3.3]), i.e.

We define the diagonal -action on by

Although we will divide out the diagonal -action on , we are still in a Morse-Bott situation due to the presence of another -action. See [Bou02, Fra04] for Floer homology in the Morse-Bott situation. Thus we choose an additional Morse-Bott function invariant under the diagonal -action such that is Morse. We denote an -family of critical points of containing by

Suppose that . We define the index by

where and stand for the Conley-Zehnder index and the Morse index respectively, see [BO09b, BO12a]. In particular if is of the first type, i.e. ,

In this case, is a function on invariant under the rotation and thus .

We also define a family of -invariant admissible compatible almost complex structures , , such that is an admissible compatible almost complex structure on and is -invariant, i.e. for . Together with a Riemannian metric on invariant under the diagonal -action, a metric on is defined by

A (negative) gradient flow line of with respect to the metric is a solution of

(2.2)

We denote the moduli space of gradient flow lines with cascades (gradient flow lines of ) from to for by

That is, such that

  • s are solutions of (2.2);

  • and satisfy

    where (resp. ) is the (un)stable set of a critical manifold of ;

  • and , satisfy

    where is the negative gradient flow of .

We divide out the -action on defined by shifting cascades in the -variable. Then we have the moduli space of gradient flow lines with unparametrized cascades denoted by

We note that solutions of (2.2) are equivariant under the diagonal -action, that is if solves (2.2), then so does . Since is invariant under the diagonal -action as well, the moduli space carries a free -action. We denote the quotient by

It turns out that this moduli space is a smooth manifold of dimension

for a generic . For the detailed transversality analysis we refer to [BO10]. We define the -equivariant chain group by the -vector space generated by -families of critical points of of -index .

The boundary operator is defined by

where by we mean a signed (via the coherent orientations) count of the number of the finite set . Then and thus we are able to define

Taking direct limits, the -equivariant symplectic homology of is defined by

As the notation indicates, the homology depends only on . Here the direct limit of with respect to the embedding is taken as follows. A Morse-Bott function extends to a Morse-Bott function for so that in a tubular neighborhood of in where is the normal coordinate. This induces a direct system over . In order to define the negative/positive part of -equivariant symplectic homology, we consider

where . That is, resp. is generated by type 1) resp. type 2) critical points of , see the property (v) of . Since the action values decrease along negative gradient flow lines, there exist associated boundary operators induced by , and hence we are able to define the negative/positive part of the -equivariant symplectic homology of .

2.2. Morse-Bott spectral sequence


This subsection is devoted to observe that bad orbits do not contribute to -equivariant symplectic homology which is certainly expected to be true in -equivariant theory. This is clearly true for contact homology and proofs of the present paper may become more transparent if we use contact homology. Nevertheless we use -equivariant symplectic homology since contact homology is still problematic due to transversality issues. To see this feature in -equivariant symplectic homology, we use a Morse-Bott spectral sequence. We refer to [Fuk96] for detailed explanation about the Morse-Bott spectral sequence. We should mention that this approach was used by [FSvK12] to study the non-existence of a displaceable exact contact embedding of Brieskorn manifolds.

There is a Morse-Bott spectral sequence which converges to whose first page is given by

where is a orientation rational bundle of and where is the set of nonconstant closed orbits of . We note that if is a -fold cover of a simple closed orbit, is the infinite dimensional lens space . We recall that parities of Conley-Zehnder indices of all even/odd multiple covers of a simple closed orbits are the same, i.e.

See [Vit89, Ust99] for instance. A closed orbit is called bad if for a simple closed orbit and some (if fact, ) and the parity of disagrees with the parity of . A closed orbit which is not bad is called good. If is a good orbit, the twist bundle is trivial and vanishes except degree zero. If is a bad orbit, is the orientation bundle of and vanishes for every degree, see [Vit89]. Therefore only good closed orbits contribute to the first page of the Morse-Bott spectral sequence and thus to the positive part of -equivariant symplectic homology as well. Note that if and hence the Morse-Bott spectral sequence stabilizes at the second page, i.e. .

Remark 2.1.

As the Morse-Bott spectral sequence shows, only with (i.e. ) contributes to .

2.3. Resonance identity


Following [vK05] we define the mean Euler characteristic by

if the limit exists. The limit exists if is homologically bounded, i.e. , are uniformly bounded. From the observation of the previous subsection we know the first page of the Morse-Bott spectral sequence converging to is given by

where is the set of good closed orbits contractible in . Since the mean Euler characteristic of is the same as that of , we have

where is the set of good closed orbits of Conley-Zehnder indices in . Let be the mean Conley-Zehnder index of which will be explained in the next section. From , see [SZ92], we have

Suppose that . Then there exist constants such that if and only if

(2.3)

We recall that nonconstant closed orbits of correspond to closed Reeb orbits after reparametrization and their Conley-Zehnder indices are the same. We abbreviate by the set of simple closed Reeb orbits contractible in whose multiple covers are all good and by the set of simple closed Reeb orbits contractible in whose even multiple covers are bad. Then (2.3) implies the following proposition. This idea is essentially identical to [GK10].

Proposition 2.2.

Let be homologically bounded. Assume that there are only finitely many simple closed Reeb orbits on and their mean Conley-Zehnder indices are positive. Then we have

(2.4)
Proof.

From (2.3), we have

3. Index Iteration formula

In the present section, we first recall the Conley-Zehnder index of a closed Reeb orbit and then briefly explain how the Conley-Zehnder index varies under iteration. For detailed explanation we refer to Long’s book [Lon02], see also [CZ84, SZ92, Sal99, Gut12]. For the sake of compatibility, we will adopt the notation and terminology of [Lon02].

Let be the space of symplectic matrices and be a subset which consists of nondegenerate elements, i.e.

We observe that where

An element is called elliptic if the spectrum is contained in the unit circle . Since we are interested in the nondegenerate case, i.e. , . The elliptic height of is defined by the total algebraic multiplicity of all eigenvalues of in and denoted by . On the other hand if , i.e. , is called hyperbolic.

We abbreviate

For , we join to

by a path . We recall that there exists a continuous map

which is uniquely characterized by the naturality, the determinant, and the normalization properties. The map induces an isomorphism between fundamental groups and . For any path