Equivariant wrapped Floer homology and symmetric periodic Reeb orbits
The aim of this article is to apply a Floer theory to study symmetric periodic Reeb orbits. We define positive equivariant wrapped Floer homology using a (anti-)symplectic involution on a Liouville domain and investigate its algebraic properties. By a careful analysis of index iterations, we obtain a non-trivial lower bound on the minimal number of geometrically distinct symmetric periodic Reeb orbits on a certain class of real contact manifolds. This includes non-degenerate real dynamically convex starshaped hypersurfaces in which are invariant under complex conjugation. As a result, we give a partial answer to the Seifert conjecture on brake orbits in the contact setting.
Key words and phrases:equivariant wrapped Floer homology, symmetric periodic Reeb orbit, Seifert conjecture, brake orbit
2010 Mathematics Subject Classification:53D40, 37C27, 37J05
- 1 Introduction
- 2 Wrapped Floer homology
3 Equivariant wrapped Floer homology
- 3.1 -complexes
- 3.2 Equivariant Morse homology
- 3.3 Definition of equivariant wrapped Floer homology
- 3.4 Periodic family of Floer data
- 3.5 Leray-Serre type spectral sequence
- 3.6 Positive equivariant wrapped Floer homology
- 3.7 Morse-Bott spectral sequence
- 3.8 Generators of are Reeb chords
- 3.9 Computing positive equivariant wrapped Floer homology
- 4 On the minimal number of symmetric periodic Reeb orbits
A conjecture of Seifert.
Consider a mechanical Hamiltonian system in associated with a Hamiltonian of the form
where the matrix is symmetric and positive-definite for each and is a smooth function of . Note that this Hamiltonian is invariant under complex conjugation which is an anti-symplectic involution, i.e., satisfies and , where is the standard symplectic form on .
Let be a regular value of . A -periodic Hamiltonian orbit is called a brake orbit if . In , Seifert proved that if is real analytic and the projection of to the position space is bounded and homeomorphic to a closed unit ball, then there exists a brake orbit on . Under these assumptions, he then conjectured the following:
|There exist at least geometrically distinct brake orbits on .|
We remark that the lower bound in the Seifert conjecture is optimal. For example, the Hamiltonian system
on has precisely geometrically distinct brake orbits on each positive energy level.
We state some related results on brake orbits. Below, a Hamiltonian is an arbitrary smooth function on . It is said to be even in if it is invariant under .
(Rabinowitz, ) If is even in , is a regular value, and for all , then there exists a brake orbit on .
(Szulkin, ) If satisfies the assumptions of Rabinowitz and is -pinched, then the Seifert conjecture holds.
(Long-Zhang-Zhu, ) If is even in , the regular energy level is strictly convex, and is also invariant under the antipodal map , then there exist at least two brake orbits on .
(Liu-Zhang, ) Under the assumptions of Long-Zhang-Zhu, the Seifert conjecture holds.
(Giambò-Giannoni-Piccione, ) The Seifert conjecture holds for the case .
The Seifert conjecture in contact geometry.
In this article, we study the Seifert conjecture in the contact setting. A real contact manifold is a triple , where is a (co-oriented) contact manifold with a contact form and is an anti-contact involution, meaning that
The Reeb vector field is then anti-invariant under the involution , i.e.,
Assume that the fixed point set , which is a Legendrian submanifold of , is nonempty. The flow of the Reeb vector field satisfies
A smooth integral curve of the Reeb vector field satisfying the boundary condition is called a Reeb chord. In view of equation (1.2), associated to each Reeb chord is a symmetric periodic Reeb orbit
see Figure 1.
In particular, any Reeb chord comes in a pair and there is a one-to-one correspondence between the set of such pairs and the set of symmetric periodic Reeb orbits.
If the regular energy level is starshaped (with respect to the origin) which is invariant under complex conjugation , then the standard Liouville one-form on restricts to a contact form on . Moreover, the restriction defines an anti-contact involution on . Consequently, is a real contact manifold. The corresponding Legendrian submanifold is given by which implies that brake orbits are symmetric periodic Reeb orbits on . In this case, the existence theorems of Seifert  and Rabinowitz  follow immediately from the Arnold chord conjecture proved by Mohnke .
We translate the Seifert conjecture into the language of contact geometry:
There exist at least geometrically distinct symmetric periodic Reeb orbits on any real contact manifold .
In this article, we apply a Floer theory to this conjecture, taking up an approach given by Liu and Zhang [30, 32]. In their theory, they use strict convexity of a hypersurface twice as pointed out in : First, the Clarke dual action functional, which exists only in the strictly convex case, is used to obtain information on the interval on which the indices of brake orbits lie. Secondly, when the hypersurface is strictly convex, then the index of brake orbits behaves well under iterations. Borrowing an idea of Gutt and Kang , we show that under a weaker assumption the approach of Liu and Zhang works well in the framework of a Floer theory and that the index of symmetric periodic Reeb orbits behaves well under iterations, provided that the contact form is non-degenerate.
Now let a hypersurface be compact and starshaped with respect to the origin. Assume that it is invariant under an anti-contact involution . Recall that the standard Liouville form restricts to a contact form on . Then the triple is a real contact manifold. The contact form is called real dynamically convex if the Maslov indices of all Reeb chords satisfy suitable lower bounds, see Section 4.2. One of our main results is the following assertion.
Let be a compact starshaped hypersurface which is invariant under . Assume that the contact form is non-degenerate and real dynamically convex. Then there exist at least geometrically distinct and simple symmetric periodic Reeb orbits on .
This theorem in particular implies that if a compact starshaped hypersurface in invariant under an anti-contact involution is non-degenerate and real dynamically convex, then the contact Seifert conjecture holds. Without the non-degeneracy assumption, Liu and Zhang  prove that if is strictly convex and invariant under complex conjugation and the antipodal map, then the contact Seifert conjecture holds. In Theorem 1.2, we obtain the same result under weaker assumptions, but under the additional assumption of non-degeneracy, in order to apply Floer theory.
In Theorem 4.12 we prove the same assertion for a broader class of real contact manifolds and Theorem 1.2 is obtained as a corollary. As an immediate corollary of the methods used in the proof of Theorem 1.2 we obtain the following result.
In addition to the assumptions of Theorem 1.2, assume that there exist precisely geometrically distinct and simple symmetric periodic Reeb orbits on . Then their indices are all different.
In [30, Conjecture 1.1] Liu and Zhang conjecture that if is a strictly convex hypersurface which is invariant under complex conjugation, then there exist either precisely or infinitely many geometrically distinct symmetric periodic Reeb orbits. Note that non-degeneracy is not assumed. The conjecture is proved for the case under a weaker assumption by Frauenfelder-Kang [11, Theorem 2.5] via holomorphic curve techniques. More precisely, they proved that if is a starshaped dynamically convex hypersurface in which is invariant under , then there are either two or infinitely many symmetric periodic Reeb orbits. Recall that convexity implies dynamical convexity, see . Their theorem says, in particular, that the existence of any nonsymmetric periodic Reeb orbit ensures infinitely many symmetric periodic Reeb orbits. This phenomenon can also be found in the study of symmeric periodic points of reversible maps. Indeed, any area-preserving map defined on the open unit disk in the complex plane which is reversible with respect to complex conjugation must admit a symmetric periodic point. If there is more than one periodic point, which is possibly nonsymmetric, then there have to be infinitely many symmetric periodic points, see .
We now further assume that is invariant under a contact involution which commutes with , i.e., , and . For a Reeb chord , the corresponding symmetric periodic Reeb orbit is called doubly symmetric if , see Figure 2.
Our next result is a slight generalization of [30, Theorem 1.2].
Let be a compact starshaped hypesurface which is invariant under both and . Assume further that is non-degenerate and real dynamically convex. Then there exist at least geometrically distinct and simple symmetric periodic Reeb orbits on , where is the number of geometrically distinct and simple symmetric periodic Reeb orbits which are not doubly symmetric.
In addition to the assumptions of Theorem 1.5, assume that there exist precisely geometrically distinct and simple symmetric periodic Reeb orbits on . Then they are all doubly symmetric periodic Reeb orbits and their indices are all different.
In the following we explain two main ingredients to prove our theorems: equivariant wrapped Floer homology and analysis of index iterations.
Equivariant wrapped Floer homology.
Our main tool is a variant of Lagrangian Floer theory, namely equivariant wrapped Floer homology. Let be a Liouville domain with a Liouville form and an admissible Lagrangian , meaning that is exact and intersects the boundary in a Legendrian submanifold, see Section 2.1.1 for definitions. This is a basic setup that one considers to define (non-equivariant) wrapped Floer homology, see for example , .
For an equivariant theory, we additionally put a -symmetry on given by a symplectic or an anti-symplectic involution under which is invariant. We then have an induced -action on the space of paths in with end points on the Lagrangian , see equation (3.7). This allows us to define a -equivariant version of wrapped Floer homology, which is an open string analogue of the -equivariant symplectic homology in , .
We are mainly interested in the case when is given by the fixed point set of an anti-symplectic involution. In this case, we say that is a real Liouville domain. If the anti-symplectic involution is exact, then the contact boundary with the induced contact form is a real contact manifold. Examples are starshaped domains in invariant under complex conjugation .
In Section 3.3, we define equivariant wrapped Floer homology groups for both symplectic and anti-symplectic involutions. In view of , one can develop equivariant Floer theory in two different flavors. This is basically due to the fact that the Borel construction, see Section 3.2.1, admits several geometric structures, for example, it can be seen as a quotient space by an action or as a total space of a fiber bundle, see [7, Section 1]. The construction of the equivariant theory given in this paper takes the second point of view, and this matches the construction in  where an equivariant Lagrangian Floer theory with symplectic involutions is outlined. See also  for a general theory of Floer homology of families. As pointed out in  and , a technical benefit of this choice is that the analysis on Floer trajectories is easier to deal with. We also provide a description of equivariant Morse homology in Section 3.2. This hopefully makes many ideas of the constructions in equivariant wrapped Floer homology transparent.
For applications to multiplicity results of symmetric periodic Reeb orbits, we need to observe that the generators of positive equivariant wrapped Floer homology correspond to (-pairs of) Reeb chords on the contact type boundary. Even though equivariant wrapped Floer homology is designed to have this property, it is not directly visible from its definition. To identify generators with Reeb chords, we construct a Morse-Bott spectral sequence in equivariant wrapped Floer homology, which might be of independent interest, in Section 3.7. If the contact form on the boundary is chord-non-degenerate, see Section 3.8, the first page of the Morse-Bott spectral sequence is generated by (-pairs of) Reeb chords and hence so is the resulting homology, see Corollary 3.35. A similar technique was used in  to show that positive -equivariant symplectic homology is generated by periodic Reeb orbits.
We next compute equivariant wrapped Floer homology for the Liouville domains and the Lagrangians in question to detect sufficiently many generators. For this purpose, we establish algebraic properties of equivariant wrapped Floer homology, notably Leray-Serre type long exact sequences, see Corollary 3.28. A version of such a long exact sequence in symplectic homology is given in [7, Theorem 1.2]. The idea is that a special choice of Floer data, which we call periodic family of admissible Floer data in Section 3.4, simplifies the equivariant Floer chain complex, see Lemma 3.25, so that we have a -complex structure on it, see Section 3.1. A -complex structure on a chain complex then algebraically produces a Leray-Serre type spectral sequence for the resulting -equivariant homology, see Lemma 3.5.
It turns out that if the non-equivariant wrapped Floer homology vanishes, then the positive equivariant wrapped Floer homology has enough generators for our multiplicity results, see Corollary 3.40 and Corollary 3.42. Note that in Liouville domains with vanishing symplectic homology such as subcritical Stein domains, the wrapped Floer homology of any admissible Lagrangian vanishes, see Remark 2.16. In Section 2.2, we also prove a vanishing property of wrapped Floer homology under a displaceability condition on contact type boundaries and Lagrangians. A similar vanishing result for the symplectic homology of displaceable contact hypersurfaces can be found in .
Gutt-Hutchings  defined a sequence of symplectic capacities using positive equivariant symplectic homology. Based on the equivariant wrapped Floer homology developed in this paper, we are constructing a real version of symplectic capacities . This has applications to symplectic embedding problems in real Liouville domains.
Analysis of index iterations.
The main issue in applying Floer theory to multiplicity questions is to distinguish iterated orbits from simple ones. Even though the equivariant wrapped Floer homology groups have infinitely many generators, it is not obvious that they are represented by (geometrically distinct) symmetric periodic Reeb orbits. Indeed, iterations of a single symmetric periodic Reeb orbit might represent distinct generators of the homology groups. In order to avoid this, we have to establish a suitable condition on the Maslov index of Reeb chords: First of all, we impose a condition on the contact form on a real contact manifold to satisfy a certain lower bound of the Maslov index. Such a contact form is called real dynamically convex, see Definition 4.4. Our terminology is motivated by the fact that a strictly convex hypersurface in which is invariant under complex conjugation is real dynamically convex, see Theorem 4.5. Assuming real dynamical convexity, we prove by a careful analysis based on the properties of the Robbin-Salamon index and the formula for the Hörmander index given in [8, Formula 2.10] that the Maslov index is increasing under the iterations, see Theorem 4.6. This together with the common index jump theorem, which is proved by Liu and Zhang [30, Theorem 1.5], gives rise to a powerful tool for counting geometrically distinct symmetric periodic Reeb orbits in terms of the homology computations.
The authors would like to thank Urs Frauenfelder for suggesting an extension of the main theorems to the displaceable case, Felix Schlenk for reading the draft of this paper and Chungen Liu for sending us his papers. The authors cordially thank the Institut de Mathématiques at Neuchâtel, the Department of Mathematics at ETH Zürich, and the Mathematisches Institut at Giessen for their warm hospitality. JK is supported by the Swiss Government Excellence Scholarship. MK is supported by SFB/TRR 191 “Symplectic Structures in Geometry, Algebra and Dynamics” funded by DFG.
2. Wrapped Floer homology
2.1. Recollection of wrapped Floer homology
We recall the definition of wrapped Floer homology. As we will apply the index iteration theory, the homology is equipped with a -grading which comes from the Maslov index. Basically, our convention are those of . We refer to [1, Section 3] and [40, Section 4] for more detailed descriptions.
2.1.1. Geometric setup
Let be a Liouville domain. By definition, is a compact manifold with boundary such that is symplectic and the associated Liouville vector field , which is defined by the condition , is pointing outwards along the boundary. The restricted 1-form becomes a contact form on the boundary . We denote by the Reeb vector field of on characterized by the conditions
Attaching an infinite cone, we obtain the completion of as a noncompact manifold
equipped with the extended Liouville form
where stands for the coordinate on . We denote by the symplectic form on . Note that the Liouville vector field restricts to on .
A Lagrangian in is called admissible if it has the following properties:
is transverse to the boundary and is a Legendrian, i.e., ;
is exact, i.e., is an exact 1-form; and
the Liouville vector field of is tangent to near its boundary.
The third condition allows us to complete an admissible Lagrangian to a noncompact Lagrangian in :
2.1.2. Hamiltonians and Reeb chords
Given a Hamiltonian , the associated Hamiltonian vector field is defined by the equation . We denote by the Hamiltonian flow of . Its time-1 map is called the Hamiltonian diffeomorphism of . An orbit of the Hamiltonian vector field meeting the boundary condition is called a Hamiltonian chord of . The Hamiltonian chords naturally correspond to the intersection points of via the assignment . We say that a Hamiltonian chord is contractible if the class is trivial. We denote by the set of all contractible Hamiltonian chords of . A Hamiltonian chord is called non-degenerate if it satisfies
A Hamiltonian is called non-degenerate if all Hamiltonian chords of are non-degenerate, equivalently, and intersect transversally. Let be a contact manifold and let be a Legendrian. A Reeb chord of length (with respect to ) is an orbit of the Reeb vector field meeting the boundary condition . A Reeb chord is called non-degenerate if it satisfies
where is the Reeb flow of . We denote by the set of all lengths of Reeb chords.
A time-independent Hamiltonian is called admissible if on and is of the form on the symplectization with , and . Here is called the slope of .
We denote by the space of admissible Hamiltonians on . For any admissible Hamiltonian , a -small generic perturbation on is an admissible Hamiltonian which is non-degenerate, see [1, Lemma 8.1]. All Hamiltonian chords are contained in the compact domain as .
2.1.3. Maslov index of Hamiltonian chords
In order to define the Maslov index of Hamiltonian chords, we additionally assume that the Maslov class of the Lagrangian vanishes. For example, this assumption is satisfied if vanishes on and .
Let be the half-disk with real part . A Hamiltonian chord is contractible if and only if it admits a capping half-disk, namely, a map satisfying for . A capping half-disk of yields a symplectic trivialization of :
such that for where is the horizontal Lagrangian subspace in . Such a trivialization is called an adapted symplectic trivialization of . We write
for the linearization of the Hamiltonian flow along with respect to the trivialization .
The Maslov index of a contractible Hamiltonian chord is
where denotes the Robbin-Salamon index, see Section 4.1.1.
Since the Maslov class vanishes, is independent of the choices involved, see [27, Lemma 2.1].
One can define the Maslov index using any Lagrangian subspace in other than . The Maslov index is independent of the choice of .
We define the index of by .
If is non-degenerate, then we have by [41, Theorem 2.4].
2.1.4. Action functional
be the space of contractible paths of class with ends in . Recall that is exact, so for some . We fix any such . The action functional of a Hamiltonian is defined by
where is a capping half-disk of . We write for the set of critical points of . Then . An -compatible almost complex structure on is said to be of contact type (or SFT-like) if it satisfies the condition on . We abbreviate
An element is called admissible. In particular, for an admissible we can apply the maximum principle for solutions to Floer-type equations, see [40, Lemma 4.5].
Now we fix a non-degenerate admissible Hamiltonian and . For with we let be the space of Floer strips consisting of satisfying
This space carries a free -action which is given by translation in the -variable. Its quotient
Let . The filtered wrapped Floer chain complex is defined by
The Floer differential is defined by counting rigid Floer strips
where denotes the count modulo two. For a generic choice of , the differential is well-defined and satisfies . We define the filtered wrapped Floer homology of the Hamiltonian as the homology of the chain complex
For generic pairs with , one can define a continuation map by counting rigid -dependent Floer strips, see [40, Section 3.2] for the construction. Using continuation maps one shows that does not depend on the choice of and only depends on the slope of , see [40, Lemma 3.1]. We therefore omit from the notation. We define the filtered wrapped Floer homology of a Lagrangian by
where the direct limit is taken over admissible Hamiltonians using continuation maps. If we call the wrapped Floer homology of .
2.1.5. Positive wrapped Floer homology
Fix which is smaller than the minimum length of Reeb chords from to . We define the positive wrapped chain complex as the quotient chain complex
equipped with the induced differential. Following the usual limit procedure, the resulting homology group, denoted by , is called positive wrapped Floer homology of the Lagrangian . There exists a short exact sequence
which induces a long exact sequence
Taking the direct limit with respect to , we obtain the tautological exact sequence
where we have used the fact that is isomorphic to , see [48, Proposition 1.3].
2.2. Vanishing of wrapped Floer homology
In this section, we show that (non-equivariant) wrapped Floer homology vanishes under a displaceability assumption. In , using the notion of leaf-wise intersections, it is proved that the symplectic homology of a Liouville domain vanishes if its boundary is displaceable. We carry out this idea in wrapped Floer homology.
In what follows, is a Liouville domain, is a contact type boundary, is an admissible Lagrangian, and is a Legendrian submanifold in . The following notion of displaceability is taken from .
We say that is displaceable from in if there exists a compactly supported Hamiltonian diffeomorphism on such that .
starshaped hypersurfaces in are displaceable from the real Lagrangian .
If is displaceable from in , then is displaceable from in by the same displacing Hamiltonian diffeomorphism.
Let be a Hamiltonian diffeomorphism displacing . Note that the hypersurface separates into two connected components: the bounded piece and the unbounded piece . The image also separates into the bounded piece and the other. Since is connected and unbounded, cannot entirely be contained in the bounded piece . Since is bounded by , it follows that if , then . ∎
2.2.2. Filtered wrapped Floer homology and a tweaked action functional
Let be an admissible Hamiltonian of slope and take any compactly supported Hamiltonian . Following  we define a tweaked action functional by
where is a non-negative smooth function with for and , and is a smooth monotone function such that and . The following lemma is straightforward.
The differential of is given by
Consequently, if and only if solves the equation
Critical points of the tweaked action functional are closely related to the notion of relative leaf-wise intersections in .
For generic in the -topology, the tweaked functional is Morse.
Denote the Floer homology of the tweaked action functional by .
It follows from a standard argument using continuation maps with respect to a homotopy between and the constant function that ; we refer to [23, Section 2.2] for details. On the other hand, since is an admissible Hamiltonian of slope , we have . ∎
2.2.3. Vanishing of wrapped Floer homology
We would like to prove the following vanishing property.
If is displaceable from in , then the inclusion map
is a zero map for all , where is defined as in (2.5).
An immediate corollary is the following.
If is displaceable from in , the wrapped Floer homology vanishes.
If is displaceable from itself, then we already know that vanishes for any admissible Lagrangian . In this case vanishes as shown in . By the result in [40, Theorem 6.17] the wrapped Floer homology is a module over for any admissible Lagrangian . It follows that the wrapped Floer homology vanishes.
For a given compactly supported Hamiltonian , we define
Here denotes the cylindrical coordinate, and if is supported only in , then we put by convention. From now on, we always assume that is displaceable from and that is a displacing Hamiltonian diffeomorphism.
If , then we have . In particular, if the set is empty.
For each , there exists as large as one likes such that there is no critical point of with . Consequently, the filtered chain complex and hence the homology vanishes.
For notational convenience, we denote , and we choose without loss of generality for . By Lemma 2.17, if , then . In particular, is a solution of the equation where denotes the Reeb vector field. Abbreviating we compute
Since , the term is bounded from above by a constant which does not depend on . It follows that
We then take such that . This completes the proof. ∎
We now prove the vanishing property. This will basically be done by showing that the inclusion factors through which vanishes for suitable . Define the displacement energy of from in by
Proof of Theorem 2.14.
Choosing a generic homotopy such that and , we have continuation maps on filtered homology groups
for each . We remark that the action shift by (or ) is due to the energy consumption property of parametrized Floer solutions which we count to define the continuation maps, see Remark 2.19. By composing the two maps, we have a map (note that )
and by the usual homotopy of homotopies argument in Floer theory, we have that
where is the map induced by inclusion at the chain level.
Taking as in Lemma 2.18, we have . Therefore the map
vanishes for all and . We finally take the direct limit along , and conclude that the inclusion
also vanishes. ∎
For a solution of the parametrized Floer equation , the energy
3. Equivariant wrapped Floer homology
As an algebraic preliminary to develop equivariant theories, we briefly outline the notion of -complex. The parallel notion of -complex can be found in .
A -graded chain complex with -coefficients is called a -complex if it admits the additional datum of a sequence of maps
such that has degree , i.e., , and satisfies the relations