Reeb Orbits Research Articles

Symmetric periodic Reeb orbits on the sphere

A long standing conjecture in Hamiltonian Dynamics states that every contact form on the standard contact sphere S 2 n + 1 S^{2n+1} has at least n + 1 n+1 simple periodic Reeb orbits. In this work, we consider a refinement of this problem when the contact form has a suitable symmetry and we ask if there are at least n + 1 n+1 simple symmetric periodic orbits. We show that there is at least one symmetric periodic orbit for any contact form and at least two symmetric closed orbits whenever the contact form is dynamically convex.

On a theorem by Schlenk

In this paper we prove a generalisation of Schlenk’s theorem about the existence of contractible periodic Reeb orbits on stable, displaceable hypersurfaces in symplectically aspherical, geometrically bounded, symplectic manifolds, to a forcing result for contractible twisted periodic Reeb orbits. We make use of holomorphic curve techniques for a suitable generalisation of the Rabinowitz action functional in the stable case in order to prove the forcing result. As in Schlenk’s theorem, we derive a lower bound for the displacement energy of the displaceable hypersurface in terms of the action value of such periodic orbits. The main application is a forcing result for noncontractible periodic Reeb orbits on quotients of certain symmetric star-shaped hypersurfaces. In this case, the lower bound for the displacement energy is explicitly given by the difference of the two periods. This theorem can be applied to many physical systems including the Hénon–Heiles Hamiltonian and Stark–Zeeman systems. Further applications include a new proof of the well-known fact that the displacement energy is a relative symplectic capacity on R2n\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$${\\mathbb {R}}^{2n}$$\\end{document} and that the Hofer metric is indeed a metric.

Elliptic Reeb orbit on some real projective three-spaces via ECH

We prove the existence of an elliptic Reeb orbit for some contact forms on the real projective three space \mathbb{R}P^{3} . The main ingredient of the proof is the existence of a distinguished pseudoholomorphic curve in the symplectization given by the U map on ECH. Also, we check that the first value on the ECH spectrum coincides with the smallest action of null-homologous orbit sets for 1/4 -pinched Riemannian metrics. This action coincides with twice the length of a shortest closed geodesic. In addition, we compute the ECH spectrum for the irrational Katok metric example.

Contact three-manifolds with exactly two simple Reeb orbits

Contact three-manifolds with exactly two simple Reeb orbits

Existence of a positive hyperbolic Reeb orbit in three spheres with finite free group actions

Existence of a positive hyperbolic Reeb orbit in three spheres with finite free group actions

On the structure of Besse convex contact spheres

We consider convex contact spheres Y Y all of whose Reeb orbits are closed. Any such Y Y admits a stratification by the periods of closed Reeb orbits. We show that Y Y “resembles” a contact ellipsoid: any stratum of Y Y is an integral homology sphere, and the sequence of Ekeland-Hofer spectral invariants of Y Y coincides with the full sequence of action values, each one repeated according to its multiplicity.

Analysis of contact Cauchy–Riemann maps III: Energy, bubbling and Fredholm theory

In [Y.-G. Oh and R. Wang, Analysis of contact Cauchy-Riemann maps I: A priori [Formula: see text] estimates and asymptotic convergence, Osaka J. Math. 55(4) (2018) 647–679; Y.-G. Oh and R. Wang, Analysis of contact Cauchy-Riemann maps II: Canonical neighborhoods and exponential convergence for the Morse-Bott case, Nagoya Math. J. 231 (2018) 128–223], the authors studied the nonlinear elliptic system [Formula: see text] without involving symplectization for each given contact triad [Formula: see text], and established the a priori [Formula: see text] elliptic estimates and proved the asymptotic (subsequence) convergence of the map [Formula: see text] for any solution, called a contact instanton, on [Formula: see text] under the hypothesis [Formula: see text] and [Formula: see text]. The asymptotic limit of a contact instanton is a ‘spiraling’ instanton along a ‘rotating’ Reeb orbit near each puncture on a punctured Riemann surface [Formula: see text]. Each limiting Reeb orbit carries a ‘charge’ arising from the integral of [Formula: see text]. In this paper, we further develop analysis of contact instantons, especially the [Formula: see text] estimate for [Formula: see text] (or the [Formula: see text]-estimate), which is essential for the study of compactification of the moduli space and the relevant Fredholm theory for contact instantons. In particular, we define a Hofer-type off-shell energy [Formula: see text] for any pair [Formula: see text] with a smooth map [Formula: see text] satisfying [Formula: see text], and develop the bubbling-off analysis and prove an [Formula: see text]-regularity result. We also develop the relevant Fredholm theory and carry out index calculations (for the case of vanishing charge).

S1$S^1$‐equivariant contact homology for hypertight contact forms

In a previous paper, we showed that the original definition of cylindrical contact homology, with rational coefficients, is valid on a closed three-manifold with a dynamically convex contact form. However, we did not show that this cylindrical contact homology is an invariant of the contact structure. In the present paper, we define ‘nonequivariant contact homology’ and ‘ S 1 $S^1$ -equivariant contact homology’, both with integer coefficients, for a contact form on a closed manifold in any dimension with no contractible Reeb orbits. We prove that these contact homologies depend only on the contact structure. Our construction uses Morse–Bott theory and is related to the positive S 1 $S^1$ -equivariant symplectic homology of Bourgeois-Oancea. However, instead of working with Hamiltonian Floer homology, we work directly in contact geometry, using families of almost complex structures. When cylindrical contact homology can also be defined, it agrees with the tensor product of the S 1 $S^1$ -equivariant contact homology with Q ${\mathbb {Q}}$ . We also present examples showing that the S 1 $S^1$ -equivariant contact homology contains interesting torsion information. In a subsequent paper, we will use obstruction bundle gluing to extend the above story to closed three-manifolds with dynamically convex contact forms, which in particular will prove that their cylindrical contact homology has a lift to integer coefficients which depends only on the contact structure.

An algebraic approach to the algebraic Weinstein conjecture

How does one measure the failure of Hochschild homology to commute with colimits? Here, I relate this question to a major open problem about dynamics in contact manifolds—the assertion that Reeb orbits exist and are detected by symplectic homology. More precisely, I show that for polarizably Weinstein fillable contact manifolds, said property is equivalent to the failure of Hochschild homology to commute with certain colimits of representation categories of tree quivers. So as to be intelligible to algebraists, I try to include or black-box as much of the geometric background as possible.

Diffeomorphism type via aperiodicity in Reeb dynamics

We characterise boundary shaped disc like neighbourhoods of certain isotropic submanifolds in terms of aperiodicity of Reeb flows. We prove uniqueness of homotopy and diffeomorphism type of such contact manifolds assuming non-existence of short periodic Reeb orbits.

On contact invariants of non-simply connected Gorenstein toric contact manifolds

The first two authors showed in~\cite{AM1} how the Conley-Zehnder index of any contractible periodic Reeb orbit of a non-degenerate toric contact form on a good toric contact manifold with zero first Chern class, i.e. a Gorenstein toric contact manifold, can be explicitly computed using moment map data. In this paper we show that the same explicit method can be used to compute Conley-Zehnder indices of non-contractible periodic Reeb orbits. Under appropriate conditions, the (finite) number of such orbits in a given free homotopy class and with a given index is a contact invariant of the underlying contact manifold. We compute these invariants for two sets of examples that satisfy these conditions: $5$-dimensional contact manifolds that arise as unit cosphere bundles of $3$-dimensional lens spaces, and $2n+1$-dimensonal Gorenstein contact lens spaces. As illustrative contact topology applications, we show that diffeomorphic lens spaces might not be contactomorphic and that there are homotopy classes of diffeomorphisms of some lens spaces that do not contain any contactomorphism.

Contact forms with large systolic ratio in arbitrary dimensions

If a contact form on a (2n+1)-dimensional closed contact manifold admits closed Reeb orbits, then its systolic ration is defined to be the quotient of (n+1)th power of the shortest period of Reeb orbits by the contact volume. We prove that every co-orientable contact structure on any closed contact manifold admits a contact form with arbitrarily systolic ratio. This statement generalizes the recent result of Abbondandolo et al. in dimension three to higher dimensions. We extend the plug construction of Abbondandolo et. al. to any dimension, by means of generalizing the hamiltonian disc maps studied by the authors to the symplectic ball of any dimension. The plug is a mapping torus and it is equipped with a special contact form so that one can use it to modify a given contact form if the Reeb flow leads to a circle bundle on a large portion of the given contact manifold. Inserting the plug sucks up the contact volume while the minimal period remains the same. Following the ideas of Abbondandolo et al. and using Giroux's theory of Liouville open books, we show that any co-orientable contact structure is defined by a contact form, which is suitable to be modified via inserting plugs.

Reeb orbits that force topological entropy

Abstract We develop a forcing theory of topological entropy for Reeb flows in dimension three. A transverse link L in a closed contact $3$ -manifold $(Y,\xi )$ is said to force topological entropy if $(Y,\xi )$ admits a Reeb flow with vanishing topological entropy, and every Reeb flow on $(Y,\xi )$ realizing L as a set of periodic Reeb orbits has positive topological entropy. Our main results establish topological conditions on a transverse link L, which imply that L forces topological entropy. These conditions are formulated in terms of two Floer theoretical invariants: the cylindrical contact homology on the complement of transverse links introduced by Momin [A. Momin. J. Mod. Dyn.5 (2011), 409–472], and the strip Legendrian contact homology on the complement of transverse links, introduced by Alves [M. R. R. Alves. PhD Thesis, Université Libre de Bruxelles, 2014] and further developed here. We then use these results to show that on every closed contact $3$ -manifold that admits a Reeb flow with vanishing topological entropy, there exist transverse knots that force topological entropy.

Equivariant basic cohomology under deformations

There is a natural way to deform a Killing foliation with non-closed leaves, due to Ghys and Haefliger--Salem, into a closed foliation, i.e., a foliation whose leaves are all closed. Certain transverse geometric and topological properties are preserved under these deformations, as previously shown by the authors. For instance, the basic Euler characteristic is invariant. In this article we show that the equivariant basic cohomology ring structure is preserved under these deformations, which in turn leads to a sufficient algebraic condition (namely, equivariant formality) for the Betti numbers of basic cohomology to be preserved as well. In particular, this is true for the deformation of the Reeb orbit foliation of a $K$-contact manifold. Another consequence is that there is a universal bound on the sum of basic Betti numbers of any equivariantly formal, positively curved Killing foliation of a given codimension. We also show that a Killing foliation with negative transverse Ricci curvature is closed. If the transverse sectional curvature is negative we show, furthermore, that its fundamental group has exponential growth. Finally, we obtain a transverse generalization of Synge's theorem to Killing foliations.

Dynamical convexity and closed orbits on symmetric spheres

Our main theme here is the dynamics of Reeb flows with symmetries on the standard contact sphere. We introduce the notion of strong dynamical convexity for contact forms invariant under a group action, supporting the standard contact structure, and prove that in dimension 2n+1 any such contact form satisfying a condition slightly weaker than strong dynamical convexity has at least n+1 simple closed Reeb orbits. For contact forms with antipodal symmetry, we prove that strong dynamical convexity is a consequence of ordinary convexity. In dimension 5 or greater, we construct examples of antipodally symmetric, dynamically convex contact forms which are not strongly dynamically convex, and thus not contactomorphic to convex ones via a contactomorphism commuting with the antipodal map. Finally, we relax this condition on the contactomorphism furnishing a condition that has nonempty C1-interior.

Equivariant wrapped Floer homology and symmetric periodic Reeb orbits

AbstractThe 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 star-shaped hypersurfaces in ${\mathbb {R}}^{2n}$ 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.

Equidistributed periodic orbits of $C^\infty$-generic three-dimensional Reeb flows

We prove that, for a $C^\infty$-generic contact form $\lambda$ adapted to a given contact distribution on a closed three-manifold, there exists a sequence of periodic Reeb orbits which is equidistributed with respect to $d\lambda$. This is a quantitative refinement of the $C^\infty$-generic density theorem for three-dimensional Reeb flows, which was previously proved by the author. The proof is based on the volume theorem in embedded contact homology (ECH) by Cristofaro-Gardiner, Hutchings, Ramos, and inspired by the argument of Marques-Neves-Song, who proved a similar equidistribution result for minimal hypersurfaces. We also discuss a question about generic behavior of periodic Reeb orbits representing ECH homology classes, and give a partial affirmative answer to a toy model version of this question which concerns boundaries of star-shaped toric domains.

On manifolds with infinitely many fillable contact structures

We introduce the notion of asymptotically finitely generated contact structures, which states essentially that the Symplectic Homology in a certain degree of any filling of such contact manifolds is uniformly generated by only finitely many Reeb orbits. This property is used to generalize a famous result by Ustilovsky: We show that in a large class of manifolds (including all unit cotangent bundles and all Weinstein fillable contact manifolds with torsion first Chern class) each carries infinitely many exactly fillable contact structures. These are all different from the ones constructed recently by Lazarev. Along the way, the construction of Symplectic Homology is made more general. Moreover, we give a detailed exposition of Cieliebak’s Invariance Theorem for subcritical handle attaching, where we provide explicit Hamiltonians for the squeezing on the handle.

A local contact systolic inequality in dimension three

Let \alpha be a contact form on a connected closed three-manifold \Sigma . The systolic ratio of \alpha is defined as \rho_{\mathrm {sys}}(\alpha) := \frac {1}{\mathrm {Vol}(\alpha)} T_{\mathrm {min}} (\alpha)^2 , where T_{\mathrm {min}} (\alpha) and {\mathrm {Vol}} (\alpha) denote the minimal period of periodic Reeb orbits and the contact volume. The form \alpha is said to be Zoll if its Reeb flow generates a free S^1 -action on \Sigma . We prove that the set of Zoll contact forms on \Sigma locally maximises the systolic ratio in the C^3 -topology. More precisely, we show that every Zoll form \alpha_{\ast} admits a C^3 -neighbourhood \mathcal U in the space of contact forms such that \rho_{\mathrm {sys}}(\alpha) \leq \rho_{\mathrm {sys}}(\alpha_{\ast}) for every \alpha \in \mathcal U , with equality if and only if \alpha is Zoll.

The action spectrum characterizes closed contact 3-manifolds all of whose Reeb orbits are closed

A classical theorem due to Wadsley implies that, on a connected contact manifold all of whose Reeb orbits are closed, there is a common period for the Reeb orbits. In this paper we show that, for any Reeb flow on a closed connected 3-manifold, the following conditions are actually equivalent: (1) every Reeb orbit is closed; (2) all closed Reeb orbits have a common period; (3) the action spectrum has rank 1. We also show that, on a fixed closed connected 3-manifold, a contact form with an action spectrum of rank 1 is determined (up to pull-back by diffeomorphisms) by the set of minimal periods of its closed Reeb orbits.