Closed Contact Manifold Research Articles

On the topology of loops of contactomorphisms and Legendrians in non-orderable manifolds

We study the global topology of the space L of loops of contactomorphisms of a non-orderable closed contact manifold (M2n+1,ξ). We filter L by a quantitative measure of the “positivity” of the loops and describe the topology of L in terms of the subspaces of the filtration. In particular, we show that the homotopy groups of L are subgroups of the homotopy groups of the subspace of positive loops L+. We obtain analogous results for the space of loops of Legendrian submanifolds in (M2n+1,ξ).

Reeb flows with small contact volume and large return time to a global section

In this paper, we show that any co-oriented closed contact manifold of dimension at least five admits a contact form such that the contact volume is arbitrarily small but the Reeb flow admits a global hypersurface of section with the property that the minimal period on the boundary of the hypersurface and the first return time in the interior of the hypersurface are bounded below. An immediate consequence of this statement is that every co-oriented contact structure on any closed manifold admits a contact form with arbitrarily large systolic ratio. This generalizes the result of Abbondandolo et al. in dimension three to higher dimensions. The proof of the main result is inductive and uses the result of Abbondandolo et al. on large systolic ratio in dimension three in its basis step. The essential construction in the proof relies on the Giroux correspondence in higher dimensions.

Entropy collapse versus entropy rigidity for Reeb and Finsler flows

On every closed contact manifold there exist contact forms with volume one whose Reeb flows have arbitrarily small topological entropy. In contrast, for many closed manifolds there is a uniform positive lower bound for the topological entropy of (not necessarily reversible) normalized Finsler geodesic flows.

Bordered Floer homology and contact structures

Abstract We introduce a contact invariant in the bordered sutured Heegaard Floer homology of a three-manifold with boundary. The input for the invariant is a contact manifold $(M, \xi , \mathcal {F})$ whose convex boundary is equipped with a signed singular foliation $\mathcal {F}$ closely related to the characteristic foliation. Such a manifold admits a family of foliated open book decompositions classified by a Giroux correspondence, as described in [LV20]. We use a special class of foliated open books to construct admissible bordered sutured Heegaard diagrams and identify well-defined classes $c_D$ and $c_A$ in the corresponding bordered sutured modules. Foliated open books exhibit user-friendly gluing behavior, and we show that the pairing on invariants induced by gluing compatible foliated open books recovers the Heegaard Floer contact invariant for closed contact manifolds. We also consider a natural map associated to forgetting the foliation $\mathcal {F}$ in favor of the dividing set and show that it maps the bordered sutured invariant to the contact invariant of a sutured manifold defined by Honda–Kazez–Matić.

Positive mass theorem and the CR Yamabe equation on 5-dimensional contact spin manifolds

We consider the CR Yamabe equation with critical Sobolev exponent on a closed contact manifold M of dimension 2n+1. The problem of finding solutions with minimum energy has been resolved for all dimensions except for dimension 5 (n=2). In this paper we prove the existence of minimum energy solutions in the 5-dimensional case when M is spin. The proof is based on a positive mass theorem built up through a spinorial approach.

A note on Lagrangian intersections and Legendrian Cobordism

Let Λ,Λ′ be a pair of closed Legendrian submanifolds in a closed contact manifold (Y,ξ=Ker(α)) related by a Legendrian cobordism W⊂(C×Y,ξ˜=Ker(−yḍx+α)). In this note, we show that in the hypertight setting, if Λ intersects for reasons of Floer homology a closed pre-Lagrangian P⊂Y which is either weakly exact or monotone, then so does Λ′.

Lorentzian distance functions in contact geometry

An important tool to analyse the causal structure of a Lorentzian manifold is given by the Lorentzian distance function. We define a class of Lorentzian distance functions on the group of contactomorphisms of a closed contact manifold depending on the choice of a contact form. These distance functions are continuous with respect to the Hofer norm for contactomorphisms defined by Shelukhin [The Hofer norm of a contactomorphism, J. Symplectic Geom. 15 (2017) 1173–1208] and finite if and only if the group of contactomorphisms is orderable. To prove this, we show that intervals defined by the positivity relation are open with respect to the topology induced by the Hofer norm. For orderable Legendrian isotopy classes we show that the Chekanov-type metric defined in [D. Rosen and J. Zhang, Chekanov’s dichotomy in contact topology, Math. Res. Lett. 27 (2020) 1165–1194] is nondegenerate. In this case, similar results hold for a Lorentzian distance functions on Legendrian isotopy classes. This leads to a natural class of metrics associated to a globally hyperbolic Lorentzian manifold such that its Cauchy hypersurface has a unit co-tangent bundle with orderable isotopy class of the fibres.

Iso-contact embeddings of manifolds in co-dimension $2$

The purpose of this article is to study co-dimension $2$ iso-contact embeddings of closed contact manifolds. We first show that a closed contact manifold $(M^{2n-1}, \xi_M)$ iso-contact embeds in a contact manifold $(N^{2n+1}, \xi_N),$ provided $M$ contact embeds in $(N, \xi_N)$ with a trivial normal bundle and the contact structure induced on $M$ via this embedding is homotopic as an almost-contact structure to $\xi_M.$ We apply this result to first establish that a closed contact $3$--manifold having no $2$--torsion in its second integral cohomology iso-contact embeds in the standard contact $5$--sphere if and only if the first Chern class of the contact structure is zero. Finally, we discuss iso-contact embeddings of closed simply connected contact $5$--manifolds.

Fredholm theory for pseudoholomorphic curves with brake symmetry

We study the pseudoholomorphic curves with brake symmetry in symplectization of a closed contact manifold. We introduce the pseudoholomorphic curves with brake symmetry and the corresponding moduli space. Then we get the virtual dimension of the moduli space.

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.

On the spectral characterization of Besse and Zoll Reeb flows

A closed contact manifold is called Besse when all its Reeb orbits are closed, and Zoll when they have the same minimal period. In this paper, we provide a characterization of Besse contact forms for convex contact spheres and Riemannian unit tangent bundles in terms of S1-equivariant spectral invariants. Furthermore, for restricted contact type hypersurfaces of symplectic vector spaces, we give a sufficient condition for the Besse property via the Ekeland–Hofer capacities.

Two closed orbits for non-degenerate Reeb flows

AbstractWe prove that every non-degenerate Reeb flow on a closed contact manifold M admitting a strong symplectic filling W with vanishing first Chern class carries at least two geometrically distinct closed orbits provided that the positive equivariant symplectic homology of W satisfies a mild condition. Under further assumptions, we establish the existence of two geometrically distinct closed orbits on any contact finite quotient of M. Several examples of such contact manifolds are provided, like displaceable ones, unit cosphere bundles, prequantisation circle bundles, Brieskorn spheres and toric contact manifolds. We also show that this condition on the equivariant symplectic homology is preserved by boundary connected sums of Liouville domains. As a byproduct of one of our applications, we prove a sort of Lusternik–Fet theorem for Reeb flows on the unit cosphere bundle of not rationally aspherical manifolds satisfying suitable additional assumptions.

On some examples and constructions of contact manifolds

The first goal of this paper is to construct examples of higher dimensional contact manifolds with specific properties. Our main results in this direction are the existence of tight virtually overtwisted closed contact manifolds in all dimensions and the fact that every closed contact 3-manifold, which is not (smoothly) a rational homology sphere, contact-embeds with trivial normal bundle inside a hypertight closed contact 5-manifold. This uses known construction procedures by Bourgeois (on products with tori) and Geiges (on branched covering spaces). We pass from these procedures to definitions; this allows to prove a uniqueness statement in the case of contact branched coverings, and to study the global properties (such as tightness and fillability) of the results of both constructions without relying on any auxiliary choice in the procedures. A second goal allowed by these definitions is to study relations between these constructions and the notions of supporting open book, as introduced by Giroux, and of contact fiber bundle, as introduced by Lerman. For instance, we give a definition of Bourgeois contact structures on flat contact fiber bundles which is local, (strictly) includes the results of the Bourgeois construction, and allows to recover an isotopy class of supporting open books on the fibers. This last point relies on a reinterpretation, inspired by an idea by Giroux, of supporting open books in terms of pairs of contact vector fields.

Rigid constellations of closed Reeb orbits

We use Hamiltonian Floer theory to recover and generalize a classic rigidity theorem of Ekeland and Lasry. That theorem can be rephrased as an assertion about the existence of multiple closed Reeb orbits for certain tight contact forms on the sphere that are close, in a suitable sense, to the standard contact form. We first generalize this result to Reeb flows of contact forms on prequantization spaces that are suitably close to Boothby–Wang forms. We then establish, under an additional nondegeneracy assumption, the same rigidity phenomenon for Reeb flows on any closed contact manifold. A natural obstruction to obtaining sharp multiplicity results for closed Reeb orbits is the possible existence of fast closed orbits. To complement the existence results established here, we also show that the existence of such fast orbits cannot be precluded by any condition which is invariant under contactomorphisms, even for nearby contact forms.

The Hofer norm of a contactomorphism

We show that the $L^{\infty}$-norm of the contact Hamiltonian induces a non-degenerate right-invariant metric on the group of contactomorphisms of any closed contact manifold. This contact Hofer metric is not left-invariant, but rather depends naturally on the choice of a contact form $\alpha,$ whence its restriction to the subgroup of $\alpha$-strict contactomorphisms is bi-invariant. The non-degeneracy of this metric follows from an analogue of the energy-capacity inequality. We show furthermore that this metric has infinite diameter in a number of cases by investigating its relations to previously defined metrics on the group of contact diffeomorphisms. We study its relation to Hofer's metric on the group of Hamiltonian diffeomorphisms, in the case of prequantization spaces. We further consider the distance in this metric to the Reeb one-parameter subgroup, which yields an intrinsic formulation of a small-energy case of Sandon's conjecture on the translated points of a contactomorphism. We prove this Chekanov-type statement for contact manifolds admitting a strong exact filling.

Polyfolds, cobordisms, and the strong Weinstein conjecture

We prove the strong Weinstein conjecture for closed contact manifolds that appear as the concave boundary of a symplectic cobordism admitting an essential local foliation by holomorphic spheres.

Chern–Weil theory and the group of strict contactomorphisms

In this paper we study the groups of contactomorphisms of a closed contact manifold from a topological viewpoint. First we construct examples of contact forms on spheres whose Reeb flow has a dense orbit. Then we show that the unitary group [Formula: see text] is homotopically essential in the group of contactomorphisms of the standard contact sphere [Formula: see text] and present a proof of the homotopy equivalence [Formula: see text]. In the second part of the paper we focus on the group of strict contactomorphisms — using the framework of Chern–Weil theory we introduce and study contact characteristic classes analogous to the Reznikov Hamiltonian classes in symplectic topology. We carry out several explicit calculations illustrating the nontriviality of these contact characteristic classes.

Isometry-invariant geodesics and the fundamental group

We prove that on closed Riemannian manifolds with infinite abelian, but not cyclic, fundamental group, any isometry that is homotopic to the identity possesses infinitely many invariant geodesics. We conjecture that the result remains true if the fundamental group is infinite cyclic. We also formulate a generalization of the isometry-invariant geodesics problem, and a generalization of the celebrated Weinstein conjecture: on a closed contact manifold with a selected contact form, any strict contactomorphism that is contact-isotopic to the identity possesses an invariant Reeb orbit.

Morse relations and Fredholm deformations of v-convex contact forms

We address in this paper the Fredholm and compactness issues for the variational problem (J,C β ), Bahri (Pitman Research Notes in Mathematics Series No. 173. Scientific and Technical, London, 1988), Bahri (C. R. Acad. Sci. Paris 299, Serie I, 15, 757–760, 1984). We prove that the intersection operator restricted to periodic orbits of the Reeb vector-field ∂per does not mix with the intersection operator ∂∞ of the critical points at infinity. The Fredholm issues are extensively discussed in the Introduction and solved in Bahri (Arab J Maths, 2014). We also address in this paper the issue of existence of periodic orbits for three-dimensional Reeb vector-fields, the Weinstein conjecture (Weinstein, J Differ Equ 33:353–358, 1979) on S3, solved in dimension 3 throughout the works of Rabinowitz (Commun Pure Appl Math 31:157–184, 1978) and Hofer (Invent Math 114:515–563, 1993); see also Hutchings (Proc. 2010 ICM 46:1022–1041, 2010) and Taubes (Geom Topol 11:2117–2202, 2007) for the full Weinstein conjecture in dimension 3. Following our previous work (Bahri, Adv Nonlinear Stud 8:1–17, 2008), we devise a new method to find these periodic orbits when they are of odd index. We conjecture that this method, when combined with the other results described above about the intersection operator, gives rise to a homology that is specific of the contact structure and that is invariant by deformation. The existence result, as derived here, is weaker than the one announced by Taubes (Geom Topol 11:2117–2202, 2007). After appropriate generalization, it provides a new proof, via variational theory, of the Weinstein conjecture on three-dimensional closed contact manifolds with finite fundamental group.

A remark on the Reeb flow for spheres

We prove the non-triviality of the Reeb flow for the standard contact spheres S 2n+1 , n 6 3, inside the fundamental group of their contactomorphism group. The argument uses the existence of homotopically non-trivial 2-spheres in the space of contact structures of a 3-Sasakian manifold. Let (M,ξ) be a closed contact manifold. Consider the space C(M,ξ) of contact structures isotopic to ξ. This space has been studied in special cases. See (El) for the 3-sphere and (Bo), (Ge) for torus bundles. In the present note we prove the non-triviality of its second homotopy group for 3-Sasakian manifolds, see (BG). Theorem 1. Let (M,ξ) be a 3-Sasakian manifold, then rk(π2(C(M,ξ))) ≥ 1. Let (S 4n+3 ,ξ0 = kerα0) be the standard contact sphere with the standard contact form. The non- trivial spheres in C(S 4n+3 ,ξ0) allow us to answer a question posed in (Gi):