The Weinstein conjecture for stable Hamiltonian structures

Abstract

We use the equivalence between embedded contact homology and Seiberg-Witten Floer homology to obtain the following improvements on the Weinstein conjecture. Let Y be a closed oriented connected 3-manifold with a stable Hamiltonian structure, and let R denote the associated Reeb vector field on Y. We prove that if Y is not a T^2-bundle over S^1, then R has a closed orbit. Along the way we prove that if Y is a closed oriented connected 3-manifold with a contact form such that all Reeb orbits are nondegenerate and elliptic, then Y is a lens space. Related arguments show that if Y is a closed oriented 3-manifold with a contact form such that all Reeb orbits are nondegenerate, and if Y is not a lens space, then there exist at least three distinct embedded Reeb orbits.

39 pages

References in corpus (5)

• The Seiberg-Witten equations and the Weinstein conjecture
• Relative asymptotic behavior of pseudoholomorphic half-cylinders
• Intersection theory of punctured pseudoholomorphic curves
• Embedded contact homology and Seiberg-Witten Floer cohomology I
• Reeb vector fields and open book decompositions

Cited by in corpus (36)

• Proof of the Arnold chord conjecture in three dimensions II
• HF=HM IV: The Seiberg-Witten Floer homology and ech correspondence
• Recent progress on symplectic embedding problems in four dimensions
• Torsion contact forms in three dimensions have two or infinitely many Reeb orbits
• A dynamical characterization of universally tight lens spaces
• Embedded contact homology and open book decompositions
• Graph cobordisms and Heegaard Floer homology
• Contact Homology of Orbit Complements and Implied Existence
• First steps in stable Hamiltonian topology
• Contact three-manifolds with exactly two simple Reeb orbits
• A note on the stationary Euler equations of hydrodynamics
• Seiberg-Witten and Gromov invariants for self-dual harmonic 2-forms
• Steady Euler flows and Beltrami fields in high dimensions
• On cobordism maps on periodic Floer homology
• On the minimal number of periodic orbits on some hypersurfaces in ${\mathbb{R}}^{2n}$
• Taming the pseudoholomorphic beasts in $\mathbb{R}\times (S^1\times S^2)$
• An elementary alternative to ECH capacities
• The asymptotics of ECH capacities
• PFH spectral invariants and $C^{\infty }$ closing lemmas
• Stable Hamiltonian structures in dimension three are supported by open books
• An observation concerning uniquely ergodic vector fields on 3-manifolds
• Equivariant symplectic homology and multiple closed Reeb orbits
• Symplectic embeddings of products
• Hyperbolicity, irrationality exponents and the eta invariant
• Ricci Curvature, Reeb Flows and Contact 3-Manifolds
• Counting periodic orbits of vector fields over smooth closed manifolds
• An overtwisted disk in a virtual contact structure and the Weinstein conjecture
• A Simple Proof On Poincaré Conjecture
• Contact topology and electromagnetism: The Weinstein conjecture and Beltrami-Maxwell fields
• Some Fundamental Theorems in Mathematics
• Periodic orbits of analytic Euler fields on 3-manifolds
• Real hypersurfaces in the complex quadric with Reeb parallel structure Jacobi operator
• Giroux correspondence, confoliations, and symplectic structures on S^1 x M
• Periodic orbits of Euler vector fields on 3-manifolds
• Taubes's proof of the Weinstein conjecture in dimension three
• An exposition of the equivalence of Heegaard Floer homology and embedded contact homology