#### Volume 13, issue 2 (2009)

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The Weinstein conjecture for stable Hamiltonian structures

### Michael Hutchings and Clifford Henry Taubes

Geometry & Topology 13 (2009) 901–941
##### 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.

##### Keywords
dynamical system, Seiberg–Witten, Floer homology
##### Mathematical Subject Classification 2000
Primary: 57R17, 57R57, 53D40
Secondary: 57R58