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.