The Weinstein conjecture for stable Hamiltonian structures

TL;DR

This paper proves the existence of closed Reeb orbits on certain 3-manifolds with stable Hamiltonian structures, extending the Weinstein conjecture using advanced homological methods and classifying manifolds with specific Reeb orbit properties.

Contribution

It establishes new cases of the Weinstein conjecture for stable Hamiltonian structures and classifies 3-manifolds with particular Reeb orbit characteristics.

Findings

If Y is not a T^2-bundle over S^1, then the Reeb vector field has a closed orbit.

Y is a lens space if all Reeb orbits are nondegenerate and elliptic.

If all Reeb orbits are nondegenerate and Y is not a lens space, then there are at least three distinct embedded Reeb orbits.

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.