  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.
