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The Weinstein conjecture for stable Hamiltonian structures

Michael Hutchings and Clifford Henry Taubes

Geometry & Topology 13 (2009) 901–941

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 T2–bundle over S1, 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.

dynamical system, Seiberg–Witten, Floer homology
Mathematical Subject Classification 2000
Primary: 57R17, 57R57, 53D40
Secondary: 57R58
Received: 21 September 2008
Revised: 8 December 2008
Accepted: 20 November 2008
Published: 8 January 2200
Proposed: Yasha Eliashberg
Seconded: Peter Ozsvath, Leonid Polterovich
Michael Hutchings
Mathematics Department
970 Evans Hall
University of California
Berkeley, CA 94720
Clifford Henry Taubes
Mathematics Department
Harvard University
Cambridge, MA 02138