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G&T Monographs |
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The Weinstein conjecture for stable Hamiltonian structures
Michael Hutchings and Clifford Henry Taubes
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Geometry & Topology 13 (2009)
901–941
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Abstract
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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.
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Keywords
dynamical system, Seiberg–Witten,
Floer homology
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Mathematical Subject Classification
Primary: 53D40, 57R17, 57R57
Secondary: 57R58
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Publication
Received: 21 September 2008
Revised: 8 December 2008
Accepted: 20 November 2008
Published: 8 January 2200
Proposed: Yasha Eliashberg
Seconded: Peter Ozsvath, Leonid Polterovich
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