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Hutchings, Michael; Taubes, Clifford Henry

The Weinstein conjecture for stable Hamiltonian structures. (English) Zbl 1169.53065

Geom. Topol. 13, No. 2, 901-941 (2009).
Authors’ 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.
Reviewer: Karl Heinz Dovermann (Honolulu)

Cited in 2 Reviews
Cited in 34 Documents

MSC:

53D40 Symplectic aspects of Floer homology and cohomology
57R17 Symplectic and contact topology in high or arbitrary dimension
57R57 Applications of global analysis to structures on manifolds
57R58 Floer homology

Keywords:

dynamical system; Seiberg-Witten; Floer homology; lens space
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References:

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