Dense existence of periodic Reeb orbits and ECH spectral invariants

Kei  Irie

doi:10.3934/jmd.2015.9.357

Article Contents

2015, Volume 9: 357-363. Doi: 10.3934/jmd.2015.9.357

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Dense existence of periodic Reeb orbits and ECH spectral invariants

Kei Irie^1,

1.
Research Institute for Mathematical Sciences, Kyoto University, Kyoto 606-8502

Received: June 30, 2015

Revised: July 31, 2015

Published: November 2015

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Abstract

In this paper, we prove: (1) for any closed contact three-manifold with a $C^\infty$-generic contact form, the union of periodic Reeb orbits is dense; (2) for any closed surface with a $C^\infty$-generic Riemannian metric, the union of closed geodesics is dense. The key observation is a $C^\infty$-closing lemma for three-dimensional Reeb flows, which follows from the fact that the embedded contact homology (ECH) spectral invariants recover the volume.

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Mathematics Subject Classification: Primary: 37J45; Secondary: 53D42, 53D25.

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References

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