Dense existence of periodic Reeb orbits and ECH spectral invariants

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 $C^{\infty }$-closing lemma for 3D Reeb flows, which follows from the fact that the embedded contact homology (ECH) spectral invariants recover the volume.

arXiv comment7 pages, minor revisions and a reference added