Dense existence of periodic Reeb orbits and ECH spectral invariants

TL;DR

This paper proves that for generic contact forms and Riemannian metrics on closed manifolds, the sets of periodic Reeb orbits and closed geodesics are dense, using ECH spectral invariants and a $C^ ext{infty}$-closing lemma.

Contribution

It establishes the density of periodic orbits and geodesics for generic structures, connecting ECH spectral invariants with volume recovery.

Findings

Density of periodic Reeb orbits in generic contact three-manifolds

Density of closed geodesics in generic Riemannian surfaces

ECH spectral invariants recover volume

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.