Remarks about the C∞-closing lemma for 3-dimensional Reeb flows

Abstract

We prove two refinements of the $C^{\mathrm{\infty }}$-closing lemma for 3-dimensional Reeb flows, which was proved by the author as an application of spectral invariants of Embedded Contact Homology (ECH). Specifically, we prove the following two results: (i) for a $C^{\mathrm{\infty }}$-generic contact form on any closed 3-manifold, the union of periodic Reeb orbits representing ECH homology classes is dense; (ii) a certain real-analytic version of the $C^{\mathrm{\infty }}$-closing lemma for 3-dimensional Reeb flows. A few questions and conjectures related to these results are also discussed.