We prove two refinements of the -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 -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 -closing lemma for 3-dimensional Reeb flows. A few questions and conjectures related to these results are also discussed.
"Remarks about the -closing lemma for 3-dimensional Reeb flows." Kyoto J. Math. 61 (2) 305 - 322, June 2021. https://doi.org/10.1215/21562261-2021-0003