Contact homology and higher dimensional closing lemmas

TL;DR

This paper introduces contact homology techniques to address the smooth closing lemma for Reeb flows in all dimensions, proving a key conjecture for ellipsoids and extending to other examples.

Contribution

It develops new methods using contact homology for the closing lemma in higher dimensions and proves a significant conjecture for Reeb flows on ellipsoids.

Findings

Proves the strong closing lemma for Reeb flows on ellipsoids.

Extends methods to other Reeb flows and examples.

Introduces contact homology as a tool in this context.

Abstract

We develop methods for studying the smooth closing lemma for Reeb flows in any dimension using contact homology. As an application, we prove a conjecture of Irie, stating that the strong closing lemma holds for Reeb flows on ellipsoids. Our methods also apply to other Reeb flows, and we illustrate this for a class of examples introduced by Albers-Geiges-Zehmisch.