Equidistributed Closed Geodesics on Closed Finsler and Riemannian Surfaces

TL;DR

This paper proves that for generic metrics on closed surfaces, there exists a sequence of nondegenerate closed geodesics that are evenly distributed, using contact homology and variational methods.

Contribution

It establishes the existence of equidistributed closed geodesics on all closed surfaces for various metrics, extending previous results to a broader setting.

Findings

Existence of equidistributed nondegenerate closed geodesics on closed surfaces.

Application of contact homology volume property in geodesic distribution.

Extension of equidistribution results to Finsler and Riemannian metrics.

Abstract

In this paper, we establish the existence of an equidistributed sequence of nondegenerate closed geodesics for generic Finsler, symmetric Finsler and Riemannian metrics on every closed surface. The proof relies on the volume property of embedded contact homology, established by Cristofaro-Gardiner, Hutchings and Ramos, along with specific local variational constructions and transversality arguments. Our approach is motivated by Irie's equidistribution result in [19] for three-dimensional Reeb flows and the analogous result presented by Marques, Neves and Song [27] for embedded minimal hypersurfaces.