On the homology length spectrum of surfaces

TL;DR

This paper studies the asymptotic behavior of the number of length-minimizing closed geodesics in homology classes on surfaces with Finsler metrics, with a focus on hyperbolic surfaces.

Contribution

It introduces an analysis of the homology length spectrum for surfaces with Finsler metrics, extending understanding beyond classical Riemannian cases.

Findings

Asymptotic growth rate of homology-minimizing geodesics established

Results specialized to hyperbolic surfaces

Insights into the structure of geodesic multicurves in homology classes

Abstract

On a surface with a Finsler metric, we investigate the asymptotic growth of the number of closed geodesics of length less than $L$ which minimize length among all geodesic multicurves in the same homology class. An important class of surfaces which are of interest to us are hyperbolic surfaces.