Meromorphic $L^2$ functions on flat surfaces

TL;DR

This paper establishes a quantitative bound on the spectral gap for the Teichmüller flow, linking geometric properties of flat surfaces to dynamical stability and ergodic behavior.

Contribution

It provides a new estimate for the spectral gap of the Teichmüller flow based on geometric quantities, enhancing understanding of hyperbolicity and ergodicity in flat surface dynamics.

Findings

Bound on infinitesimal spectral gap in terms of flat systole

Strengthened results on unique ergodicity of measured foliations

Estimate for spectral gaps of pseudo-Anosov homeomorphisms

Abstract

We prove a quantitative version of the non-uniform hyperbolicity of the Teichm\"uller geodesic flow. Namely, at each point of any Teichm\"uller flow line, we bound the infinitesimal spectral gap for variations of the Hodge norm along the flow line in terms of an easily estimated geometric quantity on the flat surface, which is greater than or equal to the flat systole. As applications, we strengthen results of Trevi\~no and Smith regarding unique ergodicity of measured foliations, and give an estimate for the spectral gaps of pseudo-Anosov homeomorphisms based on the location of their axes in the moduli space of quadratic differentials.