Meromorphic $L^2$ functions on flat surfaces

Abstract

We prove a quantitative version of the non-uniform hyperbolicity of the Teichmüller geodesic flow. Namely, at each point of any Teichmüller 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 Treviño 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.

Preliminary version