Quadratic differentials and foliations on infinite Riemann surfaces

TL;DR

This paper characterizes parabolic infinite Riemann surfaces via quadratic differentials, proves the density of Jenkins-Strebel differentials, and extends Kerckhoff's Teichmüller metric formula to this setting.

Contribution

It extends the Hubbard-Masur theorem and establishes new characterizations and density results for quadratic differentials on infinite Riemann surfaces.

Findings

Characterization of parabolic surfaces using quadratic differentials

Density of Jenkins-Strebel differentials in the space of integrable quadratic differentials

Extension of Kerckhoff's formula for the Teichmüller metric to infinite surfaces

Abstract

We prove that an infinite Riemann surface $X$ is parabolic ($X\in O_G$) if and only if the union of the horizontal trajectories of any integrable holomorphic quadratic differential that are cross-cuts is of zero measure. Then we establish the density of the Jenkins-Strebel differentials in the space of all integrable quadratic differentials when $X\in O_G$ and extend Kerckhoff's formula for the Teichm\"uller metric in this case. Our methods depend on extending to infinite surfaces the Hubbard-Masur theorem describing which measured foliations can be realized by horizontal trajectories of integrable holomorphic quadratic differentials.