Visualizing the unit ball for the Teichm\"uller metric

TL;DR

This paper introduces a practical method to compute the Teichmüller metric's norm on the cotangent space of Riemann surface moduli spaces, using period calculations of abelian double covers, and visualizes the unit ball for a specific genus zero surface.

Contribution

It presents an accessible computational approach for the Teichmüller metric via period matrices, enabling visualization and verification of theoretical results.

Findings

Successfully visualized the unit sphere in the cotangent space.

Confirmed Royden's theorem for the example surface.

Provided an implementable method for algebraic curves.

Abstract

We describe a method to compute the norm on the cotangent space to the moduli space of Riemann surfaces associated to the Finsler Teichm\"uller metric. Our method involves computing the periods of abelian double covers and is easy to implement for Riemann surfaces presented as algebraic curves using existing tools for approximating period matrices of plane algebraic curves. We illustrate our method by depicting the unit sphere in the cotangent space to moduli space at a particular surface of genus zero with five punctures and by corroborating the proof of a theorem of Royden's for our example.