Visualizing the Unit Ball for the Teichmüller Metric

ABSTRACT

We describe a method to compute the norm on the cotangent space to the moduli space of Riemann surfaces associated to the Finsler Teichmüller 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.

KEYWORDS:

• Riemann surfaces
• Teichmuller theory

2000 AMS Subject Classification:

• 32G15
• 30F60
• 30F30

Acknowledgments

The author would like to thank C. T. McMullen, A. Epstein and the referee for helpful suggestions.

Funding

The research for this article was supported in part by grant DMS-1103654 from the National Science Foundation.