Jonathan Tsai, CUHK, Hong Kong

A Schwarz-Christoffel formula for covering maps of Riemann surfaces

Abstract

The famous Schwarz-Christoffel formula is a formula for calculating the conformal map of the unit disc onto a domain bounded by a polygon in the complex plane. In this talk, we will discuss a generalization of the Schwarz-Christoffel formula to domains bounded by trajectory arcs of certain quadratic differentials. Then we will see that, by defining suitable quadratic differentials on the covering space of a Riemann surface, we will obtain a version of the Schwarz-Christoffel formula for covering maps on Riemann surfaces. We will then discuss some applications of this result.