Toshiyuki Sugawa, Tohoku U., Japan

Hyperbolic metric with conic singularities on Riemann surfaces and its applications

Abstract

We first review fundamentals of hyperbolic metric on Riemann surfaces which admits conic singularities. In this talk, we will be mainly concerned with the simplest case when the Riemann surface is the complex projective line (Riemann sphere) and when the number of singularities is three. In this case, the density function of the metric can be explicitly described in terms of hypergeometric functions. As an application, we will give some refinements of Schottky and Landau theorems. This talk is based on the joint work with Daniela Kraus and Oliver Roth (University of Wuerzburg).