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). |