Function Model of the Teichmüller space of a closed hyperbolic Riemann Surface

Abstract

We introduce a function model for the Teichmüller space of a closed hyperbolic Riemann surface. Then we introduce a new metric by using the maximum norm on the function space on the Teichmüller space. We prove that the identity map from the Teichmüller space equipped with the usual Teichmüller metric to the Teichmüller space equipped with this new metric is uniformly continuous. Furthermore, we also prove that the inverse of the identity, that is, the identity map from the Teichmüller space equipped with this new metric to the Teichmüller space equipped with the usual Teichmüller metric, is continuous. Therefore, the topology induced by the new metric is just the same as the topology induced by the usual Teichmüller metric on the Teichmüller space. We give a remark about the pressure metric and the Weil-Petersson metric.