Function Model of the Teichm\"uller space of a closed hyperbolic Riemann Surface

TL;DR

This paper introduces a new function-based model and metric for the Teichm"uller space of a closed hyperbolic Riemann surface, establishing their topological equivalence with the classical Teichm"uller metric.

Contribution

It presents a novel function model and a maximum norm-based metric for Teichm"uller space, proving their topological compatibility with the traditional metric.

Findings

The new metric is topologically equivalent to the Teichm"uller metric.

The identity map between the two metric spaces is uniformly continuous.

The inverse map is continuous, ensuring topological equivalence.

Abstract

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