Teichm\"uller extremal maps on infinite Riemann surfaces

TL;DR

This paper provides a criterion to determine when a class in the Teichmüller space of an infinite Riemann surface admits a Teichmüller extremal map, extending understanding of extremal problems in complex analysis.

Contribution

It establishes a necessary and sufficient condition for the existence of Teichmüller extremal maps on infinite Riemann surfaces, generalizing previous finite cases.

Findings

Criterion for extremal maps on infinite surfaces

Extension of Teichmüller theory to infinite Riemann surfaces

Characterization of extremal classes in Teichmüller space

Abstract

Let $X=\mathbb{D}/\Gamma$ be an arbitrary Riemann surface. We establish a necessary and sufficient criterion for $[f]\in T(X)$ to have a Teichm\"uller-type extremal map.