On A Fully Nonlinear Equation in Relativistic Teichm\"{u}ller Theory

TL;DR

This paper studies a Monge-Ampère equation in Relativistic Teichmüller Theory, providing new estimates, alternative proofs of classical theorems, and insights into the structure of Teichmüller space.

Contribution

It offers new estimates for a Monge-Ampère equation, alternative proofs of the parametrization of Teichmüller space and its classical properties, enhancing understanding of the theory.

Findings

Basic estimates for the Monge-Ampère equation in the theory.

Alternative proof of the Teichmüller space parametrization.

Proof of properness of an energy function on Teichmüller space.

Abstract

We obtain basic estimates for a Monge-Amp\`{e}re equation introduced by Moncrief in the study of the Relativistic Teichm\"{u}ller Theory. We then give another proof of the parametrization of the Teichm\"uller space obtained by Moncrief. Our approach provides yet another proof of the classical Teichm\"{u}ller theorem that the Teichm\"uller space of a compact oriented surface of genus $g(\Sigma)>1$ is diffeomorphic to the disk of dimension $6g(\Sigma)-6$. We also give another proof of properness of a certain energy function on the Teichm\"uller space.