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

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.