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

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 $g(\mathrm{\Sigma })>1$ is diffeomorphic to the disk of dimension $6g(\mathrm{\Sigma })-6$. We also give another proof of properness of a certain energy function on the Teichm"uller space.