  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.
