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We obtain basic estimates for a Monge-Amp\`{e}re equation introduced byMoncrief in the study of the Relativistic Teichm\"{u}ller Theory. We then giveanother proof of the parametrization of the Teichm\"uller space obtained byMoncrief. Our approach provides yet another proof of the classicalTeichm\"{u}ller theorem that the Teichm\"uller space of a compact orientedsurface 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 energyfunction on the Teichm\"uller space.

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