LIOUVILLE EQUATION AND SCHOTTKY PROBLEM

An Ansatz for the Poincare metric on compact Riemann surfaces is propo sed. This implies that the Liouville equation reduces to an equation r esembling a nonchiral analogous of the higher genus relationships (KP equation) arising within the framework of Schottky's problem solution. This approach connects uniformization (Fuchsian groups) and moduli sp ace theories with KP hierarchy. Besides its mathematical interest, the Ansatz has some applications within the framework of quantum Riemann surfaces arising in 2D gravity.