Liouville Equation And Schottky Problem

TL;DR

This paper proposes an Ansatz linking the Liouville equation, uniformization, and KP hierarchy, offering new insights into the Schottky problem and applications in quantum Riemann surfaces and 2D gravity.

Contribution

It introduces a novel Ansatz that simplifies the Liouville equation and connects uniformization, moduli space, and KP hierarchy, advancing the understanding of the Schottky problem.

Findings

Liouville equation reduces to a KP-like equation

Connects uniformization with KP hierarchy

Applications in quantum Riemann surfaces and 2D gravity

Abstract

An Ansatz for the Poincar\'e metric on compact Riemann surfaces is proposed. This implies that the Liouville equation reduces to an equation resembling a non chiral analogous of the higher genus relationships (KP equation) arising in the framework of Schottky's problem solution. This approach connects uniformization (Fuchsian groups) and moduli space theories with KP hierarchy. Besides its mathematical interest, the Ansatz has some applications in the framework of quantum Riemann surfaces arising in 2D gravity.