On the solution of Liouville equation

TL;DR

This paper provides a concise and rigorous proof for the existence and uniqueness of solutions to the Liouville equation with sources on various compact Riemann surfaces, including the sphere and higher genus cases.

Contribution

It offers a new, rigorous proof for the existence and uniqueness of solutions to the Liouville equation with sources on complex surfaces.

Findings

Proved existence of solutions on the sphere and higher genus surfaces.

Established uniqueness of solutions under specified conditions.

Applicable to elliptic and parabolic Liouville equations.

Abstract

We give a short and rigorous proof of the existence and uniqueness of the solution of Liouville equation with sources, both elliptic and parabolic, on the sphere and on all higher genus compact Riemann surfaces.