Classification results of Liouville equations and rigidity of Riemannian surfaces

TL;DR

This paper classifies solutions to the Liouville equation on Riemannian surfaces with nonnegative Ricci curvature and establishes rigidity results for the manifold under optimal asymptotic conditions.

Contribution

It provides a classification of solutions and rigidity results for Riemannian surfaces under new optimal asymptotic assumptions, extending classical finite curvature conditions.

Findings

Classified all solutions to the Liouville equation under specified conditions.

Established rigidity results for the ambient Riemannian manifold.

Assumptions are shown to be optimal and differ from classical finite total curvature conditions.

Abstract

We study the Liouville equation $\triangle u+e^{2u} =0$ in a Riemannian surface $(M, g)$ with nonnegative $Ricci$ curvature. Under some asymptotic lower bound assumptions, we classify all the solutions to this equation, meanwhile we obtain the rigidity results for the ambient manifold. Note that our assumptions are optimal in some sense and differ from the classical assumption of finite total curvature.