Liouville equations on complete surfaces with nonnegative Gauss curvature

TL;DR

This paper classifies solutions to the Liouville equation on complete surfaces with nonnegative Gauss curvature, showing they are either on the Euclidean plane or flat cylinder, based on their decay behavior.

Contribution

It provides a complete classification of finite total curvature solutions on such surfaces, linking asymptotic decay to the surface's geometry.

Findings

Solutions on Euclidean plane decay slowly

Solutions on flat cylinder decay linearly

Complete classification of solutions based on asymptotic behavior

Abstract

We study finite total curvature solutions of the Liouville equation $\Delta u+e^{2u}=0$ on a complete surface $(M,g)$ with nonnegative Gauss curvature. It turns out that the asymptotic behavior of the solution separates two extremal cases: on the one end, if the solution decays not too fast, then $(M,g)$ must be isometric to the standard Euclidean plane; on the other end, if $(M,g)$ is isometric to the flat cylinder $\mathbb{S}^1\times \mathbb{R}$, then solutions must decay linearly and are completely classified.