Liouville equations on complete surfaces with nonnegative Gauss curvature

Xiaohan Cai and Mijia Lai

Vol. 332 (2024), No. 1, 23–37

DOI: 10.2140/pjm.2024.332.23

Abstract

We study finite total curvature solutions of the Liouville equation $\mathrm{\Delta }u+e^{2u}=0$ on a complete surface $(M,g)$ with nonnegative Gauss curvature. It turns out that the asymptotic behavior of the solution separates into 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 $S^1\times ℝ$, then solutions must decay linearly and can be completely classified.

Keywords

Liouville equation, complete surfaces with nonnegative Gauss curvature

Mathematical Subject Classification

Primary: 35B40, 35B53, 53C45

Milestones

Received: 5 September 2023

Revised: 19 June 2024

Accepted: 18 October 2024

Published: 20 November 2024