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arXiv · 2309.01956

Liouville equations on complete surfaces with nonnegative Gauss curvature

Xiaohan Cai·Mijia Lai
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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.

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Xiaohan Cai, Mijia Lai. 2023-09-05. Liouville equations on complete surfaces with nonnegative Gauss curvature. https://doi.org/10.2140/pjm.2024.332.23

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