Liouville equations on complete surfaces with nonnegative Gauss curvature

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 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.