Liouville equations on complete surfaces with nonnegative Gauss curvature

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.