  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.
