A note on the Gaussian curvature on noncompact surfaces

We give a short proof of the following fact. Let $\mathrm{\Sigma }$ be a connected, finitely connected, noncompact manifold without boundary. If $g$ is a complete Riemannian metric on $\mathrm{\Sigma }$ whose Gaussian curvature $K$ is nonnegative at infinity, then $K$ must be integrable. In particular, we obtain a new short proof of the fact that if $\mathrm{\Sigma }$ admits a complete metric whose Gaussian curvature is nonnegative and positive at one point, then $\mathrm{\Sigma }$ is diffeomorphic to ${\mathbb{R}}^2$.