A note on the Gaussian curvature on noncompact surfaces

TL;DR

This paper provides a concise proof that on noncompact surfaces with complete metrics and nonnegative Gaussian curvature at infinity, the curvature is integrable, leading to a topological classification of such surfaces.

Contribution

It offers a new, simplified proof of the integrability of Gaussian curvature and the topological characterization of noncompact surfaces with nonnegative curvature.

Findings

Gaussian curvature is integrable under given conditions

Noncompact surfaces with positive curvature at one point are diffeomorphic to 

Provides a shorter proof of a classical topological result

Abstract

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