A new proof of Huber’s theorem on differential geometry in the large

Abstract

In this paper we give a new, and shorter, proof of Huber’s theorem [Theorem 13 in Huber (Comment Math Helve 32:13–72, 1958)] which affirms that for a connected open Riemann surface endowed with a complete conformal Riemannian metric, if the negative part of its Gaussian curvature has finite mass, then the Riemann surface is homeomorphic to the interior of a compact surface with boundary, and thus it has finite topological type. We will also show that such Riemann surface is parabolic [Theorem 15 in Huber (Comment Math Helve 32:13–72, 1958)].