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

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 connected open Riemann surface endowed with complete conformal Riemannian metric, if the negative part its Gaussian curvature has finite mass, then is homeomorphic to interior compact boundary, thus it topological type. We will also show such parabolic 15 1958)].