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

TL;DR

This paper presents a shorter proof of Huber's theorem, establishing that certain complete Riemann surfaces with finite negative curvature mass are topologically finite and parabolic, enhancing understanding of their geometric structure.

Contribution

It provides a new, more concise proof of Huber's theorem, clarifying the topological and parabolic properties of specific Riemann surfaces.

Findings

Riemann surface is homeomorphic to a compact surface with boundary

Such surfaces have finite topological type

They are parabolic

Abstract

In this paper we give a new, and shorter, proof of Huber's theorem 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.