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 a connected open Riemann surface endowed with a complete conformal Riemannian metric, if the negative part of its Gaussian curvature has finite mass, th...

Full description

Saved in:
Bibliographic Details
Published inGeometriae dedicata Vol. 217; no. 3
Main Author Zhou, Chen
Format Journal Article
LanguageEnglish
Published Dordrecht Springer Netherlands 01.06.2023
Springer Nature B.V
Subjects
Algebraic Geometry
Convex and Discrete Geometry
Differential Geometry
Hyperbolic Geometry
Mathematics
Mathematics and Statistics
Original Paper
Projective Geometry
Riemann surfaces
Theorem proving
Theorems
Topology
Conformal metric
Finite topological type
Potential theory
30F20
Gaussian curvature
53C21
Online AccessGet full text

Cover

More Information
Summary: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)].
ISSN:0046-5755
1572-9168
DOI:10.1007/s10711-023-00769-z