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

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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)].

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Acknowledgements

I would like to thank Professor Mohan Ramachandran for giving me this interesting problem and also providing many enlightening and helpful suggestions about it. His instruction through both face-to-face conversations and emails is crucial for me to find a new proof of Huber’s theorem.

Funding

Research is supported by the following two foundations: China Postdoctoral Science Foundation. Grant No. 2021M693676. National Natural Science Foundation of China. Grant No. 12101620.

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  1. Sun Yat-sen University, Zhuhai, Guangdong, China

    Chen Zhou

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  1. Chen Zhou
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Correspondence to Chen Zhou.

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Zhou, C. A new proof of Huber’s theorem on differential geometry in the large. Geom Dedicata 217, 48 (2023). https://doi.org/10.1007/s10711-023-00769-z

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