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A Gap Theorem for Free Boundary Minimal Surfaces in Geodesic Balls of Hyperbolic Space and Hemisphere

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Abstract

In this paper we provide a pinching condition for the characterization of the totally geodesic disk and the rotational annulus among minimal surfaces with free boundary in geodesic balls of three-dimensional hyperbolic space and hemisphere. The pinching condition involves the length of the second fundamental form, the support function of the surface, and a natural potential function in hyperbolic space and hemisphere.

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Acknowledgements

The authors would like to thank the referees for careful reading of the paper and for the valuable suggestions and comments which made this paper better and more readable. They wish to thank Professor Ben Andrews for his interest in this work. The second author is also grateful to Professor Jaigyoung Choe for discussions on minimal surfaces in hyperbolic space at the workshop on Nonlinear and Geometric Partial Differential Equations at Kioloa Campus of ANU, 2016, Australia. The first author was supported by NSFC Grant No. 11671224. The second author was supported by a postdoctoral fellowship funded via ARC Laureate Fellowship FL150100126.

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Authors and Affiliations

  1. Department of Mathematical Sciences, Tsinghua University, Beijing, 100084, People’s Republic of China

    Haizhong Li

  2. Mathematical Sciences Institute, Australian National University, Canberra, ACT, 2601, Australia

    Changwei Xiong

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  1. Haizhong Li
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  2. Changwei Xiong
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Correspondence to Changwei Xiong.

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Li, H., Xiong, C. A Gap Theorem for Free Boundary Minimal Surfaces in Geodesic Balls of Hyperbolic Space and Hemisphere. J Geom Anal 28, 3171–3182 (2018). https://doi.org/10.1007/s12220-017-9953-6

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  • DOI: https://doi.org/10.1007/s12220-017-9953-6

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