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Free boundary minimal surfaces in the unit ball : recent advances and open questions

Martin Man-chun Li Chinese University of Hong Kong

Differential Geometry Geometric Analysis and Geometric Topology mathscidoc:2001.10006

arXiv preprint arXiv:1907.05053, 2019.7
In this survey, we discuss some recent results on free boundary minimal surfaces in the Euclidean unit-ball. The subject has been a very active field of research in the past few years due to the seminal work of Fraser and Schoen on the extremal Steklov eigenvalue problem. We review several different techniques of constructing examples of embedded free boundary minimal surfaces in the unit ball. Next, we discuss some uniqueness results for free boundary minimal disks and the conjecture about the uniqueness of critical catenoid. We also discuss several Morse index estimates for free boundary minimal surfaces. Moreover, we describe estimates for the first Steklov eigenvalue on such free boundary minimal surfaces and various smooth compactness results. Finally, we mention some sharp area bounds for free boundary minimal submanifolds and related questions.
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  • to appear in Proceedings of the first annual meeting of the ICCM
@inproceedings{martin2019free,
  title={Free boundary minimal surfaces in the unit ball : recent advances and open questions},
  author={Martin Man-chun Li},
  url={http://archive.ymsc.tsinghua.edu.cn/pacm_paperurl/20200103164644640713620},
  booktitle={arXiv preprint  arXiv:1907.05053},
  year={2019},
}
Martin Man-chun Li. Free boundary minimal surfaces in the unit ball : recent advances and open questions. 2019. In arXiv preprint arXiv:1907.05053. http://archive.ymsc.tsinghua.edu.cn/pacm_paperurl/20200103164644640713620.
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