Symmetry of hypersurfaces and the Hopf Lemma

YanYan Li; YanYan Li

doi:10.3934/mine.2023084

Mathematics in Engineering

2023, Volume 5, Issue 5: 1-9. doi: 10.3934/mine.2023084

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Symmetry of hypersurfaces and the Hopf Lemma

YanYan Li ^,

Department of Mathematics, Rutgers University, 110 Frelinghuysen Road, Piscataway, NJ 08854, USA

Received: 09 May 2022 Revised: 19 February 2023 Accepted: 05 April 2023 Published: 26 April 2023

A classical theorem of A. D. Alexandrov says that a connected compact smooth hypersurface in Euclidean space with constant mean curvature must be a sphere. We give exposition to some results on symmetry properties of hypersurfaces with ordered mean curvature and associated variations of the Hopf Lemma. Some open problems will be discussed.
- symmetry,
- hypersurfaces,
- mean curvature,
- higher order curvature,
- Hopf Lemma
Citation: YanYan Li. Symmetry of hypersurfaces and the Hopf Lemma[J]. Mathematics in Engineering, 2023, 5(5): 1-9. doi: 10.3934/mine.2023084

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Abstract

A classical theorem of A. D. Alexandrov says that a connected compact smooth hypersurface in Euclidean space with constant mean curvature must be a sphere. We give exposition to some results on symmetry properties of hypersurfaces with ordered mean curvature and associated variations of the Hopf Lemma. Some open problems will be discussed.

References

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