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Research Article | Open Access

Symmetry of hypersurfaces and the Hopf Lemma

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

Keywords

symmetryhypersurfacesmean curvaturehigher order curvatureHopf Lemma

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Mathematics in Engineering
Volume 5 Issue 5,
October 2023
Pages 1-9
DOI: 10.3934/mine.2023084

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Cite this article:
Li Y. Symmetry of hypersurfaces and the Hopf Lemma. Mathematics in Engineering, 2023, 5(5): 1-9. https://doi.org/10.3934/mine.2023084

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Received: 09 May 2022
Revised: 19 February 2023
Accepted: 05 April 2023
Published: 15 October 2023
©2023 the Author(s), licensee AIMS Press.

This is an open access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/4.0)