Symmetry of hypersurfaces and the Hopf Lemma

TL;DR

This paper explores symmetry properties of hypersurfaces with constant or ordered mean curvature, extending classical results like Alexandrov's theorem, and discusses variations of the Hopf Lemma and related open problems.

Contribution

It provides an exposition on symmetry results for hypersurfaces with ordered mean curvature and introduces variations of the Hopf Lemma, highlighting open problems in the field.

Findings

Connected compact hypersurfaces with constant mean curvature are spheres

Symmetry properties extend to hypersurfaces with ordered mean curvature

Discussion of open problems related to the Hopf Lemma variations

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.