Symmetry of hypersurfaces and the Hopf Lemma

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

• symmetry
• hypersurfaces
• mean curvature
• higher order curvature
• hopf lemma