Stability from rigidity via umbilicity

Abstract

We consider a range of geometric stability problems for hypersurfaces of spaceforms. One of the key results is an estimate relating the distance to a geodesic sphere of an embedded hypersurface with integral norms of the traceless Hessian operator of a level set function for the open set bounded by the hypersurface. As application, we give a unified treatment of many old and new stability problems arising in geometry and analysis. Those problems ask for spherical closeness of a hypersurface, given a geometric constraint. Examples include stability in Alexandroff's soap bubble theorem in space forms, Serrin's overdetermined problem, a Steklov problem involving the bi-Laplace operator and non-convex Alexandroff-Fenchel inequalities.

34 pages. Major revision of the first version: More explicit exponents, relaxed restrictions on constants

References in corpus (4)

• A fully-nonlinear flow and quermassintegral inequalities in the sphere
• Stability for Serrin's problem and Alexandroff's theorem in warped product manifolds
• Quantitative oscillation estimates for almost-umbilical closed hypersurfaces in Euclidean space
• Spectral geometry of the Steklov problem

Cited by in corpus (1)

• On compact embbeded Weingarten hypersurfaces in warped products