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arXiv · 2103.07178

Stability from rigidity via umbilicity

Julian Scheuer
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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.

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Julian Scheuer. 2021-03-12. Stability from rigidity via umbilicity. https://doi.org/10.1515/acv-2023-0119

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