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Stability from rigidity via umbilicity

  • Julian Scheuer EMAIL logo
Published/Copyright: April 28, 2024

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.


Communicated by Francesca Da Lio


Award Identifier / Grant number: SCHE 1879/3-1

Funding statement: This work was partially funded by the “Deutsche Forschungsgemeinschaft” (DFG, German research foundation); Project “Quermassintegral preserving local curvature flows”; Grant number SCHE 1879/3-1.

Acknowledgements

This work was made possible through a research scholarship the author received from the DFG and which was carried out at Columbia University in New York. The author would like to thank the DFG, Columbia University and especially Professor Simon Brendle for their support.

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Received: 2023-10-30
Accepted: 2024-03-10
Published Online: 2024-04-28
Published in Print: 2025-04-01

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