Stability from rigidity via umbilicity

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.

💡 Research Summary

The paper develops a unified quantitative framework for a broad class of geometric stability problems concerning hypersurfaces in space forms (Euclidean, hyperbolic, and spherical). The central idea is to control the Hausdorff distance between an embedded closed hypersurface M and a geodesic sphere S by an Lᵖ‑norm of the traceless part of the Hessian of a level‑set function that defines the domain Ω bounded by M.

The authors first prove a “level‑set stability” theorem (Theorem 1.1). Let (N, ḡ) be a conformally flat domain, M a smooth closed hypersurface, and U a one‑sided neighbourhood of M foliated by the level sets of a C² function f with f|_M=0 and ∇̄f≠0. Assuming a uniform bound C₀ on the surface area, the Lᵖ‑norm of the second fundamental form of the level sets, and the C⁰‑norm of the conformal factor, they show that if the Cauchy–Schwarz deficit
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