2021/03/12 by Julian Scheuer, Scheuer, Julian · 1 citation
Engineering · Mathematics · #Advanced Numerical Analysis Techniques #Analysis of PDEs (math.AP) #Differential Geometry (math.DG) #FOS: Mathematics #Geometric Analysis and Curvature Flows #Point processes and geometric inequalities
paper · doi:10.48550/arxiv.2103.07178
openalex publication_date 2021/03/12 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
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.