Stability of the Wulff shape with respect to anisotropic curvature functionals

TL;DR

This paper establishes quantitative stability estimates for Wulff shapes under anisotropic curvature functionals, linking geometric deviations to the $L^p$-norm of the traceless $F$-Hessian of a foliation function.

Contribution

It provides new stability results for anisotropic geometric inequalities and boundary problems using a novel estimate involving the traceless $F$-Hessian.

Findings

Quantitative stability for anisotropic Heintze-Karcher inequality

Stability results for the anisotropic Alexandrov problem

Applications to Serrin-type overdetermined boundary value problems

Abstract

For a function $f$ which foliates a one-sided neighbourhood of a closed hypersurface $M$, we give an estimate of the distance of $M$ to a Wulff shape in terms of the $L^{p}$-norm of the traceless $F$-Hessian of $f$, where $F$ is the support function of the Wulff shape. This theorem is applied to prove quantitative stability results for the anisotropic Heintze-Karcher inequality, the anisotropic Alexandrov problem, as well as for the anisotropic overdetermined boundary value problem of Serrin-type.