Stability of the Wulff shape with respect to anisotropic curvature functionals

Abstract

For a function f which foliates a one-sided neighborhood 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.