Stability of the Wulff shape with respect to anisotropic curvature functionals

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.

arXiv comment24 pages

References in corpus (5)

• Bubbling with $L^2$-almost constant mean curvature and an Alexandrov-type theorem for crystals
• Alexandrov's theorem for anisotropic capillary hypersurfaces in the half-space
• Absence of bubbling phenomena for non convex anisotropic nearly umbilical and quasi Einstein hypersurfaces
• Stability for Serrin's problem and Alexandroff's theorem in warped product manifolds
• Capillary Schwarz symmetrization in the half-space