Quantitative stability for anisotropic nearly umbilical hypersurfaces

Abstract

We prove a qualitative and a quantitative stability of the following rigidity theorem: an anisotropic totally umbilical closed hypersurface is the Wulff shape. Consider $n \geq 2$, $p\in (1, \, +\infty)$ and $\Sigma$ an $n$-dimensional, closed hypersurface in $\mathbb{R}^{n+1}$, boundary of a convex, open set. We show that if the $L^p$ norm of the trace-free part of the anisotropic second fundamental form is small, then $\Sigma$ must be $W^{2, \, p}$-close to the Wulff shape, with a quantitative estimate.