Quantitative Stability for Anisotropic Nearly Umbilical Hypersurfaces

Abstract

We prove qualitative and quantitative stability of the following rigidity theorem: the only anisotropic totally umbilical closed hypersurface is the Wulff shape. Consider $$n \ge 2$$ , $$p\in (1, \, +\infty )$$ and $$\Sigma $$ an n-dimensional, closed hypersurface in $$\mathbb {R}^{n+1}$$ , which is the 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.