A $W^{2,p}$-estimate for nearly umbilical hypersurfaces

Abstract

Let $n\ge 2$, $p\in (1,+\infty )$ be given and let $\Sigma$ be a $n$-dimensional, closed hypersurface in ${\mathbb{R}}^{n+1}$. Denote by $A$ its second fundamental form, and by $\mathring{A}$ the tensor $A-\frac{1}{n}{A}_i^{i}g$ where $g=\delta {|}_{\Sigma }$.Assuming that $\Sigma$ is the boundary of a convex, open set we prove that if the $L^p$-norm of $\mathring{A}$ is small, then $\Sigma$ must be $W^{2,p}$-close to a sphere, with a quantitative estimate.