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.