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

@inproceedings{Gioffr2016A,
  title={A \$W^\{2, \, p\}\$-estimate for nearly umbilical hypersurfaces},
  author={Stefano Gioffr{\`e}},
  year={2016}
}
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. 
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