A sharp quantitative version of Alexandrov's theorem via the method of moving planes

We prove the following quantitative version of the celebrated Soap Bubble Theorem of Alexandrov. Let $S$ be a $C^2$ closed embedded hypersurface of ${\mathbb{R}}^{n+1}$, $n\ge 1$, and denote by $osc(H)$ the oscillation of its mean curvature. We prove that there exists a positive $\epsilon$, depending on $n$ and upper bounds on the area and the $C^2$-regularity of $S$, such that if $osc(H)\le \epsilon$ then there exist two concentric balls $B_{r_i}$ and $B_{r_e}$ such that $S\subset {\bar{B}}_{r_e}\setminus B_{r_i}$ and $r_e-r_i\le Cosc(H)$, with $C$ depending only on $n$ and upper bounds on the surface area of $S$ and the $C^2$ regularity of $S$. Our approach is based on a quantitative study of the method of moving planes and the quantitative estimate on $r_e-r_i$ we obtain is optimal. As a consequence of this theorem, we also prove that if $osc(H)$ is small then $S$ is diffeomorphic to a sphere and give a quantitative bound which implies that $S$ is $C^1$-close to a sphere.