# Journal of the European Mathematical Society

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**Volume 20, Issue 2, 2018, pp. 261–299**

**DOI: 10.4171/JEMS/766**

Published online: 2018-01-31

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

Giulio Ciraolo^{[1]}and Luigi Vezzoni

^{[2]}(1) Università di Palermo, Italy

(2) Università di Torino, Italy

We prove the following quantitative version of the celebrated *Soap Bubble Theorem* of Alexandrov. Let $S$ be a $C^2$ closed embedded hypersurface of $\mathbb{R}^{n+1}$, $n\geq1$, and denote by osc$(H)$ the oscillation of its mean curvature. We prove that there exists a positive $\varepsilon$, depending on $n$ and upper bounds on the area and the $C^2$-regularity of $S$, such that if osc$(H) \leq \varepsilon$ then there exist two concentric balls $B_{r_i}$ and $B_{r_e}$ such that $S \subset \overline{B}_{r_e} \setminus B_{r_i}$ and $r_e -r_i \leq C \, \mathrm {osc}(H)$, 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, 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.

*Keywords: *Alexandrov Soap Bubble Theorem, method of moving planes, stability, mean curvature, pinching

Ciraolo Giulio, Vezzoni Luigi: A sharp quantitative version of Alexandrov's theorem via the method of moving planes. *J. Eur. Math. Soc.* 20 (2018), 261-299. doi: 10.4171/JEMS/766