### Abstract

We prove the following quantitative version of the celebrated Soap Bubble Theorem of Alexandrov. Let S be a C-2 closed embedded hypersurface of Rn+1, n >= 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) <= epsilon then there exist two concentric balls B-ri and B-re such that S subset of (B) over bar (re) \ B-ri and r(e) - r(i) <= C 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.

Original language | English |
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Pages (from-to) | 261-299 |

Number of pages | 39 |

Journal | Journal of the European Mathematical Society |

Volume | 20 |

Publication status | Published - 2018 |

### All Science Journal Classification (ASJC) codes

- Mathematics(all)
- Applied Mathematics

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## Cite this

Ciraolo, G., & Vezzoni, L. (2018). A sharp quantitative version of Alexandrov's theorem via the method of moving planes.

*Journal of the European Mathematical Society*,*20*, 261-299.