JournalsjemsVol. 20, No. 2pp. 261–299

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

  • Giulio Ciraolo

    Università di Palermo, Italy
  • Luigi Vezzoni

    Università di Torino, Italy
A sharp quantitative version of Alexandrov's theorem via the method of moving planes cover

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Abstract

We prove the following quantitative version of the celebrated Soap Bubble Theorem of Alexandrov. Let SS be a C2C^2 closed embedded hypersurface of Rn+1\mathbb{R}^{n+1}, n1n\geq1, and denote by osc(H)(H) the oscillation of its mean curvature. We prove that there exists a positive ε\varepsilon, depending on nn and upper bounds on the area and the C2C^2-regularity of SS, such that if osc(H)ε(H) \leq \varepsilon then there exist two concentric balls BriB_{r_i} and BreB_{r_e} such that SBreBriS \subset \overline{B}_{r_e} \setminus B_{r_i} and reriCosc(H)r_e -r_i \leq C \, \mathrm {osc}(H), with CC depending only on nn and upper bounds on the surface area of SS and the C2C^2 regularity of SS. Our approach is based on a quantitative study of the method of moving planes, and the quantitative estimate on rerir_e-r_i we obtain is optimal.

As a consequence, we also prove that if osc(H)(H) is small then SS is diffeomorphic to a sphere, and give a quantitative bound which implies that SS is C1C^1-close to a sphere.

Cite this article

Giulio Ciraolo, Luigi Vezzoni, A sharp quantitative version of Alexandrov's theorem via the method of moving planes. J. Eur. Math. Soc. 20 (2018), no. 2, pp. 261–299

DOI 10.4171/JEMS/766