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We prove the following quantitative version of the celebrated Soap Bubble Theorem of Alexandrov. Let be a closed embedded hypersurface of , , and denote by osc the oscillation of its mean curvature. We prove that there exists a positive , depending on and upper bounds on the area and the -regularity of , such that if osc then there exist two concentric balls and such that and , with depending only on and upper bounds on the surface area of and the regularity of . Our approach is based on a quantitative study of the method of moving planes, and the quantitative estimate on we obtain is optimal.
As a consequence, we also prove that if osc is small then is diffeomorphic to a sphere, and give a quantitative bound which implies that is -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