Geodesic
A sharp quantitative version of Alexandrov's theorem via the method of moving planes
Journal of the European Mathematical Society, Tome 20 (2018) no. 2, pp. 261-299.

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We prove the following quantitative version of the celebrated Soap Bubble Theorem of Alexandrov. Let S be a C2 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 ε, depending on n and upper bounds on the area and the C2-regularity of S, such that if osc(H)≤ε then there exist two concentric balls Bri​​ and Bre​​ such that S⊂Bre​​∖Bri​​ and re​−ri​≤Cosc(H), with C depending only on n and upper bounds on the surface area of S and the C2 regularity of S. Our approach is based on a quantitative study of the method of moving planes, and the quantitative estimate on re​−ri​ we obtain is optimal.
DOI : 10.4171/jems/766
Classification : 35-XX, 53-XX
Keywords: Alexandrov Soap Bubble Theorem, method of moving planes, stability, mean curvature, pinching 
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     author = {Giulio Ciraolo and Luigi Vezzoni},
     title = {A sharp quantitative version of {Alexandrov's} theorem via the method of moving planes},
     journal = {Journal of the European Mathematical Society},
     pages = {261--299},
     publisher = {mathdoc},
     volume = {20},
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     year = {2018},
     doi = {10.4171/jems/766},
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Giulio Ciraolo; Luigi Vezzoni. A sharp quantitative version of Alexandrov's theorem via the method of moving planes. Journal of the European Mathematical Society, Tome 20 (2018) no. 2, pp. 261-299. doi : 10.4171/jems/766. http://geodesic.mathdoc.fr/articles/10.4171/jems/766/

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