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.

A sharp quantitative version of Alexandrov's theorem via the method of moving planes / G. Ciraolo, L. Vezzoni. - In: JOURNAL OF THE EUROPEAN MATHEMATICAL SOCIETY. - ISSN 1435-9855. - 20:2(2018), pp. 261-299. [10.4171/JEMS/766]

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

G. Ciraolo;
2018

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.
Alexandrov Soap Bubble Theorem; method of moving planes; stability; mean curvature; pinching
Settore MAT/05 - Analisi Matematica
2018
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Utilizza questo identificativo per citare o creare un link a questo documento: https://hdl.handle.net/2434/675071
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