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

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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 ≤ C osc(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. As a consequence of this theorem, 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 C¹-close to a sphere.