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

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.