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- Title
A sharp quantitative version of Alexandrov's theorem via the method of moving planes.
- Authors
Ciraolo, Giulio; Vezzoni, Luigi
- Abstract
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 e, 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, 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.
- Subjects
MATHEMATICS theorems; HYPERSURFACES; CURVATURE; DISBUDDING; QUANTITATIVE research
- Publication
Journal of the European Mathematical Society (EMS Publishing), 2018, Vol 20, Issue 2, p261
- ISSN
1435-9855
- Publication type
Article
- DOI
10.4171/JEMS/766