# On compact hypersurfaces with almost constant mean curvature

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Giulio Ciraolo

created by daniele on 02 Jun 2015

modified on 18 Jun 2015

13 jul 2015
-- 14:30

Torino

Differential Geometry Seminars at Università di Torino

**Abstract.**

Alexandrov’s theorem asserts that spheres are the only closed compact embedded hypersurfaces with constant mean curvature in R^{n.} In this talk we will discuss some quantitative versions of Alexandrov’s theorem, i.e. we will consider a hypersurface with mean curvature close to a constant and quantitatively describe its proximity to a ball or a collection of tangent balls of equal radii in terms of the oscillation of the mean curvature.

We will also discuss these issues for the nonlocal mean curvature introduced by Caffarelli and Souganidis, showing a remarkable rigidity property of the nonlocal problem which prevents bubbling phenomena and proving the proximity to a single sphere.

References:

1. G. Ciraolo, L. Vezzoni. A sharp quantitative version of Alexandrov's theorem via the method of moving planes. Preprint.(arXiv:1501.07845)

2. G. Ciraolo, A. Figalli, F. Maggi, M. Novaga. Rigidity and sharp stability estimates for hypersurfaces with constant and almost-constant nonlocal mean curvature. Preprint. (arXiv:1503.00653)

3. G. Ciraolo, F. Maggi. On the shape of compact hypersurfaces with almost constant mean curvature. Preprint. (arXiv:1503.06674)