Quantitative Studies of Alexadrov's Theorem

Abstract: Alexandrov's soap bubbles Theorem states that the spheres are the only closed, connected, and embedded hypersurfaces with constant mean curvature in the Euclidean space. The theorem holds true also in the so-called space forms and for more general functions of the principal curvatures.

In the talk we will present the classical result by Alexandrov together with two proofs: the original one based on the, nowadays called, method of moving planes and another one based on integral inequalities. Then we will show a quantitative stability result for hypersurfaces with almost constant mean curvature. In particular, we will consider hypersurfaces, satisfying the so-called uniform touching ball condition, whose mean curvature is close to a constant and we will quantitatively describe, in terms of the oscillation of the mean curvature, the closedness to a single ball.

This is based on a joint work with G. Ciraolo and L. Vezzoni.

differential geometry

Audience: researchers in the topic

( video )

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The 2nd Geometric Analysis Festival

Series comments: Please, submit your questions to speakers at forms.gle/F8z5LjfNNwt3DD4i8

The recorded lecture videos are available at www.youtube.com/channel/UC2gHzqcv7CT1G3fa4a7sx-Q/playlists

Video Abstracts: youtu.be/uMBqmW4r8GE

For more details, please visit the webpage (will be updated later) at cosmogeometer.wordpress.com/geometric-analysis

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