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arXiv · 1503.00469

Curves and surfaces with constant nonlocal mean curvature: meeting Alexandrov and Delaunay

Xavier Cabre·Mouhamed Moustapha Fall·Joan Solà-Morales·Tobias Weth
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Abstract

We are concerned with hypersurfaces of $\mathbb{R}^N$ with constant nonlocal (or fractional) mean curvature. This is the equation associated to critical points of the fractional perimeter under a volume constraint. Our results are twofold. First we prove the nonlocal analogue of the Alexandrov result characterizing spheres as the only closed embedded hypersurfaces in $\mathbb{R}^N$ with constant mean curvature. Here we use the moving planes method. Our second result establishes the existence of periodic bands or "cylinders" in $\mathbb{R}^2$ with constant nonlocal mean curvature and bifurcating from a straight band. These are Delaunay type bands in the nonlocal setting. Here we use a Lyapunov-Schmidt procedure for a quasilinear type fractional elliptic equation.

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Xavier Cabre, Mouhamed Moustapha Fall, Joan Solà-Morales, Tobias Weth. 2015-03-02. Curves and surfaces with constant nonlocal mean curvature: meeting Alexandrov and Delaunay. https://arxiv.org/abs/1503.00469

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