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


This is a joint work with Mouhamed M. Fall, Joan Sol- Morales and Tobias Weth. It concerns hypersurfaces of RN with con- stant 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 ana- logue of the Alexandrov result characterizing spheres as the only closed embedded hypersurfaces in RN with constant mean curvature. Here we use the moving planes method. Our second result establishes the existence of periodic bands or “cylinders” in R2 with constant nonlo- cal 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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