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

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