Delaunay hypersurfaces with constant nonlocal mean curvature

We study hypersurfaces of ${\mathbb{R}}^N$ with constant nonlocal (or fractional) mean curvature. This is the equation associated to critical points of the fractional perimeter functional under a volume constraint. We establish the existence of a smooth branch of periodic cylinders in ${\mathbb{R}}^N$, $N\ge 2$, all of them with the same constant nonlocal mean curvature, and bifurcating from a straight cylinder. These are Delaunay type cylinders in the nonlocal setting. The proof uses the Crandall-Rabinowitz theorem applied to a quasilinear type fractional elliptic equation.