Delaunay hypersurfaces with constant nonlocal mean curvature

Abstract

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 $\mathbb{R}^N$, $N\geq 2$, 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.