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

TL;DR

This paper studies hypersurfaces with constant nonlocal mean curvature, proving a nonlocal analogue of Alexandrov's sphere characterization and constructing Delaunay-type periodic bands in the plane.

Contribution

It establishes the first nonlocal Alexandrov-type result and constructs nonlocal Delaunay-type solutions bifurcating from straight bands.

Findings

Spheres are the only closed embedded hypersurfaces with constant nonlocal mean curvature.

Existence of periodic nonlocal mean curvature bands in the plane bifurcating from straight bands.

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.