Near-sphere lattices with constant nonlocal mean curvature

TL;DR

This paper constructs and analyzes unbounded sets with constant nonlocal mean curvature, formed by unions of a bounded domain and its lattice translations, revealing their asymptotic shapes and bifurcation from the sphere.

Contribution

It introduces a new class of CNMC sets composed of periodic unions of near-spheres, expanding understanding of nonlocal curvature phenomena and bifurcation structures.

Findings

Constructed CNMC sets as unions of a domain and lattice translations.

Identified a $C^2$ branch of solutions emanating from the sphere.

Determined the asymptotic shape of these sets as the parameter grows.

Abstract

We are concerned with unbounded sets of $\mathbb{R}^N$ whose boundary has constant nonlocal (or fractional) mean curvature, which we call CNMC sets. This is the equation associated to critical points of the fractional perimeter functional under a volume constraint. We construct CNMC sets which are the countable union of a certain bounded domain and all its translations through a periodic integer lattice of dimension $M\leq N$. Our CNMC sets form a $C^2$ branch emanating from the unit ball alone and where the parameter in the branch is essentially the distance to the closest lattice point. Thus, the new translated near-balls (or near-spheres) appear from infinity. We find their exact asymptotic shape as the parameter tends to infinity.