Nonlocal Delaunay surfaces

TL;DR

This paper constructs nonlocal Delaunay surfaces that minimize a periodic nonlocal perimeter functional, revealing their structure and behavior for small volumes, with explicit bounds and asymptotic properties.

Contribution

It introduces a new class of nonlocal Delaunay surfaces, extending classical geometric concepts to a nonlocal setting with explicit quantitative analysis.

Findings

Surfaces are close to periodic arrays of balls for small volume

Explicit bounds on the concentration of mass in the constructed surfaces

Surfaces exhibit periodic and cylindrically symmetric properties

Abstract

We construct codimension 1 surfaces of any dimension that minimize a periodic nonlocal perimeter functional among surfaces that are periodic, cylindrically symmetric and decreasing. These surfaces may be seen as a nonlocal analogue of the classical Delaunay surfaces (onduloids). For small volume, most of their mass tends to be concentrated in a periodic array and the surfaces are close to a periodic array of balls (in fact, we give explicit quantitative bounds on these facts).