Existence and symmetry of periodic nonlocal-CMC surfaces via variational methods

TL;DR

This paper proves the existence of periodic nonlocal constant mean curvature surfaces using variational methods, demonstrating symmetry and monotonicity properties that address previous open problems in the field.

Contribution

It provides the first variational proof of periodic nonlocal-CMC surfaces, establishing their existence, symmetry, and monotonicity, extending classical results to nonlocal settings.

Findings

Existence of periodic nonlocal-CMC surfaces proven.

Surfaces are cylindrically symmetric and monotone.

Addresses and solves an open problem regarding minimizers.

Abstract

This paper provides the first variational proof of the existence of periodic nonlocal-CMC surfaces. These are nonlocal analogues of the classical Delaunay cylinders. More precisely, we show the existence of a set in $\mathbb{R}^n$ which is periodic in one direction, has a prescribed (but arbitrary) volume within a slab orthogonal to that direction, has constant nonlocal mean curvature, and minimizes an appropriate periodic version of the fractional perimeter functional under the volume constraint. We show, in addition, that the set is cylindrically symmetric and, more significantly, that it is even as well as nonincreasing on half its period. This monotonicity property solves an open problem and an obstruction which arose in an earlier attempt, by other authors, to show the existence of minimizers.