Periodic Delaunay cylinders with constant anisotropic nonlocal mean curvature

TL;DR

This paper proves the existence and symmetry of periodic surfaces with constant anisotropic nonlocal mean curvature, extending classical Delaunay results to a nonlocal and anisotropic setting, with implications for geometric analysis.

Contribution

It generalizes Delaunay's classical results to anisotropic nonlocal mean curvature surfaces, establishing existence, symmetry, and bifurcation properties in this new setting.

Findings

Existence of periodic surfaces with constant anisotropic nonlocal mean curvature.

Symmetry properties of these surfaces are established.

Construction of Delaunay near-cylinders bifurcating from straight cylinders.

Abstract

In this article we prove existence and symmetry properties of periodic surfaces of revolution with constant anisotropic nonlocal mean curvature, generalizing a classical result of Delaunay to the anisotropic nonlocal setting. First, by studying the corresponding periodic isoperimetric problem, under natural assumptions on the kernel, we use rearrangement inequalities to extend a periodic version of the Wulff inequality to the nonlocal setting. This leads to the existence and symmetry properties of minimizers for every given volume in each period, thus generalizing the results of Cabr\'e, Csat\'o, and Mas to the anisotropic case. Second, under the same hypotheses on the kernel, we prove the existence of a one-parameter family of Delaunay near-cylinders in $\mathbb{R}^2$ bifurcating from a straight cylinder and having each constant anisotropic mean curvature. This extends the results of Cabr\'e, Fall, Sol\`a-Morales, and Weth to the anisotropic case. The stability of these near-cylinders will be studied in a forthcoming paper.