Multiply-periodic hypersurfaces with constant nonlocal mean curvature

TL;DR

This paper investigates hypersurfaces with fractional mean curvature in Euclidean space, establishing the existence of multiply-periodic solutions that bifurcate from hyperplanes, advancing understanding of nonlocal geometric variational problems.

Contribution

It introduces new existence results for multiply-periodic hypersurfaces with constant nonlocal mean curvature using bifurcation techniques.

Findings

Existence of smooth multiply-periodic hypersurfaces with constant fractional mean curvature.

Bifurcation from parallel hyperplanes demonstrated.

Use of local inversion methods in nonlocal geometric analysis.

Abstract

We study hypersurfaces with fractional mean curvature in N-dimensional Euclidean space. These hypersurfaces are critical points of the fractional perimeter under a volume constraint. We use local inversion arguments to prove existence of smooth branches of multiply-periodic hypersurfaces bifurcating from suitable parallel hyperplanes.