Rigidity and sharp stability estimates for hypersurfaces with constant and almost-constant nonlocal mean curvature

TL;DR

This paper proves that smooth bounded sets with constant nonlocal mean curvature are spheres and establishes a sharp stability estimate linking the Lipschitz constant of the curvature to the set's proximity to a sphere.

Contribution

It provides a rigidity result for nonlocal mean curvature and introduces a sharp stability inequality controlling the shape deviation from a sphere.

Findings

Boundaries with constant nonlocal mean curvature are spheres.

The Lipschitz constant of the nonlocal mean curvature controls the $C^2$-distance to a sphere.

A sharp decay rate in the stability inequality is established.

Abstract

We prove that the boundary of a (not necessarily connected) bounded smooth set with constant nonlocal mean curvature is a sphere. More generally, and in contrast with what happens in the classical case, we show that the Lipschitz constant of the nonlocal mean curvature of such a boundary controls its $C^2$-distance from a single sphere. The corresponding stability inequality is obtained with a sharp decay rate.