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

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.

Rigidity of critical points for a nonlocal Ohta–Kawasaki energy

“…This and (21) give that η s (E) is small if so is c o , hence we are in the position to apply Theorem 1.5 in [8].…”

Short time existence of the classical solution to the fractional mean curvature flow

“…It is proven in [18] that the ball is the solution to the fractional isoperimetric problem, and we even have the sharp quantification of the isoperimetric inequality [17,19]. Related result is the generalization of the Alexandroff theorem [4,13], where the authors prove that the ball is the only smooth compact set with constant fractional mean curvature. Note that one does not need to assume the set to be connected, which is in contrast to the classical Alexandroff theorem.…”

Near-sphere lattices with constant nonlocal mean curvature

“…Moreover, since |θ − σ| 2 = 2(1 − σ · θ), we may rewrite the first integral in (5.5) to obtain the equality 6) which will be used in the proof of Lemma 5.1.…”

“…Thus, the new translated near-balls (or near-spheres) appear from infinity. We point out that it is necessary to consider infinite lattices in this problem -a finite disjoint union of two or more bounded sets cannot have constant NMC by the Alexandrov type rigidity result in [3,6].…”