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

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 {C^{2}} -distance from a single sphere. The corresponding stability inequality is obtained with a sharp decay rate. [ABSTRACT FROM AUTHOR]