Local and nonlocal almost-constant mean curvature hypersurfaces

Alexandrov's theorem asserts that a (bounded, embedded) constant mean curvature (cmc) hypersurface must be a sphere. It is well-known that if this condition is relaxed and the mean curvature is just assumed to be close to a constant, then the corresponding hypersurfaces does not need to be close to a sphere. Indeed any family of nearby spheres with equal radii connected by short catenoidal necks can be slightly perturbed to obtain examples of almost-cmc hypersurfaces (Kapouleas, Butscher, Mazzeo).

We show that these examples actually capture the only possible behavior of almost-cmc hypersurfaces, by proving various quantitative bounds on the distance between an almost-cmc hypersurface and a collection of tangent spheres of equal radii in terms of their mean curvature oscillation. This is a joint work with G. Ciraolo (U Palermo).

We next discuss these issues for the nonlocal mean curvature introduced by Caffarelli and Souganidis, showing in particular a remarkable rigidity property of the nonlocal problem which prevents bubbling phenomena, in other words, every nonlocal almost-cmc hypersurface must be close to a single sphere. This is a joint work with G. Ciraolo, A. Figalli (UT Austin) and M. Novaga (U Pisa).