Stability of the ball under volume preserving fractional mean curvature flow

Abstract

We consider the volume constrained fractional mean curvature flow of a nearly spherical set, and prove long time existence and asymptotic convergence to a ball. The result applies in particular to convex initial data, under the assumption of global existence. Similarly, we show exponential convergence to a constant for the fractional mean curvature flow of a periodic graph.

Long time behaviour of the discrete volume preserving mean curvature flow in the flat torus

“…Building upon recent developments in the study of geometric flows, we are able to extend to all dimensions the aforementioned results on the dynamical stability of strictly stable sets in the flat torus, both for the surface diffusion flow and the volume preserving mean curvature flow. Specifically, we will employ a quantitative Alexandrov-type estimate recently established in [9], based on prior results in the Euclidean setting [32] (see also [10,7] for similar result in the nonlocal setting). Regarding our result, simply assuming the initial set to be close in the C 1,1 -topology to a strictly stable set, we obtain global existence and asymptotic convergence of both the flows to (a translated of) the underlying stable set.…”