Asymptotic of the discrete volume-preserving fractional mean curvature flow via a nonlocal quantitative Alexandrov theorem

Abstract

We characterize the long time behaviour of a discrete-in-time approximation of the volume preserving fractional mean curvature flow. In particular, we prove that the discrete flow starting from any bounded set of finite fractional perimeter converges exponentially fast to a single ball if the dimension N≤7 and the fractional esponent s≈1, or for any s∈(0,1) when N=2. As an intermediate result we establish a fractional quantitative Alexandrov type estimate for normal deformations of a ball. Finally, we provide existence for flat flows as limit points of the discrete flow when the time discretization parameter tends to zero. Furthermore, analogous results for the classical perimeter are obtained in the limit s→1.