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

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