Inserted: 27 jul 2016
Last Updated: 30 may 2018
Journal: Proc. Amer. Math. Soc.
In this paper we consider the evolution of sets by a fractional mean curvature flow. Our main result states that for any dimension $n \geq 2$, there exists an embedded surface in \mathbb Rn evolving by fractional mean curvature flow, which developes a singularity before it can shrink to a point. When $n \geq 3$ this result generalizes the analogue result of Grayson for the classical mean curvature flow. Interestingly, when $n = 2$, our result provides instead a counterexample in the nonlocal framework to the well known Grayson Theorem, which states that any smooth embedded curve in the plane evolving by (classical) MCF shrinks to a point.