Neckpinch singularities in fractional mean curvature flows

In this paper we consider the evolution of boundaries of sets by a fractional mean curvature flow. We show that for any dimension n >=2, there exist embedded hypersurfaces in R^n which develop a singularity without shrinking to a point. Such examples are well known for the classical mean curvature flow for n >=3. Interestingly, when n = 2, our result provides instead a counterexample in the nonlocal framework to the well-known Grayson’s Theorem, which states that any smooth embedded curve in the plane evolving by (classical) MCF shrinks to a point. The essential step in our construction is an estimate which ensures that a suitably small perturbation of a thin strip has positive fractional curvature at every boundary point.