Eleonora Cinti, Carlo Sinestrari and Enrico Valdinoci
Neckpinch singularities in fractional mean curvature flows
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ABSTRACT.  In this paper we consider the evolution of sets by a fractional mean curvature flow. 
Our main result states that there exists an embedded surface evolving by fractional mean curvature flow, which developes a singularity before it can shrink to a point. 
When the ambient dimension n is greater or equal to 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.