*Accepted Paper*

**Inserted:** 27 jul 2016

**Last Updated:** 30 may 2018

**Journal:** Proc. Amer. Math. Soc.

**Year:** 2016

**Abstract:**

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 R^{n} 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.

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