Neckpinch singularities in fractional mean curvature flows

Abstract

In this paper we consider the evolution of sets by a fractional mean curvature flow. Our main result states that for any dimension $n>2$, 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 $n>3$ 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.