## Abstract

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.

Original language | English |
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Pages (from-to) | 2637-2646 |

Number of pages | 10 |

Journal | Proceedings of the American Mathematical Society |

Volume | 146 |

Issue number | 6 |

DOIs | |

Publication status | Published - 2018 |

Externally published | Yes |