### 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 |
---|---|

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 |

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### Cite this

*Proceedings of the American Mathematical Society*,

*146*(6), 2637-2646. https://doi.org/10.1090/proc/14002

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*Proceedings of the American Mathematical Society*, vol. 146, no. 6, pp. 2637-2646. https://doi.org/10.1090/proc/14002

**Neckpinch singularities in fractional mean curvature flows.** / Cinti, Eleonora; Sinestrari, Carlo; Valdinoci, Enrico.

Research output: Contribution to journal › Article

TY - JOUR

T1 - Neckpinch singularities in fractional mean curvature flows

AU - Cinti, Eleonora

AU - Sinestrari, Carlo

AU - Valdinoci, Enrico

PY - 2018

Y1 - 2018

N2 - 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 Rn 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.

AB - 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 Rn 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.

KW - And phrases

KW - Fractional mean curvature flow

KW - Fractional perimeter

UR - http://www.scopus.com/inward/record.url?scp=85044356840&partnerID=8YFLogxK

U2 - 10.1090/proc/14002

DO - 10.1090/proc/14002

M3 - Article

VL - 146

SP - 2637

EP - 2646

JO - Proceedings of the American Mathematical Soceity

JF - Proceedings of the American Mathematical Soceity

SN - 0002-9939

IS - 6

ER -