## Abstract

We show that a mean curvature flow starting from a compact, smoothly embedded hypersurface M ⊆ ℝ^{n + 1} remains unique past singularities, provided the singularities are of mean convex type, i.e., if around each singular point, the surface moves in one direction. Specifically, the level set flow of M does not fatten if all singularities are of mean convex type. We further show that assumptions of the theorem hold provided all blowup flows are of the kind appearing in a mean convex flow, i.e., smooth, multiplicity 1, and convex. Our results generalize the well-known fact that the level set flow of a mean convex initial hypersurface M does not fatten. They also provide the first instance where nonfattening is concluded from local information around the singular set or from information about the singularity profiles of a flow.

Original language | American English |
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Pages (from-to) | 558-580 |

Number of pages | 23 |

Journal | Communications on Pure and Applied Mathematics |

Volume | 73 |

Issue number | 3 |

DOIs | |

State | Published - 1 Mar 2020 |

Externally published | Yes |

### Bibliographical note

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