## Abstract

In this paper we discuss a simple relation, which was previously missed, between the high co-dimensional isoperimetric problem of finding a filling with small volume to a given cycle and extinction estimates for singular, high co-dimensional, mean curvature flow. The utility of this viewpoint is first exemplified by two results which, once casted in the light of this relation, are almost self-evident. The first is a genuine, 5-line proof, for the isoperimetric inequality for k-cycles in R^{n}, with a constant differing from the optimal constant by a factor of only √k, as opposed to a factor of k^{k} produced by all of the other soft methods. The second is a 3-line proof of a lower bound for extinction for arbitrary co-dimensional, singular, mean curvature flows starting from cycles, generalizing the main result of Giga and Yama-uchi (1993). We then turn to use the above-mentioned relation to prove a bound on the parabolic Hausdorff measure of the space-time track of high co-dimensional, singular, mean curvature flow starting from a cycle, in terms of the mass of that cycle. This bound is also reminiscent of a Michael-Simon isoperimetric inequality. To prove it, we are led to study the geometric measure theory of Euclidean rectifiable sets in parabolic space and prove a co-area formula in that setting. This formula, the proof of which occupies most of this paper, may be of independent interest.

Original language | English |
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Pages (from-to) | 4367-4383 |

Number of pages | 17 |

Journal | Transactions of the American Mathematical Society |

Volume | 369 |

Issue number | 6 |

DOIs | |

State | Published - 2017 |

Externally published | Yes |

### Bibliographical note

Publisher Copyright:© 2017 American Mathematical Society.