Isoperimetric properties of the mean curvature flow

Hershkovits, Or

doi:10.1090/tran/7039

Isoperimetric properties of the mean curvature flow
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Trans. Amer. Math. Soc. 369 (2017), 4367-4383 Request permission

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 $\mathbb {R}^n$, with a constant differing from the optimal constant by a factor of only $\sqrt {k}$, 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.

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