#### Volume 21, issue 1 (2017)

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Mean curvature flow of Reifenberg sets

### Or Hershkovits

Geometry & Topology 21 (2017) 441–484
##### Abstract

In this paper, we prove short time existence and uniqueness of smooth evolution by mean curvature in ${ℝ}^{n+1}$ starting from any $n$–dimensional $\left(\epsilon ,R\right)$–Reifenberg flat set with $\epsilon$ sufficiently small. More precisely, we show that the level set flow in such a situation is non-fattening and smooth. These sets have a weak metric notion of tangent planes at every small scale, but the tangents are allowed to tilt as the scales vary. As for every $n$ this class is wide enough to include some fractal sets, we obtain unique smoothing by mean curvature flow of sets with Hausdorff dimension larger than $n$, which are additionally not graphical at any scale. Except in dimension one, no such examples were previously known.

##### Keywords
mean curvature flow, Reifenberg sets, Reifenberg flat, non-fattening
Primary: 53C44
##### Publication
Received: 19 May 2015
Accepted: 26 December 2015
Published: 10 February 2017
Proposed: Tobias H. Colding
Seconded: Bruce Kleiner, John Lott
##### Authors
 Or Hershkovits Department of Mathematics Stanford University 380 Serra Mall Stanford, CA 94305 United States