## Geometry & Topology

### Mean curvature flow of Reifenberg sets

Or Hershkovits

#### Abstract

In this paper, we prove short time existence and uniqueness of smooth evolution by mean curvature in $ℝn+1$ starting from any $n$–dimensional $(ε,R)$–Reifenberg flat set with $ε$ 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.

#### Article information

Source
Geom. Topol., Volume 21, Number 1 (2017), 441-484.

Dates
Accepted: 26 December 2015
First available in Project Euclid: 16 November 2017

https://projecteuclid.org/euclid.gt/1510859138

Digital Object Identifier
doi:10.2140/gt.2017.21.441

Mathematical Reviews number (MathSciNet)
MR3608718

Zentralblatt MATH identifier
1358.53066

#### Citation

Hershkovits, Or. Mean curvature flow of Reifenberg sets. Geom. Topol. 21 (2017), no. 1, 441--484. doi:10.2140/gt.2017.21.441. https://projecteuclid.org/euclid.gt/1510859138

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