Geometry & Topology

Mean curvature flow of Reifenberg sets

Or Hershkovits

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In this paper, we prove short time existence and uniqueness of smooth evolution by mean curvature in n+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

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

Received: 19 May 2015
Accepted: 26 December 2015
First available in Project Euclid: 16 November 2017

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Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 53C44: Geometric evolution equations (mean curvature flow, Ricci flow, etc.)

mean curvature flow Reifenberg sets Reifenberg flat non-fattening


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

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