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Mean curvature flow of Reifenberg sets

Or Hershkovits

Geometry & Topology 21 (2017) 441–484

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.

mean curvature flow, Reifenberg sets, Reifenberg flat, non-fattening
Mathematical Subject Classification 2010
Primary: 53C44
Received: 19 May 2015
Accepted: 26 December 2015
Published: 10 February 2017
Proposed: Tobias H. Colding
Seconded: Bruce Kleiner, John Lott
Or Hershkovits
Department of Mathematics
Stanford University
380 Serra Mall
Stanford, CA 94305
United States