Mean curvature flow of Reifenberg sets

Hershkovits, Or

doi:10.2140/gt.2017.21.441

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Mean curvature flow of Reifenberg sets

Or Hershkovits

Geometry & Topology 21 (2017) 441–484

DOI: 10.2140/gt.2017.21.441

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 $(ε, R)$ –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.

Keywords

mean curvature flow, Reifenberg sets, Reifenberg flat, non-fattening

Mathematical Subject Classification 2010

Primary: 53C44

References

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

Volume 21, issue 1 (2017)