Mean curvature flow of Reifenberg sets

Or Hershkovits*

*Corresponding author for this work

Research output: Contribution to journalArticlepeer-review

3 Scopus citations


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.

Original languageAmerican English
Pages (from-to)441-484
Number of pages44
JournalGeometry and Topology
Issue number1
StatePublished - 10 Feb 2017
Externally publishedYes

Bibliographical note

Publisher Copyright:
© 2017, Mathematical Sciences Publishers. All rights reserved.


  • Mean curvature flow
  • Non-fattening
  • Reifenberg flat
  • Reifenberg sets


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