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.
Bibliographical noteFunding Information:
The author wishes to thank his advisor Bruce Kleiner for suggesting the problem, as well as for many useful discussions. He also wishes to thank the anonymous referee for comments and suggestions. He further wishes to thank Jacobus Portegies for carefully reading and commenting on an earlier version of this note, as well as for his many suggestions. The author wishes to thank Robert Haslhofer, for his many suggestions and comments about an earlier version of this notes, and Jeff Cheeger for his generous support during the months in which this project was initiated. The author was partially supported by NSF grants DMS 1406407 and DMS 1105656.
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- Mean curvature flow
- Reifenberg flat
- Reifenberg sets