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Abstract
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In this paper, we prove short time existence and uniqueness of smooth evolution by mean curvature
in
starting from
any
–dimensional
–Reifenberg
flat set with
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
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
,
which are additionally not graphical at any scale. Except in dimension one, no such
examples were previously known.
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Keywords
mean curvature flow, Reifenberg sets, Reifenberg flat,
non-fattening
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Mathematical Subject Classification 2010
Primary: 53C44
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Publication
Received: 19 May 2015
Accepted: 26 December 2015
Published: 10 February 2017
Proposed: Tobias H. Colding
Seconded: Bruce Kleiner, John Lott
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