On Ilmanen's multiplicity-one conjecture for mean curvature flow with type-I mean curvature
JOURNAL OF THE EUROPEAN MATHEMATICAL SOCIETY(2022)
Abstract
In this paper, we show that if the mean curvature of a closed smooth embedded mean curvature flow in R3 is of type-I, then the rescaled flow at the first finite singular time converges smoothly to a self-shrinker flow with multiplicity one. This result confirms Ilmanen's multiplicityone conjecture under the assumption that the mean curvature is of type-I. As a corollary, we show that the mean curvature at the first singular time of a closed smooth embedded mean curvature flow in R3 is at least of type-I.
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Key words
Mean curvature flow, multiplicity-one conjecture, self-shrinker
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