In this paper, we show that if the mean curvature of a closed smooth embedded mean curvature flow in is of type-, 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 multiplicity-one conjecture under the assumption that the mean curvature is of type-. As a corollary, we show that the mean curvature at the first singular time of a closed smooth embedded mean curvature flow in is at least of type-.
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Haozhao Li, Bing Wang, On Ilmanen’s multiplicity-one conjecture for mean curvature flow with type- mean curvature. J. Eur. Math. Soc. 24 (2022), no. 1, pp. 37–135DOI 10.4171/JEMS/1090