On Ilmanen’s multiplicity-one conjecture for mean curvature flow with type-$I$ mean curvature

In this paper, we show that if the mean curvature of a closed smooth embedded mean curvature flow in {\mathbb R}^3 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 multiplicity-one 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 {\mathbb R}^3 is at least of type- I .