Journal of the European Mathematical Society


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Published online first: 2021-06-08
DOI: 10.4171/JEMS/1090

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

Haozhao Li[1] and Bing Wang[2]

(1) University of Science and Technology of China, Hefei, China
(2) University of Science and Technology of China, Hefei, China

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-$I$.

Keywords: Mean curvature flow, multiplicity-one conjecture, self-shrinker

Li Haozhao, Wang Bing: On Ilmanen’s multiplicity-one conjecture for mean curvature flow with type-$I$ mean curvature. J. Eur. Math. Soc. Electronically published on June 8, 2021. doi: 10.4171/JEMS/1090 (to appear in print)