Abstract Given a constant $$k>1$$ k > 1 , let Z be the family of round spheres radius $${{\,\mathrm{artanh}\,}}(k^{-1})$$ artanh ( - ) in hyp...

In this paper, we show that if the mean curvature of a closed smooth embedded flow in ${\\mathbb R}^3$ is type-$I$, then rescaled at first finite singular time converges smoothly to self-shrinker with multiplicity one. This result confirms Ilmanen's multiplicity-one conjecture under assumption type-$I$. As corollary, least