Singularities of mean curvature flow with bounded mean curvature and Morse index

We study the multiplicity of the singularities of mean curvature flow with bounded mean curvature and Morse index.

For $3\leq n\leq 6$, we show that either the mean curvature or the Morse index blows up at the first singular time for a closed smooth embedded mean curvature flow in $\mathbb{R}^{n+1}$.