  Consider a family of smooth immersions $F(\cdot,t): M^n\to \mathbb{R}^{n+1}$
of closed hypersurfaces in $\mathbb{R}^{n+1}$ moving by the mean curvature flow
$\frac{\partial F(p,t)}{\partial t} = -H(p,t)\cdot \nu(p,t)$, for $t\in [0,T)$.
We prove that the mean curvature blows up at the first singular time $T$ if all
singularities are of type I. In the case $n = 2$, regardless of the type of a
possibly forming singularity, we show that at the first singular time the mean
curvature necessarily blows up provided that either the Multiplicity One
Conjecture holds or the Gaussian density is less than two. We also establish
and give several applications of a local regularity theorem which is a
parabolic analogue of Choi-Schoen estimate for minimal submanifolds.
