  In dimensions $n \geq 5$, we prove a canonical neighborhood theorem for the
mean curvature flow of compact $n$-dimensional submanifolds in $\mathbb{R}^N$
satisfying a pinching condition $|A|^2 < c|H|^2$ for $c = \min \{
\frac{3(n+1)}{2n(n+2)},\frac{1}{n-2}\}.$
