  We consider the evolution by mean curvature of smooth $n$-dimensional
submanifolds in $\mathbb{R}^{n+k}$ which are compact and quadratically pinched.
We will be primarily interested in flows of high codimension, the case $k\geq
2$. We prove that our submanifold is asymptotically convex, that is the first
eigenvalue of the second fundamental form in the principal mean curvature
direction blows up at a strictly slower rate than the mean curvature vector. We
use this convexity estimate to show that at a singular time of the flow, there
exists a rescaling that converges to a smooth codimension-one limiting flow
which is convex and moves by translation.
