Singularity models of pinched solutions of mean curvature flow in higher codimension

We consider ancient solutions to the mean curvature flow in ${\mathbb{R}}^{n+1}$ ($n\ge 3$) that are weakly convex, uniformly two-convex, and satisfy derivative estimates $|\nabla A| \leq \gamma_1 |H|^2, |\nabla^2 A| \leq \gamma_2 |H|^3$. We show that such solutions are noncollapsed. As an application, in arbitrary codimension, we consider compact $n$-dimensional ($n \geq 5$)solutionstothemeancurvatureflowin$$\mathbb{R}$^N$ that satisfy the pinching condition $\mathrm{\mid }H\mathrm{\mid }>0$ and $\mathrm{\mid }A{\mathrm{\mid }}^2<c(n)\mathrm{\mid }H{\mathrm{\mid }}^2$, $c(n) = \min{\frac{1}{n-2}, \frac{3(n+1)}{2n(n+2)}}$. We conclude that any blow-up model at the first singular time must be a codimension one shrinking sphere, shrinking cylinder, or translating bowl soliton.