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

Abstract

We consider ancient solutions to the mean curvature flow in Rn + 1 (n ≥ 3) that are weakly convex, uniformly two-convex, and satisfy two pointwise derivative estimates | ∇ A | ≤ γ 1 | H | 2, | ∇ 2 A | ≤ γ 2 | H |3. We show that such solutions are noncollapsed. As an application, in arbitrary codimension, we consider compact n-dimensional (n ≥ 5) solutions to the mean curvature flow in R N that satisfy the pinching condition | A |2 < c | H | 2 for a suitable constant c = c (n). 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.