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

Abstract

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