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Singularity models of pinched solutions of mean curvature flow in higher codimension (1910.03968v2)
Published 9 Oct 2019 in math.DG
Abstract: We consider ancient solutions to the mean curvature flow in $\mathbb{R}{n+1}$ ($n \geq 3$) that are weakly convex, uniformly two-convex, and satisfy derivative estimates $|\nabla A| \leq \gamma_1 |H|2, |\nabla2 A| \leq \gamma_2 |H|3$. We show that such solutions are noncollapsed. As an application, in arbitrary codimension, we consider compact $n$-dimensional ($n \geq 5$) solutions to the mean curvature flow in $\mathbb{R}N$ that satisfy the pinching condition $|H| > 0$ and $|A|2 < c(n) |H|2$, $c(n) = \min{\frac{1}{n-2}, \frac{3(n+1)}{2n(n+2)}}$. 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.