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

TL;DR

This paper classifies certain ancient solutions to mean curvature flow in higher codimension, showing they are noncollapsed and that blow-up models at singularities are well-understood solitons.

Contribution

It establishes noncollapsing for weakly convex, uniformly two-convex ancient solutions in higher codimension and classifies blow-up models under pinching conditions.

Findings

Ancient solutions are noncollapsed.

Blow-up models are shrinking spheres, cylinders, or translating solitons.

Results extend classification to higher codimension cases.

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 $|\nabla A| \leq \gamma_1 |H|^2, |\nabla^2 A| \leq \gamma_2 |H|^3$. We show that such solutions are noncollapsed. As an application, in arbitrary codimension, we consider compact $n$-dimensional ($n \geq 5$) 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.