Pinched Ancient Solutions to the High Codimension Mean Curvature Flow

TL;DR

This paper proves that certain ancient solutions to high codimension mean curvature flow are necessarily shrinking spheres, simplifying previous convergence proofs under a natural pinching condition.

Contribution

It establishes that compact ancient solutions satisfying a specific pinching condition must be shrinking spheres, extending understanding of ancient solutions in high codimension mean curvature flow.

Findings

Ancient solutions with a pinching condition are shrinking spheres.

Simplifies the proof of convergence to round points for such solutions.

Provides a classification result for ancient solutions in high codimension.

Abstract

We study solutions of high codimension mean curvature flow defined for all negative times, usually referred to as ancient solutions. We show that any compact ancient solution whose second fundamental form satisfies a certain natural pinching condition must be a family of shrinking spheres. Andrews and Baker have shown that initial submanifolds satisfying this pinching condition, which generalises the notion of convexity, converge to round points under the flow. As an application, we use our result to simplify their proof.