Pinched ancient solutions to the high codimension mean curvature flow

摘要

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