  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.
