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 (J Differ Geom 85(3):357–395, 2010) 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.

中文翻译：

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高维平均曲率流的压缩古代解

我们研究为所有负时间定义的高维平均曲率流的解，通常称为古代解。我们表明，任何第二个基本形式都满足一定自然捏合条件的紧凑型古代解决方案，都必须是收缩球族。Andrews和Baker（J Differ Geom 85（3）：357–395，2010）表明，满足此捏合条件的初始子流形（泛化了凸性的概念）在流下收敛到圆点。作为应用程序，我们使用我们的结果来简化其证明。