Abstract:It is a fundamental open problem for the mean curvature flow, and in fact for many partial differential equations, whether or not all blowup limits are selfsimilar. In this short note, we prove that for the mean curvature flow of mean convex surfaces all limit flows are selfsimilar (static, shrinking, or translating) if and only if there are only finitely many spherical singularities. More generally, using the solution of the mean convex neighborhood conjecture for neck singularities, we establish a local version of this equivalence for neck singularities in arbitrary dimension. In particular, we see that the ancient ovals occur as limit flows if and only if there is a sequence of spherical singularities converging to a neck singularity.

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中文翻译：

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关于限流自相似性的注释

摘要：对于平均曲率流，以及实际上对于许多偏微分方程而言，这是所有爆破极限是否都是自相似的，这是一个基本的开放问题。在此简短说明中，我们证明，对于且仅当球面奇点有限时，所有凸曲面的平均曲率流的所有极限流都是自相似的（静态，收缩或平移）。更一般地，使用颈部奇异点的平均凸邻域猜想的解，我们在任意维度上建立了针对脖子奇异点的等价性的局部版本。尤其是，我们看到，当且仅当存在一系列球形奇点收敛到颈部奇点时，古代椭圆形才作为限流出现。

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