We first bound the codimension of an ancient mean curvature flow by the entropy. As a consequence, all blowups lie in a Euclidean subspace whose dimension is bounded by the entropy and dimension of the evolving submanifolds. This drastically reduces the complexity of the system. We use this in a major application of our new methods to give the first general bounds on generic singularities of surfaces in arbitrary codimension.

We also show sharp bounds for codimension in arguably some of the most important situations of general ancient flows. Namely, we prove that in any dimension and codimension any ancient flow that is cylindrical at \(-\infty \) must be a flow of hypersurfaces in a Euclidean subspace. This extends well-known classification results to higher codimension.

The bound on the codimension in terms of the entropy is a special case of sharp bounds for spectral counting functions for shrinkers and, more generally, ancient flows. Shrinkers are solutions that evolve by scaling and are the singularity models for the flow.

We show rigidity of cylinders as shrinkers in all dimension and all codimension in a very strong sense: Any shrinker, even in a large dimensional space, that is sufficiently close to a cylinder on a large enough, but compact, set is itself a cylinder. This is an important tool in the theory and is key for regularity; cf. (Colding and Minicozzi II in preprint, 2020).

我们首先通过熵来约束古代平均曲率流的余维。结果，所有爆炸都位于欧几里得子空间中，该子空间的大小受进化子流形的熵和大小限制。这大大降低了系统的复杂性。我们将其用于新方法的主要应用中，以给出任意余维曲面的通用奇异性的第一个一般边界。

在一般古代水流的某些最重要情况下，我们也可以证明维数的界线。即，我们证明，在任何维度和余维上，在\（-\ infty \）处呈圆柱形的任何古代流动都必须是欧几里得子空间中超表面的流动。这将众所周知的分类结果扩展到较高的维数。

用熵表示的余维边界是对收缩器（更常见的是古代流量）的频谱计数函数的尖锐边界的特例。收缩器是通过缩放而演变的解决方案，并且是流的奇异模型。

我们在很强的意义上显示圆柱体在所有维度和所有维度上的刚度：任何收缩器，即使在较大的尺寸空间中，也足够接近圆柱体，但足够大，但结构紧凑，本身就是圆柱体。这是理论上的重要工具，是规律性的关键。cf. （Colding and Minicozzi II在预印本中，2020年）。