Complexity of parabolic systems
Complexity of parabolic systems
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DOI：
10.1007/s10240-020-00117-x
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发表时间：
2019-03
期刊：
影响因子：
--
通讯作者：
T. Colding;W. Minicozzi
中科院分区：
文献类型：
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作者：
T. Colding;W. Minicozzi
我们首先用熵来界定一个古老的平均曲率流的余维。因此，所有的爆破都位于一个欧氏子空间中，其维数由演化子流形的熵和维数所限定。这大大降低了系统的复杂性。我们用这个在我们的新方法的一个主要应用程序，给第一个一般的边界上的一般奇点的表面在任意的余维。我们还显示尖锐的界限余维可以说是一些最重要的情况下，一般古代流。也就是说，我们证明了在任意维数和余维数下，任何古代流，如果是圆柱的，那么它一定是欧氏子空间中的超曲面流。这扩展了著名的分类结果，以更高的cowimension.The约束的余维的熵是一个特殊的情况下，尖锐的界限的频谱计数功能收缩，更一般地说，古代的流量。收缩器是通过缩放而演化的解，并且是流动的奇异模型。我们在所有维度和所有余维度上以非常强的意义证明了圆柱体作为收缩器的刚性：任何收缩器，即使在大维度空间中，在足够大但紧凑的集合上足够接近圆柱体，本身就是圆柱体。这是理论中的一个重要工具，也是规律性的关键;参见。（Colding和Minicozzi II预印本，2020年）。
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 atmust 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).