A sharp lower bound for the entropy of closed hypersurfaces up to dimension six
A sharp lower bound for the entropy of closed hypersurfaces up to dimension six
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六维以下闭合超曲面熵的急剧下界
DOI：
10.1007/s00222-016-0659-3
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发表时间：
2014
影响因子：
3.1
通讯作者：
Lu Wang
中科院分区：
文献类型：
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作者：
J. Bernstein;Lu Wang
熵是一个自然的几何量，它度量了$\mathbb R^{n+1}$$Rn+1中超曲面的复杂性。它沿平均曲率流沿着是不增加的，因此在分析这种流的动力学中起着重要作用。在（Colding等人，J Differ Geom 95（1）：53-69，2013），Colding-Ilmanen-Minicozzi-White证明了在$\mathbb R^{n+1}$$Rn+1中的平均曲率流的闭光滑自收缩解的类中，熵在圆球处唯一地最小化。他们证明，对于$2\le n\le 6$2 ≤n≤6，圆球面使所有闭超曲面的熵最小化。利用适当的弱平均曲率流，我们证明了他们的猜想。对于这些维数，我们的方法也给出了一个新的证明Colding等人的主要结果（J Differ Geom 95（1）：53-69，2013），并将其结论扩展到紧致奇异自收缩解。
The entropy is a natural geometric quantity which measures the complexity of a hypersurface in $$\mathbb R^{n+1}$$Rn+1. It is non-increasing along the mean curvature flow and so plays a significant role in analyzing the dynamics of this flow. In (Colding et al., J Differ Geom 95(1):53–69, 2013), Colding–Ilmanen–Minicozzi-White showed that within the class of closed smooth self-shrinking solutions of the mean curvature flow in $$\mathbb R^{n+1}$$Rn+1, the entropy is uniquely minimized at the round sphere. They conjectured that, for $$2\le n\le 6$$2≤n≤6, the round sphere minimizes the entropy among all closed hypersurfaces. Using an appropriate weak mean curvature flow, we prove their conjecture. For these dimensions, our approach also gives a new proof of the main result of Colding et al. (J Differ Geom 95(1):53–69, 2013) and extends its conclusions to compact singular self-shrinking solutions.
平均曲率流中紧凑切线流的独特性
DOI：
10.1515/crelle-2012-0070
发表时间：
2014
期刊：
arXiv: Differential Geometry
影响因子：
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作者：
F. Schulze
通讯作者：
F. Schulze