Structure theory of metric measure spaces with lower Ricci curvature bounds
Structure theory of metric measure spaces with lower Ricci curvature bounds
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DOI：
10.4171/jems/874
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发表时间：
2014-05
影响因子：
2.6
通讯作者：
Andrea Mondino;A. Naber
中科院分区：
文献类型：
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作者：
Andrea Mondino;A. Naber
证明了满足有限维Ricci曲率下界且Sobolev空间W1，2为Hilbert的度量测度空间（X，d，m）是可求长的.也就是说，RCD空间是可求长的，特别是对于m-a. e。点的切锥是唯一的，欧氏维数最多为N。证明是基于一个极大函数的论点结合了原来的几乎分裂定理通过估计的梯度的过剩。我们还证明了关于过剩函数的一个锐积分Abresh-Gromoll型不等式和关于过剩函数的梯度的一个Abresh-Gromoll型不等式。即使在平稳的环境中，
We prove that a metric measure space (X,d,m) satisfying finite dimensional lower Ricci curvature bounds and whose Sobolev space W1,2 is Hilbert is rectifiable. That is, a RCD∗(K,N)-space is rectifiable, and in particular for m-a.e. point the tangent cone is unique and euclidean of dimension at most N. The proof is based on a maximal function argument combined with an original Almost Splitting Theorem via estimates on the gradient of the excess. We also show a sharp integral Abresh–Gromoll type inequality on the excess function and an Abresh–Gromoll-type inequality on the gradient of the excess. The argument is new even in the smooth setting