Isoperimetric rigidity and distributions of 1-Lipschitz functions
Isoperimetric rigidity and distributions of 1-Lipschitz functions
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DOI：
10.1016/j.aim.2019.04.043
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发表时间：
2018-01
影响因子：
1.7
通讯作者：
Hiroki Nakajima;T. Shioya
中科院分区：
文献类型：
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作者：
Hiroki Nakajima;T. Shioya
证明了如果测地度量空间满足等周轮廓的比较条件，且可观测方差最大，则该空间被极小测地线分片，其中可观测方差定义为空间上1-Lipschitz函数的方差的上确界。我们的结果可以看作是Cheeger-Gromoll的分裂定理和程的最大直径定理的变形。作为应用，我们得到了一个新的具有正Bakry-Émery Ricci曲率的完备加权黎曼流形的等距分裂定理。
We prove that if a geodesic metric measure space satisfies a comparison condition for isoperimetric profile and if the observable variance is maximal, then the space is foliated by minimal geodesics, where the observable variance is defined to be the supremum of the variance of 1-Lipschitz functions on the space. Our result can be considered as a variant of Cheeger-Gromoll's splitting theorem and also of Cheng's maximal diameter theorem. As an application, we obtain a new isometric splitting theorem for a complete weighted Riemannian manifold with a positive Bakry-Émery Ricci curvature.