Concentration, Ricci curvature, and eigenvalues of Laplacian
Concentration, Ricci curvature, and eigenvalues of Laplacian
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浓度、Ricci 曲率和拉普拉斯特征值
DOI：
10.1007/s00039-013-0215-x
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发表时间：
2013
期刊：
影响因子：
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通讯作者：
Kei Funano and Takashi Shioya
中科院分区：
文献类型：
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作者：
笹平 裕史;笹平 裕史;Tetsuya Hosaka;笹平 裕史;Tetsuya Hosaka;笹平 裕史;Naotsugu Chinen and Tetsuya Hosaka;Tetsuya Hosaka and Yonghuo Xiao;Tetsuya Hosaka;Tetsuya Hosaka;Tetsuya Hosaka;Tetsuya Hosaka;Kei Funano;Kei Funano;Kei Funano;Kei Funano and Takashi Shioya
在本文中，我们研究了度量测度空间的集中行为。我们证明了曲率维数条件相对于格罗莫夫浓度拓扑的稳定性。作为一个应用，在 Bakry-Émery Ricci 曲率的非负性下，我们证明了闭黎曼流形的加权拉普拉斯算子的第 k 个特征值由第一个特征值的常数倍决定，其中该常数仅取决于 k 并且与流形的维数无关。
In this paper we study the concentration behavior of metric measure spaces. We prove the stability of the curvature-dimension condition with respect to the concentration topology due to Gromov. As an application, under the nonnegativity of Bakry–Émery Ricci curvature, we prove that thekth eigenvalue of the weighted Laplacian of a closed Riemannian manifold is dominated by a constant multiple of the first eigenvalue, where the constant depends only onkand is independent of the dimension of the manifold.