Lower bound of coarse Ricci curvature on metric measure spaces and eigenvalues of Laplacian
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In this paper,we investigate the coarse Ricci curvature on metric spaces with random walks. There exists no canonical random walk on metric space with a reference measure. However, we prove that a Bishop–Gromov inequality gives a lower bound of coarse Ricci curvature with respect to a random walk called an \(r\)-step random walk. The lower bound does not coincide with the constant corresponding to curvature in Bishop–Gromov inequality. As a corollary, we obtain a lower bound of coarse Ricci curvature with respect to an \(r\)-step random walk for a metric measure space satisfying the curvature-dimension condition. Moreover we give an important example, Heisenberg group, which does not satisfy the curvature-dimension condition for any constant but has a lower bound of coarse Ricci curvature. We also have an estimate of the eigenvalues of the Laplacian by a lower bound of coarse Ricci curvature.
KeywordsCoarse Ricci curvature Bishop–Gromov inequality Eigenvalues Laplacian The curvature-dimension condition
Mathematics Subject Classification (2000)51F99 53C23 58J65 53C17
The author is grateful to Professor Nicola Gigli for pointing out Proposition 3.3,Professor Shin-ichi Ohta for helpful comments, Professor Kazuhiro Kuwae for valuable comments and Professor Takashi Shioya for reading this paper and giving useful advices. The author is partly supported by the Grant-in-Aid for JSPS Fellows, The Ministry of Education, Culture, Sports, Science and Technology, Japan.
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