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Concentration, Ricci Curvature, and Eigenvalues of Laplacian

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Abstract

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 the kth 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 on k and is independent of the dimension of the manifold.

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Authors and Affiliations

  1. Research Institute for Mathematical Sciences, Kyoto University, Kyoto, 606-8502, Japan

    Kei Funano

  2. Mathematical Institute, Tohoku University, Sendai, 980-8578, Japan

    Takashi Shioya

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  1. Kei Funano
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  2. Takashi Shioya
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Correspondence to Takashi Shioya.

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The authors are partially supported by a Grant-in-Aid for Scientific Research from the Japan Society for the Promotion of Science.

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Funano, K., Shioya, T. Concentration, Ricci Curvature, and Eigenvalues of Laplacian. Geom. Funct. Anal. 23, 888–936 (2013). https://doi.org/10.1007/s00039-013-0215-x

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  • DOI: https://doi.org/10.1007/s00039-013-0215-x

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