Concentration, Ricci Curvature, and Eigenvalues of Laplacian

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15 Citations (Scopus)

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.

Original languageEnglish
Pages (from-to)888-936
Number of pages49
JournalGeometric and Functional Analysis
Volume23
Issue number3
DOIs
Publication statusPublished - 2013 Jun

Keywords

  • Concentration of measure phenomenon
  • Ricci curvature
  • convergence of mm-spaces
  • eigenvalues of Laplacian

ASJC Scopus subject areas

  • Analysis
  • Geometry and Topology

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