2013/07/15 by Kei Funano, Funano, Kei
Computer Science · Mathematics · #Advanced Mathematical Modeling in Engineering #Differential Geometry (math.DG) #FOS: Mathematics #Geometric Analysis and Curvature Flows #Geometry and complex manifolds #Nonlinear Partial Differential Equations #Numerical methods in inverse problems
paper · pdf · doi:10.48550/arxiv.1307.3919
openalex publication_date 2013/07/15 · openalex created_date 2022/10/01 · openalex updated_date 2026/07/28
We investigate the distribution of eigenvalues of the weighted Laplacian on\nclosed weighted Riemannian manifolds of nonnegative Bakry- 'Emery Ricci\ncurvature. We derive some universal inequalities among eigenvalues of the\nweighted Laplacian on such manifolds. These inequalities are quantitative\nversions of the previous theorem by the author with Shioya. We also study some\ngeometric quantity, called multi-way isoperimetric constants, on such manifolds\nand obtain similar universal inequalities among them. Multi-way isoperimetric\nconstants are generalizations of the Cheeger constant. Extending and following\nthe heat semigroup argument by Ledoux and E. Milman, we extend the Buser-Ledoux\nresult to the k-th eigenvalue and the k-way isoperimetric constant. As a\nconsequence the k-th eigenvalue of the weighted Laplacian and the k-way\nisoperimetric constant are equivalent up to polynomials of k on closed\nweighted manifolds of nonnegative Bakry- 'Emery Ricci curvature.\n