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We investigate the distribution of eigenvalues of the weighted Laplacian onclosed weighted Riemannian manifolds of nonnegative Bakry-\'Emery Riccicurvature. We derive some universal inequalities among eigenvalues of theweighted Laplacian on such manifolds. These inequalities are quantitativeversions of the previous theorem by the author with Shioya. We also study somegeometric quantity, called multi-way isoperimetric constants, on such manifoldsand obtain similar universal inequalities among them. Multi-way isoperimetricconstants are generalizations of the Cheeger constant. Extending and followingthe heat semigroup argument by Ledoux and E. Milman, we extend the Buser-Ledouxresult to the $k$-th eigenvalue and the $k$-way isoperimetric constant. As aconsequence the $k$-th eigenvalue of the weighted Laplacian and the $k$-wayisoperimetric constant are equivalent up to polynomials of $k$ on closedweighted manifolds of nonnegative Bakry-\'Emery Ricci curvature.

Eigenvalues of Laplacian and multi-way isoperimetric constants on weighted Riemannian manifolds