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

Abstract

We investigate the distribution of eigenvalues of the weighted Laplacian on closed weighted Riemannian manifolds of nonnegative Bakry-\'Emery Ricci curvature. We derive some universal inequalities among eigenvalues of the weighted Laplacian on such manifolds. These inequalities are quantitative versions of the previous theorem by the author with Shioya. We also study some geometric quantity, called multi-way isoperimetric constants, on such manifolds and obtain similar universal inequalities among them. Multi-way isoperimetric constants are generalizations of the Cheeger constant. Extending and following the heat semigroup argument by Ledoux and E. Milman, we extend the Buser-Ledoux result to the $k$-th eigenvalue and the $k$-way isoperimetric constant. As a consequence the $k$-th eigenvalue of the weighted Laplacian and the $k$-way isoperimetric constant are equivalent up to polynomials of $k$ on closed weighted manifolds of nonnegative Bakry-\'Emery Ricci curvature.