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

TL;DR

This paper establishes universal inequalities relating eigenvalues of the weighted Laplacian and multi-way isoperimetric constants on weighted Riemannian manifolds with nonnegative Bakry-Émery Ricci curvature, extending previous results.

Contribution

It introduces new universal inequalities for eigenvalues and isoperimetric constants, extending heat semigroup methods to higher eigenvalues and isoperimetric constants.

Findings

Eigenvalues and isoperimetric constants are equivalent up to polynomial factors in k.

Universal inequalities among eigenvalues of the weighted Laplacian.

Extension of Buser-Ledoux results to k-th eigenvalues and k-way isoperimetric constants.

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.