[Paper Review] Eigenvalues of Laplacian and multi-way isoperimetric constants on weighted Riemannian manifolds

Kei Funano|arXiv (Cornell University)|Jul 15, 2013

Geometric Analysis and Curvature Flows42 references19 citations

TL;DR

This paper establishes quantitative universal inequalities for eigenvalues of the weighted Laplacian and multi-way isoperimetric constants on closed weighted Riemannian manifolds with nonnegative Bakry-Émery Ricci curvature. Using heat semigroup techniques and concentration of measure arguments, it proves that the $k$-th eigenvalue and the $k$-way isoperimetric constant are equivalent up to polynomial factors in $k$, with explicit exponential dependence on $k$ in the bounds.

ABSTRACT

We investigate the distribution of eigenvalues of the weighted Laplacian on closed weighted Riemannian manifolds of nonnegative Bakry-Émery 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-Émery Ricci curvature.

Motivation & Objective

• To derive universal, quantitative inequalities among eigenvalues of the weighted Laplacian on closed weighted Riemannian manifolds with nonnegative Bakry-Émery Ricci curvature.
• To extend Buser-Ledoux-type inequalities to the $k$-th eigenvalue and $k$-way isoperimetric constants, generalizing the Cheeger constant.
• To establish dimension-free bounds relating the $k$-th eigenvalue and the $k$-way isoperimetric constant via observable diameter and concentration of measure.
• To investigate the stability and polynomial dependence of eigenvalue and isoperimetric ratios in $k$ under nonnegative curvature conditions.

Proposed method

• Adapts the heat semigroup method of Ledoux and E. Milman to analyze higher-order spectral and isoperimetric quantities.
• Introduces the concept of $k$-separation to generalize concentration of measure and link it to eigenvalue and isoperimetric constants.
• Uses Davies-Gaffney heat kernel estimates and spectral asymptotics to control the decay of heat content over sets.
• Applies the observable diameter $ ext{ObsDiam}_{\mathbb{R}}((M,\mu); -\kappa)$ as a dimension-free substitute for diameter, connecting it to eigenvalues via concentration inequalities.
• Employs iterative volume decay estimates via $h_k(M,\mu)$, the $k$-way isoperimetric constant, to bound the measure of neighborhoods of unions of sets.
• Derives exponential bounds in $k$ using iterative application of the isoperimetric inequality with a constant $c$ independent of dimension.

Experimental results

Research questions

• RQ1 Can the ratio $\lambda_k(M,\mu)/\lambda_1(M,\mu)$ be bounded by a polynomial in $k$ for closed weighted Riemannian manifolds with nonnegative Bakry-Émery Ricci curvature?
• RQ2 Is the $k$-way isoperimetric constant $h_k(M,\mu)$ equivalent to the $k$-th eigenvalue $\lambda_k(M,\mu)$ up to polynomial factors in $k$?
• RQ3 Can the exponential dependence on $k$ in the eigenvalue ratio be improved to a polynomial bound, or is $\exp(ck)$ optimal?
• RQ4 Does the stability of eigenvalues and isoperimetric constants hold under the concentration topology or for Alexandrov spaces with nonnegative curvature?
• RQ5 Can E. Milman’s concentration-to-spectral inequality be extended to $k$-separation and $k$-th eigenvalues in general CD$(0,\infty)$ spaces?

Key findings

• The $k$-th eigenvalue $\lambda_k(M,\mu)$ is bounded above by $\exp(ck)\lambda_1(M,\mu)$ for some universal constant $c>0$, independent of dimension.
• The $k$-way isoperimetric constant $h_k(M,\mu)$ satisfies $\lambda_k(M,\mu) \lesssim \exp(ck) h_k(M,\mu)^2$, showing equivalence up to $\exp(ck)$ factors.
• The observable diameter satisfies $\text{ObsDiam}_{\mathbb{R}}((M,\mu); -\kappa) \leq \frac{\exp(ck)}{\sqrt{\lambda_k(M,\mu)}} \log \frac{1}{\kappa}$, extending Gromov-V. Milman's first eigenvalue result.
• For any $k$, the ratio $\lambda_k(M,\mu)/\lambda_1(M,\mu)$ is bounded by $\exp(ck)$, and the same holds for $h_k(M,\mu)/h_1(M,\mu)$, both independent of dimension.
• The proof relies on iterative volume decay via $h_k(M,\mu)$, showing that if $k+1$ sets are not $k$-separated, then their union has small neighborhood measure.
• The results extend to convex domains with $C^2$ boundary under Neumann conditions, with identical proof structure.