Sturm-Liouville Estimates for the Spectrum and Cheeger Constant

TL;DR

This paper refines bounds on the Laplacian spectrum of manifolds using Sturm-Liouville theory, improving understanding of eigenvalues in relation to the Cheeger constant and generalizing Agol's inequality.

Contribution

It transforms Agol's implicit differential equation into a solvable Riemann form and extends Buser's inequality to higher eigenvalues via Sturm-Liouville analysis.

Findings

Provides explicit upper bounds for higher eigenvalues of manifolds.

Generalizes Agol's inequality using Sturm-Liouville theory.

Compares eigenvalue asymptotics with classical results.

Abstract

Buser's inequality gives an upper bound on the first non-zero eigenvalue of the Laplacian of a closed manifold M in terms of the Cheeger constant h(M). Agol later gave a quantitative improvement of Buser's inequality. Agol's result is less transparent since it is given implicitly by a set of equations, one of which is a differential equation Agol could not solve except when M is three-dimensional. We show that a substitution transforms Agol's differential equation into the Riemann differential equation. Then, we give a proof of Agol's result and also generalize it using Sturm-Liouville theory. Under the same assumptions on M, we are able to give upper bounds on the higher eigenvalues of M, \lambda_k(M), in terms of the eigenvalues of a Sturm-Liouville problem which depends on h(M). We then compare the Weyl asymptotic of \lambda_k(M) given by the works of Cheng, Gromov, and B\'erard-Besson-Gallot to the asymptotics of our Sturm-Liouville problems given by Atkinson-Mingarelli.