Sturm-Liouville Estimates for the Spectrum and Cheeger Constant
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.
Keywords
Cite
@article{arxiv.1308.5936,
title = {Sturm-Liouville Estimates for the Spectrum and Cheeger Constant},
author = {Brian Benson},
journal= {arXiv preprint arXiv:1308.5936},
year = {2016}
}
Comments
41 pages, 7 figures. Answered the question from v1 in the negative; see Example 1.5. Some changes in numbering conventions of equations and examples to avoid confusion that occurred in v1. Improvements to the writing of the proof of Proposition 2.1. Lemma 3.5 now references the slicing lemma, which was omitted in v1. Some typos were fixed and references added from v1