Sturm-Liouville estimates for the spectrum and Cheeger constant

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 , λ_k(M), in terms of the eigenvalues of a Sturm-Liouville problem which depends on h(M). We then compare the Weyl asymptotic of λ_k(M) given by the works of Cheng, Gromov, and Berard-Besson-Gallot to the asymptotics of our Sturm-Liouville problems given by Atkinson-Mingarelli.

Name thesis:degree_name

Ph.D.

Level thesis:degree_level

Dissertation

Discipline thesis:degree_discipline

Mathematics

Grantor

University of Illinois at Urbana-Champaign

Year dc:date

2014

Author dc:creator

• Benson, Brian

Contributors dc:contributor

• Dunfield, Nathan M.
• Laugesen, Richard S.
• Alexander, Stephanie B.
• Leininger, Christopher J.

• Cheeger constant
• spectrum of Laplacian
• eigenvalues of closed Riemannian manifolds
• Buser's inequality

Statement dc:rights

• Copyright 2014 Brian Benson

Language dc:language

en

Handle dc:identifier

http://hdl.handle.net/2142/50697

OAI identifier oai:identifier

oai:www.ideals.illinois.edu:2142/50697