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Title:Sturm-Liouville estimates for the spectrum and Cheeger constant
Author(s):Benson, Brian
Director of Research:Dunfield, Nathan M.
Doctoral Committee Chair(s):Laugesen, Richard S.
Doctoral Committee Member(s):Dunfield, Nathan M.; Alexander, Stephanie B.; Leininger, Christopher J.
Department / Program:Mathematics
Degree Granting Institution:University of Illinois at Urbana-Champaign
Subject(s):Cheeger constant
spectrum of Laplacian
eigenvalues of closed Riemannian manifolds
Buser's inequality
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 , 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.
Issue Date:2014-09-16
Rights Information:Copyright 2014 Brian Benson
Date Available in IDEALS:2014-09-16
Date Deposited:2014-08

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