Sturm-Liouville estimates for the spectrum and Cheeger constant

Benson, Brian

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https://hdl.handle.net/2142/50697 Copy

Description

Title

Sturm-Liouville estimates for the spectrum and Cheeger constant

Author(s)

Benson, Brian

Issue Date

2014-09-16

Director of Research (if dissertation) or Advisor (if thesis)

Dunfield, Nathan M.

Doctoral Committee Chair(s)

Laugesen, Richard S.

Committee Member(s)

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

Department of Study

Mathematics

Discipline

Mathematics

Degree Granting Institution

University of Illinois at Urbana-Champaign

Degree Name

Ph.D.

Degree Level

Dissertation

Keyword(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.

Graduation Semester

2014-08

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

Copyright and License Information

Copyright 2014 Brian Benson

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