Eigenvalue Inequalities for a Family of Spherically Symmetric Riemannian Manifolds

Author

Julie Miker, University of KentuckyFollow

Date Available

12-14-2011

Year of Publication

2009

Degree Name

Doctor of Philosophy (PhD)

Document Type

Dissertation

College

Arts and Sciences

Department

Mathematics

First Advisor

Dr. Peter Hislop

Abstract

This thesis considers two isoperimetric inequalities for the eigenvalues of the Laplacian on a family of spherically symmetric Riemannian manifolds. The Payne-Pólya-Weinberger Conjecture (PPW) states that for a bounded domain Ω in Euclidean space Rⁿ, the ratio λ1(Ω)/λ0(Ω) of the first two eigenvalues of the Dirichlet Laplacian is bounded by the corresponding eigenvalue ratio for the Dirichlet Laplacian on the ball B_Ωof equal volume. The Szegö-Weinberger inequality states that for a bounded domain Ω in Euclidean space Rⁿ, the first nonzero eigenvalue of the Neumann Laplacian μ1(Ω) is maximized on the ball B_Ω of the same volume. In the first three chapters we will look at the known work for the manifolds Rⁿ and Hⁿ. Then we will take a family a spherically symmetric manifolds given by Rⁿ with a spherically symmetric metric determined by a radially symmetric function f. We will then give a PPW-type upper bound for the eigenvalue gap, λ1(Ω) − λ0(Ω), and the ratio λ1(Ω)/λ0(Ω) on a family of symmetric bounded domains in this space. Finally, we prove the Szegö-Weinberger inequality for this same class of domains.

Recommended Citation

Miker, Julie, "Eigenvalue Inequalities for a Family of Spherically Symmetric Riemannian Manifolds" (2009). University of Kentucky Doctoral Dissertations. 783.
https://uknowledge.uky.edu/gradschool_diss/783