Eigenvalues of the Laplacian for certain Riemannian metrics on S2 and S3
Abstract
In this dissertation, we begin by characterizing the left-invariant Riemannian metrics on S3 possessing a positive definite Ricci tensor. Of these metrics, for the ones that further satisfy Ric g ≥ 2g, we compare the eigenvalues of the associated Laplace operator, Δg, with the eigenvalues of the Laplace operator for the standard Euclidean metric on S3 with constant sectional curvature 1. We then introduce certain conformal, rotationally symmetric, real-analytic perturbations of the standard Euclidean metric on S 2, and study the perturbed eigen-values of the Laplace operators for the metrics sufficiently close to the Euclidean metric. The motivation for this study has been to answer a problem regarding the vector space of harmonic functions with growth conditions on Riemannian manifolds of dimensions three and four having non-negative Ricci curvature.
Degree
Ph.D.
Advisors
Donnelly, Purdue University.
Subject Area
Mathematics
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