Use this URL to cite or link to this record in EThOS: http://ethos.bl.uk/OrderDetails.do?uin=uk.bl.ethos.631609
Title: On the second variation of the spectral zeta function of the Laplacian on homogeneous Riemanniann manifolds
Author: Omenyi, Louis Okechukwu
ISNI:       0000 0004 5357 4948
Awarding Body: Loughborough University
Current Institution: Loughborough University
Date of Award: 2014
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Abstract:
The spectral zeta function, introduced by Minakshisundaram and Pleijel in [36] and denoted by ζg(s), encodes important spectral information for the Laplacian on Riemannian manifolds. For instance, the important notions of the determinant of the Laplacian and Casimir energy are defined via the spectral zeta function. On homogeneous manifolds, it is known that the spectral zeta function is critical with respect to conformal metric perturbations, (see e.g Richardson ([47]) and Okikiolu ([41])). In this thesis, we compute a second variation formula of ζg(s) on closed homogeneous Riemannian manifolds under conformal metric perturbations. It is well known that the quadratic form corresponding to this second variation is given by a certain pseudodifferential operator that depends meromorphically on s. The symbol of this operator was analysed by Okikiolu in ([42]). We analyse it in more detail on homogeneous spaces, in particular on the spheres Sn. The case n = 3 is treated in great detail. In order to describe the second variation we introduce a certain distributional integral kernel, analyse its meromorphic properties and the pole structure. The Casimir energy defined as the finite part of ζg(-½) on the n-sphere and other points of ζg(s) are used to illustrate our results. The techniques employed are heat kernel asymptotics on Riemannian manifolds, the associated meromorphic continuation of the zeta function, harmonic analysis on spheres, and asymptotic analysis.
Supervisor: Not available Sponsor: Not available
Qualification Name: Thesis (Ph.D.) Qualification Level: Doctoral
EThOS ID: uk.bl.ethos.631609  DOI: Not available
Keywords: Spectral zeta function ; Riemannian manifold ; Meromorphic continuation ; Harmonic analysis
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