Date of Award

2015

Document Type

Thesis

Degree Name

Bachelors

Department

Natural Sciences

First Advisor

McDonald, Patrick

Keywords

Mathematics, Dirichlet Laplacian, Dirichlet Spectrum, Poisson Spectrum

Area of Concentration

Mathematics

Abstract

Let M be a Riemannian manifold. A domain functional on M is a real valued function on the collection of smoothly bounded domains of M with compact closure. A domain functional is a geometric invariant if it is invariant under the action of the isometry group associated to M. We survey literature associated to two important sequences of geometric invariants, the Dirichlet spectrum and the Poisson spectrum, paying close attention to the principal Dirichlet eigenvalue and the torsional rigidity, or the first element of each sequence, respectively. We study domain functionals in the context of surfaces of revolution. Our main result is an explicit formula for the variation of the logarithm of the determinant of the Dirchlet Laplacian on a surface of revolution.

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