Harmonic forms under metric and topological perturbations
Kerofsky, Louis Joseph
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https://hdl.handle.net/2142/22489
Description
Title
Harmonic forms under metric and topological perturbations
Author(s)
Kerofsky, Louis Joseph
Issue Date
1995
Doctoral Committee Chair(s)
Tondeur, Philippe
Department of Study
Mathematics
Discipline
Mathematics
Degree Granting Institution
University of Illinois at Urbana-Champaign
Degree Name
Ph.D.
Degree Level
Dissertation
Keyword(s)
Mathematics
Language
eng
Abstract
We construct examples illustrating various aspects of Hodge theory on Riemannian manifolds. We consider the relationship between the harmonic forms on a product space and the harmonic forms on the factors as well as the harmonic forms on a connected sum and the harmonic forms on the summands. Algebraic topology provides relations between the cohomology groups of these spaces via the Kunneth formula and the connected sum formula. We consider the problem of describing these relationships in terms of DeRham cohomology and in terms of Hodge theory. For a connected sum, we use a localization method to produce DeRham representatives on the connected sum from DeRham representatives on the summands. We give an example of a Riemannian connected sum, the double torus, where it is impossible to extend a harmonic form on one summand to the entire connected sum.
Studying Hodge theory on warped products, we give an explicit solution for the harmonic one forms on the torus embedded in three space with its induced metric. We consider examples of harmonic forms on the same smooth manifold equipped with different metrics. In general, a localized change in metric alters the harmonic representative of a fixed cohomology class globally. We give a surprising example of two metrics on the torus which differ only inside of a ball such that the harmonic forms representing the DeRham class of dx in each metric are identical outside the ball. We utilize the heat equation determined by the Laplacian to examine these examples more carefully.
Finally, possible applications of the methods introduced in this work are discussed. Areas of application include spectral theory, Dehn surgery, the metric dependence of harmonic forms, and the study of warped product spaces.
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