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Title:Norms on cohomology, harmonic forms, eigenvalues and minimal surfaces in hyperbolic manifolds
Author(s):Han, Xiaolong
Director of Research:Dunfield, Nathan
Doctoral Committee Chair(s):Laugesen, Richard
Doctoral Committee Member(s):Hirani, Anil; Albin, Pierre
Department / Program:Mathematics
Discipline:Mathematics
Degree Granting Institution:University of Illinois at Urbana-Champaign
Degree:Ph.D.
Genre:Dissertation
Subject(s):norms of cohomology
hyperbolic manifolds
eigenvalues
Abstract:We bound the L2-norm of an L2 harmonic 1-form in an orientable cusped hyperbolic 3-manifold M by its topological complexity, measured by the Thurston norm, up to a constant depending on M. It generalizes two inequalities of Brock-Dunfield. We also study the sharpness of the inequalities in the closed and cusped cases, using the interaction of minimal surfaces and harmonic forms. We unify various results by defining two functionals on orientable closed and cusped hyperbolic-manifolds, and formulate several questions and conjectures. Using similar decomposition principles, we also obtain results on eigenvalues of infinite volume geometrically finite hyperbolic manifolds.
Issue Date:2021-04-19
Type:Thesis
URI:http://hdl.handle.net/2142/110435
Rights Information:Copyright 2021 Xiaolong Han
Date Available in IDEALS:2021-09-17
Date Deposited:2021-05


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