Norms on cohomology, harmonic forms, eigenvalues and minimal surfaces in hyperbolic manifolds
Han, Xiaolong
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https://hdl.handle.net/2142/110435
Description
Title
Norms on cohomology, harmonic forms, eigenvalues and minimal surfaces in hyperbolic manifolds
Author(s)
Han, Xiaolong
Issue Date
2021-04-19
Director of Research (if dissertation) or Advisor (if thesis)
Dunfield, Nathan
Doctoral Committee Chair(s)
Laugesen, Richard
Committee Member(s)
Hirani, Anil
Albin, Pierre
Department of Study
Mathematics
Discipline
Mathematics
Degree Granting Institution
University of Illinois at Urbana-Champaign
Degree Name
Ph.D.
Degree Level
Dissertation
Keyword(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.
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