Norms on cohomology, harmonic forms, eigenvalues and minimal surfaces in hyperbolic manifolds

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.

Name thesis:degree_name

Ph.D.

Level thesis:degree_level

Dissertation

Discipline thesis:degree_discipline

Mathematics

Grantor

University of Illinois at Urbana-Champaign

Year dc:date

2021

Author dc:creator

• Han, Xiaolong

Contributors dc:contributor

• Dunfield, Nathan
• Laugesen, Richard
• Hirani, Anil
• Albin, Pierre

• norms of cohomology
• hyperbolic manifolds
• eigenvalues

Statement dc:rights

• Copyright 2021 Xiaolong Han

Language dc:language

en

Handle dc:identifier

http://hdl.handle.net/2142/110435

OAI identifier oai:identifier

oai:www.ideals.illinois.edu:2142/110435