Microlocal analysis of asymptotically complex hyperbolic manifolds and their conformal contact boundaries

This thesis presents a study of asymptotically complex hyperbolic manifolds and their natural conformal boundaries, compact CR contact manifolds, using the tools of microlocal analysis in the broad sense. In the first chapter, we study the behavior of the heat kernel of the Hodge Laplacian on a contact manifold endowed with a family of Riemannian metrics that blow up the directions transverse to the contact distribution. We apply this to analyze the behavior of global spectral invariants such as the $\eta$-invariant and the determinant of the Laplacian. In particular, we prove that contact versions of the relative $\eta$-invariant and the relative analytic torsion are equal to their Riemannian analogs and hence topological. In the second chapter, we study the behavior of the resolvent and wave kernel of the Laplacian on asymptotically complex hyperbolic manifolds. In particular, applying the methods introduced in [Va17] in the real setting, we obtain non-trapping estimates for the resolvent, deriving an effective version of the main result of [EpMeMe91]. We also show that the wave group on such manifolds belongs to an appropriate class of Fourier integral operators and analyze its trace. We prove that the singularities of its trace are contained in the set of lengths of closed geodesics and obtain an asymptotic expansion for the trace at time zero. Each chapter can be treated as self-contained and read independently of the other. They were initially written separately and submitted for publication.

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

• Quan, Hadrian

Contributors dc:contributor

• Albin, Pierre
• Tyson, Jeremy
• Laugesen, Richard
• Dunfield, Nathan

• Global Analysis
• Contact Manifolds
• Complex Hyperbolic Manifolds
• Microlocal Analysis

Statement dc:rights

• Copyright 2021 Hadrian Quan

Language dc:language

en

Handle dc:identifier

http://hdl.handle.net/2142/110560

OAI identifier oai:identifier

oai:www.ideals.illinois.edu:2142/110560