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 Title: Microlocal analysis of asymptotically complex hyperbolic manifolds and their conformal contact boundaries Author(s): Quan, Hadrian Director of Research: Albin, Pierre Doctoral Committee Chair(s): Tyson, Jeremy Doctoral Committee Member(s): Laugesen, Richard; Dunfield, Nathan Department / Program: Mathematics Discipline: Mathematics Degree Granting Institution: University of Illinois at Urbana-Champaign Degree: Ph.D. Genre: Dissertation Subject(s): Global Analysis Contact Manifolds Complex Hyperbolic Manifolds Microlocal Analysis Abstract: 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. Issue Date: 2021-04-23 Type: Thesis URI: http://hdl.handle.net/2142/110560 Rights Information: Copyright 2021 Hadrian Quan Date Available in IDEALS: 2021-09-17 Date Deposited: 2021-05
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