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Spectral Asymptotics on Degenerating Hyperbolic 3-Manifolds
Józef Dodziuk, City University of New York, NY, and Jay Jorgenson, Oklahoma State University, Stillwater, OK

Memoirs of the American Mathematical Society
1998; 75 pp; softcover
Volume: 135
ISBN-10: 0-8218-0837-0
ISBN-13: 978-0-8218-0837-5
List Price: US$48
Individual Members: US$28.80
Institutional Members: US$38.40
Order Code: MEMO/135/643
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In this volume, the authors study asymptotics of the geometry and spectral theory of degenerating sequences of finite volume hyperbolic manifolds of three dimensions. Thurston's hyperbolic surgery theorem asserts the existence of non-trivial sequences of finite volume hyperbolic three manifolds which converge to a three manifold with additional cusps. In the geometric aspect of their study, the authors use the convergence of hyperbolic metrics on the thick parts of the manifolds under consideration to investigate convergence of tubes in the manifolds of the sequence to cusps of the limiting manifold.

In the spectral theory aspect of the work, they prove convergence of heat kernels. They then define a regularized heat trace associated to any finite volume, complete, hyperbolic three manifold, and study its asymptotic behavior through degeneration. As an application of the analysis of the regularized heat trace, they study asymptotic behavior of the spectral zeta function, determinant of the Laplacian, Selberg zeta function, and spectral counting functions through degeneration.

The authors' methods are an adaptation to three dimensions of the earlier work of Jorgenson and Lundelius who investigated the asymptotic behavior of spectral functions on degenerating families of finite area hyperbolic Riemann surfaces.


Graduate students and research mathematicians working in global analysis, analysis on manifolds.

Table of Contents

  • Introduction
  • Review of hyperbolic geometry
  • Convergence of heat kernels
  • Infinite cylinder estimates
  • Heat kernels and regularized heat traces
  • Degenerating heat traces
  • Poisson kernel estimates
  • Analysis of trace integrals
  • Convergence of regularized heat traces
  • Long time asymptotics
  • Spectral zeta functions
  • Selberg zeta functions
  • Hurwitz-type zeta functions
  • Asymptotics of spectral measures
  • Eigenvalue counting problems
  • Convergence of spectral projections
  • Bibliography
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