# Spectral Asymptotics on Degenerating Hyperbolic 3-Manifolds

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*Józef Dodziuk; Jay Jorgenson*

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.

#### Table of Contents

# Table of Contents

## Spectral Asymptotics on Degenerating Hyperbolic 3-Manifolds

- Contents vii8 free
- Introduction 110 free
- §1- Review of hyperbolic geometry 514 free
- §2. Convergence of heat kernels 1019
- §3. Infinite cylinder estimates 1726
- §4. Heat kernels and regularized heat traces 2231
- §5. Degenerating heat traces 2837
- §6. Poisson kernel estimates 3241
- §7. Analysis of trace integrals 3746
- §8. Convergence of regularized heat traces 4150
- §9. Long time asymptotics 4352
- §10. Spectral zeta functions 4958
- §11- Selberg zeta functions 5261
- §12. Hurwitz- type zeta functions 5564
- §13. Asymptotics of spectral measures 5766
- §14. Eigenvalue counting problems 5867
- §15. Convergence of spectral projections 6877

- Bibliography 7382