**Memoirs of the American Mathematical Society**

1998;
75 pp;
Softcover

MSC: Primary 58; 11; 57;
Secondary 35

Print ISBN: 978-0-8218-0837-5

Product Code: MEMO/135/643

List Price: $48.00

Individual Member Price: $28.80

**Electronic ISBN: 978-1-4704-0232-7
Product Code: MEMO/135/643.E**

List Price: $48.00

Individual Member Price: $28.80

# 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

#### Readership

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