eBook ISBN:  9781470402327 
Product Code:  MEMO/135/643.E 
List Price:  $48.00 
MAA Member Price:  $43.20 
AMS Member Price:  $28.80 
eBook ISBN:  9781470402327 
Product Code:  MEMO/135/643.E 
List Price:  $48.00 
MAA Member Price:  $43.20 
AMS Member Price:  $28.80 

Book DetailsMemoirs of the American Mathematical SocietyVolume: 135; 1998; 75 ppMSC: Primary 58; 11; 57; Secondary 35
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 nontrivial 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.
ReadershipGraduate students and research mathematicians working in global analysis, analysis on manifolds.

Table of Contents

Chapters

Introduction

1. Review of hyperbolic geometry

2. Convergence of heat kernels

3. Infinite cylinder estimates

4. Heat kernels and regularized heat traces

5. Degenerating heat traces

6. Poisson kernel estimates

7. Analysis of trace integrals

8. Convergence of regularized heat traces

9. Long time asymptotics

10. Spectral zeta functions

11. Selberg zeta functions

12. Hurwitztype zeta functions

13. Asymptotics of spectral measures

14. Eigenvalue counting problems

15. Convergence of spectral projections


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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 nontrivial 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.

Chapters

Introduction

1. Review of hyperbolic geometry

2. Convergence of heat kernels

3. Infinite cylinder estimates

4. Heat kernels and regularized heat traces

5. Degenerating heat traces

6. Poisson kernel estimates

7. Analysis of trace integrals

8. Convergence of regularized heat traces

9. Long time asymptotics

10. Spectral zeta functions

11. Selberg zeta functions

12. Hurwitztype zeta functions

13. Asymptotics of spectral measures

14. Eigenvalue counting problems

15. Convergence of spectral projections