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Spectral Asymptotics on Degenerating Hyperbolic 3-Manifolds
 
Józef Dodziuk City University of New York (CUNY), New York
Jay Jorgenson Oklahoma State University, Stillwater
Spectral Asymptotics on Degenerating Hyperbolic 3-Manifolds
eBook ISBN:  978-1-4704-0232-7
Product Code:  MEMO/135/643.E
List Price: $48.00
MAA Member Price: $43.20
AMS Member Price: $28.80
Spectral Asymptotics on Degenerating Hyperbolic 3-Manifolds
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Spectral Asymptotics on Degenerating Hyperbolic 3-Manifolds
Józef Dodziuk City University of New York (CUNY), New York
Jay Jorgenson Oklahoma State University, Stillwater
eBook ISBN:  978-1-4704-0232-7
Product Code:  MEMO/135/643.E
List Price: $48.00
MAA Member Price: $43.20
AMS Member Price: $28.80
  • Book Details
     
     
    Memoirs of the American Mathematical Society
    Volume: 1351998; 75 pp
    MSC: 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 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.

    Readership

    Graduate 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. Hurwitz-type zeta functions
    • 13. Asymptotics of spectral measures
    • 14. Eigenvalue counting problems
    • 15. Convergence of spectral projections
  • Requests
     
     
    Review Copy – for publishers of book reviews
    Permission – for use of book, eBook, or Journal content
    Accessibility – to request an alternate format of an AMS title
Volume: 1351998; 75 pp
MSC: 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 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.

Readership

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. Hurwitz-type zeta functions
  • 13. Asymptotics of spectral measures
  • 14. Eigenvalue counting problems
  • 15. Convergence of spectral projections
Review Copy – for publishers of book reviews
Permission – for use of book, eBook, or Journal content
Accessibility – to request an alternate format of an AMS title
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