Spectral asymptotics on degenerating hyperbolic 3-manifolds / Józef Dodziuk, Jay Jorgenson.

Format:

Book

Author/Creator:

Dodziuk, Józef, 1947- author.

Jorgenson, Jay, author.

Series:

Memoirs of the American Mathematical Society ; no. 643.

Memoirs of the American Mathematical Society, 0065-9266 ; number 643

Language:

English

Subjects (All):

Geometry, Hyperbolic.

Hyperbolic spaces.

Spectral theory (Mathematics).

Asymptotic expansions.

Physical Description:

1 online resource (90 p.)

Edition:

1st ed.

Place of Publication:

Providence, Rhode Island : American Mathematical Society, [1998]

Language Note:

English

Summary:

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 assets 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 convergentce of tubes in the manifolds of the sequence to cusps of the limiting manifold. In the specral theory aspect of the work, they prove convergence of heat kernels. They then define a regualrized heat race associated to any finite volume, complete, hyperbolic three manifold, and study its asymptotic behaviour through degeneration. As an application of the analysis of the regularized heat trace, they study asymptotic behaviours 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 behaviour of spectral functions on degenerating families of finite area hyperbolic Riemann surfaces.

Contents:

""Contents""; ""Introduction""; ""Â1- Review of hyperbolic geometry""; ""Â2. Convergence of heat kernels""; ""Â3. Infinite cylinder estimates""; ""Â4. Heat kernels and regularized heat traces""; ""Â5. Degenerating heat traces""; ""Â6. Poisson kernel estimates""; ""Â7. Analysis of trace integrals""; ""Â8. Convergence of regularized heat traces""; ""Â9. Long time asymptotics""; ""Â10. Spectral zeta functions""; ""Â11- Selberg zeta functions""; ""Â12. Hurwitz- type zeta functions""; ""Â13. Asymptotics of spectral measures""; ""Â14. Eigenvalue counting problems""

""Â15. Convergence of spectral projections""""Bibliography""

Notes:

"September 1998, volume 135, number 643 (third of 5 numbers)."

Includes bibliographical references (pages 73-75).

Description based on print version record.

ISBN:

1-4704-0232-7