Memoirs of the American Mathematical Society 1998; 75 pp; softcover Volume: 135 ISBN10: 0821808370 ISBN13: 9780821808375 List Price: US$48 Individual Members: US$28.80 Institutional Members: US$38.40 Order Code: MEMO/135/643
 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. Readership Graduate students and research mathematicians working in global analysis, analysis on manifolds. Table of Contents  Introduction
 Review of hyperbolic geometry
 Convergence of heat kernels
 Infinite cylinder estimates
 Heat kernels and regularized heat traces
 Degenerating heat traces
 Poisson kernel estimates
 Analysis of trace integrals
 Convergence of regularized heat traces
 Long time asymptotics
 Spectral zeta functions
 Selberg zeta functions
 Hurwitztype zeta functions
 Asymptotics of spectral measures
 Eigenvalue counting problems
 Convergence of spectral projections
 Bibliography
