SPECTRAL ASYMPTOTICS ON DEGENERATING HYPERBOLIC 3-MANIFOLDS

In this memoir we study asymptotics of the geometry and spectral theor y of degenerating sequences of finite volume hyperbolic manifolds of t hree dimensions. Thurston's hyperbolic surgery theorem asserts the exi stence of nontrivial sequences of finite volume hyperbolic three manif olds which converge to a three manifold with additional cusps. In the geometric aspect of our study, we use the convergence of hyperbolic me trics on the thick parts of the manifolds under consideration to inves tigate convergence of tubes in the manifolds of the sequence to cusps of the limiting manifold. In the spectral theory aspect of our work, w e prove convergence of heat kernels. We then define a regularized heat trace associated to any finite volume, complete, hyperbolic three man ifold, and study its asymptotic behavior through degeneration. As an a pplication of our analysis of the regularized heat trace, we study asy mptotic behavior of the spectral zeta function, determinant of the Lap lacian, Selberg zeta function, and spectral counting functions through degeneration. Our methods are an adaptation to three dimensions of th e earlier work of Jorgenson and Lundelius who investigated the asympto tic behavior of spectral functions on degenerating families of finite area hyperbolic Riemann surfaces.