A REGULARIZED HEAT TRACE FOR HYPERBOLIC RIEMANN SURFACES OF FINITE-VOLUME

Let M denote a hyperbolic Riemann surface of finite volume, and let K- M(t,x,y) be the heat kernel associated to the hyperbolic Laplacian whi ch acts on the space of smooth functions on M. If M is compact, then w e have the equality [GRAPHICS] where {lambda(n)} is the set of eigenva lues of the Laplacian. If M is not compact, then it is well known that the heat kernel exists yet is not of trace class. In this paper we wi ll define a regularized heat trace associated to any hyperbolic Rieman n surface of finite volume, compact or noncompact. After we have defin ed the regularized heat trace, we study the asymptotic behavior of the regularized heat trace on a family of degenerating hyperbolic Riemann surfaces. Our results involve pointwise convergence and uniformity of asymptotic expansions in the pinching parameters. In particular, we s tudy uniformity of long time asymptotics of tile regularized heat trac e minus the contribution from the small eigenvalues by analyzing the P oisson kernel and Dirichlet heat kernel in a finite cylindrical neighb orhood of tile pinching geodesics. As applications of our results, we are able to study asymptotic expansions of the Selberg zeta function a nd spectral zeta function on degenerating families, both improving kno wn results in the compact setting and proving new results in the non-c ompact situation. Results from this article have been extended to the setting of degenerating hyperbolic three manifolds of finite volume In [DJ1] and [DJ2].