A regularized heat trace for hyperbolic Riemann surfaces of finite volume

Abstract

Let M denote a hyperbolic Riemann surface of finite volume, and let KM(t,x,y) be the heat kernel associated to the hyperbolic Laplacian which acts on the space of smooth functions on M. If M is compact, then we have the equality¶¶M∫KM(t, x, x)dμ(x) = ∞n=0Σe-λnt,¶where {λn} is the set of eigenvalues 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 will define a regularized heat trace associated to any hyperbolic Riemann surface of finite volume, compact or non-compact. After we have defined 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 study uniformity of long time asymptotics of the regularized heat trace minus the contribution from the small eigenvalues by analyzing the Poisson kernel and Dirichlet heat kernel in a finite cylindrical neighborhood of the pinching geodesics. As applications of our results, we are able to study asymptotic expansions of the Selberg zeta function and spectral zeta function on degenerating families, both improving known results in the compact setting and proving new results in the non-compact situation. Results from this article have been extended to the setting of degenerating hyperbolic three manifolds of finite volume in [DJ1] and [DJ2].