The asymptotic behavior of Green's functions for degenerating hyperbolic surfaces

Abstract

Given a degenerating family of Riemann surfaces of genus g > 2, it is a natural and important question to study their spectral behaviors, in particular, Green's functions, during degeneration (see [9, 10, 17, 11, 13 and 5]). By the uniformization theorem, any Riemann surface of genus g > 2 has a canonical metric of constant curvature 1 . From now on, we call a Riemann surface with this canonical metric a hyperbolic surface, and a degenerating family of hyperbolic surfaces is denoted by St (l> 0). The degeneration of St is caused by the pinching of several simple, closed geodesics 71(l), ..., 7,,(1) on St, that is, the lengths of these geodesics go to zero as l--+0. Let At be the Laplacian of St, then the kernel of At consists of functions which are constant on each connected component of St, and At is invertible on the orthogonal complement of the kernel. The Schwartz kernel of this inverse is called the Green function of St, and denoted by G(z, w; t) for z, w~Sl. We would like to study the asymptotic behavior of G(z, w; I) as l ~O. In order to compare functions on S O and St, we use the harmonic map of infinite energy ~ : So --* S~\{71 (l) . . . . ,7,,(l)} constructed by Wolf El4]. Intuitively speaking, this map ~z opens up each node (a pair of cusps) of So into a simple closed pinching geodesic of SF. At first, we assume that there is only one pinching geodesic ~'(l)=7~(/) of length l on S~. If 7'([) separates St, then So has two components Se, ~ and So,z, of genus g~, gz respectively.