The asymptotic behavior of Green's functions for degenerating hyperbolic surfaces (Q1319320)

Let \(S_ l\) \((l\geq 0)\) be a degenerating family of hyperbolic surfaces. In particular, for \(l > 0\), \(S_ l\) is a compact surface with a metric of constant curvature \(-1\), but \(S_ 0\) is a complete and non-compact with a metric of constant curvature \(-1\) and of finite area. For each \(l \geq 0\), the Laplacian \(\Delta_ l\) of \(S_ l\) is invertible on the orthogonal complement of constant functions, and the Schwartz kernel of this inverse is called the Green function of \(S_ l\), and denoted by \(G(z,w;l)\) for \(z,w \in S_ l\). We get the asymptotic behavior of Green's function \(G(z,w;l)\) of \(S_ l\) as \(l\to 0\) in terms of the lengths of the pinching geodesics of the family \(S_ l\) and the topology of the limit surface.