The asymptotic behavior of Green's functions for quasi-hyperbolic metrics on degenerating Riemann surfaces

In this article, we consider a family of compact Riemann surfaces of genus q ≥ 2 degenerating to a Riemann surface of genus q-1 with a non-separating node. We show that the Green's functions associated to a continuous family of quasi-hyperbolic metrics on such degenerating Riemann surfaces simply degenerate to that on the smooth part of the noded Riemann surface.