The asymptotic behavior of Green's functions for quasi-hyperbolic metrics on degenerating Riemann surfaces
Let \(N\) be a punctured Riemann surface and \(ds^2\) a Hermitian metric on \(N\) which is of moderate growth near the punctures. Then there exists a unique Green's function for \(N\) with respect to \(ds^2\) satisfying the standard conditions and a suitable growth condition near the punctures (Theorem 1). The main aim of the paper under review is to study the behaviour of the Green's function for a family of compact Riemann surfaces of genus \(g\geq 2\) degenerating to a Riemann surface of genus \(g-1\) with a non-separating node. The authors show that the Green's functions associated to such a continuous family of so-called quasi-hyperbolic matrics simply degenerate to that on the smooth part of the noded surface (Theorem 2).
scientific article; zbMATH DE number 1054316