A note on continuity of Green's functions on Riemann surfaces
Let R be an arbitrary Riemann surface admitting Green's functions, and denote by g(\(\cdot,r)\) the Green function with the pole \(r\in R\). Set \(U(r,M)=\{s\in R:\) \(g(s,r)>M\}\) where M is positive. The main results of this paper are as follows: (a) Let q be a point on R. Take M so large that U(q,M) is simply connected. Then it holds that \[ \| dg(\cdot,q')- dg(\cdot,q)\|_{R-U(q,M)}<6\cdot e^{M+4} \exp (-g(q',q)) \] for every \(q'\in U(q,M+4).\) (b) Let p and q be two distinct points on R. Take M so large that U(q,M) is contained in R-\(\{\) \(p\}\) and simply connected. Then it holds that \[ | g(q',p)-g(q,p)| <5M^{1/2} e^{M+4} \exp (-g(q',q)) \] for every \(q'\in U(q,M+4)\).
scientific article; zbMATH DE number 4029927