Harmonic differential with prescribed singularities

Abstract

1. Throughout this paper we denote by R an open Riemann surface and by Ro a relatively compact subdomain of R with the relative boundary aRo consisting of a finite number of mutually disjoint closed analytic Jordan curves. The open set R, = R Po can be considered to be a neighborhood of the ideal boundary 3 of R. For the sake of simplicity, we denote by a the common relative boundary aRo = aR and we fix the orientation of a positively with respect to the domain Ro. A harmonic differential a defined on R1 = R1 u a is called a harmonic singularity