Generalizations of Picard’s theorem for Riemann surfaces

Abstract

Abstract.Let D be a plane domain, E C D a compact set of capacity zero, and / a holomorphic mapping of D\E into a hyperbolic Riemann surface W . Then there is a Riemann surface W1 containing W such that f extends to a holomorphic mapping of D into W . The same conclusion holds if hyperbolicity is replaced by the assumption that the genus of W be at least two. Furthermore, there is quite a general class of sets of positive capacity which are removable in the above sense for holomorphic mappings into Riemann surfaces of positive genus, except for tori.