A theorem on mappings into Riemann surfaces of infinite genus

Abstract

1. Consider complex analytic mappings f of a noncompact Riemann surface R into a compact or noncompact Riemann surface S of finite or infinite genus. If the integrated Euler characteristic E(Q) of an exhausting subregion Q c R does not grow more rapidly than the characteristic function C(Q) off, then the defect sum is finite [2]. In the general case there can be infinitely many Picard points and nothing is known about the set of defect points. We shall show that the defect points under every f are so scarce that