Conformal mapping of Riemann surfaces and the classical theory of univalent functions

Abstract

Analytic mappings between Riemann surfaces are very natural objects in complex analysis. Corresponding to the classical univalent functions we have the class of injective holomorphic mappings — i.e., conformal embeddings — of a Riemann surface into another. We find indeed a number of analogies between them. On the other hand, because of the non-planarity of the domain surface we face some new problems which we have never encountered in the classical theory. We discuss various problems concerning the conformal embeddings.