Mapping by analytic functions. I. Conformal mapping of multiply-connected domains

Abstract

1. Methods employed. In the theory of analytic functions of one complex variable it is possible to obtain many results of far-reaching importance by means of relatively few general theorems, for example, Riemann's mapping theorem and the lemma of Schwarz. The existence of a non-Euclidean metric which is invariant with respect to conformal mapping follows from Riemann's mapping theorem and is in many instances equivalent to it. In the case of a simply-connected domain B with more than one boundary point, the distance (in this sense) between two points a and b of B is defined as the hyperbolic distance between the images of a and b in the unit circle. By means of this metric it is possible to give to Schwarz's lemma a new and useful formulation, the lemma of Schwarz-Pick. Completely avoiding Riemann's theorem, Bergman succeeded in showing that an invariant Hermitian metric can be derived from a quite different approach and can be generalized to the case of several complex variables. This is important, since Poincare has shown that it is not in general possible to map one domain in 2n-dimensional space on another by means of n analytic functions of n complex variables (pseudo-conformal mapping), so that an immediate generalization of classical function theoretic methods is not possible. Further, in the case of mappings into schlicht domains, Bergman has obtained results which include the lemma of Schwarz-Pick but which can also be applied to the case of several complex variables. However, if those methods originally employed to provide results in pseudo-conformal mapping are specialized to the theory of analytic functions of one complex variable, they do more than merely provide known results: they constitute methods for the investigation of many questions in conformal mapping of multiply-connected domains. Further, theorems that we may obtain in the case of one variable by these methods suggest analogous theorems for pseudo-conformal mapping. It is this dual role which makes both the theorems obtained and the methods employed take on added importance (1).