On canonical conformal maps of multiply connected domains

Abstract

In a recent paper [II], J. L. Walsh has established the existence of two new canonical domains for the conformal mapping of multiply connected regions. The canonical maps may be characterized by the fact that they transform certain functions into functions which are extendible harmonically to the entire plane, with the exception of a finite number of points, one in each of the components of the complement of the canonical domain. Two other existence proofs have been given, by H. Grunsky [6] and J. A. Jenkins [7]. All of these proofs rely on uniformization. In part I of the present paper we give a geometric and constructive proof of existence, based primarily on potential-theoretic considerations. In part II we consider the relation between the new canonical mapping functions and extremal problems, and show thereby that the maps are natural generalizations of those which take, respectively, a doubly connected onto an annulus and simply connected onto the unit circle. Finally, in part III, by specializing the existence proof to the case of a doubly connected domain, we obtain a new construction procedure for the conformal map of such a onto an annulus, and show it to converge more rapidly than one commonly used at present. The author is deeply grateful to Professors J. L. Walsh, L. V. Ahlfors, and R. Osserman for their inspiration and help. I. Existence of the canonical map. Throughout this discussion let the term harmonic measure of an annular domain be taken to mean the measure of the outer boundary curve with respect to the domain; that is, the function which is in the and takes the values 0, 1 on the inner and outer boundary curves respectively. Also, let the term contour denote an analytic Jordan curve. THEOREM 1. Let C1 be a Jordan curve in the z-plane, and D its exterior. Let