On conformal maps of infinitely connected Dirichlet regions

Abstract

Let D be a plane region of arbitrary connectivity (> 1) for which the Dirichlet problem is solvable. There exists a conformal map of D onto a region bounded by two level loci of H, a nontrivial harmonic measure. H is essentially the difference of two logarithmic potentials. The two measures involved are mutually singular probability measures. Further properties of these measures, and of H, are derived. The special case in which D is of connectivity 2 is the classical theorem which states that an annular region is conformally equivalent to a region bounded by two circles. The case in which D is of finite connectivity was treated by J. L. Walsh in 1956. A similar generalization of the Riemann mapping theorem is also established. Finally, converses of the above results are also valid. In 1956, J. L. Walsh showed that a plane region bounded by finitely many, mutually disjoint Jordan curves could be mapped conformally onto a region bounded by two level loci of a function harmonic in the plane with the exception of a finite number of poles, one in each component of the complement of the region [11]. This is presented below as Theorem 1, where -r log T(Z) is harmonic, and the region A is defined by 0 < log T(Z) < 1. A special case of Theorem 1 is the classical theorem which states that a doubly connected region can be mapped conformally onto an annulus. Furthermore, Theorem 3 below, due also to J. L. Walsh, is a similar generalization of the Riemann mapping theorem. At least three other proofs of these theorems have since been given (H. Grunsky, [4]; J. A. Jenkins, [7]; H. J. Landau, [8]). It is the object of this paper to show in Theorems 2 and 4 below that the theorems of Walsh extend further to Dirichlet regions of arbitrary connectivity. Finally, Theorem 5 below can be regarded as a converse of Theorem 2. The existence of the conformal map in Theorem 2 characterizes the Dirichlet region, and thus may prove useful in the study of such regions. Here, a Dirichlet region is one for which the Dirichlet problem is solvable. See [3, p. 53] for a totally unrelated definition of Presented in part to the Society, January 27, 1963, under the title On conformal maps of regions of infinite connectivity; received by the editors November 20, 1969. AMS 1969 subject classifications. Primary 3040; Secondary 3075, 3110, 3115.