Conformal Invariants and Prime Ends

Abstract

Introduction. From the point of view of conformal mapping, it is unsatisfactory to consider the individual points of the boundary of a simply connected region as the primitive constituents of this boundarv. When such a region is mapped conformally onto the unit disk, in accordance with the Riemann mapping theorem, the points of the unit circumference correspond, indeed, to the prime ends of the region. These prime ends are here considered as equivalence classes of sequences of points of the region under a relation P*. The principal tool we employ to introduce this relation is extremal length, a conformally invariant quantity associated with a family of curves. Its definition was first given by Ahlfors and Beurling ([2], [3]). The term prinle end originated with Caratheodory [4], who initiated the systematic study of the structure of the boundary of a simply connected region. His approach was topological, and it dealt with conceptssubregions, crosscuts, etc.-which are defined with reference to the given region. It is easily seen that with Caratheodory's definition the prime ends of the unit disk correspond to points on the circumference in the sense that an equivalence class is associated with each boundary point and that two sequences belong to the same class if and only if they converge to the same point of the circumference. The problem that arises under his approach, however, and this is solved by one of Caratheodory's fundamental theorems, is that one must show that prime ends are preserved under conformal mapping. Lindelif [6] circumvented this difficulty by defining prime ends by reference to the conformal map of the disk onto the region; namely in terms of the set of indetermination or cluster set. However, his method does not obviate an explicit analysis of the topological situation in the region itself. Lindelof also obtained the following result, which is related to the classification of prime ends. Caratheodory had distinguished two classes of points, principal and subsidiary (see Section 2. 7, below), on the point set of a prime end, i. e. on the