On the Continuation of a Riemann Surface

Abstract

Let F denote a Riemann surface in the sense of Weyl-Rado [15, 17]. If there exists another Riemann surface G such that F admits a (1, 1) directly conformal map onto a proper part, F', of G, then F is said to be continuable and G is said to be a continuation of F; otherwise, F is said to be non-continuable or maximal. The closed Riemann surfaces are maximal. Rado [14] has shown by example that there exist open maximal Riemann surfaces. We remark that every open topological surface admits a topological continuation. Later Bochner [1] established by appeal to the well-ordering hypothesis and transfinite induction that every continuable Riemann surface admits a maximal continuation. Shortly thereafter, the question of characterizing the continuable Riemann surfaceswas considered by de Possel [10, 11], first, in two notes which are fragmentary in character and do not contain any indication of proofs, and later, in his thesis [12], where he gives a characterization of continuable Riemann surfaces in terms of sets of maximal type [12, p. 4] and the topological structure of the surface. We shall consider the family (D consisting of Riemann surfaces, G, which are continuations of a given continuable Riemann surface, F, and of F itself. Let it be assumed that F does not admit the Riemann sphere or a closed Riemann surface of genus one as a continuation save when the contrary is mentioned. Under these hypotheses, it will be shown that an explicitly given subset, (Fe, of (F may be defined in a natural manner to be an V-space in the sense of Frechet [3] and that so defined (Do is compact. With the aid of this result, the theorem of Bochner may be established without appeal to transfinite induction. Problems concerning the exhibition of a maximal continuation of a given continuable Riemann surface, and the existence of a maximal continuation with specified properties to be stated in the course of the present paper find an appropriate setting in the study of the structure of (D and (D . These questions have not been treated hitherto.