Concerning the uniformization of certain Riemann surfaces allied to the inverse cosine and inverse gamma surfaces
Mac Lane, Gerald R.
Doctor of Philosophy
The present paper will prove three different classes of surfaces parabolic, each class containing members exhibiting an indirectly critical point. In chapter II it is shown that all symmetric surfaces "like" that defined by w = cos z are parabolic. The method employed also yields the precise form of the corresponding entire function. The converse is then proved: i.e., all entire functions of this form map the punched plane onto a symmetric semi-cosinic surface. Chapters III and IV follow an analogous scheme for the symmetric surfaces modelled after those defined by w = 1/Gamma (z) and w = cosz. Chapter V contains a brief discussion of the fundamental theorem connecting the type problem with functions of bounded eccentricity. In Chapter VI this method is used to show that all surfaces of the above three classes are parabolic, whether symmetric or not. The question of the form of the corresponding entire function in the unsymmetric cases is as yet unanswered. The method employed in Chapters II, III, IV is that of approximation by elliptic surfaces. How far it is capable of extension to other surfaces depends, roughly speaking, on how much may be determined about the numerical nature of rational functions with prescribed surfaces.