Analytic Functions on Some Riemann Surfaces, II

Abstract

To Professor Kinjiro Kunugi on the occasion of his 60th birthday 1. In their paper [12], Toda and the author have concerned themselves in the following THEOREM OF KURAMOCHI. Let R be a hyperbolic Riemann surface of the class OHB(OHD, resp.). Then, for any compact subset K of R such that R-K is connected, R-K as an open Riemann surface belongs to the class O A B(O AΌ ,resp.) (Kuramochi [4]).They have raised there the question as to whether there exists a hyperbolic Riemann surface, which has no Martin or Royden boundary point with positive harmonic measure and has yet the same property as stated in Theorem of Kuramochi, and given a positive answer to the Martin part of this question.The main purpose of this paper is to show that the Royden part is also answered in the positive. In the sequel, we shall investigate covering properties of analytic functions on Riemann surfaces of the class O ADy which was introduced by Kuroda in his paper [6], give an extension of the D part of Theorem of Kuramochi and, using this extension, give an example of a Riemann surface which answers the Royden part of the question in the positive.2. Let R be a Riemann surface and let G be a domain on R with smooth relative boundary dG clustering nowhere in R. For simplicity, we shall call such a domain G a subregion of R. If G admits no non-constant single-valued analytic function with a finite Dirichlet integral and with real part vanishing continuously on its relative boundary 'dG, we say that G belongs to the class SO AD .Let w = f(p) be a non-constant single-valued analytic function in a subregion G with relative boundary dG of a Riemann surface R. We suppose that Received May 10, 1963. Therefore at any ideal boundary point P of R, the cluster set Cκ-κ(f, P)can not be total and hence, by Theorem 2, it reduces to one point, .since the finiteness of the spherical area is derived from the finiteness of the Dirichlet integral. Our proof is now complete. con,m(p) increases as ra-* co, so that ω n .m(P) tends to a harmonic function ω n {p). Now let w tend to infinity. Then ω n (p) decreases and tends to a non-R -Ro, and we say that the harmonic measure of an ideal boundary point P of R is zero or positive according as the first or the second case occurs, respectively.Of course this property of P does not depend on the choice of the exhaustion {R n } of R.Now we have an extension of Theorem of Kuramochi stated in § 1. such that each boundary component of the relative boundary of A rounds the hole of (Ci, C 2 ) that is, According to Nakai, we say that a Riemann surface R is an almost finite Riemann surface or that R is of almost finite genus, if there exists a sequence {An} of annuli in R satisfying the following condition:(5) An^Hn, where {H n } is the totality of handles in R, Further we denote by γ n the closed Jordan curve in A n which divides A n into two annuli A n ,i and A n ,2 such that mod i4«,i = mod A n ,2 = mod Aj2.Nakai's theorem can be derived from the following four propositions. PROPOSITION 4. Let u be an HP-minimal f...