Hostname: page-component-78c5997874-4rdpn Total loading time: 0 Render date: 2024-11-03T08:45:22.676Z Has data issue: false hasContentIssue false
  • English
  • Français

Theory of Meromorphic Functions on an Open Riemann Surface with Null Boundary

Published online by Cambridge University Press:  22 January 2016

Masatsugu Tsuji
Masatsugu Tsuji*
Affiliation:
Mathematical Institute, Tokyo University
Rights & Permissions [Opens in a new window]

Extract

Core share and HTML view are not available for this content. However, as you have access to this content, a full PDF is available via the ‘Save PDF’ action button.

In the former paper, I have developped a theory of meromorphic functions in a neighbourhood of a bounded closed set E of logarithmic capacity zero, by means of Evans’ potential fnnction u(z), which tends to ∞, when z tends to any point of E. It is not known, whether such a potential function exists on an open Riemann surface with null boundary, but by a substitute of Evans’ function. we shall develop the similar theory of meromorphic functions on an open Riemann surface with null boundary.

Type
Research Article
Information
Nagoya Mathematical Journal , Volume 6 , October 1953 , pp. 137 - 150
Copyright
Copyright © Editorial Board of Nagoya Mathematical Journal 1953

References

1) Tsuji, M.: On the behaviour of a mernmorphic function in the neighbourhood of a closed set of capacity zero. Proc. Imp. Acad. 18 (1942)Google Scholar. Tsuji, M.: Theory of meromorphic functions in an neighbourhood of a closed set of capacity zero. Jap. Journ. Math. 19 (1944-48)Google Scholar.

2) Nagai, Y.: On the behaviour of the boundary of Riemann surfaces, II. Proc. Japan Acrid. 26 (1950)Google Scholar. Yûjôbô, Z.: On the Riemann surfaces, no Green’s function of which exists. Mathematica Japonicae. II, No. 2 (1951)Google Scholar. Tsuji, M.: Some metrical theorems on Fuchsian groups. Kodai Math. Seminar Reports. Nos. 4-5 (1950)Google Scholar. Mori, A.: On Riemann surfaces on which no bounded harmonic function exists. Journ. Math. Soc. Japan. 3 (1951)Google Scholar.

3) Noshiro, K.: Open Riemann surface with null boundary. Nagoya Math. Journ. 3 (1951)Google Scholar. Z, Yûjôbô. 1. c. 2).

4) Gross, W.: Über die Singularitaten analytischer Funktionen. Monatshefte f, Math. u. Phys. 29 (1918)Google Scholar.

5) K. Noshiro: 1. c. 3).

6) Myrberg, P. J.: Die Kapazitôt der singularen Menge der linearen Gruppen. Ann. Acad. Fenn. Ser. A. Math.-Phys. 10 (1941)Google Scholar. Tsuji, M.: On the uniformization of an algebraic function of genus p≧2. Tohoku Math. Journ. 3 (1951)Google Scholar.

7) K. Noshiro: 1. c. 3).

8) Ahlfors, L.: Zur Theorie der Überlagerungsflächen. Acta Math. 65 (1935)Google Scholar.

9) L. Ahlfors: 1. c. 8).

10) Nevanlinna, R.: Uber der Existenz von beschränkten Potentialfunktionen auf Flachen von unendlichem Geschlecht. Math. Zeits. 52 (1950)Google Scholar.

11) Bader, R. and Pareau, M.: Domaines non-compacts et classification des surfaces de Riemann. C.R. 232 (1951)Google Scholar. Mori, A.: On the existence of harmonic functions on a Riemann surface. Journ. Fac. Sci. Tokyo Univ. Section I, Vol. VI, Part 4 (1951)Google Scholar.

12) Sario, L.: Über Riemannsche Fläche mit hebbarem Rand. Ann. Acad. Fenn. A. I. 50 (1948)Google Scholar.

13) Tsuji, M.: On a regular function which is of constant absolute value on the boundary of an infinite domain. Tohoku Math. Journ. 3 (1951)Google Scholar.

14) M. Tsuji: 1. c. 13).

15) Dinghas, A.: Eine Bemerkung zur Ahlforsschen Theorie der Überlagerungsflächen. Math. Zeits. 44 (1936).Google Scholar

16) Y. Nagai: 1. c. 2), 1. Yûjôbô: 1. c. 2). M. Tsuji: 1. c. 2). A. Mori: 1. c. 2).