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Open Riemann Surface with Null Boundary

Published online by Cambridge University Press:  22 January 2016

Kiyoshi Noshiro
Kiyoshi Noshiro*
Affiliation:
Mathematical Institute, Nagoya University
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Recently the writer has obtained some results concerning meromorphic or algebroidal functions with the set of essential singularities of capacity zero, with an aid of a theorem of Evans. In the present paper, suggested from recent interesting papers of Sario and Pfluger, the writer will extend his results to single-valued analytic functions defined on open abstract Riemann surfaces with null boundary in the sense of Nevanlinna, using a lemma instead of Evans’ theorem.

Type
Research Article
Information
Nagoya Mathematical Journal , Volume 3 , October 1951 , pp. 73 - 79
Copyright
Copyright © Editorial Board of Nagoya Mathematical Journal 1951

References

1) Noshiro, K.: [1] Contributions to the theory of the singularities of analytic functions, Jap. Journ. of Math. 19 (1948), pp, 299–327 Google Scholar; [2] Note on the cluster sets of analytic functions, Journ. Math. Soc. Japan, 1 (1950), pp. 275-281; [3] A theorem on the cluster sets of pseudo-analytic functions, Nagoya Math. Journ. 1 (1950), pp. 83-89.

2) Evans, G. C.: Potentials and positively infinite singularities of harmonic functions, Monatshefte fur Math, und Phys. 43 (1936), pp. 419–424.CrossRefGoogle Scholar

3) Sario, Leo: [1] Über Riemannsche Flachen mit hebbarem Rand, Ann. Acad, Sci. Fenn. A. 1. 50 (1948), 79 ppGoogle Scholar. [2] Sur les problémes du type des surfaces de Riemann, Comptes Rendus, Paris, 229 (1949), pp. 1109-1111; [3] Questions d’existence au voisinage de la frontiere d’une surface de Riemann, Comptes Rendus, Paris, 230 (1950), pp. 269-271.

4) Pfluger, A.: Über das Anwachsen eindautiger analytischer Funktion auf offenen Riemannschen Flache, Ann. Acad. Sci. Fenn. A. I. 64 (1949), 18 pp.Google Scholar

5) Nevanlinna, R.: Quadratisch integrierbare Differentiale auf einer Riemannschen Mannigfaltigkeit, Ann. Acad. Sci. Fenn. A. I. 1 (1941), 34 pp.Google Scholar

6) For the definition of modulus, cf. Sario: loc. cit. [3]; Pfluger: loc. cit.

7) Cf. Sario: loc. cit. [1], p. 11.

8) Sario stated only the sufficient condition. Cf. loc. cit. [3]; R. Nevanlinna: loe. cit Moreover, Sario remarked that a graph K of finite length can be constructed by a suitable choice of an exhaustion of F, in the case when F is simply connected and of parabolic type. Cf. loc. cit. [2].

9) Z. Yûjôbô reported this result at the annual meeting of the Math. Soc. of Japan in 1948. However, his proof has teen published nowhere. Tsuji, M.: On the behaviour of a meromorphic function in the neighbourhood of a closed set of capacity zero, Proc. Imp. Acad. 18 (1942), pp. 213–219.CrossRefGoogle Scholar

10) Gross, W.: Über die Singularitäten analytischer Funktionen, Monatshefte für Math, und Phys. 29 (1918), pp. 1–47.Google Scholar

11) Evidently the niveau curve CΛ coincides with Γn when λ-rn (n=0, 1],…).

12) Stoïlow, S.: Sur les singularités des fonctions analytiques multiformes dont la surface de Riemann a sa frontière de mesure harmonique nuli, Mathematica, 19 (1943), pp. 126–138.Google Scholar

13) Nagai, Y.: On the behaviour of the boundary of Riemann surfaces, II, Proc. Jap. Acad. 26 (1950), pp. 10–16 CrossRefGoogle Scholar; Tsuji, M.: Some metrical theorems on Fuchsian groups, Kôdai Math. Sem. Rep. Nos. 4-5 (1950), pp. 89–93 CrossRefGoogle Scholar; A. Mori: On Riemann surfaces, on which no bounded harmonic function exists, which will appear in Journ. Math. Soc. Japan.

14) Virtanen, K. I.: Über die Existent von beschränkten harmonischen Funktionen auf offenen Riemannschen Flächen, Ann. Acad. Sci. Fenn. A. I. 75 (1950), 8 pp.Google Scholar

15) Ahlfors, L.: Zur Theorie der Überlagerungsflächen, Acta Math. 65 (1935), pp. 157–194.CrossRefGoogle Scholar

16) Cf. Noshiro: loc. cit. [1], p. 307. A. Mori kindly remarked that in the case when <I> has a finite number of sheets, the assertion is directly proved by the fact that a bounded closed set of capacity zero is of linear measure zero.

17) Cf. Noshiro: loc. cit [1], p. 310.

18) Compare with Noshiro: loc. cit. [1], Theorem 3, p. 315 and Theorem 4, p. 327.

19) Pfluger, A.: Sur une propriété de l’application quasi-conforme d’une surface de Riemarn ouverte, Comptes Rendus, Paris, 227 (1948), pp. 25–26.Google Scholar