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Some Theorems on Open Riemann Surfaces

Published online by Cambridge University Press:  22 January 2016

Masatsugu Tsuji
Masatsugu Tsuji*
Affiliation:
Mathematical Institute, Tokyo University
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Let F be an open Riemann surface spread over the z-plane. We say that F is of positive or null boundary, according as there exists a Green’s function on F or not, Let u(z) be a harmonic function on F

and be its Dirichlet integral As R. Nevanlinna proved, if F is of null boundary, there exists no one-valued non-constant harmonic function on F5 whose Dirichlet integral is finite, This Nevanlinna’s theorem was proved very simply by Kuroda.

Type
Research Article
Information
Nagoya Mathematical Journal , Volume 3 , October 1951 , pp. 141 - 145
Copyright
Copyright © Editorial Board of Nagoya Mathematical Journal 1951

References

1) (a) R. Nevanlinna: Quadratisch integrierbare Differentiale auf einer Riemannschen Mannigfaltigkeit. Annales Acad. Sci, Fenn, Series A, Mathematica-Physica 1 (1941). (b) Über das Anwachsen des Dirichletintegrals- einer analytischen Funktion auf einer offenen Riemannschen Fläche. Annales Acad. Sci. Fenn. Series A, Mathematica-Physica 45 (1948)

2) T. Kuroda; Some remarks on an open Riemann surface. To appear in the Tohoku Math. Journ.

3) R. Nevanlinna. I.c. 1) (a).

4) R. Nevanlinna. I.c. 1) (a).

5) c. f. R. Nevanlinna. I.c. l) (a).

6) Heins, M.: The conformal mapping of simply connected Riemann surfaces. Annals of Math. 50 (1949).CrossRefGoogle Scholar