Analytic Functions pp 308-320 | Cite as

# The Type of a Riemann Surface

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## Abstract

We restrict ourselves at first to the class considered in XI, § 2, of simply connected Riemann surfaces *F* _{ q } whose branch points project onto a finite number *q* of image points *w* _{1}, .... *w* _{ q }. These surfaces can be represented concretely by the line complexes defined in the place just mentioned. To each half sheet of *F* _{ q } there corresponds a node point in the line complex. By means of the line segments connecting the node points, the plane is decomposed into *elementary regions* which correspond one-to-one to the branch points of *F* in such a way that one elementary region with 2m sides is associated with a branch point of order *m* — 1. Corresponding in the complex to the sheets which are unramified over the base points are two-sided figures (double 1ine segments).

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## References

- 1.L. Ahlfors [9]. I am indebted to aletter from Mr. Ahlfors for the presentation given in the text.Google Scholar
- 1.Cf. L. Ahlfors [9].Google Scholar
- 1.This result is obvious for
*a*= 0. If*a*≠ 0 one has to transform*w*by the rotation of the sphere that takes*a*into 0.Google Scholar - 1.This visualization of Kobayashi’s theorem I likewise owe to a communication from Mr. Ahlfors.Google Scholar
- 1.This theorem was first proved by the author [8] in a way whose natural extension the technique of Kobayashi can be considered.Google Scholar
- 2.For surfaces with nothing but algebraic branch points there is a closely related theorem of Ahlfors [4]. In this regard also cf. A. Speiser [2], [3], E. Ullrich [3].Google Scholar
- 3.P. J. Myrberg [2]. For newer results concerning the type problem cf. H.Wittich [7], [8], O. Teichmüller [5], F. E. Ullrich [1], R. Nevanlinna [15], Le-Van [2] (in the last article one finds an extensive bibliography on the type problem), C. Blanc [1], [2], Z. Kobayashi [2], P. Laasonen [1].Google Scholar