The Type of a Riemann Surface

Part of the Die Grundlehren der mathematischen Wissenschaften in Einzeldarstellungen book series (GL, volume 162)


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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  1. 1.
    L. Ahlfors [9]. I am indebted to aletter from Mr. Ahlfors for the presentation given in the text.Google Scholar
  2. 1.
    Cf. L. Ahlfors [9].Google Scholar
  3. 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
  4. 1.
    This visualization of Kobayashi’s theorem I likewise owe to a communication from Mr. Ahlfors.Google Scholar
  5. 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
  6. 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
  7. 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

Copyright information

© Springer-Verlag Berlin Heidelberg 1970

Authors and Affiliations

  1. 1.Academy of FinlandFinland

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