The Ahlfors Theory of Covering Surfaces

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


In the preceding discussion of the value distribution theory our interest has been increasingly directed toward the properties of the Riemann surface F onto which the disk |z| < R ≦ ∞ is conformally mapped by means of the given meromorphic function w = w(z). The following was essential to our investigation; 1. A metric was introduced on the surface F (e.g., a spherical metric in the case of the first main theorem, a non-euclidean [or still more general] metric to obtain the second main theorem). 2. To master the properties of the entire open surface F we had to define a set of approximating surfaces by means of which F was exhausted; primarily we have made use of the images F r of the disks |z| ≦ r < R. 3. The mapping zw is single-valued and conformai.


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  1. 1.
    That means mapped onto one another topologic ally.Google Scholar
  2. 1.
    These are simple curves that close on the surface.Google Scholar
  3. 1.
    L. Ahlfors [10], p. 168.Google Scholar
  4. 2.
    ϱ is the larger of the numbers ϱ and zero. The theorem is obviously trivial for ϱ 0 ≦ 0; it can thus be assumed in the following that ϱ 0 > 0.Google Scholar
  5. 3.
    A surface is called planar if it is decomposed by any simple polygon (ring cut). Ahlfors [10] has also considered nonplanar surfaces.Google Scholar
  6. 1.
    Henceforth every such number will be denoted by h.Google Scholar
  7. 2.
    Here we shall not consider the case of a nonplanar surface.Google Scholar
  8. 1.
    We have already introduced this notion in XI, § 2.Google Scholar
  9. 1.
    G. Valiron [6], A. Bloch [1], E. Landau [2].Google Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 1970

Authors and Affiliations

  1. 1.Academy of FinlandFinland

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