Analytic Functions pp 321-354 | Cite as

# The Ahlfors Theory of Covering Surfaces

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## Abstract

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 *z* → *w* is single-valued and conformai.

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## References

- 1.That means mapped onto one another topologic ally.Google Scholar
- 1.These are simple curves that close on the surface.Google Scholar
- 1.L. Ahlfors [10], p. 168.Google Scholar
- 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 - 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
- 1.Henceforth every such number will be denoted by
*h*.Google Scholar - 2.Here we shall not consider the case of a nonplanar surface.Google Scholar
- 1.We have already introduced this notion in XI, § 2.Google Scholar
- 1.