# The Ahlfors Theory of Covering Surfaces

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

## 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 zw is single-valued and conformai.

## References

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.
9. 1.
G. Valiron [6], A. Bloch [1], E. Landau [2].Google Scholar