The Ahlfors Theory of Covering Surfaces
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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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- 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 , 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  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.G. Valiron , A. Bloch , E. Landau .Google Scholar