A conformal Riemannian metric is defined by a length element ds which is expressed in conformal local coordinates z as ds = ρ(z) |dz|, where ρ is a smooth positive function with isolated zeros.
Let X and Y be two bordered Riemann surfaces, and suppose that Y is equipped with a smooth (including the boundary) conformal metric σ(z) dz.
Suppose now that Z is an open Riemann surface, for example the complex plane or the unit disc, and let Z be equipped with a conformal metric ds.
We say that (Z,ds) is regularly exhaustible if there is an increasing sequence of bordered surfaces Dj contained in Z with their closures, whose union in Z, and such that Ahlfors proved that the complex plane with arbitrary conformal metric is regularly exhaustible.
Take, for example f(z) = ℘(Kz), where K > 0 is arbitrarily large, and ℘ is the Weierstrass elliptic function satisfying the differential equation All preimages of the four points e1,e2,e3,∞ are multiple, so if we take four discs with disjoint closures around these points, there will be no region which is mapped on any of these discs homeomorphically.
[2] [3] [4] Simplified proofs of the Second Main Theorem can be found in the papers of Toki[5] and de Thelin.
[6] A simple proof of the Five Island Theorem, not relying on Ahlfors' theory, was developed by Bergweiler.