This is a joint work with Mario Bonk from Michigan/Bonn.
Let S be the Riemann sphere, realized as the unit sphere in R3 and f: C -> S a non-constantmeromorphic function in the complex plane. Then continuous branchesof f -1 exist in spherical discs of spherical radii arbitrarily closeto arctan square root of 8 approximates 70º32'. This constant is best possible,the extremal function is a Weierstrass p corresponding to a hexagonal lattice. Thisresult implies the Ahlfors Five Island Theorem, as well as Picard's Theorem.
Earlier estimates of this extremal radius are due to Ahlfors (1932) 45º, who used hisFive Islands Theorem, and Pommerenke (1970) 60º.
If, in addition, all critical points of the function f are multiple, then asimilar statement holds with radii arbitrarily close to pi/2, which is also best possible.This improves earlier results of Pommerenke, Peschi, and Minda.
The proof is based on considerations of singular surfaces associated withmeromorphic functions. We show that singularities of such surfaces can be smoothened while keeping the large-scale uniform hyperbolicity.