Abstract:
The main results of the paper are two effective versions of the Riemann mapping theorem. The first, uniform version is based on the constructive proof of the Riemann mapping
theorem by Bishop and Bridges and formulated in the computability framework developed
by Kreitz and Weihrauch. It states which topological information precisely one needs about
a nonempty, proper, open, connected, and simply connected subset of the complex plane
in order to compute a description of a holomorphic bijection from this set onto the unit
disk, and vice versa, which topological information about the set can be obtained from a
description of a holomorphic bijection. The second version, which is derived from the first
by considering the sets and the functions with computable descriptions, characterizes the
subsets of the complex plane for which there exists a computable holomorphic bijection onto
the unit disk. This solves a problem posed by Pour-El and Richards. We also show that
this class of sets is strictly larger than a class of sets considered by Zhou, which solves an
open problem posed by him. In preparation, recursively enumerable open subsets and closed
subsets of Euclidean spaces are considered and several effective results in complex analysis
are proved.