54.3 Riemann Mapping Theorem
From the open mapping theorem analytic functions map regions to other regions
or else to single points. The Riemann mapping theorem states that for every
simply connected region, Ω which is not equal to all of ℂ there exists an analytic
function, f such that f
and in addition to this,
is one to one. The
proof involves several ideas which have been developed up to now. The proof is
based on the following important theorem, a case of Montel’s theorem.
beginning, note that the Riemann mapping theorem is a classic example of a
major existence theorem. In mathematics there are two sorts of questions, those
related to whether something exists and those involving methods for finding it.
The real questions are often related to questions of existence. There is a long
and involved history for proofs of this theorem. The first proofs were based on
the Dirichlet principle and turned out to be incorrect, thanks to Weierstrass
who pointed out the errors. For more on the history of this theorem, see Hille
The following theorem is really wonderful. It is about the existence of a subsequence
having certain salubrious properties. It is this wonderful result which will give the
existence of the mapping desired. The other parts of the argument are technical details to
set things up and use this theorem.