is a non-empty simply connected open subset of the complex number plane
Such a map may be interpreted as preserving the shape of any sufficiently small figure, while possibly rotating and scaling (but not reflecting) it.
Later, David Hilbert was able to prove that, to a large extent, the Dirichlet principle is valid under the hypothesis that Riemann was working with.
(namely, that it is a Jordan curve) which are not valid for simply connected domains in general.
The first rigorous proof of the theorem was given by William Fogg Osgood in 1900.
He proved the existence of Green's function on arbitrary simply connected domains other than
[6] Carathéodory's proof used Riemann surfaces and it was simplified by Paul Koebe two years later in a way that did not require them.
Another proof, due to Lipót Fejér and to Frigyes Riesz, was published in 1922 and it was rather shorter than the previous ones.
the following conditions are equivalent:[10] (1) ⇒ (2) because any continuous closed curve, with base point
By approximation γ is in the same homotopy class as a rectangular path on the square grid of length
The reasoning follows a "northeast argument":[11][12] in the non self-intersecting path there will be a corner
As a consequence of the Riemann mapping theorem, every simply connected domain in the plane is homeomorphic to the unit disk.
and bounded by finitely many Jordan contours, there is a unique univalent function
Jenkins (1958), on his book on univalent functions and conformal mappings, gave a treatment based on the work of Herbert Grötzsch and René de Possel from the early 1930s; it was the precursor of quasiconformal mappings and quadratic differentials, later developed as the technique of extremal metric due to Oswald Teichmüller.
In a theorem on "boundary variation" (to distinguish it from "interior variation"), he derived a differential equation and inequality, that relied on a measure-theoretic characterisation of straight-line segments due to Ughtred Shuttleworth Haslam-Jones from 1936.
[27][28][29] Schiff (1993) gave a proof of uniformization for parallel slit domains which was similar to the Riemann mapping theorem.
: this is an immediate consequence of Grönwall's area theorem, applied to the family of univalent functions
can be uniformized by a horizontal parallel slit conformal mapping take
To check that is the required parallel slit transformation, suppose reductio ad absurdum that
[32][33][34] The proof of the uniqueness of the conformal parallel slit transformation is given in Goluzin (1969) and Grunsky (1978).
and isolated points (the images of other the inverse of the Joukowsky transform under the other parallel horizontal slits).
is not in one of these lines, Cauchy's argument principle shows that the number of solutions of
This can be proved using regularity properties of solutions of the Dirichlet boundary value problem, which follow either from the theory of Sobolev spaces for planar domains or from classical potential theory.
Other methods for proving the smooth Riemann mapping theorem include the theory of kernel functions[37] or the Beltrami equation.
Computational conformal mapping is prominently featured in problems of applied analysis and mathematical physics, as well as in engineering disciplines, such as image processing.
In the early 1980s an elementary algorithm for computing conformal maps was discovered.
in the plane, the algorithm computes an explicit conformal map of the unit disk onto a region bounded by a Jordan curve
This algorithm converges for Jordan regions[38] in the sense of uniformly close boundaries.
The algorithm was discovered as an approximate method for conformal welding; however, it can also be viewed as a discretization of the Loewner differential equation.
[39] The following is known about numerically approximating the conformal mapping between two planar domains.