Riemann mapping theorem


In complex analysis, the Riemann mapping theorem states that if U is a non-empty simply connected open subset of the complex number plane C which is not all of C, then there exists a biholomorphic mapping f from U onto the open unit disk
This mapping is known as a Riemann mapping.
Intuitively, the condition that U be simply connected means that U does not contain any “holes”. The fact that f is biholomorphic implies that it is a conformal map and therefore angle-preserving. Intuitively, such a map preserves the shape of any sufficiently small figure, while possibly rotating and scaling it.
Henri Poincaré proved that the map f is essentially unique: if z0 is an element of U and φ is an arbitrary angle, then there exists precisely one f as above such that f = 0 and such that the argument of the derivative of f at the point z0 is equal to φ. This is an easy consequence of the Schwarz lemma.
As a corollary of the theorem, any two simply connected open subsets of the Riemann sphere which both lack at least two points of the sphere can be conformally mapped into each other.

History

The theorem was stated by Bernhard Riemann in 1851 in his PhD thesis. Lars Ahlfors wrote once, concerning the original formulation of the theorem, that it was “ultimately formulated in terms which would defy any attempt of proof, even with modern methods”. Riemann's flawed proof depended on the Dirichlet principle, which was considered sound at the time. However, Karl Weierstrass found that this principle was not universally valid. 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. However, in order to be valid, the Dirichlet principle needs certain hypotheses concerning the boundary of U which are not valid for simply connected domains in general. Simply connected domains with arbitrary boundaries were first treated by.
The first proof of the theorem is due to Constantin Carathéodory, who published it in 1912. His proof used Riemann surfaces and it was simplified by Paul Koebe two years later in a way which 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. In this proof, like in Riemann's proof, the desired mapping was obtained as the solution of an extremal problem. The Fejér–Riesz proof was further simplified by Alexander Ostrowski and by Carathéodory.

Importance

The following points detail the uniqueness and power of the Riemann mapping theorem:
Given U and a point z0 in U, we want to construct a function f which maps U to the unit disk and z0 to 0. For this sketch, we will assume that U is bounded and its boundary is smooth, much like Riemann did. Write
where g = u + iv is some holomorphic function with real part u and imaginary part v. It is then clear that z0 is the only zero of f. We require |f| = 1 for z ∈ ∂U, so we need
on the boundary. Since u is the real part of a holomorphic function, we know that u is necessarily a harmonic function; i.e., it satisfies Laplace's equation.
The question then becomes: does a real-valued harmonic function u exist that is defined on all of U and has the given boundary condition? The positive answer is provided by the Dirichlet principle. Once the existence of u has been established, the Cauchy–Riemann equations for the holomorphic function g allow us to find v. Once u and v have been constructed, one has to check that the resulting function f does indeed have all the required properties.

Uniformization theorem

The Riemann mapping theorem can be generalized to the context of Riemann surfaces: If U is a non-empty simply-connected open subset of a Riemann surface, then U is biholomorphic to one of the following: the Riemann sphere, C or D. This is known as the uniformization theorem.

Smooth Riemann mapping theorem

In the case of a simply connected bounded domain with smooth boundary, the Riemann mapping function and all its derivatives extend by continuity to the closure of the domain. 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 or the Beltrami equation.

Algorithms

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. Given points in the plane, the algorithm computes an explicit conformal map of the unit disk onto a region bounded by a Jordan curve with This algorithm converges for Jordan regions in the sense of uniformly close boundaries. There are corresponding uniform estimates on the closed region and the closed disc for the mapping functions and their inverses. Improved estimates are obtained if the data points lie on a curve or a K-quasicircle. 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.
The following is known about numerically approximating the conformal mapping between two planar domains.
Positive results:
Negative results: