In complex analysis, the Riemann mapping theorem states that if is a non-empty simply connected open subset of the complex number plane which is not all of , then there exists a biholomorphic (bijective and holomorphic) mapping from onto the open unit disk
This mapping is known as a Riemann mapping.
Intuitively, the condition that be simply connected means that does not contain any “holes”. The fact that 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 (but not reflecting) it.
Henri Poincaré proved that the map is essentially unique: if is an element of and φ is an arbitrary angle, then there exists precisely one as above with the additional properties that maps into and that the argument of the derivative of at the point 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 each lack at least two points of the sphere) can be conformally mapped into each other (because conformal equivalence is an equivalence relation).
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The theorem was stated (under the assumption that the boundary of is piecewise smooth) 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 (whose name was created by Riemann himself), 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 which are not valid for simply connected domains in general. Simply connected domains with arbitrary boundaries were first treated by William Fogg Osgood (1900).
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 Leopold 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.
The following points detail the uniqueness and power of the Riemann mapping theorem:
Given and , we want to construct a function which maps to the unit disk and to . For this sketch, we will assume that is bounded and its boundary is smooth, much like Riemann did. Write
where is some (to be determined) holomorphic function with real part and imaginary part . It is then clear that z0 is the only zero of f. We require for on the boundary of , so we need
on the boundary. Since is the real part of a holomorphic function, we know that is necessarily a harmonic function, i.e. it satisfies Laplace's equation.
The question then becomes: does a real-valued harmonic function exist that is defined on all of 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 allow us to find (this argument depends on the assumption that be simply connected). Once and have been constructed, one has to check that the resulting function does indeed have all the required properties.
The Riemann mapping theorem can be generalized to the context of Riemann surfaces: If U is a simply-connected open subset of a Riemann surface, then U is biholomorphic to one of the following: the Riemann sphere, the complex plane or the open unit disk. This is known as the uniformization theorem.