Riemann mapping theorem

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Riemann mapping theorem

In complex analysis, the Riemann mapping theorem states that if $U$ is a simply connected open subset of the complex number plane $Bbb C$ which is not all of $Bbb C$, then there exists a biholomorphic (bijective and holomorphic) mapping $f,$ from $U,$ onto open unit disk $D,$.

:$f : U ightarrow D ,$

where

:$D=\left\{zin \left\{Bbb C\right\} :|z|<1\right\}$

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 (but not reflecting) it.

Henri Poincaré proved that the map $f$ is essentially unique: if $z_0$ is an element of $U$ and φ is an arbitrary angle, then there exists precisely one $f$ as above with the additional properties that $f$ maps $z_0$ into $0$ and that the argument of the derivative of $f$ at the point $z_0$ 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).

History

The theorem was stated (under the assumption that the boundary of $U$ 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 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 $U$ which are not valid for simply connected domains in general. Simply connected domains with arbitrary boundaries were first treated in 1900 (by W. F. Osgood).

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.

Why is this theorem impressive?

To better understand how unique and powerful the Riemann mapping theorem is, consider the following facts:

* Even relatively simple Riemann mappings, say a map from the interior of a circle to the interior of a square, have no explicit formula using only elementary functions.
* Simply connected open sets in the plane can be highly complicated, for instance the boundary can be a nowhere differentiable fractal curve of infinite length, even if the set itself is bounded. The fact that such a set can be mapped in an "angle-preserving" manner to the nice and regular unit disc seems counter-intuitive.
* The analog of the Riemann mapping theorem for doubly connected domains is not true. In fact, there are no conformal maps between annuli except inversion and multiplication by constants, so the annulus { $z$ : 1 < $|z|$ < 2 } is not conformally equivalent to the annulus { $z$ : 1 < $|z|$ < 4 } (as can be proven using extremal length). However, any doubly connected domain except the punctured plane is conformally equivalent to some annulus { $z$ : r < $|z|$ < 1 } with 0 ≤r<1.
* The analog of the Riemann mapping theorem in three real dimensions or above is not even remotely true. In fact, the family of conformal maps in three dimensions is very poor, and essentially contains only Möbius transformations.
* Even if we allow arbitrary homeomorphisms in higher dimensions, we can find contractible manifolds that are not homeomorphic to the ball, such as the Whitehead continuum.
* The Riemann mapping theorem is the easiest way to prove that any two simply connected domains in the plane are homeomorphic. Even though the class of continuous functions is infinitely larger than that of conformal maps, it is not easy to construct a one-to-one function onto the disk knowing only that the domain is simply connected.

A proof sketch

Given $U$ and $z_0$, we want to construct a function $f$ which maps $U$ to the unit disk and $z_0$ to $0$. For this sketch, we will assume that $U$ is bounded and its boundary is smooth, much like Riemann did. Write:$f\left(z\right)=\left(z-z_0\right)exp\left(g\left(z\right)\right) ,!$where $g=u+iv$ is some (to be determined) holomorphic function with real part $u$ and imaginary part $v$. It is then clear that "z"0 is the only zero of "f". We require $|f\left(z\right)|=1$ for $z$ on the boundary of $U$, so we need $u\left(z\right)=-log|z-z_0|$ 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 $v$ (this argument depends on the assumption that $U$ be simply connected). 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 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.

Bibliography

*John B. Conway, "Functions of one complex variable", Springer-Verlag, 1978, ISBN 0-387-90328-3
*John B. Conway, "Functions of one complex variable II", Springer-Verlag, 1995, ISBN 0-387-94460-5
*Reinhold Remmert, "Classical topics in complex function theory", Springer-Verlag, 1998, ISBN 0-387-98221-3
*Bernhard Riemann, " [http://www.emis.de/classics/Riemann/Grund.pdf Grundlagen für eine allgemeine Theorie der Functionen einer veränderlichen complexen Grösse] ", Göttingen, 1851

* [http://planetmath.org/encyclopedia/ProofOfRiemannMappingTheorem.html Proof of the Riemann mapping theorem] , from PlanetMath

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