In complex analysis, the Riemann mapping theorem states that if U is a non-empty simply connected open subset of the complex number plane \mathbb{C} which is not all of \mathbb{C}, then there exists a biholomorphic mapping f (i.e. a bijective holomorphic mapping whose inverse is also holomorphic) 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. Such a map may be interpreted as preserving the shape of any sufficiently small figure, while possibly rotating and scaling (but not reflecting) it. Henri Poincaré proved that the map f is unique up to rotation and recentering: if z_0 is an element of U and \phi is an arbitrary angle, then there exists precisely one f as above such that f(z_0)=0 and such that the argument of the derivative of f at the point z_0 is equal to \phi. 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 (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 flawed proof depended on the Dirichlet principle (which was named 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 (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 \mathbb{C} itself; this established the Riemann mapping theorem. Constantin Carathéodory gave another proof of the theorem in 1912, which was the first to rely purely on the methods of function theory rather than potential theory. His proof used Montel's concept of normal families, which became the standard method of proof in textbooks. Carathéodory continued in 1913 by resolving the additional question of whether the Riemann mapping between the domains can be extended to a homeomorphism of the boundaries (see Carathéodory's theorem). 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. 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:

Proof via normal families

Simple connectivity

Theorem. For an open domain the following conditions are equivalent: (1) ⇒ (2) because any continuous closed curve, with base point a\in G, can be continuously deformed to the constant curve a. So the line integral of over the curve is 0. (2) ⇒ (3) because the integral over any piecewise smooth path \gamma from a to z can be used to define a primitive. (3) ⇒ (4) by integrating along \gamma from a to x to give a branch of the logarithm. (4) ⇒ (5) by taking the square root as where f is a holomorphic choice of logarithm. (5) ⇒ (6) because if \gamma is a piecewise closed curve and f_n are successive square roots of z-w for w outside G, then the winding number of about w is 2^n times the winding number of \gamma about 0. Hence the winding number of \gamma about w must be divisible by 2^n for all n, so it must equal 0. (6) ⇒ (7) for otherwise the extended plane can be written as the disjoint union of two open and closed sets A and B with \infty\in B and A bounded. Let \delta>0 be the shortest Euclidean distance between A and B and build a square grid on \mathbb{C} with length \delta/4 with a point a of A at the centre of a square. Let C be the compact set of the union of all squares with distance from A. Then and \partial C does not meet A or B: it consists of finitely many horizontal and vertical segments in G forming a finite number of closed rectangular paths. Taking C_i to be all the squares covering A, then equals the sum of the winding numbers of C_i over a, thus giving 1. On the other hand the sum of the winding numbers of \gamma_j about a equals 1. Hence the winding number of at least one of the \gamma_j about a is non-zero. (7) ⇒ (1) This is a purely topological argument. Let \gamma be a piecewise smooth closed curve based at z_0\in G. By approximation γ is in the same homotopy class as a rectangular path on the square grid of length \delta>0 based at z_0; such a rectangular path is determined by a succession of N consecutive directed vertical and horizontal sides. By induction on N, such a path can be deformed to a constant path at a corner of the grid. If the path intersects at a point z_1, then it breaks up into two rectangular paths of length <N, and thus can be deformed to the constant path at z_1 by the induction hypothesis and elementary properties of the fundamental group. The reasoning follows a "northeast argument": in the non self-intersecting path there will be a corner z_0 with largest real part (easterly) and then amongst those one with largest imaginary part (northerly). Reversing direction if need be, the path go from z_0-\delta to z_0 and then to for n\geq1 and then goes leftwards to w_0-\delta. Let R be the open rectangle with these vertices. The winding number of the path is 0 for points to the right of the vertical segment from z_0 to w_0 and -1 for points to the right; and hence inside R. Since the winding number is 0 off G, R lies in G. If z is a point of the path, it must lie in G; if z is on \partial R but not on the path, by continuity the winding number of the path about z is -1, so z must also lie in G. Hence. But in this case the path can be deformed by replacing the three sides of the rectangle by the fourth, resulting in two less sides (with self-intersections permitted).

Riemann mapping theorem

Definitions. A family {\cal F} of holomorphic functions on an open domain is said to be normal if any sequence of functions in {\cal F} has a subsequence that converges to a holomorphic function uniformly on compacta. A family {\cal F} is compact if whenever a sequence f_n lies in {\cal F} and converges uniformly to f on compacta, then f also lies in {\cal F}. A family {\cal F} is said to be locally bounded if their functions are uniformly bounded on each compact disk. Differentiating the Cauchy integral formula, it follows that the derivatives of a locally bounded family are also locally bounded. G , after Ahlfors. Let be the supremum of f'(a) for. Pick with f_n'(a) tending to M. By Montel's theorem, passing to a subsequence if necessary, f_n tends to a holomorphic function f uniformly on compacta. By Hurwitz's theorem, f is either univalent or constant. But f has f(a)=0 and f'(a)>0. So M is finite, equal to f'(a)>0 and. It remains to check that the conformal mapping f takes G onto D. If not, take c\neq0 in and let H be a holomorphic square root of on G. The function H is univalent and maps G into D. Let Remark. As a consequence of the Riemann mapping theorem, every simply connected domain in the plane is homeomorphic to the unit disk. If points are omitted, this follows from the theorem. For the whole plane, the homeomorphism gives a homeomorphism of \mathbb{C} onto D.

Parallel slit mappings

Koebe's uniformization theorem for normal families also generalizes to yield uniformizers f for multiply-connected domains to finite parallel slit domains, where the slits have angle \theta to the x -axis. Thus if G is a domain in containing \infty and bounded by finitely many Jordan contours, there is a unique univalent function f on G with near \infty, maximizing and having image f(G) a parallel slit domain with angle \theta to the x -axis. The first proof that parallel slit domains were canonical domains for in the multiply connected case was given by David Hilbert in 1909. , 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. Menahem Schiffer gave a treatment based on very general variational principles, summarised in addresses he gave to the International Congress of Mathematicians in 1950 and 1958. 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. Haslam-Jones' proof was regarded as difficult and was only given a satisfactory proof in the mid-1970s by Schober and Campbell–Lamoureux. gave a proof of uniformization for parallel slit domains which was similar to the Riemann mapping theorem. To simplify notation, horizontal slits will be taken. Firstly, by Bieberbach's inequality, any univalent function with z in the open unit disk must satisfy |c|\leq2. As a consequence, if is univalent in |z|>R, then. To see this, take S>R and set for z in the unit disk, choosing b so the denominator is nowhere-vanishing, and apply the Schwarz lemma. Next the function is characterized by an "extremal condition" as the unique univalent function in z>R of the form that maximises : this is an immediate consequence of Grönwall's area theorem, applied to the family of univalent functions f(zR)/R in z>1. To prove now that the multiply connected domain can be uniformized by a horizontal parallel slit conformal mapping take R large enough that \partial G lies in the open disk |z|<R. For S>R, univalency and the estimate imply that, if z lies in G with |z|\leq S, then. Since the family of univalent f are locally bounded in, by Montel's theorem they form a normal family. Furthermore if f_n is in the family and tends to f uniformly on compacta, then f is also in the family and each coefficient of the Laurent expansion at \infty of the f_n tends to the corresponding coefficient of f. This applies in particular to the coefficient: so by compactness there is a univalent f which maximizes. To check that is the required parallel slit transformation, suppose reductio ad absurdum that f(G)=G_1 has a compact and connected component K of its boundary which is not a horizontal slit. Then the complement G_2 of K in is simply connected with. By the Riemann mapping theorem there is a conformal mapping such that h(G_2) is \mathbb{C} with a horizontal slit removed. So we have that and thus by the extremality of f. Therefore,. On the other hand by the Riemann mapping theorem there is a conformal mapping mapping from |w|>S onto G_2. Then By the strict maximality for the slit mapping in the previous paragraph, we can see that, so that. The two inequalities for are contradictory. The proof of the uniqueness of the conformal parallel slit transformation is given in and. Applying the inverse of the Joukowsky transform h to the horizontal slit domain, it can be assumed that G is a domain bounded by the unit circle C_0 and contains analytic arcs C_i and isolated points (the images of other the inverse of the Joukowsky transform under the other parallel horizontal slits). Thus, taking a fixed a\in G, there is a univalent mapping with its image a horizontal slit domain. Suppose that F_1(w) is another uniformizer with The images under F_0 or F_1 of each C_i have a fixed y -coordinate so are horizontal segments. On the other hand, is holomorphic in G. If it is constant, then it must be identically zero since F_2(a)=0. Suppose F_2 is non-constant, then by assumption F_2(C_i) are all horizontal lines. If t is not in one of these lines, Cauchy's argument principle shows that the number of solutions of F_2(w)=t in G is zero (any t will eventually be encircled by contours in G close to the C_i's). This contradicts the fact that the non-constant holomorphic function F_2 is an open mapping.

Sketch proof via Dirichlet problem

Given U and a point z_0\in U, 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 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(z)|=1 for, 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 (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 non-empty simply-connected open subset of a Riemann surface, then U is biholomorphic to one of the following: the Riemann sphere, the complex plane \mathbb{C}, or the unit disk 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 \gamma 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 C^1 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: n )-machine). Furthermore, the algorithm computes the value of \phi(w) with precision as long as Negative results: n ) with a linear time overhead. In other words, #P is poly-time reducible to computing the conformal radius of a set.