# Riemann mapping theorem

(7) (1) This is a purely topological argument. Let γ be a piecewise smooth closed curve based at z0 in G. By approximation γ is in the same homotopy class as a rectangular path on the square grid of length δ > 0 based at z0; 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 z1, then it breaks up into two rectangular paths of length < N, so can be deformed to the constant path at z1 by the induction hypothesis and elementary properties of the fundamental group. The reasoning follows a "northeast argument":[10][11] in the non self-intersecting path there will be a corner z0 with largest real part (easterly) and then amongst those one with largest imaginary part (northerly). Reversing direction if need be, the path go from z0 − δ to z0 and then to w0 = z0 − i n δ for n ≥ 1 and then goes leftwards to w0 − δ. 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 z0 to w0 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 ∂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 R ∪ ∂R ⊂ G. But in this case the path can be deformed by replacing the three sides of the rectangle by the fourth, resulting in 2 less sides. (Self-intersections are permitted.) Riemann mapping theorem Weierstrass's convergence theorem. The uniform limit on compacta of a sequence of holomorphic functions is holomorphic; similarly for derivatives. This is an immediate consequence of Morera's theorem for the first statement. Cauchy's integral formula gives a formula for the derivatives which can be used to check that the derivatives also converge uniformly on compacta.[12] Hurwitz's theorem. If a sequence of nowhere-vanishing holomorphic functions on an open domain has a uniform limit on compacta, then either the limit is identically zero or the limit is nowhere-vanishing. If a sequence of univalent holomorphic functions on an open domain has a uniform limit on compacta, then either the limit is constant or the limit is univalent. If the limit function is non-zero, then its zeros have to be isolated. Zeros with multiplicities can be counted by the winding number (2 i π)−1 ∫C g(z)−1 g‘(z) dz for a holomorphic function g. Hence winding numbers are continuous under uniform limits, so that if each function in the sequence has no zeros nor can the limit. For the second statement suppose that f(a) = f(b) and set gn(z) = fn(z) − fn(a). These are nowhere-vanishing on a disk but g(z) = f(z) − f(a) vanishes at b, so g must vanish identically.[13] Definitions. A family {displaystyle {cal {F}}} of holomorphic functions on an open domain is said to be normal if any sequence of functions in {displaystyle {cal {F}}} has a subsequence that converges to a holomorphic function uniformly on compacta. A family {displaystyle {cal {F}}} is compact if whenever a sequence fn lies in {displaystyle {cal {F}}} and converges uniformly to f on compacta, then f also lies in {displaystyle {cal {F}}} . A family {displaystyle {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.[14][15] Montel's theorem. Every locally bounded family of holomorphic functions in a domain G is normal. Let fn be a totally bounded sequence and chose a countable dense subset wm of G. By locally boundedness and a "diagonal argument", a subsequence can be chosen so that gn is convergent at each point wm. It must be verified that this sequence of holomorphic functions converges on G uniformly on each compactum K. Take E open with K ⊂ E such that the closure of E is compact and contains G. Since the sequence (gn′) is locally bounded, |gn′| ≤ M on E. By compactness, if δ > 0 is taken small enough, finitely many open disks Dk of radius δ > 0 are required to cover K while remaining in E. Desde {estilo de exibição g_{n}(b)-g_{n}(uma)=int_{uma}^{b}g_{n}^{melhor }(z),dz} , |gn(uma) − gn(b)| ≤ M |a − b| ≤ 2 δ M. Now for each k choose some wi in Dk where gn(wi) converge, taking n and m so large to be within δ of its limit. Then for z in Dk, {estilo de exibição |g_{n}(z)-g_{m}(z)|leq |g_{n}(z)-g_{n}(W_{eu})|+|g_{n}(W_{eu})-g_{m}(W_{eu})|+|g_{m}(W_{1})-g_{(}z)|leq 4Mdelta +2delta .} Hence the sequence (gn) forms a Cauchy sequence in the uniform norm on K as required.[16][17] Riemann mapping theorem. If G is a simply connected domain ≠ ℂ and a lies in G, there is a unique conformal mapping f of G onto the unit disk D normalized such that f(uma) = 0 and f ′(uma) > 0. Uniqueness follows because of f and g satisfied the same conditions h = f ∘ g−1 would be a univalent holomorphic map of the unit disk with h(0) = 0 and h‘(0) >0. But by the Schwarz lemma, the univalent holomorphic maps of the unit disk onto itself are given by the Möbius transformations k(z) = eiθ(z − α)/(1 − α* z) com |uma| < 1. So h must be the identity map and f = g. To prove existence, take {displaystyle {cal {F}}} to be the family of holomorphic univalent mappings f of G into the open unit disk D with f(a) = 0 and f ‘(a) > 0. It is a normal family by Montel's theorem. By the characterization of simple-connectivity, for b in ℂ G there is a holomorphic branch of the square root {estilo de exibição h(z)={quadrado {z-b}}} in G. It is univalent and h(z1) ≠ − h(z2) for z1 and z2 in G. Since h(G) must contain a closed disk Δ with centre h(uma) and radius r > 0, no points of −Δ can lie in h(G). Let F be the unique Möbius transformation taking ℂ −Δ onto D with the normalization F(h(uma)) = 0 and F′(h(uma)) > 0. By construction F ∘ h is in {estilo de exibição {cal {F}}} , de modo a {estilo de exibição {cal {F}}} is non-empty. The method of Koebe is to use an extremal function to produce a conformal mapping solving the problem: in this situation it is often called the Ahlfors function of G, after Ahlfors.[18] Deixar 0 < M ≤ ∞ be the supremum of f′(a) for f in {displaystyle {cal {F}}} . Pick fn in {displaystyle {cal {F}}} with fn′(a) tending to M. By Montel's theorem, passing to a subsequence if necessary, fn 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′(uma) > 0 and f lies in {estilo de exibição {cal {F}}} . It remains to check that the conformal mapping f takes G onto D. If not, take c ≠ 0 in D f(G) and let H be a holomorphic square root of (f(z) − c)/(1 − c*f(z)) on G. The function H is univalent and maps G into D. Let F(z) = eiθ(H(z) − H(uma))/(1 − H(uma)*H(z)) where H′(uma)/|H′(uma)| = e−iθ. Then F lies in {estilo de exibição {cal {F}}} and a routine computation shows that F′(uma) = H′(uma) / (1 − |H(uma)|2) = f′(uma) (|c| +|c|−1)/2 > f′(uma) = M. This contradicts the maximality of M, so that f must take all values in D.[19][20][21] Observação. 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 φ(z) = z/(1 + |z|) gives a homeomorphism of 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 θ to the x-axis. Thus if G is a domain in ℂ ∪ {∞} containing ∞ and bounded by finitely many Jordan contours, there is a unique univalent function f on G with f(z) = z−1 + a1 z + a2 z2 ⋅⋅⋅ near ∞, maximizing Re e −2i θ a1 and having image f(G) a parallel slit domain with angle θ to the x-axis.[22][23][24] The first proof that parallel slit domains were canonical domains for in the multiply connected case was given by David Hilbert in 1909. 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.[25] Menahem Schiffer gave a treatment based on very general variational principles, summarised in addresses he gave to the International Congress of Mathematicians in 1950 e 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.[26][27][28] Schiff (1993) 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. Primeiramente, by Bieberbach's inequality, any univalent function g(z) = z + c z2 + ··· with z in the open unit disk must satisfy |c| ≤ 2. Como consequência, if f(z) = z + a0 + a1 z–1 + ··· is univalent in | z | > R, então | f(z) – a0 | ≤ 2 | z |: take S > R, set g(z) = S [f(S/z) – b]–1 for z in the unit disk, choosing b so the denominator is nowhere-vanishing, and apply the Schwarz lemma. Next the function fR(z) = z + R2/z is characterized by an "extremal condition" as the unique univalent function in z > R of the form z + a1 z–1 + ··· that maximises Re a1: this is an immediate consequence of Grönwall's area theorem, applied to the family of univalent functions f(z R) / R in z > 1.[29][30] To prove now that the multiply connected domain G ⊂ ℂ ∪ {∞} can be uniformized by a horizontal parallel slit conformal mapping f(z) = z + a1 z–1 + ···, take R large enough that ∂G lies in the open disk |z| < R. For S > R, univalency and the estimate | f(z) | ≤ 2 |z| imply that, if z lies in G with | z | ≤ S, então | f(z) | ≤ 2S. Since the family of univalent f are locally bounded in G {∞}, by Montel's theorem they form a normal family. Furthermore if fn 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 ∞ of the fn tends to the corresponding coefficient of f. This applies in particular to the coefficient: so by compactness there is a univalent f which maximizes Re a1. To check that f(z) = z + a1 + ⋅⋅⋅ is the required parallel slit transformation, suppose reductio ad absurdum that f(G) = G1 has a compact and connected component K of its boundary which is not a horizontal slit. Then the complement G2 of K in ℂ ∪ {∞} is simply connected with G2 ⊃ G1. By the Riemann mapping theorem there is a conformal mapping h(W) = w + b1 w−1 + ⋅⋅⋅ such that h(G2) is ℂ with a horizontal slit removed. So h(f(z)) = z + (a1 + b1)z−1 + ⋅⋅⋅ and hence Re (a1 + b1) ≤ Re a1 by the extremality of f. Thus Re b1 ≤ 0. On the other hand by the Riemann mapping theorem there is a conformal mapping k(W) = w + c0 + c1 w−1 + ⋅⋅⋅ from |W| > S onto G2. Então f(k(W)) − c0 = w + (a1 + c1) w−1 + ⋅⋅⋅. By the strict maximality for the slit mapping in the previous paragraph Re c1 < Re (b1 + c1), so that Re b1 > 0. The two inequalities for Re b1 are contradictory.[31][32][33] The proof of the uniqueness of the conformal parallel slit transformation is given in Goluzin (1969) and Grunsky (1978). 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 C0 and contains analytic arcs Ci and isolated points (the images of other the inverse of the Joukowsky transform under the other parallel horizontal slits). Desta forma, taking a fixed a in G, there is a univalent mapping F0(W) = h ∘ f (W) = (W - uma)−1 + a1 (w − a) + a2(w − a)2 + ⋅⋅⋅ with image a horizontal slit domain. Suppose that F1(W) is another uniformizer with F1(W) = (W - uma)−1 + b1 (w − a) + b2(w − a)2 + ⋅⋅⋅. The images under F0 or F1 of each Ci have a fixed y-coordinate so are horizontal segments. On the other hand F2(W) = F0(W) − F1(W) is holomorphic in G. If it is constant, then it must be identically zero since F2(uma) = 0. Suppose F2 is non-constant. Then by assumption F2(Ci) are all horizontal lines. If t is not in one of these lines, Cauchy's argument principle shows that the number of solutions of F2(W) = t in G is zero (any t will eventually be encircled by contours in G close to the Ci's). This contradicts the fact that the non-constant holomorphic function F2 is an open mapping.[34] Sketch proof via Dirichlet problem 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 {estilo de exibição f(z)=(z-z_{0})e^{g(z)}} where g = u + iv is some (to be determined) 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(z)| = 1 for z ∈ ∂U, so we need {estilo de exibição você(z)=-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; ou seja, 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.[35] 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[36] 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 {estilo de exibição z_{0},ldots ,z_{n}} in the plane, the algorithm computes an explicit conformal map of the unit disk onto a region bounded by a Jordan curve {gama de estilo de exibição } com {estilo de exibição z_{0},ldots ,z_{n}in gamma .} This algorithm converges for Jordan regions[37] 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 {estilo de exibição C^{1}} curve or a K-quasicircle. The algorithm was discovered as an approximate method for conformal welding; Contudo, it can also be viewed as a discretization of the Loewner differential equation.[38] The following is known about numerically approximating the conformal mapping between two planar domains.[39] Positive results: There is an algorithm A that computes the uniformizing map in the following sense. Deixar {estilo de exibição Omega } be a bounded simply-connected domain, e {displaystyle w_{0}in Omega .} ∂Ω is provided to A by an oracle representing it in a pixelated sense (ou seja, if the screen is divided to {estilo de exibição 2 ^{n}times 2^{n}} pixels, the oracle can say whether each pixel belongs to the boundary or not). Then A computes the absolute values of the uniformizing map {estilo de exibição phi :(Ómega ,W_{0})para (D,0)} with precision {estilo de exibição 2 ^{-n}} in space bounded by {displaystyle Ccdot n^{2}} and time {estilo de exibição 2 ^{O(n)}} , where C depends only on the diameter of {estilo de exibição Omega } e {estilo de exibição d(W_{0},Ômega parcial ).} Além disso, the algorithm computes the value of φ(W) with precision {estilo de exibição 2 ^{-n}} as long as {estilo de exibição |phi (W)|<1-2^{-n}.} Moreover, A queries ∂Ω with precision of at most {displaystyle 2^{-O(n)}.} In particular, if ∂Ω is polynomial space computable in space {displaystyle n^{a}} for some constant {displaystyle ageq 1} and time {displaystyle T(n)<2^{O(n^{a})},} then A can be used to compute the uniformizing map in space {displaystyle Ccdot n^{max(a,2)}} and time {displaystyle 2^{O(n^{a})}.} There is an algorithm A′ that computes the uniformizing map in the following sense. Let {displaystyle Omega } be a bounded simply-connected domain, and {displaystyle w_{0}in Omega .} Suppose that for some {displaystyle n=2^{k},} ∂Ω is given to A′ with precision {displaystyle {tfrac {1}{n}}} by {displaystyle O(n^{2})} pixels. Then A′ computes the absolute values of the uniformizing map {displaystyle phi :(Omega ,w_{0})to (D,0)} within an error of {displaystyle O(1/n)} in randomized space bounded by {displaystyle O(k)} and time polynomial in {displaystyle n=2^{k}} (that is, by a BPL(n)-machine). Furthermore, the algorithm computes the value of {displaystyle phi (w)} with precision {displaystyle {tfrac {1}{n}}} as long as {displaystyle |phi (w)|<1-{tfrac {1}{n}}.} Negative results: Suppose there is an algorithm A that given a simply-connected domain {displaystyle Omega } with a linear-time computable boundary and an inner radius > 1/2 and a number {estilo de exibição m} computes the first {displaystyle 20n} digits of the conformal radius {estilo de exibição r(Ómega ,0),} then we can use one call to A to solve any instance of a #SAT(n) with a linear time overhead. Em outras palavras, #P is poly-time reducible to computing the conformal radius of a set. Consider the problem of computing the conformal radius of a simply-connected domain {estilo de exibição Omega ,} where the boundary of {estilo de exibição Omega } is given with precision {estilo de exibição 1/n} by an explicit collection of {estilo de exibição O(n^{2})} pixels. Denote the problem of computing the conformal radius with precision {displaystyle 1/n^{c}} por {displaystyle CONF(n,n^{c}).} Então, {displaystyle MAJ_{n}} is AC0 reducible to {displaystyle CONF(n,n^{c})} para qualquer {estilo de exibição 0

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