Riemann mapping theorem

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”.{{Citation | last=Ahlfors | first=Lars | author-link=Lars Ahlfors | title=Developments of the Theory of Conformal Mapping and Riemann Surfaces Through a Century | journal=Contributions to the Theory of Riemann Surfaces | editor1=L. Ahlfors | editor2=E. Calabi | editor3=M. Morse | editor4=L. Sario | editor5=D. Spencer | year=1953 | pages=3–4}} 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 $\backslash mathbb\{C\}$ itself; this established the Riemann mapping theorem.For the original paper, see {{harvnb|Osgood|1900}}. For accounts of the history, see {{harvnb|Walsh|1973|pp=270–271}}; {{harvnb|Gray|1994|pp=64–65}}; {{harvnb|Greene|Kim|2017|p=4}}. Also see {{harvnb|Carathéodory|1912|p=108|loc=footnote **}} (acknowledging that {{harvnb|Osgood|1900}} had already proven 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.{{harvnb|Gray|1994|pp=78–80}}, citing {{harvnb|Carathéodory|1912}} His proof used Montel's concept of normal families, which became the standard method of proof in textbooks.{{harvnb|Greene|Kim|2017|p=1}} 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).{{harvnb|Gray|1994|pp=80–83}}

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.{{Cite web |title=What did Riemann Contribute to Mathematics? Geometry, Number Theory and Others |url=https://www.researchgate.net/publication/344401528}}

The following points detail the uniqueness and power of the Riemann mapping theorem:

• Even relatively simple Riemann mappings (for example 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. One such example is the Koch curve.{{cite journal |last1=Lakhtakia |first1=Akhlesh |last2=Varadan |first2=Vijay K. |last3=Messier |first3=Russell |title=Generalisations and randomisation of the plane Koch curve |journal=Journal of Physics A: Mathematical and General |date=August 1987 |volume=20 |issue=11 |pages=3537–3541 |doi=10.1088/0305-4470/20/11/052}} 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 more complicated domains is not true. The next simplest case is of doubly connected domains (domains with a single hole). Any doubly connected domain except for the punctured disk and the punctured plane is conformally equivalent to some annulus with
• The analogue of the Riemann mapping theorem in three or more real dimensions is not true. The family of conformal maps in three dimensions is very poor, and essentially contains only Möbius transformations (see Liouville's theorem).
• Even if arbitrary homeomorphisms in higher dimensions are permitted, contractible manifolds can be found that are not homeomorphic to the ball (e.g., the Whitehead continuum).
• The analogue of the Riemann mapping theorem in several complex variables is also not true. In $\backslash mathbb\{C\}^n$ ($n\; \backslash ge\; 2$), the ball and polydisk are both simply connected, but there is no biholomorphic map between them.{{harvnb|Remmert|1998}}, section 8.3, p. 187

{{main|Normal families}}

Theorem. For an open domain $G\backslash subset\backslash mathbb\{C\}$ the following conditions are equivalent:See

• {{harvnb|Ahlfors|1978}}
• {{harvnb|Beardon|1979}}
• {{harvnb|Conway|1978}}
• {{harvnb|Gamelin|2001}}

1. $G$ is simply connected;
2. the integral of every holomorphic function $f$ around a closed piecewise smooth curve in $G$ vanishes;
3. every holomorphic function in $G$ is the derivative of a holomorphic function;
4. every nowhere-vanishing holomorphic function $f$ on $G$ has a holomorphic logarithm;
5. every nowhere-vanishing holomorphic function $g$ on $G$ has a holomorphic square root;
6. for any $w\backslash notin\; G$, the winding number of $w$ for any piecewise smooth closed curve in $G$ is $0$;
7. the complement of $G$ in the extended complex plane $\backslash mathbb\{C\}\backslash cup\backslash \{\backslash infty\backslash \}$ is connected.

(1) ⇒ (2) because any continuous closed curve, with base point $a\backslash in\; G$, can be continuously deformed to the constant curve $a$. So the line integral of $f\backslash ,\backslash mathrm\{d\}z$ over the curve is $0$.

(2) ⇒ (3) because the integral over any piecewise smooth path $\backslash gamma$ from $a$ to $z$ can be used to define a primitive.

(3) ⇒ (4) by integrating $f^\{-1\}\backslash ,\backslash mathrm\{d\}f/\backslash mathrm\{d\}z$ along $\backslash gamma$ from $a$ to $x$ to give a branch of the logarithm.

(4) ⇒ (5) by taking the square root as $g(z)=\backslash exp(f(x)/2)$ where $f$ is a holomorphic choice of logarithm.

(5) ⇒ (6) because if $\backslash gamma$ is a piecewise closed curve and $f\_n$ are successive square roots of $z-w$ for $w$ outside $G$, then the winding number of $\backslash gamma$ about $w$ is $2^n$ times the winding number of $f\_n\backslash circ\backslash gamma$ about $0$. Hence the winding number of $\backslash gamma$ about $w$ must be divisible by $2^n$ for all $n$, so it must equal $0$.

(6) ⇒ (7) for otherwise the extended plane $\backslash mathbb\{C\}\backslash cup\backslash \{\backslash infty\backslash \}\backslash setminus\; G$ can be written as the disjoint union of two open and closed sets $A$ and $B$ with $\backslash infty\backslash in\; B$ and $A$ bounded. Let $\backslash delta>0$ be the shortest Euclidean distance between $A$ and $B$ and build a square grid on $\backslash mathbb\{C\}$ with length $\backslash delta/4$ 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 $\backslash leq\backslash delta/4$ from $A$. Then $C\backslash cap\; B=\backslash varnothing$ and $\backslash partial\; C$ 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 $\backslash gamma\_j\backslash in\; G$. Taking $C\_i$ to be all the squares covering $A$, then $\backslash frac\{1\}\{2\backslash pi\}\backslash int\_\{\backslash partial\; C\}\backslash mathrm\{d\}\backslash mathrm\{arg\}(z-a)$ 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 $\backslash gamma\_j$ about $a$ equals $1$. Hence the winding number of at least one of the $\backslash gamma\_j$ about $a$ is non-zero.

(7) ⇒ (1) This is a purely topological argument. Let $\backslash gamma$ be a piecewise smooth closed curve based at $z\_0\backslash in\; G$. By approximation γ is in the same homotopy class as a rectangular path on the square grid of length $\backslash delta>0$ 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

• Weierstrass' 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.{{harvnb|Gamelin|2001}}

• 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 $\backslash frac\{1\}\{2\backslash pi\; i\}\backslash int\_Cg^\{-1\}(z)g\text{'}(z)\backslash mathrm\{d\}z$ 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 $g\_n(z)=f\_n(z)-f\_n(a)$. These are nowhere-vanishing on a disk but $g(z)=f(z)-f(a)$ vanishes at $b$, so $g$ must vanish identically.{{harvnb|Gamelin|2001}}

Definitions. A family $\{\backslash cal\; F\}$ of holomorphic functions on an open domain is said to be normal if any sequence of functions in $\{\backslash cal\; F\}$ has a subsequence that converges to a holomorphic function uniformly on compacta.

A family $\{\backslash cal\; F\}$ is compact if whenever a sequence $f\_n$ lies in $\{\backslash cal\; F\}$ and converges uniformly to $f$ on compacta, then $f$ also lies in $\{\backslash cal\; F\}$. A family $\{\backslash cal\; F\}$ 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.{{harvnb|Duren|1983}}{{harvnb|Jänich|1993}}

• Montel's theorem. Every locally bounded family of holomorphic functions in a domain $G$ is normal.

::Let $f\_n$ be a totally bounded sequence and chose a countable dense subset $w\_m$ of $G$. By locally boundedness and a "diagonal argument", a subsequence can be chosen so that $g\_n$ is convergent at each point $w\_m$. It must be verified that this sequence of holomorphic functions converges on $G$ uniformly on each compactum $K$. Take $E$ open with $K\backslash subset\; E$ such that the closure of $E$ is compact and contains $G$. Since the sequence $\backslash \{g\_n\text{'}\backslash \}$ is locally bounded, $|g\_n\text{'}|\backslash leq\; M$ on $E$. By compactness, if $\backslash delta>0$ is taken small enough, finitely many open disks $D\_k$ of radius $\backslash delta>0$ are required to cover $K$ while remaining in $E$. Since

:::$g\_n(b)\; -\; g\_n(a)=\; \backslash int\_a^b\; g\_n^\backslash prime(z)\backslash ,\; dz$,

::we have that $|g\_n(a)-g\_n(b)|\backslash leq\; M|a-b|\backslash leq2\backslash delta\; M$. Now for each $k$ choose some $w\_i$ in $D\_k$ where $g\_n(w\_i)$ converges, take $n$ and $m$ so large to be within $\backslash delta$ of its limit. Then for $z\backslash in\; D\_k$,

:::$|g\_n(z)\; -\; g\_m(z)|\; \backslash leq\; |g\_n(z)\; -\; g\_n(w\_i)|\; +\; |g\_n(w\_i)\; -\; g\_m(w\_i)|\; +\; |g\_m(w\_1)\; -\; g\_m(z)|\backslash leq\; 4M\backslash delta\; +\; 2\backslash delta.$

::Hence the sequence $\backslash \{g\_n\backslash \}$ forms a Cauchy sequence in the uniform norm on $K$ as required.{{harvnb|Duren|1983}}{{harvnb|Jänich|1993}}

• Riemann mapping theorem. If $G\backslash neq\backslash mathbb\{C\}$ is a simply connected domain and $a\backslash in\; G$, there is a unique conformal mapping $f$ of $G$ onto the unit disk $D$ normalized such that $f(a)=0$ and $f\text{'}(a)>0$.

::Uniqueness follows because if $f$ and $g$ satisfied the same conditions, $h=f\backslash circ\; g^\{-1\}$ would be a univalent holomorphic map of the unit disk with $h(0)=0$ and $h\text{'}(0)>0$. But by the Schwarz lemma, the univalent holomorphic maps of the unit disk onto itself are given by the Möbius transformations

:::$k(z)=e^\{i\backslash theta\}(z-\backslash alpha)/(1-\backslash overline\{\backslash alpha\}\; z)$

::with . So $h$ must be the identity map and $f=g$.

::To prove existence, take $\{\backslash cal\; F\}$ to be the family of holomorphic univalent mappings $f$ of $G$ into the open unit disk $D$ with $f(a)=0$ and $f\text{'}(a)>0$. It is a normal family by Montel's theorem. By the characterization of simple-connectivity, for $b\backslash in\backslash mathbb\{C\}\backslash setminus\; G$ there is a holomorphic branch of the square root $h(z)=\backslash sqrt\{z\; -b\}$ in $G$. It is univalent and $h(z\_1)\backslash neq-h(z\_2)$ for $z\_1,z\_2\backslash in\; G$. By the open mapping theorem, $h(G)$ contains a closed disk $\backslash Delta$; say with centre $h(a)$ and radius $r>0$. Thus no points of $-\backslash Delta$ can lie in $h(G)$. Let $F$ be the unique Möbius transformation taking $\backslash mathbb\{C\}\backslash setminus-\backslash Delta$ onto $D$ with the normalization $F(h(a))=0$ and $(F\; \backslash circ\; h)\text{'}(a)=F\text{'}(h(a))\backslash cdot\; h\text{'}(a)>0$. By construction $F\backslash circ\; h$ is in $\{\backslash cal\; F\}$, so that $\{\backslash cal\; F\}$ 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 {{math|G}}, after Ahlfors.{{harvnb|Gamelin|2001|page=309}} Let onto $D$. If not, take $c\backslash neq0$ in $D\backslash setminus\; f(G)$ and let $H$ be a holomorphic square root of $(f(z)-c)/(1-\backslash overline\{c\}f(z))$ on $G$. The function $H$ is univalent and maps $G$ into $D$. Let

:::$F(z)=\backslash frac\{e^\{i\backslash theta\}(H(z)-H(a))\}\{1-\backslash overline\{H(a)\}H(z)\},$

::where $H\text{'}(a)/|H\text{'}(a)|=e^\{-i\backslash theta\}$. Then $F\backslash in\{\backslash cal\; F\}$ and a routine computation shows that

:::$F\text{'}(a)=H\text{'}(a)/(1-|H(a)|^2)=f\text{'}(a)\backslash left(\backslash sqrt$

::This contradicts the maximality of $M$, so that $f$ must take all values in $D$.{{harvnb|Duren|1983}}{{harvnb|Jänich|1993}}{{harvnb|Ahlfors|1978}}

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 $\backslash phi(z)=z/(1+|z|)$ gives a homeomorphism of $\backslash mathbb\{C\}$ onto $D$.

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 $\backslash theta$ to the {{math|x}}-axis. Thus if $G$ is a domain in $\backslash mathbb\{C\}\backslash cup\backslash \{\backslash infty\backslash \}$ containing $\backslash infty$ and bounded by finitely many Jordan contours, there is a unique univalent function $f$ on $G$ with

:$f(z)=z^\{-1\}+a\_1z+a\_2z^2+\backslash cdots$

near $\backslash infty$, maximizing $\backslash mathrm\{Re\}(e^\{-2i\backslash theta\}a\_1)$ and having image $f(G)$ a parallel slit domain with angle $\backslash theta$ to the {{math|x}}-axis.{{harvnb|Jenkins|1958|pages=77–78}}{{harvnb|Duren|1980}}{{harvnb|Schiff|1993|pages=162–166}}

The first proof that parallel slit domains were canonical domains for in the multiply connected case was given by David Hilbert in 1909. {{harvtxt|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.{{harvnb|Jenkins|1958|pages=77–78}} 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.{{harvnb|Schober|1975}}{{harvnb|Duren|1980}}{{harvnb|Duren|1983}}

{{harvtxt|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. Firstly, by Bieberbach's inequality, any univalent function

:$g(z)=z+cz^2+\backslash cdots$

with $z$ in the open unit disk must satisfy $|c|\backslash leq2$. As a consequence, if

:$f(z)=z+a\_0+a\_1z^\{-1\}+\backslash cdots$

is univalent in $|z|>R$, then $|f(z)-a\_0|\backslash leq2|z|$. To see this, take $S>R$ and 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 $f\_R(z)=z+R^2/z$ is characterized by an "extremal condition" as the unique univalent function in $z>R$ of the form $z+a\_1z^\{-1\}+\backslash cdots$ that maximises $\backslash mathrm\{Re\}(a\_1)$: this is an immediate consequence of Grönwall's area theorem, applied to the family of univalent functions $f(zR)/R$ in $z>1$.{{harvnb|Schiff|1993}}{{harvnb|Goluzin|1969|pages=210–216}}

To prove now that the multiply connected domain $G\backslash subset\backslash mathbb\{C\}\backslash cup\backslash \{\backslash infty\backslash \}$ can be uniformized by a horizontal parallel slit conformal mapping

:$f(z)=z+a\_1z^\{-1\}+\backslash cdots$,

take $R$ large enough that $\backslash partial\; G$ lies in the open disk

:$f(z)=z+a\_1+\backslash cdots$

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 $\backslash mathbb\{C\}\backslash cup\backslash \{\backslash infty\backslash \}$ is simply connected with $G\_2\backslash supset\; G\_1$. By the Riemann mapping theorem there is a conformal mapping

:$h(w)=w+b\_1w^\{-1\}+\backslash cdots,$

such that $h(G\_2)$ is $\backslash mathbb\{C\}$ with a horizontal slit removed. So we have that

:$h(f(z))=z+(a\_1+b\_1)z^\{-1\}+\backslash cdots,$

and thus $\backslash mathrm\{Re\}(a\_1+b\_1)\backslash leq\backslash mathrm\{Re\}(a\_1)$ by the extremality of $f$. Therefore, $\backslash mathrm\{Re\}(b\_1)\backslash leq0$. On the other hand by the Riemann mapping theorem there is a conformal mapping

:$k(w)=w+c\_0+c\_1w^\{-1\}+\backslash cdots,$

mapping from $|w|>S$ onto $G\_2$. Then

:$f(k(w))-c\_0=w+(a\_1+c\_1)w^\{-1\}+\backslash cdots.$

By the strict maximality for the slit mapping in the previous paragraph, we can see that , so that $\backslash mathrm\{Re\}(b\_1)>0$. The two inequalities for $\backslash mathrm\{Re\}(b\_1)$ are contradictory.{{harvnb|Schiff|1993}}{{harvnb|Goluzin|1969|pages=210–216}}{{harvnb|Nehari|1952|pages=351–358}}

The proof of the uniqueness of the conformal parallel slit transformation is given in {{harvtxt|Goluzin|1969}} and {{harvtxt|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 $C\_0$ 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 $a\backslash in\; G$, there is a univalent mapping

:$F\_0(w)=h\backslash circ\; f(w)=(w-a)^\{-1\}+a\_1(w-a)+a\_2(w-a)^2+\backslash cdots,$

with its image a horizontal slit domain. Suppose that $F\_1(w)$ is another uniformizer with

:$F\_1(w)=(w-a)^\{-1\}+b\_1(w-a)+b\_2(w-a)^2+\backslash cdots.$

The images under $F\_0$ or $F\_1$ of each $C\_i$ have a fixed {{math|y}}-coordinate so are horizontal segments. On the other hand, $F\_2(w)=F\_0(w)-F\_1(w)$ 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.{{harvnb|Goluzin|1969|pages=214−215}}

Given $U$ and a point $z\_0\backslash in\; U$, 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

:$f(z)\; =\; (z\; -\; z\_0)e^\{g(z)\},$

where $g=u+iv$ is some (to be determined) holomorphic function with real part $u$ and imaginary part $v$. It is then clear that $z\_0$ is the only zero of $f$. We require $|f(z)|=1$ for $z\backslash in\backslash partial\; U$, so we need

:$u(z)\; =\; -\backslash log|z\; -\; z\_0|$

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.{{harvnb|Gamelin|2001|pages=390–407}}

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 $\backslash mathbb\{C\}$, or the unit disk $D$. This is known as the uniformization 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{{harvnb|Bell|1992}} or the Beltrami equation.

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 $z\_0,\; \backslash ldots,\; z\_n$ in the plane, the algorithm computes an explicit conformal map of the unit disk onto a region bounded by a Jordan curve $\backslash gamma$ with $z\_0,\; \backslash ldots,\; z\_n\; \backslash in\; \backslash gamma.$ This algorithm converges for Jordan regionsA Jordan region is the interior of a Jordan curve. 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 {{math|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.{{Cite journal|doi=10.1137/060659119|title=Convergence of a Variant of the Zipper Algorithm for Conformal Mapping|journal=SIAM Journal on Numerical Analysis|volume=45|issue=6|pages=2577|year=2007|last1=Marshall|first1=Donald E.|last2=Rohde|first2=Steffen|citeseerx=10.1.1.100.2423}}

The following is known about numerically approximating the conformal mapping between two planar domains.{{Cite journal |doi= 10.1007/s11512-007-0045-x| title=On the computational complexity of the Riemann mapping|journal=Arkiv för Matematik| volume=45 |issue=2 |pages=221| year=2007|last1=Binder|first1=Ilia|last2=Braverman|first2=Mark|last3=Yampolsky|first3=Michael|arxiv=math/0505617|bibcode=2007ArM....45..221B| s2cid=14545404}}

Positive results:

• There is an algorithm A that computes the uniformizing map in the following sense. Let $\backslash Omega$ be a bounded simply-connected domain, and $w\_0\backslash in\backslash Omega$. $\backslash partial\backslash Omega$ is provided to A by an oracle representing it in a pixelated sense (i.e., if the screen is divided to $2^n\; \backslash times\; 2^n$ pixels, the oracle can say whether each pixel belongs to the boundary or not). Then A computes the absolute values of the uniformizing map $\backslash phi:(\backslash Omega,\; w\_0)\; \backslash to\; (D,\; 0)$ with precision $2^\{-n\}$ in space bounded by $Cn^2$ and time $2^\{O(n)\}$, where $C$ depends only on the diameter of $\backslash Omega$ and $d(w\_0,\; \backslash partial\backslash Omega).$ Furthermore, the algorithm computes the value of $\backslash phi(w)$ with precision $2^\{-n\}$ as long as Moreover, A queries $\backslash partial\backslash Omega$ with precision of at most $2^\{-O(n)\}.$ In particular, if $\backslash partial\backslash Omega$ is polynomial space computable in space $n^a$ for some constant $a\backslash geq\; 1$ and time then A can be used to compute the uniformizing map in space $C\backslash cdot\; n^\{\backslash max(a,2)\}$ and time $2^\{O(n^a)\}.$

• There is an algorithm A′ that computes the uniformizing map in the following sense. Let $\backslash Omega$ be a bounded simply-connected domain, and $w\_0\; \backslash in\; \backslash Omega.$ Suppose that for some $n=2^k,$ $\backslash partial\backslash Omega$ is given to A′ with precision $\backslash tfrac\{1\}\{n\}$ by $O(n^2)$ pixels. Then A′ computes the absolute values of the uniformizing map $\backslash phi:(\backslash Omega,\; w\_0)\; \backslash to\; (D,\; 0)$ within an error of $O(1/n)$ in randomized space bounded by $O(k)$ and time polynomial in $n=2^k$ (that is, by a BPL({{math|n}})-machine). Furthermore, the algorithm computes the value of $\backslash phi(w)$ with precision $\backslash tfrac\{1\}\{n\}$ as long as

Negative results:

• Suppose there is an algorithm A that given a simply-connected domain $\backslash Omega$ with a linear-time computable boundary and an inner radius $>1/2$ and a number $n$ computes the first $20\; n$ digits of the conformal radius $r(\backslash Omega,\; 0),$ then we can use one call to A to solve any instance of a #SAT({{math|n}}) with a linear time overhead. In other words, #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 $\backslash Omega,$ where the boundary of $\backslash Omega$ is given with precision $1/n$ by an explicit collection of $O(n^2)$ pixels. Denote the problem of computing the conformal radius with precision $1/n^c$ by $\backslash texttt\{CONF\}(n,n^c).$ Then, $\backslash texttt\{MAJ\}\_n$ is AC0 reducible to $\backslash texttt\{CONF\}(n,n^c)$ for any

• Measurable Riemann mapping theorem
• Schwarz–Christoffel mapping – a conformal transformation of the upper half-plane onto the interior of a simple polygon.
• Conformal radius

{{reflist|30em}}

{{Commons category|Riemann mapping}}

• {{citation|last=Ahlfors|first= Lars V.|author-link=Lars Ahlfors|title=Complex analysis. An introduction to the theory of analytic functions of one complex variable|edition=3rd|series= International Series in Pure and Applied Mathematics|publisher= McGraw-Hill|year= 1978|isbn= 0070006571}}
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• {{citation |author-link=Reinhold Remmert |last=Remmert |first=Reinhold |year=1998 |title=Classical topics in complex function theory |translator=Leslie M. Kay |publisher=Springer-Verlag |isbn=0-387-98221-3}}
• {{citation|author-link=Bernhard Riemann|last=Riemann|first=Bernhard| year=1851|url=http://www.emis.de/classics/Riemann/Grund.pdf|title= Grundlagen für eine allgemeine Theorie der Functionen einer veränderlichen complexen Grösse|location= Göttingen|lang=de}}
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• {{Citation | last1=Walsh | first1=J. L. | author-link=Joseph L. Walsh | title=History of the Riemann mapping theorem | jstor=2318448 | mr=0323996 | year=1973 | journal=The American Mathematical Monthly | issn=0002-9890 | volume=80 | issue=3 | pages=270–276 | doi=10.2307/2318448}}

• {{SpringerEOM|title=Riemann theorem|id=Riemann_theorem|first=E.P.|last= Dolzhenko}}

{{Bernhard Riemann}}

{{DEFAULTSORT:Riemann Mapping Theorem}}

Category:Theorems in complex analysis

Category:Bernhard Riemann