Editing Riemann mapping theorem (section)

=== Parallel slit mappings ===
Koebe's uniformization theorem for normal families also generalizes to yield uniformizers <math>f</math> for multiply-connected domains to finite '''parallel slit domains''', where the slits have angle <math>\theta</math> to the {{math|''x''}}-axis. Thus if <math>G</math> is a domain in <math>\mathbb{C}\cup\{\infty\}</math> containing <math>\infty</math> and bounded by finitely many Jordan contours, there is a unique univalent function <math>f</math> on <math>G</math> with
:<math>f(z)=z^{-1}+a_1z+a_2z^2+\cdots</math>
near <math>\infty</math>, maximizing <math>\mathrm{Re}(e^{-2i\theta}a_1)</math> and having image <math>f(G)</math> a parallel slit domain with angle <math>\theta</math> to the {{math|''x''}}-axis.<ref>{{harvnb|Jenkins|1958|pages=77–78}}</ref><ref>{{harvnb|Duren|1980}}</ref><ref>{{harvnb|Schiff|1993|pages=162–166}}</ref>

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 mapping]]s and [[quadratic differential]]s, later developed as the technique of [[extremal length|extremal metric]] due to [[Oswald Teichmüller]].<ref>{{harvnb|Jenkins|1958|pages=77–78}}</ref> [[Menahem Max Schiffer|Menahem Schiffer]] gave a treatment based on very general [[variational principle]]s, 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.<ref>{{harvnb|Schober|1975}}</ref><ref>{{harvnb|Duren|1980}}</ref><ref>{{harvnb|Duren|1983}}</ref>

{{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 [[Koebe quarter theorem#Bieberbach's coefficient inequality for univalent theorem|Bieberbach's inequality]], any univalent function
:<math>g(z)=z+cz^2+\cdots</math>
with <math>z</math> in the open unit disk must satisfy <math>|c|\leq2</math>. As a consequence, if
:<math>f(z)=z+a_0+a_1z^{-1}+\cdots</math>
is univalent in <math>|z|>R</math>, then <math>|f(z)-a_0|\leq2|z|</math>. To see this, take <math>S>R</math> and set
:<math>g(z)=S(f(S/z)-b)^{-1}</math>
for <math>z</math> in the unit disk, choosing <math>b</math> so the denominator is nowhere-vanishing, and apply the [[Schwarz lemma]]. Next the function <math>f_R(z)=z+R^2/z</math> is characterized by an "extremal condition" as the unique univalent function in <math>z>R</math> of the form <math>z+a_1z^{-1}+\cdots</math> that maximises <math>\mathrm{Re}(a_1)</math>: this is an immediate consequence of [[Koebe quarter theorem#Grönwall's area theorem|Grönwall's area theorem]], applied to the family of univalent functions <math>f(zR)/R</math> in <math>z>1</math>.<ref>{{harvnb|Schiff|1993}}</ref><ref>{{harvnb|Goluzin|1969|pages=210–216}}</ref>

To prove now that the multiply connected domain <math>G\subset\mathbb{C}\cup\{\infty\}</math> can be uniformized by a horizontal parallel slit conformal mapping
:<math>f(z)=z+a_1z^{-1}+\cdots</math>,
take <math>R</math> large enough that <math>\partial G</math> lies in the open disk <math>|z|<R</math>. For <math>S>R</math>, univalency and the estimate <math>|f(z)|\leq2|z|</math> imply that, if <math>z</math> lies in <math>G</math> with  <math>|z|\leq S</math>, then <math>|f(z)|\leq2S</math>. Since the family of univalent <math>f</math> are locally bounded in <math>G\setminus\{\infty\}</math>, by Montel's theorem they form a normal family. Furthermore if <math>f_n</math> is in the family and tends to <math>f</math> uniformly on compacta, then <math>f</math> is also in the family and each coefficient of the Laurent expansion at <math>\infty</math> of the <math>f_n</math> tends to the corresponding coefficient of <math>f</math>. This applies in particular to the coefficient: so by compactness there is a univalent <math>f</math> which maximizes <math>\mathrm{Re}(a_1)</math>. To check that
:<math>f(z)=z+a_1+\cdots</math>
is the required parallel slit transformation, suppose ''reductio ad absurdum'' that <math>f(G)=G_1</math> has a compact and connected component <math>K</math> of its boundary which is not a horizontal slit. Then the complement <math>G_2</math> of <math>K</math> in <math>\mathbb{C}\cup\{\infty\}</math> is simply connected with <math>G_2\supset G_1</math>. By the Riemann mapping theorem there is a conformal mapping
:<math>h(w)=w+b_1w^{-1}+\cdots,</math>
such that <math>h(G_2)</math> is <math>\mathbb{C}</math> with a horizontal slit removed. So we have that
:<math>h(f(z))=z+(a_1+b_1)z^{-1}+\cdots,</math>
and thus <math>\mathrm{Re}(a_1+b_1)\leq\mathrm{Re}(a_1)</math> by the extremality of <math>f</math>. Therefore, <math>\mathrm{Re}(b_1)\leq0</math>. On the other hand by the Riemann mapping theorem there is a conformal mapping
:<math>k(w)=w+c_0+c_1w^{-1}+\cdots,</math>
mapping from <math>|w|>S</math> onto <math>G_2</math>. Then
:<math>f(k(w))-c_0=w+(a_1+c_1)w^{-1}+\cdots.</math>
By the strict maximality for the slit mapping in the previous paragraph, we can see that <math>\mathrm{Re}(c_1)<\mathrm{Re}(b_1+c_1)</math>, so that <math>\mathrm{Re}(b_1)>0</math>. The two inequalities for <math>\mathrm{Re}(b_1)</math> are contradictory.<ref>{{harvnb|Schiff|1993}}</ref><ref>{{harvnb|Goluzin|1969|pages=210–216}}</ref><ref>{{harvnb|Nehari|1952|pages=351–358}}</ref>

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]] <math>h</math> to the horizontal slit domain, it can be assumed that <math>G</math> is a domain bounded by the unit circle <math>C_0</math> and contains analytic arcs <math>C_i</math> and isolated points (the images of other the inverse of the Joukowsky transform under the other parallel horizontal slits). Thus, taking a fixed <math>a\in G</math>, there is a univalent mapping
:<math>F_0(w)=h\circ f(w)=(w-a)^{-1}+a_1(w-a)+a_2(w-a)^2+\cdots,</math>
with its image a horizontal slit domain. Suppose that <math>F_1(w)</math> is another uniformizer with
:<math>F_1(w)=(w-a)^{-1}+b_1(w-a)+b_2(w-a)^2+\cdots.</math>
The images under <math>F_0</math> or <math>F_1</math> of each <math>C_i</math> have a fixed {{math|''y''}}-coordinate so are horizontal segments. On the other hand, <math>F_2(w)=F_0(w)-F_1(w)</math> is holomorphic in <math>G</math>. If it is constant, then it must be identically zero since <math>F_2(a)=0</math>. Suppose <math>F_2</math> is non-constant, then by assumption <math>F_2(C_i)</math> are all horizontal lines. If <math>t</math> is not in one of these lines, [[Cauchy's argument principle]] shows that the number of solutions of <math>F_2(w)=t</math> in <math>G</math> is zero (any <math>t</math> will eventually be encircled by contours in <math>G</math> close to the <math>C_i</math>'s). This contradicts the fact that the non-constant holomorphic function <math>F_2</math> is an [[open mapping]].<ref>{{harvnb|Goluzin|1969|pages=214−215}}</ref>