Riemann mapping theorem

Theorem. For an open domain ${\displaystyle G\subset \mathbb {C} }$ the following conditions are equivalent:[9]

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

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

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

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

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

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

(6) ⇒ (7) for otherwise the extended plane ${\displaystyle \mathbb {C} \cup \{\infty \}\setminus G}$ can be written as the disjoint union of two open and closed sets ${\displaystyle A}$ and ${\displaystyle B}$ with ${\displaystyle \infty \in B}$ and ${\displaystyle A}$ bounded. Let ${\displaystyle \delta >0}$ be the shortest Euclidean distance between ${\displaystyle A}$ and ${\displaystyle B}$ and build a square grid on ${\displaystyle \mathbb {C} }$ with length ${\displaystyle \delta /4}$ with a point ${\displaystyle a}$ of ${\displaystyle A}$ at the centre of a square. Let ${\displaystyle C}$ be the compact set of the union of all squares with distance ${\displaystyle \leq \delta /4}$ from ${\displaystyle A}$. Then ${\displaystyle C\cap B=\varnothing }$ and ${\displaystyle \partial C}$ does not meet ${\displaystyle A}$ or ${\displaystyle B}$: it consists of finitely many horizontal and vertical segments in ${\displaystyle G}$ forming a finite number of closed rectangular paths ${\displaystyle \gamma _{j}\in G}$. Taking ${\displaystyle C_{i}}$ to be all the squares covering ${\displaystyle A}$, then ${\displaystyle {\frac {1}{2\pi }}\int _{\partial C}\mathrm {d} \mathrm {arg} (z-a)}$ equals the sum of the winding numbers of ${\displaystyle C_{i}}$ over ${\displaystyle a}$, thus giving ${\displaystyle 1}$. On the other hand the sum of the winding numbers of ${\displaystyle \gamma _{j}}$ about ${\displaystyle a}$ equals ${\displaystyle 1}$. Hence the winding number of at least one of the ${\displaystyle \gamma _{j}}$ about ${\displaystyle a}$ is non-zero.

(7) ⇒ (1) This is a purely topological argument. Let ${\displaystyle \gamma }$ be a piecewise smooth closed curve based at ${\displaystyle z_{0}\in G}$. By approximation γ is in the same homotopy class as a rectangular path on the square grid of length ${\displaystyle \delta >0}$ based at ${\displaystyle z_{0}}$; such a rectangular path is determined by a succession of ${\displaystyle N}$ consecutive directed vertical and horizontal sides. By induction on ${\displaystyle N}$, such a path can be deformed to a constant path at a corner of the grid. If the path intersects at a point ${\displaystyle z_{1}}$, then it breaks up into two rectangular paths of length ${\displaystyle <N}$, and thus can be deformed to the constant path at ${\displaystyle z_{1}}$ 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 ${\displaystyle z_{0}}$ with largest real part (easterly) and then amongst those one with largest imaginary part (northerly). Reversing direction if need be, the path go from ${\displaystyle z_{0}-\delta }$ to ${\displaystyle z_{0}}$ and then to ${\displaystyle w_{0}=z_{0}-in\delta }$ for ${\displaystyle n\geq 1}$ and then goes leftwards to ${\displaystyle w_{0}-\delta }$. Let ${\displaystyle R}$ be the open rectangle with these vertices. The winding number of the path is ${\displaystyle 0}$ for points to the right of the vertical segment from ${\displaystyle z_{0}}$ to ${\displaystyle w_{0}}$ and ${\displaystyle -1}$ for points to the right; and hence inside ${\displaystyle R}$. Since the winding number is ${\displaystyle 0}$ off ${\displaystyle G}$, ${\displaystyle R}$ lies in ${\displaystyle G}$. If ${\displaystyle z}$ is a point of the path, it must lie in ${\displaystyle G}$; if ${\displaystyle z}$ is on ${\displaystyle \partial R}$ but not on the path, by continuity the winding number of the path about ${\displaystyle z}$ is ${\displaystyle -1}$, so ${\displaystyle z}$ must also lie in ${\displaystyle G}$. Hence ${\displaystyle R\cup \partial R\subset G}$. 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).

• 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.[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 ${\displaystyle {\frac {1}{2\pi i}}\int _{C}g^{-1}(z)g'(z)\mathrm {d} z}$ for a holomorphic function ${\displaystyle g}$. 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 ${\displaystyle f(a)=f(b)}$ and set ${\displaystyle g_{n}(z)=f_{n}(z)-f_{n}(a)}$. These are nowhere-vanishing on a disk but ${\displaystyle g(z)=f(z)-f(a)}$ vanishes at ${\displaystyle b}$, so ${\displaystyle g}$ 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 ${\displaystyle f_{n}}$ lies in ${\displaystyle {\cal {F}}}$ and converges uniformly to ${\displaystyle f}$ on compacta, then ${\displaystyle f}$ 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 ${\displaystyle G}$ is normal.

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

${\displaystyle g_{n}(b)-g_{n}(a)=\int _{a}^{b}g_{n}^{\prime }(z)\,dz}$,

we have that ${\displaystyle |g_{n}(a)-g_{n}(b)|\leq M|a-b|\leq 2\delta M}$. Now for each ${\displaystyle k}$ choose some ${\displaystyle w_{i}}$ in ${\displaystyle D_{k}}$ where ${\displaystyle g_{n}(w_{i})}$ converges, take ${\displaystyle n}$ and ${\displaystyle m}$ so large to be within ${\displaystyle \delta }$ of its limit. Then for ${\displaystyle z\in D_{k}}$,

${\displaystyle |g_{n}(z)-g_{m}(z)|\leq |g_{n}(z)-g_{n}(w_{i})|+|g_{n}(w_{i})-g_{m}(w_{i})|+|g_{m}(w_{1})-g_{m}(z)|\leq 4M\delta +2\delta .}$

Hence the sequence ${\displaystyle \{g_{n}\}}$ forms a Cauchy sequence in the uniform norm on ${\displaystyle K}$ as required.[16][17]

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

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

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

with ${\displaystyle |\alpha |<1}$. So ${\displaystyle h}$ must be the identity map and ${\displaystyle f=g}$.

To prove existence, take ${\displaystyle {\cal {F}}}$ to be the family of holomorphic univalent mappings ${\displaystyle f}$ of ${\displaystyle G}$ into the open unit disk ${\displaystyle D}$ with ${\displaystyle f(a)=0}$ and ${\displaystyle f'(a)>0}$. It is a normal family by Montel's theorem. By the characterization of simple-connectivity, for ${\displaystyle b\in \mathbb {C} \setminus G}$ there is a holomorphic branch of the square root ${\displaystyle h(z)={\sqrt {z-b}}}$ in ${\displaystyle G}$. It is univalent and ${\displaystyle h(z_{1})\neq -h(z_{2})}$ for ${\displaystyle z_{1},z_{2}\in G}$. Since ${\displaystyle h(G)}$ must contain a closed disk ${\displaystyle \Delta }$ with centre ${\displaystyle h(a)}$ and radius ${\displaystyle r>0}$, no points of ${\displaystyle -\Delta }$ can lie in ${\displaystyle h(G)}$. Let ${\displaystyle F}$ be the unique Möbius transformation taking ${\displaystyle \mathbb {C} \setminus -\Delta }$ onto ${\displaystyle D}$ with the normalization ${\displaystyle F(h(a))=0}$ and ${\displaystyle F'(h(a))>0}$. By construction ${\displaystyle F\circ h}$ is in ${\displaystyle {\cal {F}}}$, so that ${\displaystyle {\cal {F}}}$ 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] Let ${\displaystyle 0<M\leq \infty }$ be the supremum of ${\displaystyle f'(a)}$ for ${\displaystyle f\in {\cal {F}}}$. Pick ${\displaystyle f_{n}\in {\cal {F}}}$ with ${\displaystyle f_{n}'(a)}$ tending to ${\displaystyle M}$. By Montel's theorem, passing to a subsequence if necessary, ${\displaystyle f_{n}}$ tends to a holomorphic function ${\displaystyle f}$ uniformly on compacta. By Hurwitz's theorem, ${\displaystyle f}$ is either univalent or constant. But ${\displaystyle f}$ has ${\displaystyle f(a)=0}$ and ${\displaystyle f'(a)>0}$. So ${\displaystyle M}$ is finite, equal to ${\displaystyle f'(a)>0}$ and ${\displaystyle {f\in {\cal {F}}}}$. It remains to check that the conformal mapping ${\displaystyle f}$ takes ${\displaystyle G}$ onto ${\displaystyle D}$. If not, take ${\displaystyle c\neq 0}$ in ${\displaystyle D\setminus f(G)}$ and let ${\displaystyle H}$ be a holomorphic square root of ${\displaystyle (f(z)-c)/(1-{\overline {c}}f(z))}$ on ${\displaystyle G}$. The function ${\displaystyle H}$ is univalent and maps ${\displaystyle G}$ into ${\displaystyle D}$. Let

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

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

${\displaystyle F'(a)=H'(a)/(1-|H(a)|^{2})=f'(a)\left({\sqrt {|c|}}+{\sqrt {|c|^{-1}}}\right)/2>f'(a)=M.}$

This contradicts the maximality of ${\displaystyle M}$, so that ${\displaystyle f}$ must take all values in ${\displaystyle D}$.[19][20][21]

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

Koebe's uniformization theorem for normal families also generalizes to yield uniformizers ${\displaystyle f}$ for multiply-connected domains to finite parallel slit domains, where the slits have angle ${\displaystyle \theta }$ to the x-axis. Thus if ${\displaystyle G}$ is a domain in ${\displaystyle \mathbb {C} \cup \{\infty \}}$ containing ${\displaystyle \infty }$ and bounded by finitely many Jordan contours, there is a unique univalent function ${\displaystyle f}$ on ${\displaystyle G}$ with

${\displaystyle f(z)=z^{-1}+a_{1}z+a_{2}z^{2}+\cdots }$

near ${\displaystyle \infty }$, maximizing ${\displaystyle \mathrm {Re} (e^{-2i\theta }a_{1})}$ and having image ${\displaystyle f(G)}$ a parallel slit domain with angle ${\displaystyle \theta }$ 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 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.[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. Firstly, by Bieberbach's inequality, any univalent function

${\displaystyle g(z)=z+cz^{2}+\cdots }$

with ${\displaystyle z}$ in the open unit disk must satisfy ${\displaystyle |c|\leq 2}$. As a consequence, if

${\displaystyle f(z)=z+a_{0}+a_{1}z^{-1}+\cdots }$

is univalent in ${\displaystyle |z|>R}$, then ${\displaystyle |f(z)-a_{0}|\leq 2|z|}$. To see this, take ${\displaystyle S>R}$ and set

${\displaystyle g(z)=S(f(S/z)-b)^{-1}}$

for ${\displaystyle z}$ in the unit disk, choosing ${\displaystyle b}$ so the denominator is nowhere-vanishing, and apply the Schwarz lemma. Next the function ${\displaystyle f_{R}(z)=z+R^{2}/z}$ is characterized by an "extremal condition" as the unique univalent function in ${\displaystyle z>R}$ of the form ${\displaystyle z+a_{1}z^{-1}+\cdots }$ that maximises ${\displaystyle \mathrm {Re} (a_{1})}$: this is an immediate consequence of Grönwall's area theorem, applied to the family of univalent functions ${\displaystyle f(zR)/R}$ in ${\displaystyle z>1}$.[29][30]

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

${\displaystyle f(z)=z+a_{1}z^{-1}+\cdots }$,

take ${\displaystyle R}$ large enough that ${\displaystyle \partial G}$ lies in the open disk ${\displaystyle |z|<R}$. For ${\displaystyle S>R}$, univalency and the estimate ${\displaystyle |f(z)|\leq 2|z|}$ imply that, if ${\displaystyle z}$ lies in ${\displaystyle G}$ with ${\displaystyle |z|\leq S}$, then ${\displaystyle |f(z)|\leq 2S}$. Since the family of univalent ${\displaystyle f}$ are locally bounded in ${\displaystyle G\setminus \{\infty \}}$, by Montel's theorem they form a normal family. Furthermore if ${\displaystyle f_{n}}$ is in the family and tends to ${\displaystyle f}$ uniformly on compacta, then ${\displaystyle f}$ is also in the family and each coefficient of the Laurent expansion at ${\displaystyle \infty }$ of the ${\displaystyle f_{n}}$ tends to the corresponding coefficient of ${\displaystyle f}$. This applies in particular to the coefficient: so by compactness there is a univalent ${\displaystyle f}$ which maximizes ${\displaystyle \mathrm {Re} (a_{1})}$. To check that

${\displaystyle f(z)=z+a_{1}+\cdots }$

is the required parallel slit transformation, suppose reductio ad absurdum that ${\displaystyle f(G)=G_{1}}$ has a compact and connected component ${\displaystyle K}$ of its boundary which is not a horizontal slit. Then the complement ${\displaystyle G_{2}}$ of ${\displaystyle K}$ in ${\displaystyle \mathbb {C} \cup \{\infty \}}$ is simply connected with ${\displaystyle G_{2}\supset G_{1}}$. By the Riemann mapping theorem there is a conformal mapping

${\displaystyle h(w)=w+b_{1}w^{-1}+\cdots ,}$

such that ${\displaystyle h(G_{2})}$ is ${\displaystyle \mathbb {C} }$ with a horizontal slit removed. So we have that

${\displaystyle h(f(z))=z+(a_{1}+b_{1})z^{-1}+\cdots ,}$

and thus ${\displaystyle \mathrm {Re} (a_{1}+b_{1})\leq \mathrm {Re} (a_{1})}$ by the extremality of ${\displaystyle f}$. Therefore, ${\displaystyle \mathrm {Re} (b_{1})\leq 0}$. On the other hand by the Riemann mapping theorem there is a conformal mapping

${\displaystyle k(w)=w+c_{0}+c_{1}w^{-1}+\cdots ,}$

mapping from ${\displaystyle |w|>S}$ onto ${\displaystyle G_{2}}$. Then

${\displaystyle f(k(w))-c_{0}=w+(a_{1}+c_{1})w^{-1}+\cdots .}$

By the strict maximality for the slit mapping in the previous paragraph, we can see that ${\displaystyle \mathrm {Re} (c_{1})<\mathrm {Re} (b_{1}+c_{1})}$, so that ${\displaystyle \mathrm {Re} (b_{1})>0}$. The two inequalities for ${\displaystyle \mathrm {Re} (b_{1})}$ 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 ${\displaystyle h}$ to the horizontal slit domain, it can be assumed that ${\displaystyle G}$ is a domain bounded by the unit circle ${\displaystyle C_{0}}$ and contains analytic arcs ${\displaystyle C_{i}}$ and isolated points (the images of other the inverse of the Joukowsky transform under the other parallel horizontal slits). Thus, taking a fixed ${\displaystyle a\in G}$, there is a univalent mapping

${\displaystyle F_{0}(w)=h\circ f(w)=(w-a)^{-1}+a_{1}(w-a)+a_{2}(w-a)^{2}+\cdots ,}$

with its image a horizontal slit domain. Suppose that ${\displaystyle F_{1}(w)}$ is another uniformizer with

${\displaystyle F_{1}(w)=(w-a)^{-1}+b_{1}(w-a)+b_{2}(w-a)^{2}+\cdots .}$

The images under ${\displaystyle F_{0}}$ or ${\displaystyle F_{1}}$ of each ${\displaystyle C_{i}}$ have a fixed y-coordinate so are horizontal segments. On the other hand, ${\displaystyle F_{2}(w)=F_{0}(w)-F_{1}(w)}$ is holomorphic in ${\displaystyle G}$. If it is constant, then it must be identically zero since ${\displaystyle F_{2}(a)=0}$. Suppose ${\displaystyle F_{2}}$ is non-constant, then by assumption ${\displaystyle F_{2}(C_{i})}$ are all horizontal lines. If ${\displaystyle t}$ is not in one of these lines, Cauchy's argument principle shows that the number of solutions of ${\displaystyle F_{2}(w)=t}$ in ${\displaystyle G}$ is zero (any ${\displaystyle t}$ will eventually be encircled by contours in ${\displaystyle G}$ close to the ${\displaystyle C_{i}}$'s). This contradicts the fact that the non-constant holomorphic function ${\displaystyle F_{2}}$ is an open mapping.[34]