History
The theorem was stated (under the assumption that the boundary of
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”.[2] 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
(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
itself; this established the Riemann mapping theorem.[3]
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.[4] His proof used Montel's concept of normal families, which became the standard method of proof in textbooks.[5] 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).[6]
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.[7]
Proof via normal families
Simple connectivity
Theorem. For an open domain
the following conditions are equivalent:[10]
is simply connected;
- the integral of every holomorphic function
around a closed piecewise smooth curve in
vanishes;
- every holomorphic function in
is the derivative of a holomorphic function;
- every nowhere-vanishing holomorphic function
on
has a holomorphic logarithm;
- every nowhere-vanishing holomorphic function
on
has a holomorphic square root;
- for any
, the winding number of
for any piecewise smooth closed curve in
is
;
- the complement of
in the extended complex plane
is connected.
(1) ⇒ (2) because any continuous closed curve, with base point
, can be continuously deformed to the constant curve
. So the line integral of
over the curve is
.
(2) ⇒ (3) because the integral over any piecewise smooth path
from
to
can be used to define a primitive.
(3) ⇒ (4) by integrating
along
from
to
to give a branch of the logarithm.
(4) ⇒ (5) by taking the square root as
where
is a holomorphic choice of logarithm.
(5) ⇒ (6) because if
is a piecewise closed curve and
are successive square roots of
for
outside
, then the winding number of
about
is
times the winding number of
about
. Hence the winding number of
about
must be divisible by
for all
, so it must equal
.
(6) ⇒ (7) for otherwise the extended plane
can be written as the disjoint union of two open and closed sets
and
with
and
bounded. Let
be the shortest Euclidean distance between
and
and build a square grid on
with length
with a point
of
at the centre of a square. Let
be the compact set of the union of all squares with distance
from
. Then
and
does not meet
or
: it consists of finitely many horizontal and vertical segments in
forming a finite number of closed rectangular paths
. Taking
to be all the squares covering
, then
equals the sum of the winding numbers of
over
, thus giving
. On the other hand the sum of the winding numbers of
about
equals
. Hence the winding number of at least one of the
about
is non-zero.
(7) ⇒ (1) This is a purely topological argument. Let
be a piecewise smooth closed curve based at
. By approximation γ is in the same homotopy class as a rectangular path on the square grid of length
based at
; such a rectangular path is determined by a succession of
consecutive directed vertical and horizontal sides. By induction on
, such a path can be deformed to a constant path at a corner of the grid. If the path intersects at a point
, then it breaks up into two rectangular paths of length
, and thus can be deformed to the constant path at
by the induction hypothesis and elementary properties of the fundamental group. The reasoning follows a "northeast argument":[11][12] in the non self-intersecting path there will be a corner
with largest real part (easterly) and then amongst those one with largest imaginary part (northerly). Reversing direction if need be, the path go from
to
and then to
for
and then goes leftwards to
. Let
be the open rectangle with these vertices. The winding number of the path is
for points to the right of the vertical segment from
to
and
for points to the right; and hence inside
. Since the winding number is
off
,
lies in
. If
is a point of the path, it must lie in
; if
is on
but not on the path, by continuity the winding number of the path about
is
, so
must also lie in
. 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
- 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.[13]
- 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
for a holomorphic function
. 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
and set
. These are nowhere-vanishing on a disk but
vanishes at
, so
must vanish identically.[14]
Definitions. A family
of holomorphic functions on an open domain is said to be normal if any sequence of functions in
has a subsequence that converges to a holomorphic function uniformly on compacta.
A family
is compact if whenever a sequence
lies in
and converges uniformly to
on compacta, then
also lies in
. A family
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.[15][16]
- Montel's theorem. Every locally bounded family of holomorphic functions in a domain
is normal.
- Let
be a totally bounded sequence and chose a countable dense subset
of
. By locally boundedness and a "diagonal argument", a subsequence can be chosen so that
is convergent at each point
. It must be verified that this sequence of holomorphic functions converges on
uniformly on each compactum
. Take
open with
such that the closure of
is compact and contains
. Since the sequence
is locally bounded,
on
. By compactness, if
is taken small enough, finitely many open disks
of radius
are required to cover
while remaining in
. Since
,
- we have that
. Now for each
choose some
in
where
converges, take
and
so large to be within
of its limit. Then for
,
![{\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 .}](https://wikimedia.org/api/rest_v1/media/math/render/svg/40417de10b3e672bc17daafd302d29c8924beff9)
- Hence the sequence
forms a Cauchy sequence in the uniform norm on
as required.[17][18]
- Riemann mapping theorem. If
is a simply connected domain and
, there is a unique conformal mapping
of
onto the unit disk
normalized such that
and
.
- Uniqueness follows because if
and
satisfied the same conditions,
would be a univalent holomorphic map of the unit disk with
and
. 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)}](https://wikimedia.org/api/rest_v1/media/math/render/svg/98834d6f22a0eab98f0b7c9cafe42db33cc31708)
- with
. So
must be the identity map and
.
- To prove existence, take
to be the family of holomorphic univalent mappings
of
into the open unit disk
with
and
. It is a normal family by Montel's theorem. By the characterization of simple-connectivity, for
there is a holomorphic branch of the square root
in
. It is univalent and
for
. Since
must contain a closed disk
with centre
and radius
, no points of
can lie in
. Let
be the unique Möbius transformation taking
onto
with the normalization
and
. By construction
is in
, so that
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.[19] Let
be the supremum of
for
. Pick
with
tending to
. By Montel's theorem, passing to a subsequence if necessary,
tends to a holomorphic function
uniformly on compacta. By Hurwitz's theorem,
is either univalent or constant. But
has
and
. So
is finite, equal to
and
. It remains to check that the conformal mapping
takes
onto
. If not, take
in
and let
be a holomorphic square root of
on
. The function
is univalent and maps
into
. Let
![{\displaystyle F(z)={\frac {e^{i\theta }(H(z)-H(a))}{1-{\overline {H(a)))H(z))),}](https://wikimedia.org/api/rest_v1/media/math/render/svg/7e6936093cb6c3d7a38e9b303b35708920f15cc4)
- where
. Then
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.}](https://wikimedia.org/api/rest_v1/media/math/render/svg/6a51f78c4088fb7a5ccce568ded93d6832e7d0ee)
- This contradicts the maximality of
, so that
must take all values in
.[20][21][22]
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
onto
.
Parallel slit mappings
Koebe's uniformization theorem for normal families also generalizes to yield uniformizers
for multiply-connected domains to finite parallel slit domains, where the slits have angle
to the x-axis. Thus if
is a domain in
containing
and bounded by finitely many Jordan contours, there is a unique univalent function
on
with
![{\displaystyle f(z)=z^{-1}+a_{1}z+a_{2}z^{2}+\cdots }](https://wikimedia.org/api/rest_v1/media/math/render/svg/1080e5a5fe12ec5b1b5c8356d7734d334a0e95eb)
near
, maximizing
and having image
a parallel slit domain with angle
to the x-axis.[23][24][25]
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.[26] 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.[27][28][29]
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 }](https://wikimedia.org/api/rest_v1/media/math/render/svg/392b8691ae49edc71e3f52d205ca85aa85acfcde)
with
in the open unit disk must satisfy
. As a consequence, if
![{\displaystyle f(z)=z+a_{0}+a_{1}z^{-1}+\cdots }](https://wikimedia.org/api/rest_v1/media/math/render/svg/0e24d363e3aa6e428d0eeb82a3f4b00fa2192d09)
is univalent in
, then
. To see this, take
and set
![{\displaystyle g(z)=S(f(S/z)-b)^{-1))](https://wikimedia.org/api/rest_v1/media/math/render/svg/47a634c2f30f8a5c361fd1487e7df8510f5e122e)
for
in the unit disk, choosing
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
of the form
that maximises
: this is an immediate consequence of Grönwall's area theorem, applied to the family of univalent functions
in
.[30][31]
To prove now that the multiply connected domain
can be uniformized by a horizontal parallel slit conformal mapping
,
take
large enough that
lies in the open disk
. For
, univalency and the estimate
imply that, if
lies in
with
, then
. Since the family of univalent
are locally bounded in
, by Montel's theorem they form a normal family. Furthermore if
is in the family and tends to
uniformly on compacta, then
is also in the family and each coefficient of the Laurent expansion at
of the
tends to the corresponding coefficient of
. This applies in particular to the coefficient: so by compactness there is a univalent
which maximizes
. To check that
![{\displaystyle f(z)=z+a_{1}+\cdots }](https://wikimedia.org/api/rest_v1/media/math/render/svg/6992c9ea759b34281f999c08dc33b8f048d42ef5)
is the required parallel slit transformation, suppose reductio ad absurdum that
has a compact and connected component
of its boundary which is not a horizontal slit. Then the complement
of
in
is simply connected with
. By the Riemann mapping theorem there is a conformal mapping
![{\displaystyle h(w)=w+b_{1}w^{-1}+\cdots ,}](https://wikimedia.org/api/rest_v1/media/math/render/svg/6a265bb2c9b9d420050ef7b29ecf78f84e6469db)
such that
is
with a horizontal slit removed. So we have that
![{\displaystyle h(f(z))=z+(a_{1}+b_{1})z^{-1}+\cdots ,}](https://wikimedia.org/api/rest_v1/media/math/render/svg/72a9d7f9503ad9d6549e20cdd6350a32f8119104)
and thus
by the extremality of
. Therefore,
. 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 ,}](https://wikimedia.org/api/rest_v1/media/math/render/svg/0c795721a6f5b3af2815b491c0e82986929d410e)
mapping from
onto
. Then
![{\displaystyle f(k(w))-c_{0}=w+(a_{1}+c_{1})w^{-1}+\cdots .}](https://wikimedia.org/api/rest_v1/media/math/render/svg/b5916cd0c20f6c43aa0f8ec444df1f93cfc5256e)
By the strict maximality for the slit mapping in the previous paragraph, we can see that
, so that
. The two inequalities for
are contradictory.[32][33][34]
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
to the horizontal slit domain, it can be assumed that
is a domain bounded by the unit circle
and contains analytic arcs
and isolated points (the images of other the inverse of the Joukowsky transform under the other parallel horizontal slits). Thus, taking a fixed
, 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 ,}](https://wikimedia.org/api/rest_v1/media/math/render/svg/a7fb57ed9d84cd91b6d74fb5913be97caac641ae)
with its image a horizontal slit domain. Suppose that
is another uniformizer with
![{\displaystyle F_{1}(w)=(w-a)^{-1}+b_{1}(w-a)+b_{2}(w-a)^{2}+\cdots .}](https://wikimedia.org/api/rest_v1/media/math/render/svg/85a39988cf5adbebeacd6b467908a36f130d2fc3)
The images under
or
of each
have a fixed y-coordinate so are horizontal segments. On the other hand,
is holomorphic in
. If it is constant, then it must be identically zero since
. Suppose
is non-constant, then by assumption
are all horizontal lines. If
is not in one of these lines, Cauchy's argument principle shows that the number of solutions of
in
is zero (any
will eventually be encircled by contours in
close to the
's). This contradicts the fact that the non-constant holomorphic function
is an open mapping.[35]
Sketch proof via Dirichlet problem
Given
and a point
, we want to construct a function
which maps
to the unit disk and
to
. For this sketch, we will assume that U is bounded and its boundary is smooth, much like Riemann did. Write
![{\displaystyle f(z)=(z-z_{0})e^{g(z)},}](https://wikimedia.org/api/rest_v1/media/math/render/svg/7b55b1b976f2dc5ab7360c395ba0dc4307dcbbfe)
where
is some (to be determined) holomorphic function with real part
and imaginary part
. It is then clear that
is the only zero of
. We require
for
, so we need
![{\displaystyle u(z)=-\log |z-z_{0}|}](https://wikimedia.org/api/rest_v1/media/math/render/svg/4d46842fe0688e6dae86c75c16f2fff2a4d91638)
on the boundary. Since
is the real part of a holomorphic function, we know that
is necessarily a harmonic function; i.e., it satisfies Laplace's equation.
The question then becomes: does a real-valued harmonic function
exist that is defined on all of
and has the given boundary condition? The positive answer is provided by the Dirichlet principle. Once the existence of
has been established, the Cauchy–Riemann equations for the holomorphic function
allow us to find
(this argument depends on the assumption that
be simply connected). Once
and
have been constructed, one has to check that the resulting function
does indeed have all the required properties.[36]
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
with
This algorithm converges for Jordan regions[38] 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
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.[39]
The following is known about numerically approximating the conformal mapping between two planar domains.[40]
Positive results:
- There is an algorithm A that computes the uniformizing map in the following sense. Let
be a bounded simply-connected domain, and
.
is provided to A by an oracle representing it in a pixelated sense (i.e., if the screen is divided to
pixels, the oracle can say whether each pixel belongs to the boundary or not). Then A computes the absolute values of the uniformizing map
with precision
in space bounded by
and time
, where
depends only on the diameter of
and
Furthermore, the algorithm computes the value of
with precision
as long as
Moreover, A queries
with precision of at most
In particular, if
is polynomial space computable in space
for some constant
and time
then A can be used to compute the uniformizing map in space
and time ![{\displaystyle 2^{O(n^{a})}.}](https://wikimedia.org/api/rest_v1/media/math/render/svg/f5fc53fe9ead2f50dde47a2234b1abdf278c7f49)
- There is an algorithm A′ that computes the uniformizing map in the following sense. Let
be a bounded simply-connected domain, and
Suppose that for some
is given to A′ with precision
by
pixels. Then A′ computes the absolute values of the uniformizing map
within an error of
in randomized space bounded by
and time polynomial in
(that is, by a BPL(n)-machine). Furthermore, the algorithm computes the value of
with precision
as long as ![{\displaystyle |\phi (w)|<1-{\tfrac {1}{n)).}](https://wikimedia.org/api/rest_v1/media/math/render/svg/80013e0062ed0bf6d91caa176d7a3aba722c3d7d)
Negative results:
- Suppose there is an algorithm A that given a simply-connected domain
with a linear-time computable boundary and an inner radius
and a number
computes the first
digits of the conformal radius
then we can use one call to A to solve any instance of a #SAT(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
where the boundary of
is given with precision
by an explicit collection of
pixels. Denote the problem of computing the conformal radius with precision
by
Then,
is AC0 reducible to
for any ![{\displaystyle 0<c<{\tfrac {1}{2)).}](https://wikimedia.org/api/rest_v1/media/math/render/svg/38ea0f374a8d0c953af93569c010235cd8dacabd)