Riemann mapping theorem explained

In complex analysis, the Riemann mapping theorem states that if

D=\{z\inC:|z|<1\}.

This mapping is known as a Riemann mapping.[1]

Intuitively, the condition that

U

be simply connected means that

U

does not contain any “holes”. The fact that

f

is biholomorphic implies that it is a conformal map and therefore angle-preserving. Such a map may be interpreted as preserving the shape of any sufficiently small figure, while possibly rotating and scaling (but not reflecting) it.

Henri Poincaré proved that the map

f

is unique up to rotation and recentering: if

z0

is an element of

U

and

\phi

is an arbitrary angle, then there exists precisely one f as above such that

f(z0)=0

and such that the argument of the derivative of

f

at the point

z0

is equal to

\phi

. This is an easy consequence of the Schwarz lemma.

As a corollary of the theorem, any two simply connected open subsets of the Riemann sphere which both lack at least two points of the sphere can be conformally mapped into each other.

History

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”. 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

C

itself; this established the Riemann mapping theorem.[2]

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.[3] His proof used Montel's concept of normal families, which became the standard method of proof in textbooks. 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).

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.[4]

Importance

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

\{z:r<|z|<1\}

with

0<r<1

, however there are no conformal maps between annuli except inversion and multiplication by constants so the annulus

\{z:1<|z|<2\}

is not conformally equivalent to the annulus

\{z:1<|z|<4\}

(as can be proven using extremal length).

Cn

(

n\ge2

), the ball and polydisk are both simply connected, but there is no biholomorphic map between them.[6]

Proof via normal families

See main article: Normal families.

Simple connectivity

Theorem. For an open domain

G\subsetC

the following conditions are equivalent:[7]

G

is simply connected;
  1. the integral of every holomorphic function

f

around a closed piecewise smooth curve in

G

vanishes;
  1. every holomorphic function in

G

is the derivative of a holomorphic function;
  1. every nowhere-vanishing holomorphic function

f

on

G

has a holomorphic logarithm;
  1. every nowhere-vanishing holomorphic function

g

on

G

has a holomorphic square root;
  1. for any

w\notinG

, the winding number of

w

for any piecewise smooth closed curve in

G

is

0

;
  1. the complement of

G

in the extended complex plane

C\cup\{infty\}

is connected.

(1) ⇒ (2) because any continuous closed curve, with base point

a\inG

, can be continuously deformed to the constant curve

a

. So the line integral of

fdz

over the curve is

0

.

(2) ⇒ (3) because the integral over any piecewise smooth path

\gamma

from

a

to

z

can be used to define a primitive.

(3) ⇒ (4) by integrating

f-1df/dz

along

\gamma

from

a

to

x

to give a branch of the logarithm.

(4) ⇒ (5) by taking the square root as

g(z)=\exp(f(x)/2)

where

f

is a holomorphic choice of logarithm.

(5) ⇒ (6) because if

\gamma

is a piecewise closed curve and

fn

are successive square roots of

z-w

for

w

outside

G

, then the winding number of

fn\circ\gamma

about

w

is

2n

times the winding number of

\gamma

about

0

. Hence the winding number of

\gamma

about

w

must be divisible by

2n

for all

n

, so it must equal

0

.

(6) ⇒ (7) for otherwise the extended plane

C\cup\{infty\}\setminusG

can be written as the disjoint union of two open and closed sets

A

and

B

with

infty\inB

and

A

bounded. Let

\delta>0

be the shortest Euclidean distance between

A

and

B

and build a square grid on

C

with length

\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

\leq\delta/4

from

A

. Then

C\capB=\varnothing

and

\partialC

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

\gammaj\inG

. Taking

Ci

to be all the squares covering

A

, then
1
2\pi

\int\partialdarg(z-a)

equals the sum of the winding numbers of

Ci

over

a

, thus giving

1

. On the other hand the sum of the winding numbers of

\gammaj

about

a

equals

1

. Hence the winding number of at least one of the

\gammaj

about

a

is non-zero.

(7) ⇒ (1) This is a purely topological argument. Let

\gamma

be a piecewise smooth closed curve based at

z0\inG

. By approximation γ is in the same homotopy class as a rectangular path on the square grid of length

\delta>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

, and thus 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":[8] 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-\delta

to

z0

and then to

w0=z0-in\delta

for

n\geq1

and then goes leftwards to

w0-\delta

. 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

\partialR

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\cup\partialR\subsetG

. 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

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.

If the limit function is non-zero, then its zeros have to be isolated. Zeros with multiplicities can be counted by the winding number

1
2\pii
-1
\int
Cg

(z)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.

Definitions. A family

{\calF}

of holomorphic functions on an open domain is said to be normal if any sequence of functions in

{\calF}

has a subsequence that converges to a holomorphic function uniformly on compacta. A family

{\calF}

is compact if whenever a sequence

fn

lies in

{\calF}

and converges uniformly to

f

on compacta, then

f

also lies in

{\calF}

. A family

{\calF}

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.

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\subsetE

such that the closure of

E

is compact and contains

G

. Since the sequence

\{gn'\}

is locally bounded,

|gn'|\leqM

on

E

. By compactness, if

\delta>0

is taken small enough, finitely many open disks

Dk

of radius

\delta>0

are required to cover

K

while remaining in

E

. Since

gn(b)-gn(a)=

b
\int
a
\prime(z)
g
n

dz

,

we have that

|gn(a)-gn(b)|\leqM|a-b|\leq2\deltaM

. Now for each

k

choose some

wi

in

Dk

where

gn(wi)

converges, take

n

and

m

so large to be within

\delta

of its limit. Then for

z\inDk

,

|gn(z)-gm(z)|\leq|gn(z)-gn(wi)|+|gn(wi)-gm(wi)|+|gm(w1)-gm(z)|\leq4M\delta+2\delta.

Hence the sequence

\{gn\}

forms a Cauchy sequence in the uniform norm on

K

as required.

GC

is a simply connected domain and

a\inG

, there is a unique conformal mapping

f

of

G

onto the unit disk

D

normalized such that

f(a)=0

and

f'(a)>0

.

Uniqueness follows because if

f

and

g

satisfied the same conditions,

h=f\circg-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)=ei\theta(z-\alpha)/(1-\overline{\alpha}z)

with

|\alpha|<1

. So

h

must be the identity map and

f=g

.

To prove existence, take

{\calF}

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\inC\setminusG

there is a holomorphic branch of the square root

h(z)=\sqrt{z-b}

in

G

. It is univalent and

h(z1) ≠ -h(z2)

for

z1,z2\inG

. Since

h(G)

must contain a closed disk

\Delta

with centre

h(a)

and radius

r>0

, no points of

-\Delta

can lie in

h(G)

. Let

F

be the unique Möbius transformation taking

C\setminus-\Delta

onto

D

with the normalization

F(h(a))=0

and

F'(h(a))>0

. By construction

F\circh

is in

{\calF}

, so that

{\calF}

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, after Ahlfors. Let

0<M\leqinfty

be the supremum of

f'(a)

for

f\in{\calF}

. Pick

fn\in{\calF}

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'(a)>0

and

{f\in\calF}

. It remains to check that the conformal mapping

f

takes

G

onto

D

. If not, take

c ≠ 0

in

D\setminusf(G)

and let

H

be a holomorphic square root of

(f(z)-c)/(1-\overline{c}f(z))

on

G

. The function

H

is univalent and maps

G

into

D

. Let
F(z)=ei\theta(H(z)-H(a))
1-\overline{H(a)

H(z)},

where

H'(a)/|H'(a)|=e-i\theta

. Then

F\in{\calF}

and a routine computation shows that

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

M

, so that

f

must take all values in

D

.

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

\phi(x)=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

\theta

to the -axis. Thus if

G

is a domain in

C\cup\{infty\}

containing

infty

and bounded by finitely many Jordan contours, there is a unique univalent function

f

on

G

with

f(z)=z-1+a1z+a

2+ …
2z
near

infty

, maximizing

Re(e-2i\thetaa1)

and having image

f(G)

a parallel slit domain with angle

\theta

to the -axis.

The first proof that parallel slit domains were canonical domains for in the multiply connected case was given by David Hilbert in 1909., 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. 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.

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+cz2+ …

with

z

in the open unit disk must satisfy

|c|\leq2

. As a consequence, if

f(z)=z+a0+a

-1
1z

+ …

is univalent in

|z|>R

, then

|f(z)-a0|\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
2/z
f
R(z)=z+R
is characterized by an "extremal condition" as the unique univalent function in

z>R

of the form
-1
z+a
1z

+ …

that maximises

Re(a1)

: this is an immediate consequence of Grönwall's area theorem, applied to the family of univalent functions

f(zR)/R

in

z>1

.

To prove now that the multiply connected domain

G\subsetC\cup\{infty\}

can be uniformized by a horizontal parallel slit conformal mapping
-1
f(z)=z+a
1z

+ …

,take

R

large enough that

\partialG

lies in the open disk

|z|<R

. For

S>R

, univalency and the estimate

|f(z)|\leq2|z|

imply that, if

z

lies in

G

with

|z|\leqS

, then

|f(z)|\leq2S

. Since the family of univalent

f

are locally bounded in

G\setminus\{infty\}

, 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

infty

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\cup\{infty\}

is simply connected with

G2\supsetG1

. By the Riemann mapping theorem there is a conformal mapping
-1
h(w)=w+b
1w

+ … ,

such that

h(G2)

is

C

with a horizontal slit removed. So we have that

h(f(z))=z+(a1+b

-1
1)z

+ … ,

and thus

Re(a1+b1)\leqRe(a1)

by the extremality of

f

. Therefore,

Re(b1)\leq0

. On the other hand by the Riemann mapping theorem there is a conformal mapping

k(w)=w+c0+c

-1
1w

+ … ,

mapping from

|w|>S

onto

G2

. Then

f(k(w))-c0=w+(a1+c

-1
1)w

+ … .

By the strict maximality for the slit mapping in the previous paragraph, we can see that

Re(c1)<Re(b1+c1)

, so that

Re(b1)>0

. The two inequalities for

Re(b1)

are contradictory.

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). Thus, taking a fixed

a\inG

, there is a univalent mapping

F0(w)=h\circf(w)=(w-a)-1+a1(w-a)+a

2+ … ,
2(w-a)
with its image a horizontal slit domain. Suppose that

F1(w)

is another uniformizer with
-1
F
1(w)=(w-a)

+b1(w-a)+b

2+ … .
2(w-a)
The images under

F0

or

F1

of each

Ci

have a fixed -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(a)=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.

Sketch proof via Dirichlet problem

Given

U

and a point

z0\inU

, 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

f(z)=(z-

g(z)
z
0)e

,

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\in\partialU

, so we need

u(z)=-log|z-z0|

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.

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, the complex plane

C

, or the unit disk

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 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

z0,\ldots,zn

in the plane, the algorithm computes an explicit conformal map of the unit disk onto a region bounded by a Jordan curve

\gamma

with

z0,\ldots,zn\in\gamma.

This algorithm converges for Jordan regions[9] 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

C1

curve or a -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.[10]

The following is known about numerically approximating the conformal mapping between two planar domains.[11]

Positive results:

\Omega

be a bounded simply-connected domain, and

w0\in\Omega

.

\partial\Omega

is provided to A by an oracle representing it in a pixelated sense (i.e., if the screen is divided to

2n x 2n

pixels, the oracle can say whether each pixel belongs to the boundary or not). Then A computes the absolute values of the uniformizing map

\phi:(\Omega,w0)\to(D,0)

with precision

2-n

in space bounded by

Cn2

and time

2O(n)

, where

C

depends only on the diameter of

\Omega

and

d(w0,\partial\Omega).

Furthermore, the algorithm computes the value of

\phi(w)

with precision

2-n

as long as

|\phi(w)|<1-2-n.

Moreover, A queries

\partial\Omega

with precision of at most

2-O(n).

In particular, if

\partial\Omega

is polynomial space computable in space

na

for some constant

a\geq1

and time

T(n)<

O(na)
2

,

then A can be used to compute the uniformizing map in space

Cnmax(a,2)

and time
O(na)
2

.

\Omega

be a bounded simply-connected domain, and

w0\in\Omega.

Suppose that for some

n=2k,

\partial\Omega

is given to A′ with precision

\tfrac{1}{n}

by

O(n2)

pixels. Then A′ computes the absolute values of the uniformizing map

\phi:(\Omega,w0)\to(D,0)

within an error of

O(1/n)

in randomized space bounded by

O(k)

and time polynomial in

n=2k

(that is, by a BPL-machine). Furthermore, the algorithm computes the value of

\phi(w)

with precision

\tfrac{1}{n}

as long as

|\phi(w)|<1-\tfrac{1}{n}.

Negative results:

\Omega

with a linear-time computable boundary and an inner radius

>1/2

and a number

n

computes the first

20n

digits of the conformal radius

r(\Omega,0),

then we can use one call to A to solve any instance of a
  1. SAT
with a linear time overhead. In other words,
  1. P
is poly-time reducible to computing the conformal radius of a set.

\Omega,

where the boundary of

\Omega

is given with precision

1/n

by an explicit collection of

O(n2)

pixels. Denote the problem of computing the conformal radius with precision

1/nc

by

tt{CONF}(n,nc).

Then,

tt{MAJ}n

is AC0 reducible to

tt{CONF}(n,nc)

for any

0<c<\tfrac{1}{2}.

See also

Notes and References

  1. The existence of f is equivalent to the existence of a Green’s function.
  2. For the original paper, see . For accounts of the history, see ; ; . Also see (acknowledging that had already proven the Riemann mapping theorem).
  3. , citing
  4. Web site: What did Riemann Contribute to Mathematics? Geometry, Number Theory and Others .
  5. Lakhtakia . Akhlesh . Varadan . Vijay K. . Messier . Russell . Generalisations and randomisation of the plane Koch curve . Journal of Physics A: Mathematical and General . August 1987 . 20 . 11 . 3537–3541 . 10.1088/0305-4470/20/11/052.
  6. , section 8.3, p. 187
  7. See
  8. , elementary proof
  9. A Jordan region is the interior of a Jordan curve.
  10. 10.1137/060659119. Convergence of a Variant of the Zipper Algorithm for Conformal Mapping. SIAM Journal on Numerical Analysis. 45. 6. 2577. 2007. Marshall. Donald E.. Rohde. Steffen. 10.1.1.100.2423.
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