ahlfors l.v. - lectures on quasi con formal mappings (van nostrand, 1966)
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VAN NOSTRAND MATHEMATICAL STUDIES
Editors
Paul R. Halmos, The University of Michigan
Frederick W. Gehring, The University of Michigan
Paul R. Halmos - LECTURES ON BOOLEAN ALGEBRAS
Shmuel Agmon-LECTURES ON ELLIPTIC BOUNDARYVALUE PROBLEMS
Noel J. Hicks-NOTES ON DIFFERENTIAL GEOMETRY
Leopoldo Nachbin-TOPOLOGY AND ORDER
Sterling K. Berberian-NOTES ON SPECTRAL THEORY
Roger C. Lyndon-NOTES ON LOGIC
Robert R. Phelps-LECTURES ON CHOQUET'S THEOREM
L. E. Sigler-EXERCISES IN SET THEORY
George W. Mackey-LECTURES ON THE THEORY OFFUNCTIONS OF A COMPLEX VARIABLE
Lars V. Ahlfors-LECTURES ON QUASICONFORMAL MAPPINGS
Additional titles will be listed and announced as issued.
Lectures on
QUASICONFORMAL MAPPINGS
byLARS v: AHLFORS
Harvard University
Manuscript prepared with the assistance of
CLIFFORD J. EARLE Jr.Cornell University
D. VAN NOSTRAND COMPANY, INC.
PRINCETON, NEW JERSEY
TORONTO NEW YORK LONDON
D. VAN NOSTRAND COMPANY, INC.120 Alexander St., Princeton, New Jersey (Principal office)
24 West 40 Street, New York 18, New York
D. VAN NOSTRAND COMPANY, LTD.358, Kensington High Street, London, W.14, England
D. VAN NOSTRAND COMPANY (CANADA), LTD.25 Hollinger Road, Toronto 16, Canada
COPYRIGHT © 1966, BYD. VAN NOSTRAND COMPANY, INC.
Published simultaneously in Canada byD. VAN NOSTRAND COMPANY (Canada), LTD.
No reproduction in any form of this book, in whole or inpart (except for brief quotation in critical articles or reviews),may be made without written authorization from the publishers.
PRINTED IN THE UNITED STATES OF AMERICA
ACKNOWLEDGMENTS
The manuscript was prepared by Dr. Clifford Earle from
rough long hand notes of the author. He has contributed many
essential corrections, checked computations and supplied many
of the bridges that connect one fragment of thought with the next.
Without his devoted help the manuscript would never have attain
ed readable form.
In keeping with the informal character of this little vol
ume there is no index and the references are very spotty, to say
the least. The experts will know that the history of the subject
is one of slow evolution in which the authorship of ideas cannot
always be pinpointed.
The typing was excellently done by Mrs. Caroline W.
Browne in Princeton, arid financed by Air Force Grant AFOSR
393-63.
CONTENTS
CHAPTER I: DIFFERENTIABLE QUASICONFORMAL
MAPPINGS
Introduction . . . . . . . . . . . . . . . . . . . . . . . .. 1
A. The Problem and Definition of Grotzsch . . .. 2B. Solution of Grotzsch's Problem. . . . . . . . .. 6
C. Composed Mappings. . . . . . . . . . . . . . . .. 8
D. Extremal Length " 10
E. A Symmetry Principle . . . . . . . . . . . . . . .. 16
F. Dirichlet Integrals. . . . . . . . . . . . . . . . .. 17
CHAPTER II: THE GENERAL DEFINITION
A. The Geometric Approach . . . . . . . . . . . . .. 21
B. A 1-q-c. map is Conformal. . . . . . . . . . . .. 23
CHAPTER III: EXTREMAL GEOMETRIC PROPERTIES
A. Three Extremal Problems. . . . . . . . . . . . .. 35
B. Elliptic and Modular Functions. . . . . . . . .. 40
C. Mori's Theorem. . . . . . . . . . . . . . . . . . .. 47
D. Quadruplets....................... 53
CHAPTER IV: BOUNDARY CORRESPONDENCE
A. The M-condition. . . . . . . . . . . . . . . . . . .. 63
B. The Sufficiency of the M-condition. . . . . . .. 69
C. Quasi-isometry . . . . . . . . . . . . . . . . . . .. 73
D. Quasiconformal Reflection. . . . . . . . . . . .. 74
E. The Reverse Inequality. . . . . . . . . . . . . .. 81
Quasiconformal Mappings
CHAPTER V: THE MAPPING THEOREM
A. Two Integral Operators. . . . . . . . . . . . . .. 85
B. Solution of the Mapping Problem. . . . . . . .. 90
C. Dependence on Parameters . . . . . . . . . . .. 100
D. The Calder6n-Zygmund Inequality. 106
CHAPTER VI: TEICHMULLER SPACES
A. Preliminaries . . . . . . . . . . . . . . . . . . . .. 117
B. Beltrami Differentials 120
C. !!J. is Open 128
D. The Infinitesimal Approach . . . . . . . . . . .. 137
CHAPTER I
DIFFERENTIABLE QUASICONFORMAL MAPPINGS
Introduction
There are several reasons why quasiconformal mappings
have recently come to playa very active part in the theory of
analytic functions of a single complex variable.
1. The most superficial reason is that q.c. mappings are
a natural generalization of conformal mappings. If this were their
only claim they would soon have been forgotten.
2. It was noticed at an early stage that many theorems
on conformal mappings use only the quasiconformality. It is
therefore of some interest to determine when conformality is es
sential and when it is not.
3. Q.c. mappings are less rigid than conformal mappings,
and are therefore much easier to use as a tool. This was typi
cal of the utilitarian phase of the theory. For instance, it was
used to prove theorems about the conformal type of siinply con
nected Riemann surfaces (now mostly forgotten).1
2 Quasiconformal Mappings
4. Q.c. mappings play an important role in the study of
certain elliptic partial differential equations.
5. Extremal problems in q.c. mappings lead to analytic
functions connected with regions or Riemann surfaces. This
was a deep and unexpected discovery due to Teichmiiller.
6. The problem of moduli was solved with the help of
q.c. mappings. They also throw light on Fuchsian and Kleinian
groups.
7. Conformal mappings degenerate when generalized to
several variables, but q.c. mappings do not. This theory is
still in its infancy.
A. The Problem and Definition of Grotzsch
The notion of a quasiconformal mapping, but not the
name, was introduced by H. Grotzsch in 1928. If Q is a
square and R is a rectangle, not a square, there is no confor
mal mapping of Q on R which maps vertices on vertices.
Instead, Grotzsch asks for the most nearly conformal mapping
of this kind. This calls for a measure of approximate confor
mality, and in supplying such a measure Grotzsch took the
first step toward the creation of a theory of q.c. mappings.
All the work of Grotzsch was late to gain recognition,
and this particular idea was regarded as a curiosity and allowed
to remain dormant for several years. It reappears in 1935 in the
. work of Lavrentiev, but from the point of view of partial differ
ential equations. In 1936 I included a reference to the q.c. case
Differentiable Quasiconformal Mappings
in my theory of covering surfaces. From then on the notion
became generally known, and in 1937 Teichmiiller began to
prove important theorems by use of q.c. mappings, and later
theorems about q.c. mappings.
3
We return to the definition of Gr6tzsch. Let w = f(z)
(z = x + iy, w = u + iv) be a C1 homeomorphism from one re
gion to another. At a point Zo it induces a linear mapping of
the differentials
(1)du
dv
u dx + u dyx y
v dx + v dyx y
which we can also write in the complex form
(2)
with
dw = f dz + f- dz?: Z
(3) f-=';Ii(f+if)z x y
Geometrically, (1) represents an affine transformation
from the (dx, dy) to the (du, dv) plane. It maps circles a
bout the origin into similar ellipses. We wish to compute the
ratio between the axes as well as their direction.
In classical notation one writes
(4)
with
G
The eigenvalues are determined from
4 Quasiconformal Mappings
(5)
and are
\E- A P I
P G- A o
(6) \ \ _ E+G+[(E_G)2+4p2]'h1\1' 1\2 - 2
(7)
The ratio a: b of the axes is
(Al)lh = E+ G+[(E_G)2+4p 2]'h
A2 2(EG _ p 2) 'h
The complex notation is much more convenient. Let us
first note that
(8)
This gives
f-z
lh (u + v ) + j~ (v - u )x y x y
yiu - v ) + j~ (v + u )x y x y
(9) If 12 - \ f_1 2 = u v - u v = ]z z xy yx
which is the Jacobian. The Jacobian is positive for sense pre
serving and negative for sense reversing mappings. For the
moment we shall consider only the sense preserving case.
Then 1f'Z I < 1fz 1 .
It now follows immediately from (2) that
where both limits can be attained. We conclude that the ratio
of the major to the minor axis is
(11)If 1+ If-ID - z z > 1
f - 1f zl - If'Z 1
Differential Quasiconformal Mappings
This is called the dilatation at the poiJ1t z. It is often more
convenient to consider
5
(12)
related to D l by
(13)
The mapping is conformal at z if and only if Dl
The maximum is attained when the ratio
1, dl = O.
f- clZz
f zdz
is positive, the minimum when it is negative. We introduce now
the complex dilatation
(14)
with Illll
(15)
fIII = f z
zdl . The maximum corresponds to the direction
arg dz = a = 'h arg fl
the minimum to the direction a ±"/2. In the dw-plane the di
rection of the major axis is
(16)
where we have set
arg dw 'h arg v
(17)
The quantity v l may be called the second complex dilatation.
6
figure:
Quasiconformal Mappings
We will illustrate by the following self-explanatory
Observe that (3 - a = arg f z'
Definition 10 The mapping f is said to be quasiconfor
mal if D f is bounded. It is K -quasiconformal if D f S K.
The condition D f SKis equivalent to df S k =
(K -l)/(K + 1). A l-quasiconformal mapping is conformal.
Let it be said at once that the restriction to C1-mappings
is most unnatural. One of our immediate aims is to get rid of
this restriction. For the moment, however, we prefer to push
this difficulty aside.
B. Solution of Grotzsch's Problem
We pass to Grotzsch's problem and give it a precise
meaning by saying that f is most nearly conformal if sup D f
is as small as possible.
Let R, R' be two rectangles with sides a, b and a ~
b '. We may assume that a: b sa': b ' (otherwise, interchange
a and b). The mapping f is supposed to take a-sides into
a-sides and b-sides into b-sides.
Differentiable Quasiconiormal Mappings 7
60 6' Ia a'
The computation goesa a
,S f Idi(x+ iY)1 f (Iizl + liz I) dxa <
0 0
a b
a'b < f f (Iizl + li~) dx dy
o 0
lizl + lizllizl -lizl
a b
dx dy f fo 0
a b
a'b ' f f D f dx dy
o 0
or
(1) f f Dfdx dy
R
and in particular
~. ~ < sup Dfb' . b -
The minimum is attained for the affine mapping which
is given by
Hz)
8 Quasiconiormal Mappings
THEOREM 1. The affine mapping has the least maximal
and the least average dilatation.
The ratios m = alband m' = a'Ib' are called the
modhles of R and oR' (taken with an orientation). We have
proved that there exists a K-q.c. mapping of R on R' if and
only if
(2)
C. Composed Mappings
We shall determine the complex derivatives and complex
dilatations of a composed mapping g 0 i. There is the usual
trouble with the notation which is most easily resolved by in
troducing an intermediate variable ,= f(z).
The usual rules are applicable and we find
(1)(g 0 f) z = (g, 0 f) iz + (g, 0 f) 7z
(~ 0 f)- = (g 0 f) i- + (g- 0 f) 7-5 Z , Z , Z
When solved they give
(2)
where
1 -- [(g 0 f) i-1 z z
g- 0 i 1 [(g 0 f)- i, 1 z z
2 21 = Iizl - Iii I .
- (g 0 f) -7 ]z z
- (g 0 f) i-]z z
For g = i-I the formulas become
(3) (i-I) 0 i = 7-11 ,, z
Differ€f1tiable Quasicon£ormal Mappings 9
One derives, for instance,
(4) £-1/l -1 = -1/ f °f
(5)
and, on passing to the absolute values,
d -1 = df ° £-1f
In other words, inverse mappings have the same dilata
tion at corresponding points.
From (2) we obtain
(6) /l12 ° ££z /l12 0 f-/lf
lz 1 - 'ji.f/l12 0 f
If g is conformal, then /l12
(7)
o and we find
/l f .
If £ is conformal, /l f = 0 and
(8)f' 2/l o£ =(~
12 1£'[which can also be written as
(9)
In any case, the dilatation is invariant with respect to all con
formal transformations.
If we set g ° £ h we find from (6)
(10) £z /lh - /If
lz 1-'ji.f/lh
10
For the dilatation
Quasiconformal Mappings
(11)
and
d 0 fh 0 r 1
(12) log D -1 0 f == (Ilh' Il~ ,h 0 f
the non-euclidean distance (with respect to the metric ds
2 Idwl in Iwi < 1).l-lwl 2
We can obviously use sup (Ilh' Il~ as a distance be
tween the mappings f and h (the Teichmiiller distance). It
is a metric provided one identifies mappings that differ by a
conformal transformation.
The composite of a K 1-q.c. and a K2-q.c. mapping is
K1
K2
- q.c.
D. Extremal Length
Let r be a family of curves in the plane. Each y ( rshall be a countable union of open arcs, closed arcs or closed
curves, and every closed subarc shall be rectifiable. We shall
introduce a geometric quantity ,.\(r), called the extremal l€f1gth
of r, which is a sort of average minimal length. Its import
ance for our topic lies in the fact that it is invariant under con
formal mappings and quasi-invariant under q.c. mappings (the
latter means that it is multiplied by a bounded factor).
A function p, defined in the whole plane, will be called
allowable if it satisfies the following conditions:
Differentiable Quasiconformal Mappings 11
1. P? 0 and measurable.
2. A (p) = JJ p2 dx dy f, 0, 00 (the integral is over
the whole plane).
For such a p, set
Ly(p) f p IdzlY
if p is measurable on y*, Ly (p) 00 otherwise. We intro-
duce
L (p) = inf Ly (p)y~-r
and
Defini tion:
L (p)2sup -
p A(p)
for all allowable p.
We shall say that r 1 < r 2 if every y2 contains a y1
(the y 2 are fewer and longer).
Remark. Observe that r 1 C r 2 implies r 2 < r 1 !
THEOREM 2. If r 1 < r 2' then h(r 1) :S h(r2) .
Proof. If Y1 S; Y2 then
LY1
(p) < LY2
(p)
inf LY1
(p) < inf LY2
(p)
and it follows at once that h (r1) :S h(['2) .
* (as a function of arc-length)
12 Quasiconformal Mappings
Example 1. [' is the set of all arcs in a closed rec
tangle R which joins a pair of opposite sides.
For any pa
f p(x + iy) dx > L (p)
o
f f p dx dy > b L (p)
R
b2 L(p)2 < ab f f p2 dx dy < ab A(p)
R
L(p)2 < aA(p) - b
This proves A(n ~ a/b.
On the other hand, take p = 1 in R, P = 0 outside.
Then L(p) = a, A(p) = ab, hence A(n ;:: a/b. We have
provedab
Example 2. [' is the set of all arcs in an annulus
r1 < Iz I ~ r2 which join the boundary circles.
Computation:
ff p dr dO > 217 L (p )
Differentiable Quasiconformal Mappings 13
f f p2 rdrde
Equality for p = IIr .
Example 3. The module of an annulus.
Let G be a doubly connected region in the finite plane
with C1 the bounded, C2 the unbounded component of the com
plement. We say the closed curve y in G separates C1 and
C2 if Y has non-zero winding number about the points of CI'
Let r be the family of closed curves in G which separate C1
and C2
• The module M(G) A(r)-I, Consider, for example,
theannulus G = Ir1 ::; Izi :s r21,
217
L (p) :s f0
217L (p)
< fr0
p de
r f f pdrdeL (p) log (r 2 ) <1
L(p )22 r
217 log (;:) f f / rdrd{)log (/) <1
L(p )2 217<
A(p) log( r2 / r1
)
14 Quasiconformal Mappings
so Up) = 1 and A (p)
M(G) il7 log(r2 /r1)·
Once again p = 1/217r gives equality. Indeed, for any y € 1
we have
1 < !n(y, 0)1 = .L [ Jdz 1 <.L f~ - L (y)- 217 Z - 217 IZ I - p ,y y
= il7 log(r2 /r1)' We conclude that
Suppose that all y € 1 are contained in a region nand let ¢ be a K -quasiconformal mapping of n on n '. Let
l' be the image set of 1.
THEOREM 3. K-1 .\(1) ~ .\(1') ~ K .\(1).
Proof. For a given p (z) define p'V;) = 0 outside
n' and
in n'. Then
fp'ldt:[ ?: f p [dz[
y y
f f ,2 dt:d = f f 2 [¢zl + [¢zl dxdy <_ KA(p) .p S TJ n P 1¢z[ - 1¢zl
This proves .\' ?: 1\1.\, and the other inequality follows by
con!:!idering the inverse.
COROLLARY. '\(1) is a conformal invariant.
There are two important composition principles.
Differentiable Quasiconformal Mappings 15
*I. 11 + 12 = IY1 +Y21 Y1 f11 , Y2 f 12 1II. 1 1 u r 2
THEOREM 4.
a) A(11
+ 12
) > A(11
) + A(12
) ;
b) A(11
U 12
)-1 2: A(11)-1 + A(1
2)-1
if 11
, 12
lie in disjoint measurable sets.
Proof of a). We may assume that 0 < A(11
), A(12
) < 00
for otherwise the inequality is trivial.
We may normalize so that
L 1(p 1) A (p 1 )
L 2 (P2) = A (P2 )
Choose p
L(p) > L 1 (P1) + L 2(P2)
A (p) < A (P1 ) + A (P2)
L(p)2A = sup ATPT 2: A (P1 ) + A (P2 )
It follows that A 2: A 1 + A2 .
+
Proof of b). If A = A(11 U 12
) = 0 there is no
thing to prove. Consider an admissible p with L (p) > 0 and
set P1 = P on E1 , P2 = P on E2 , 0 outside (where E1
and E2 are complementary measurable sets with r 1 S E 1 '
* Y1 + Y2 means "Y 1 followed by Y2'"
16 Quasiconformal Mappings
['2 S E/ Then LI (PI) 2: L(p),
A(p) = A(PI) + A(P2)' Thus
A (p) A (PI)
L(p)2 > LI(PI)2 +
and hence
q.e.d.
E. A Symmetry Principle.
For any y let y be its reflection in the real axis,
and let y+ be obtained by reflecting the part below the real
axis and retaining the part above it (y U y = y+ U (y+)-).
The notations rand ['+ are self-explanatory.
THEOREM 5. If [' = r then ,\([') = Yi ,\(['+) .
Proof.
1. For a given P set p(z)
Then
max (p (z), P(z)) .
and
This makes
L+(p)2 ~ 2 L(p)2 S 2,\([')A(p) A(p)
and hence ,\ ([' +) S 2,\ ([' ) .
2. For given P set
Differentiable Quasiconformal Mappings 17
in upper halfplane
in lower halfplane
Then
L +(p+) = L + ( +)_ (p) L _ (p)y y + y y+y
L (p) + L_(p) > 2L(p) .y y
On the other hand
2A (p)
and hence
L(p)2L +(p+)2
< Y2y
< Y2 ,\,(r+)A (p) A (p+)
,\,(r) < Y2 ,\,(r+) .
q.e.d.
F. Dirichlet Integrals
z
Let ¢ be a K-q.c. mapping from n to n '. The
Dirichlet integral of a C 1 function u «() is
For the composite u 0 ¢ we have
18 Quasiconformal Mappings
(U 0 c/J)z (ul;, 0 c/J)c/Jz + (ul;, ° c/J) ¢z
I(u ° c/J) zl $ <iul;,l ° c/J)<ic/Jzl + 1c/J:zP
and thus
(1) D(u0c/J) S KD(u) .
Dirichlet integrals are quasi~invariant.
There is another formulation of this. We may consider
merely corresponding Jordan regions with boundaries y;' y , .
Let v on y' and v 0 c/J on y be corresponding boundary
values. There is a minimum Dirichlet integral Do(v) for func
tions with boundary values v, attained for the harmonic func
tion with these boundary values. Clearly,
(2)
One may go a step further and assume that v is given
only on part of the boundary. For instance, if v == 0 and
v == 1 on disjoint boundary arcs we get a new proof of the quasi
invariance of the module.
In order to define the Dirichlet integral it is not neces
sary to assume that u is of class C 1. Suppose that u (z) is
continuous with compact support. Thus we can form the Fourier
transform
Differentiable Quasiconformal Mappings
Ii(~, Tf) = 2~ !! ei(X~+Y1])u(x, y)dxdy
nand we know that
(UX)A = - i~1i
(I).) A = - iTfIi
It follows by the Plancherel formula that
D(u) =! !(e+Tf2)[liI2 d~dTf
and this can be taken as definition of D (u) .
19
CHAPTER II
THE GENERAL DEFINITION
A. The Geometric Approach
All mappings ¢ will be topological and sense preserv
ing f rom a region 11 to a region 11'.
Definition A. ¢ is K-q.c. if the modules of quadrilat
erals are K -quasi-invariant.
A quadrilateral is a Jordan region Q, Q c 11, together
with a pair of disjoint closed arcs on the boundary (the b-arcs).
Its module m (Q) = a/b is determined by conformal mapping
on a rectangle
a
The conjugate Q* is the same Q with the complemen-
tary arcs (the a-arcs). Clearly, m(Q*) m(Q)-l.
21
22 Quasiconformal Mappings
The condition is m (Q ') ~ Km (Q ). It clearly implies
a double inequality,
l\lm(Q) ::; m(Q') S Km(Q) .
Trivial properties:
1. If ¢ is of class C1, the definition agrees with
the earlier.
2. ¢ and ¢-l are simultaneously K-q.c.
3. The class of K-q.c. mappings is invariant under
conformal mappings.
4. The composite of a K l-q.c. and a K2-q.c. map
ping is K1
K2
-q.c.
K-quasiconformality is a local property:
THEOREM 1. If ¢ is K-q.c. in a neighborhood of
every point, then it is K-q.c. in n.
Proof.
Q -.---.. -Q
We subdivide Q into vertical strips Qj and then each
image, Q.' into horizontal strips Q .. ' with respect to the ree-l IJ
tangular structure of Q.'. Set m .. = m(Q ..) etc. Then1 IJ IJ
m = ~ mj
.1. > ~J' _1_m
i- m
ij
~. _1_,J m ..
IJ
The General Definition 23
If the subdivision is sufficiently fine we shall have mI•J• < Km .~.
- IJ
This gives m $ Km'.
LEMMA. m = m1
+ m2
only if the dividing line is
x = m1.
Proof. Let the conformal mapping functions be f1
, f2.
Set p = If1 'I in Q1' P = If2 'I in Q2' P = 0 everywhere.
else. Then, integrating over Q
f f (p 2 - 1) dx dy 0
f f (p - 1) dx dy ::: 0
But fJ(p-1)2dxdy = ![[(p2-1)-2(p-1)]dxdy $ O. Hence
p = 1 a.e. and this is possible only if f 1 = f 2 = z.
THEOREM 2. A l-q.c. map is conformal.
Proof. We must have equality everywhere in the proof
of Theorem 1. This shows that the rectangular map is the iden-
tity.
B. The Analytic Definition
We shall say that a function u(x, y) is ACL (absolutely
24 Quasiconformal Mappings
continuous on lines) in the region n if for every closed rec
tangle R c n with sides parallel to the x and y-axes,
u(x, y) is absolutely continuous on a.e. horizontal and a.e.
vertical line in R. Such a function has of course, partial de-
rivatives ux' u y a.e. in n.The definition carries over to complex valued functions.
Definition B. A topological mapping ¢ of n ia K-q.c.
if
1) ¢ is ACL in n;
2) I¢zl ~ kJ¢zl a.e. (k = K-l)K+l .
We ;;hall prove that this definition is equivalent to the
geometric definition. It follows from B that ¢ is sense pre
serving.
We shall say that ¢ is differentiable at Zo (in the
sense of Darboux) if
¢(z)-¢(zo) = ¢z(zo)(z-zo)+¢z(zo)(z -zo)+ o(\z-zoP
The first lemma we shall prove is an amazing result due to Geh
ring and Lehto.
LEMMA 1. If ¢ is topological* and has partial deriva
tives a.e., then it is differentiable a.e.
By Egoroff's theorem the limits
¢(z+M-¢(z)h
P(z + ik) - P(z )k
'" The result holds as soon as ¢ is an open mapping.
The General Definition 25
are taken on uniformly except on a set n - E of arbitrarily
small measure. It will be sufficient to prove that ¢ is differ
entiable a.e. on E.
Remark. Usually Egoroff's theorem is formulated for
sequences. We obtain (1) if we apply it to
\¢(z+h)-¢(z) IsiP -¢)z) .
0< Ih < lin hThe set E is measurable, and therefore it intersects
a.e. horizontal line in a measurable set. On such a line a.e.
point in E is a point of linear density 1. The same holds for
vertical lines. Therefore a.e. xo' Yo ( E is a point of linear
density 1 for the intersections of E with x = Xo and y = Yo'
It will be sufficient to prove that ¢ is differentiable at such a
point Zo = Xo + iyo' and for simplicity we assume Zo = O.
Because of the uniformity ¢x and ¢y are continuous
on E. Given ( > 0 we can therefore find a [) > 0 such that
Ih I < [), [k I < [) and
(2)
as soon as
z ( E.
I¢)z) - ¢x(O)1 < ([¢/z) - ¢y(O)1 < (
I¢ (z + h) - ¢ (z )h
I¢ (z + ik) - ¢ (z)k
Ix I < [), [y I < [),
We follow the argument which is used when one proves
that a function with continuous partial derivatives is differen
tiable. It is based on the identity
26 Quasiconformal Mappings
¢(x+ iy) - ¢(Q) - x¢x(O) - Y¢y(O) =
[¢(x+ iy) - ¢(x) - y ¢y (x)] + [¢(x) - ¢(Q) - x ¢x (0)]
+ [y (¢y (x) - ¢y (0))]
If x f E we are able to derive by (2) that
The same reasoning can be repeated if y f E. Thus we have
proved what we wish to prove if either x f E or y f E.
We now want to make use of the fact that 0 is a point
of linear density 1. Let ml(x) be the measure of that part of
E which lies on the segment (-x, xL Then (ml(x))/2Ixl -> I
for x -> 0, and we can choose 8 so small that
m (x) > ~Ixl1 I + f
for Ix [ < 8. For such an x the interval (-IX,x) cannot be+f
free from points of E, for if it were we should have
Ix [ 2Hml(x) ::; Ixl +I+f = IH Ix[
The same reasoning applies on the y-axis. If [z I < _8_ weI +f
conclude that there are points xl' x 2' Yl' Y2 f E with
We may then conclude that (3) holds on the perimeter of the
* We are assuming for convenience that z lies in the first
quadrant.
The General Definition 27
rectangle (~, ~) x (Y1 ' Y2 ).
To complete the reasoning we use the fact that ¢ sat
isfies the maximum principle. There is a point z* = x* + iy*
on the perimeter such thai
I¢(x+ iy) - ¢(O) - x¢ (0) - y¢ (0)1x y
$ 1¢(x*+iy*)-¢(O)-x¢x(O)-Y¢y(O)1
< 3f Iz* 1+ Ix-x* 1 I¢ (0)1 + Iy-y* 1 1¢ (0)1x y
< 3dlH)lzl + fl¢ (O)llzl + fl¢ (O)llzlx y
This is an estimate of the desired form, and the lemma is proved.
We can say a little more. If E is a Borel set in n we
define A (E) as the area of its image. This defines a locally
finite additive measure, and according to a theorem of Lebesgue
such a measure has a symmetric derivative a.e., that is,
f(z) lim .AlQ)m(Q)
when Q is a square of center z whose side tends to zero.
Moreover,
f f(z)dxdy $ A(E)
E
(we cannot yet guarantee equality). But if ¢ is differentiable
at z it is immediate that f (z) is the Jacobian, and we have
proved that the Jacobian is locally integrable.
But f = I¢z 12- I¢Z' 12 and if 2) is fulfilled we ob-
tain
28 Quasiconformal Mappings
We conclude that the partial derivatives are locally square in
tegrable.
Moreover, if h is a test function (h ( C l with com
pact support) we find at once by integration over horizontal or
vertical lines and subsequent use of Fubini's theorem
(4)
f f ¢x h dxdy
f f ¢yh dxdy
-f f ¢hx dxdy
-f f ¢hydxdy
In other words, ¢x and ¢y are distributional derivatives of ¢.
More important still is the converse:
LEMMA 2. If ¢ has locally integrable distributional
derivatives, then ¢ is ACL.
Proof. We assume the existence of integrable functions
¢l' ¢2 such that
f f¢l h dxdy -f f ¢ hx dxdy
(5)
f f ¢2 h dxdy - f f¢hydXdY
for all test functions.
Consider a rectangle RTJ = {O ::; x ::; a, 0 ::; y ::; TJI and
choose h = h (x) k (y) with support in RTJ' In
f f ¢l h (x )k (y) dx dy = - f f ¢ h' (x)k (y) dx dy
RTJ RTJ
we can first let k tend boundedly to 1. This gives
The General Definition
f f ¢I h(x)dxdy
RTJand hence a
f ¢ 1(x, TJ)h (x )dx
o
f f ¢h'(x)dxdy
RTJ
a-f ¢(x, TJ)h '(x )dx
o
for almost all TJ. Now let h "" hn
test-functions such that °< h <- n-B- .! ). We obtain
n
run through a sequence of
1 and h "" 1 on (1.,n n
¢(a, TJ) - ¢(o, TJ)(6)
af ¢I(x,TJ)dx
ofor almost all TJ. The exceptional set does depend on a, but
we may conclude that (6) holds a.e. for all rational a. By con
tinuity it is then true for all a, and we have proved that
¢(x, TJ) is absolutely continuous for almost all TJ. Moreover,
we have ¢x "" ¢1., ¢y "" ~ a.e.
In other words, B is equivalent to B',
integrable distributional derivatives which satisfy
q.e.d.
¢ has locally
It is now easy to show that B is invariant under con
formal mapping. We prove, a little more generally:
LEMMA 3. If w is a C2 topological mapping and if
¢ has locally integrable distributional derivatives, so does
¢ 0 w , and they are given by
30 Quasiconformal Mappings
(6)
Proof. We note first that
(~x ~Y) and (X~ x1J)
1J x 1J y y~ Y1J
are reciprocal matrices (at correspon ding points). This makes
( x~ x1J) = ! ( 1Jy -~y)
Y~ Y1J J -1J x ~x
with J = ~ 1J - ~ 1J ' the Jacobian of w.x y y x
For any test function how we can put
II [(¢~ 0 w)~x + (¢1J 0 w)1J)(h 0 w)dxdy
I I [(¢~ 0 w) 7+ (¢1J 0 w) 1JjHh 0 w) Jdxdy
I I (¢~Y1J - ¢1JY~) h d~ d1J
I I ¢(- (hy1J)~ + (hY~) 1J) d~ d1J
I I ¢(-h~Y1J + h1JY~) d~d1J
I I (¢ ow)(-(h c 0 w) ~x - (h 0 w) 1J x ) J dxdys J 1J J
= I I (¢ 0 w)(-(h~ 0 w) ~x - (h1J 0 w) 1J) dx dy
= - I I(¢ ow)(h oW)x dxdy
q.e.d.
The last lemma enables us to prove:
B => A .
The General Definition 31
Indeed, in view of the lemma we need only prove that a
rectangle with modulus m is mapped on a quadrilateral with
m' ~ Km, and this proof is exactly the same as in the differen
tiable case.
We shall now concentrate on proving the converse:
A=>B.
First we prove: if ¢ is q.c. in the geometric sense,
then it is ACL.
Let A (77) be the image area under the mapping ¢ of
the rectangle a ~ x ~ f3, Yo ~ y ~ 77· Because A (77) is in
creasing the derivative A '(77) exists a.e., and we shall assume
that A '(0) exists.
In the figure, let Qi be rectangles with height 77 and
base b.. Let b.' be the length of the image of b.. We wishI I I
to show, first of all: If 77 is sufficiently small, then the length
of any curve in Qi' which joins the "vertical" sides is nearly
b. '.I
To do so we first determine a polygon so thatn
b ' - !..i 2
32 Quasiconformal Mappings
Next, take TJ so small that the variation of ¢ on vertical seg
ments is < €I4n. Draw the lines through 'k that correspond
to verticals. Any transversal must intersect all these lines.
This gives a length
n
> b.' - (1
1
On using the euclidean metric we have now, if ( <min ~ bi ',
b. ,2> 1
4A.1
b. ,21 <
4A i
b.K-l.
TJ
b. ,2::; ~_1_
b.1
~b. < 4K A(TJ) • (~b.)1 - TJ 1 *
But A (TJ}/TJ'" A '(0) < 00. This shows that ~ b/ ... 0 with
~ bi' and hence ¢ is absolutely continuous.
q.e.d.
Finally, if ¢ is K-q.c. in the geometric sense, it is
easy to prove that the inequality
(k = K-l)K+l
holds at all points where ¢ is differentiable. This proves
that A => B.
We have now proved the equivalence of the geometric
and analytic definitions, and we have two choices for the analy
tic definition.
* The proof needs an obvious modification if bi
' = 00 ••
The General Definition 33
COROLLARY 1. If the topological mapping ep satis
fies epz = 0 a.e., and if ep is either ACL or has integrable
distributional derivatives, then ep is conformal.
Observe the close relationship with Weyl's lemma.
COROLLARY 2. If ep is q.c. and epz = 0 a.e., then
¢ is conformal.
Finally, we shall prove:
THEOREM 3. Under a q.c. mapping the image area is
an absolutely continuous set function. This means that null sets
are mapped on null sets, and that the image area can always be
represented by
A (E) I IJ dxdy .
E
Proof. ep = u + iv can be approximated by C2 func
tions un + ivn in the sense thet un ~ u, vn ~ v and
Consider rectangles R such that u and v are abso
lutely continuous on all sides.
IfR
[(u ) (v) - (u·) (v ) 1dx dymx ny my nx f u dv
m n
JR
34 Quasiconformal Mappings
As m, n .... 00 the double integral tends to ffRJ dx dy. For
m .... 00 the line integral tends to
ju dVn j vn du
aR aRand for n .... 00 this tends to
j v du judv
aR aR(all because u and v, and therefore UV, are absolutely con
tinuous on aR).
We have proved
j j ] dxdy
R
for these R. It is not precisely trivial, but in any case fairly
easy to prove that the line integral does represent the image
area,'and that proves the theorem.
COROLLARY. ¢z f, 0 a.e.
Otherwise there would exist a set of positive measure
which is mapped on a nullset, and consideration of the inverse
mapping would lead to a contradiction.
Remark. It can now be concluded that Dirichlet integrals
are K-quasi-invariant under arbitrary K-q.c. mappings. It is
possible to prove the same for extremal lengths.
CHAPTER III
EXTREMAL GEOMETRIC PROPERTIES
A. Three Extremal Problems
Let G be a doubly connected region in the finite plane,
C1 the bounded and Cz the unbounded component of its comple
ment. We want to find the largest value of the module M (G)
under one of the following conditions:
I. (Grotzsch) C1 is the unit disk (121 $ 1) and
Cz contains the point R > 1.
II. (Teichmiiller) C1 contains 0 and -1; Cz con
tains a point at distance P from the origin.
III. (Mori) diam(C1 n Ilzl $ In ? A; Cz con
tains the origin.
We claim that the maximum of M (G) is obtained in the
following symmetric cases.
(). ~I.
35
R Go
36 Quasiconformal Mappings
II. -1 0 P
III. A~.
Fig.!.
Case l. Let r be the family of closed curves that se
parate C1
and C2
• We know that A(r) = M(G)-I.
Compare r with the family r of closed curves that lie
in the complement of C1
U IR I, have zero winding number
about R and nonzero winding number about the origin. Evident
ly, r c r, and hence A(r) ~ A(f). But r is a symmetric
family, and hence A(r) = 71 A(r +) by our symmetry principle.
SimilarlY, if r 0 is the family r in the alleged extremal case*,
then A(r0) = 71A(r0+) .
We show that r + = r 0+. Each curve y in r has
points PI' P2 on (-00, -1) and 0, R) respectively. Say they
divide y into arcs Yl and h so that y = Yl + Y2' Then- + - + - + (- , - + -)+ H - + - + - b 1y = y 1 + Y2 = Y1 + Y2 . ere y 1 + Y2 e ongs
to r 0' and we conclude that Y+ fro+. Thus t + C r 0+'
and the opposite inclusion is trivial.
We have proved A(r) ~ A(r) = A(r0)' hence M (G) <M(Go) .
q.e.d.
* We enlarge r 0 slightly by allowing it to contain curves which
contain segments on the edges of the cut (R, (0). This is, of
course, harmless.
Extremal Geometric Properties
Case II. Let z = f(() map \([ < 1 conformallyonto C1 U G
with [(0) = O. Koebe's one-quarter theorem gives jf'(0)I ~ 4P
with equality for G = G1 .Suppose f(a) -1. The distortion
theorem gives
1 = If(a) [ < [at 1['(0)[- (1- [a\)2
< 4Pla[- (1- [a\)2
and equality is again attained when G = Gl' In other words
Ial ~ [a 11 where a1 belongs to the symmetric case. The
module M(G) equals the module between the image of C1 and
the unit circle.
Finally, by inversion and application of Case I, if [al is
given the module is largest for a line segment, and it increases
when la[ decreases. Hence G1 is extremal.
Case 1I/. Open up the plane by (= VZ. We get a figure
which is symmetric with respect to the origin
with two component images of C1 and two of C2 . The compo
sition laws tells us that M(G) ~ YiM(6') where 6' is the re
gion between C1- and C/. It is clear that equality holds in
the symmetric situation.
38 Quasiconformal Mappings
By assumption, C 1 contains points zl' z2' with
Iz 11 ::; 1, Iz2I ::; 1, Iz 1- z2I ~ '\. Let '1' '2 f C1+,
-'I' -, 2 f C1- be the corresponding points in the '-plane.
The linear transformation
w = '+ '1 '1 + '2,- '1 '1 - '2
carries (-'1' -'2) into (0, 1) and '1 .... 00, '2 .... W o where
w0= _ ( '2 + '1 )2'2- '1On setting u = ('2 + '1)/('2 -'1) one has
2('/ + '/)
'/ - '/
we get
lui __1_< £ J4-,\2
lui ,\
lui < 2 + J4-->.!.,\
Iwol < (2+~YOne verifies that equality holds for the symmetric case, and by
use of Case II it follows again that M (G) is a maximum for the
case in Figure 1 .
Extremal Geometric Properties 39
In order to confonn with the notations in Kiinzi's book *the extremal modules will be denoted
I. ~ log <1> (R)277
II.
III.
~ log 'P(p)277
~ log xC\.) .277
There are simple relations between these functions. Ob
viously, reflection of Go gives a twice as wide ring of type GI'
and one finds
(1)
Another relation is obtained by mapping the outside of the unit
circle on the outside of the segment (-1, 0). This gives
(2)
and together with (1), we find the identity
(3)
The previous computation in Case III gives, for instance,
(4)
*
x C\.) = <1> (V4+2X" + y4 - 2,\ ),\
Hans P. Kiinzi, Quasikonforme Abbildungen, in Ergebnisse der
Mathematik,Springer Verlag, Berlin, 1960.
40 Quasiconformal Mappings
(1)
B. Elliptic and Modular Functions
The elliptic integralz
w = fo
dz
maps the upper half of the normal region G1 on a rectangle
III
:8II
....._......J'O
with sidesp
a f dzy(z+ Uz(p-z)
0
(2) 00
b f dzy(z+ l)z(z-P)
P
Evidently,
(3)1
2" log 'I' (p) = 2~
This is an explicit expression, but it is not very convenient for
a study of asymptotic behavior. Anyway, we want to study the
connection with elliptic functions in much greater detail.
We recall that Weierstrass' ~-function is defined by
(4)
Extremal Geometric Properties
and satisfies the differential equation
where
41
(6)
(7)
It follows that the ek are distinct.
We set T = w 2 / wI and consider only the halfplane
1m T > O. In that halfplane
e3 -e1p(T) =
e2 -e1
is an analytic function .;, 0, 1. It is this function we wish to
study.
aT + bCT + d
P is invariant under modular transformations
with (: ~) == (~ ~) mod 2, for g;J does not
change and the wk
/2 change by full periods. The transforma
tion T' = T + 1 replaces p by 1/p, and T' = -1/T changes
pinto 1 - p. In other words
(8)peT + 1)
p( - ~ )
and these relations determine the behavior of p(T) under the
whole modular group.
For purely imaginary T the mapping by g;J is as indi-
cated:
42 Quasiconformal Mappings
~. _ $7////lII/l/1o C4)1
-2-
The image is similar to the normal region with P p/l-p,
and we find
(9) rCp) = .1. log qt (l-.-L)T7 -p
o < p < 1. It is evident from this formula that p varies mono
tonically from 0 to 1 when T goes from 0 to 00 along the
imaginary axis.
It can be seen by direct computation that p" 1 as soon
as 1m T .... 00, uniformly in the whole halfplane. If we make
use of the relations (8) it is readily seen that the correspondence
between T and p is as follows:
-:1 o :1
Extremal Geometric Properties
....... _ -t... ",- - I -
I
Fig. 2.
43
We shall let T(p) denote the branch of the inverse func
tion determined by this choice of the fundamental region. Ob-
serve the symmetries.
(10)
Relation (8) gives
(11)
Also
They yield
T(-I) ;: l+i2
T( 1-2i y'3 )+ 1 + i
T (t) ;: T (p) ±. 1
T(l-p) ;: - -+.npJ
rCp> ;: -T (p)
1 +iy'32
It is clear that eTTiT is analytic at p ;: 1 with a simple
zero. Therefore one can write
(12) p _ 1 - a e iTTT, (a > 0)
but the determination of the constant requires better knowledge
of the function ..
It will be of some importance to know how Ip -11 varies
when 1m T is kept fixed_ In other words, if we write T;: S + it
44 Quasiconformal Mappings
we would like to know the sign of a/as log Ip -11. This har
monic function is evidently zero on the lines s = 0, s = 1,
and a look at Figure 2 shows that it is positive on the half
circle from 0 to 1. This proves
is log Ip-ll > 0
in the right half of the fundamental region, provided that the max
imum principle is applicable. However, equation (12) shows
that a/as log Ip-ll .... 0 as r .... 00. Use of the relation (8)
shows that the same is true in the other corners, and therefore
the use of the maximum principle offers no difficulty.
We conclude that Ip -11 is smallest on the imaginary
axis and biggest on Re r = ± 1 .
The estimates of the function p (r) obtained by geomet
ric comparison are not strong enough, but classical explicit de
velopments are known. For the sake of completeness we shall
derive the most important formula.
The function~(z) - ~(u)
~(z) - ~(v)
has zeros at z = ± u + mWl + nW2
and poles at z = ± v + mwl + nW 2 •
A function with the same periods, zeros and poles is
00
IIn=-oo
F (z). (nw 2 ±v-z)
2m W1 _ e 1
(the product has one factor for each n and each ±). It is easy
to see that the product converges at both ends ..
Extremal Geometric Properties 45
2 17i .!!.±..zW
1 - e 1
217i y.±...zW
1 - e 1
F (z)
It will be convenient to use the notation q = el7iT
el7iW2 / WI. We separate the factor n = 0 and combine the
factors + n. This gives- 217i !!.::...Z
W1 _ e 1
217i Y.-zW
1 - e 1
00
ITn=l
+u+ Z217i - W-
1- q2n e 1
with one factor for each combination of signs.
It is clear that
so(z) - SO(u)
SO(z) - SO(v)
F(z)
F(O)
In order to compute
1 - P =e2 - e3e
3-e
1
we have to choose z = w2 /2, U = (WI + ( 2)/2. v = w/2
which gives
= q,
217iu
~e = - q,
Substitution gives
F(z) = 21+ q-l
00
IT1
(1 + ~n+2)(I+ ~n)2(I+q2n-2)
(1+ q2n+l)2 (l+q2n-l)2
46 Quasiconformal Mappings
(14)
and
F CO) =~ 1+ q-l (1 + q2n+l )2(1 + q2n-l)2II
2 2 (1 + q2n)4
00
(1 + q2n-l y1 n4q 1 1 + q2n
Finally00
( 1 + q'" )'(13) 1 - p = 16q II1 1 + q2n-l
A similar computation gives
00 (1 _q2n-l )8p = II
1 1 + q2n-l
«14) is obtained by the relation rCl/p) = rCp) ± 1 .)
We return to formula (9). It gives
(15) log 'PCP) = 17 1m T(I':P) = 17 1m T (1 + ~)
or
'PCP) qCp)-l for PpP+ 1
and, by (13),8
I-p 1 16q 00 (l+q2n )P+l ~ 1+ q2n-l
or
'PCP) 00
( 1 + q'" )'(16) 16 IIP+l 1 1 + q2n-l
(17)
Extremal Geometric Properties
This gives the basic inequality
W(p) S 16(P+ 1)
47
(18)
and
(19)
Because lI> (R )
lI>(R)-R-
W(R 2 - dh it follows further that
4 IT (_I_+_q,:..2_n__ ) 4
1 1+ q2n-l
lI>(R) < 4R
(20)
For X (A) we obtain
AX (A ) < 4( y'4+2X + y'4=2X)
AX(A) < 16 .
C. Mori's Theorem
Let , = ¢Az) be a K-q.c. mapping of Iz I < 1 onto
I( I < 1, normalized by ¢(O) = O.
Mori's theorem:
(1)
and 16 cannot be replaced by a smaller constant.
Remark. The theorem implies that ¢ satisfies a Holder
condition, and this was known earlier. It follows that ¢ has
a continuous extension to the closed disk Iz lSI. If we ap
ply the theorem to the inverse mapping we see that the exten
sion is a homeomorphism.
COROLLARY. Every q.c. mapping of a disk on a disk
can be extended to a homeomorphism of the closed disks.
48 Quasiconformal Mappings
In proving Mori's theorem we shall first assume that ephas an extension to the closed disk .. It will be quite easy to re
move this restriction.
For the proof we shall also need this lemma:
LEMMA. If ep (z) with ep (0) = 0 is K-q.c. and
homeomorphic on the boundary, then the extension obtained by
setting ep (1/ z) = 1/(¢;(z)) is a K-q.c. mapping.
Proof. It is clear that the extended mapping is K-q.c.
inside and outside the unit circle. It follows very easily that it
is ACL even on rectangles that overlap the unit circle. The K
q.c. follows.
Proof of (1). There is nothing to prove if IZ1 - z2 1 ~ 1,
say. We assume that IZ1 - z2 1 < 1.
Construct an annulus A whose inner circle has the seg
ment zl' z 2 for diameter and whose outer circle is a concentric
circle of radius Y2.
Case (i). A lies inside the unit circle..
Consider the mapping
w =
Extremal Geometric Properties
We obtain
1 log 1 < Klog ~ (111;-2~11;I;121)217 Iz2- z11 217
< K log 8217 1(2-1;1 1
49
This gives
Case (ii). A does not contain the origin. Now the image
of the inner continuum intersects 1(1 < 1 in a set with diam
eter ~ 1I; 1 - I; 21 , and the outer continuum contains the origin.
Hence
and we obtain
To rid ourselves of the hypothesis that ¢ is continuous
on Iz 1 = 1, consider the image region ~r of Iz 1 < r
--0
50 Quasiconformal Mappings
and map it conformally on !w! < 1 by t/Jr with t/J/O) = 0,
t/J/(0) > O. The function t/J/¢ (rz)) is K-q.c. and continuous
on Iz I = 1. Therefore,
1t/J/¢(rz2))-t/J/¢(rz1))1 < 16!z2- z1[1I K
Now we can let r .... 1. It is a simple matter to show that t/Jr
tends to the identity, and we obtain
< 161 z -z !l1K- 2 1
But this implies continuity on the boundary, and hence the strict
inequality is valid.
To show that 16 is the best constant we consider two
Mori regions (Case III) G (,\) and G ('\2). If they are mapped
on annuli we can map them on each other by stretching the radii
in constant ratio
Klog X ('\1)
log X ('\2)> 1
Because the unit circle is a line of symmetry it is clear that we
obtain a K-q.c. mapping which makes ,\ 1 correspond to '\2 .
We have now
/-- ....../ f'
I I \{ \
·,A1 1 I\ I I\ I /
" 1/....... __ /
/--]'/ "-I \I \I . A1 J\ /
\ /........... - '"
Extremal Geometric Properties
for small enough A2
. Thus
A2
? (A )1/K 16-f1 16 1/ K
51
and for large K the constant factor is arbitrarily close to 16.
There are of course strong consequences with regard to
normal and compact families of quasiconformal mappings.
THEOREM 1. The K~quasiconformalmappings of the
unit disk onto itself. normalized by ¢ (0) = 0, form a sequen
tiaily compact family with respect to uniform convergence.
Proof. By Mori's theorem the functions ¢ are equicon
tinuous. It follows by Ascoli's theorem that every infinite se
quence contains a uniformly convergent subsequence: ¢n -> ¢.
Because Mori's theorem can be applied to the inverse ¢n-1 it
follows that ¢ is schlicht. It is rather obviously K-q.c.
For conformal mappings it is customary to normalize by
¢(O) = 0, I¢ '(0)[ = 1. For q.c. mappings this does not make
sense: we must normalize at two points.
For an arbitrary region n we normalize by ¢ (a 1)
b1
, ¢(a2
) = b2
where of course a1
f. a2
, b1
f. b2
·
THEOREM 2. With this fixed normalization the K-q.c.
mappings in n satisfy a uniform Holder condition
11K1¢(z1)-¢(z2)1 ~ M[z1 -z21
on every compact set. The family of such mappings is sequenti
ally compact with respect to uniform convergence on compact sets.
52 Quasiconfotmal Mappings
Ptoof. For any z l' z2 f n there exists a simply con
nected region G en, not the whole plane, which contains z l'
z2' al' a2· If A is a compact set in n, then A x A can be
covered by a finite number of G x G. Hence it is sufficient to
prove the existence of M for G.
We map G and G' = ep (G) conformally on unit disks
as follows:
The compact set may be represented by Is I :::; to < 1. Mori's
theorem gives
(1 - Is I) :::; 16(1 -Iw 1)1/K
Hence Is I :::; to implies Iw I :::; Po (depending only on to)
and a similar reasoning gives f3 ~ fJ o .We know that
If we can show that (w) and s (z) satisfy uniform Lipschitz
conditions
we are through. These are familiar consequences of the distor
tion theorem.
Extremal Geometric Properties
For instance
53
Iz '(sl)1 > Colz'(O)1
laz - all < C'lz'(O)1
and hence
C'
The other inequality is easier.
It now follows that every sequence of mappings has a
convergent subsequence. To prove that the limit mapping is
schlicht we must reverse the inequalities. Although G' is now
a variable region this is possible because {3 < {3o' a number
that depends only on a (that is, on G, ai' az).
D. Quadruplets
Let (ai' az' a3, a4 ) and (b l , bz' b3 , b4 ) be two
ordered quadruples of distinct complex numbers. There exists
a conformal mapping of the whole extended plane which takes
ak
into bk
if and only if the cross-ratios are equal. If they
are not equal it is natural to consider the following problem.
Problem 1. For what K does there exist a K-q.c.
mapping which transforms one quadruple into another.
54 Quasiconformal Mappings
It was Teichmiiller who first pointed out that the problem
becomes more natural if one puts topological side conditions on
the mapping.
To illustrate, consider the following figure:
a4 aa
l.. 1..
a1 atThere exists a topological mapping of the extended
plane which takes the line 1. into 1.' and £ into £'. Obvi
ously, as a self-mapping of the sphere punctured at aI' a2 , a3,
a4
this mapping is quite different from the identity mapping (it
is not homotopic). Therefore, although the cross-ratios are
equal it makes sense to ask whether there exists a K-q.c.
mapping of this topological figure.
We shall have to make this much more precise. First of
all, there exist periods wI' w 2 such that, for T = w21wI,
P(T) =
Evidently, a linear transformation throws (aI' a2, a3, a4) into
(e 1, e2
, e3 , (0), and we may as well assume that this is the
original quadruple.
Let n be the finite '-plane punctured at e1, e2 , e3,
and let P be the z-plane with punctures at all points
Extremal Geometric Properties 55
m(w1 /2) + n (~ /2). The projection «oJ defines P as a cover
ing surface of n. In this situation we know that the fundamen
tal group F of n has a normal subgroup G which is isomor
phic to the fundamental group of P, and the group r of cover
transformations is isomorphic to F /G. We know F, G, and rexplicitly. F is a free group with three generators aI' a2 , a3 ,
representing loops around e 1, e2 , e3 . G is the least normal
subgroup that contains aI
2,a22,a
32 , (a
1a
2a
3)2, and r is
the group of all transformations z .... ± z + row 1 + nw2 . The
translation subgroup r 0 consists of all transformations z ....
z + mW I + nw2 . It is generated by ~ z = z + wI and A2 z =
z + w 2 .
* * *Consider now a second quadruple (e1 ' e2 ' e3 ' (0)
and use corresponding notations n*, F *, etc. A topological
* *mapping ¢: n .... n induces an isomorphism of F on F
*which maps G on G . This means that the covering ¢ 0 «oJ:
* * * *P .... n and «oJ: P .... n correspond to the same subgroup
G* of F*, so there is a topological mapping t/J: p .... p*
such that ¢ 0 «oJ = «oJ * 0 t/J. Clearly, Al * = t/J 0 Al 0 t/J-land A
2* = t/J 0 A
20 t/J-1 are generators of r 0*. We write
* * * * * *Al Z = Z + wI' A 2 Z = Z + w 2 and call (wI' w 2 ) the
base determined by t/J. Since ¢ does not uniquely determine
t/J, we must check the effect of a change of t/J on the base.
We may replace t/J by T* 0 t/J, T* I: r*, and we find that
the base (wI *, w 2*) either stays fixed or goes into (-wI *,
-W2*). In any case, the ratio r* = w 2*/ wI * is determined
*by ¢, and we say that two homeomorphisms of n on n are
56 Quasiconformal Mappings
equivalent if they determine the same T*. By methods which
are beyond our scope here, one can show that two maps of 11
on 11* are equivalent if and only if they are homotopic.
Problem 2. For what K does there exist a K-q.c. map
ping of 11 on 11* which is equivalent to a given CPo'
We need a preliminary calculation of extremal length. De
note by (y1 I the family of closed curves in 11 which lift to
arcs in P with endpoints z and z + CU l' In general, (my1 T
n y2 I denotes the family of curves which lift to arcs with end
points z and z + mcu 1 + ncu 2 .
LEMMA. The extremal length of (Y11 IS 211m T,
(T cu/cu 1).
Proof. We may assume that cu1
sider the segment e in the figure:
'T
/ 7-{/~----/(---,I
T. Con-
Its projection is a curve Y f (Y 11. For a given p in 11 set
p = p(&O(z))I&O'(z)l. We obtain
f p dx 2: L (p) ,
e
Extremal Geometric Properties
and we conclude that
A _2_$. 1m T
57
In the opposite direction, choose p so that p = 1. If
a curve y ( Iy 11 is lifted to P its image y leads from a point
z to z + 1. Hence the length of y is ~ 1, and we have
L (p) = 1, A (p) = Y2 1m T •
This completes the proof.
Of course, w 1 may be replaced by any cW 2 + dW 1
where (c, d) = 1. The result is that
(1) 2
1m (aT+ b )cT+d
where (~ ~) is a unimodular matrix of integers.
We are now in a position to solve Problem 2. We lift the
* *mapping ¢: U .... U to t/J: P .... P and choose the base
(w/, w/) determined by t/J. It is clear that the image under
¢ of Iy1/ is the class Iy 1*1 corresponding to w 1*. If ¢ is
K-q.c. it follows that
1\1 1m T $. 1m T* $. KIm T .
But !cY2 + dY11 is also mapped on ICY2* + dY1*1 and we
have also1m aT* + b
c? + d~ KIm aT+b
CT + d
for all unimodular transformations.
To interpret this result geometrically, let S be any uni
modular transformation and let U be an auxiliary linear trans-
58 Quasiconformal Mappings
formation which maps the unit disk Iw I < 1 on the upper half
planewith U(O) = r.
Mark the halfplanes bounded by horizontal lines through
Sr and K Sr.
We know that Sr* does not lie in the shaded part. Mapping
back by U-1 S-1 we see that U-1 r* does not lie in a
shaded circle
which is tangent to the unit circle at U-1 S-1 (00) and whose
radius depends only on K. But the points S-1 (00) are dense
on the real axis, and hence the U-1 S-1 (00) are dense on the
circle. This shows that r* is restricted to a smaller circle.
An invariant way of expressing this result is to say that the non
euclidean distance between rand r* is at most equal to that
between ib and iKb, or •
*d[r, r 1 < log K
Extremal Geometric Properties 59
THEOREM 3. There exists a K-q.c. mapping equivalent
*to ¢o if and only if d[T, T 1 ~ log K .
We have not yet proved the existence. But this is im
mediate, for we need only consider the affine mapping
(T* - 'T) Z + (T - T*) ZT ."
which is such that ljJ (-z) = -;fJ (z), ljJ (z + 1) = ljJ (z ) + 1,* .and ljJ (z + T) = ljJ (z ) + T . As such, ljJ covers a mapping ¢
*of n on n which is equivalent to ¢o' and the dilatation is
precisely ed [T, T*1 .
What about Problem I? Here it is assumed that ¢o
maps e l , e2 , e3 on e l *, e2 *, e3 * in this order. When does
¢ have the same property? Assume that ¢o is covered by
ljJo: P .... p* which determines the base (wI *, w 2 *) and that
¢ lifts to ljJ determining (CW2* + dw l *, aw2* + bw l *). It
is found that ¢ will map e l , e2, e3 on e l *, e2 *, e3 * if and
only if (~ ~) == (6 ~) mod 2 .
The set of linear transformations ~ ~~@ such that
(~ ~) is unimodular and congruent to the identity mod 2 is
the congruence subgroup (of level 2) of the modular group. The
solution to Problem 1 is therefore as follows:
THEOREM 4. There exists a K-q.c. of n on n*which preserves the order of the points e i if and only if the
non-euclidean distance from T to the nearest equivalent point
of T* under the congruence subgroup is < log K.
60 Quasiconformal Mappings
In connection with the lemma it is of some interest to
determine the linear transformation (~ ~) == (~ ~) mod 2 for
which the extremal length (1) is a minimum. We claim that this
is the case for the point T that lies in the fundamental region
IT ± Y21 < Y2, IRe TI < 1
Then
Ic T + dl 2: Ic Re T + d I 2: Id I - Ic Iand
ICT + dl
> 7'21 c I - II d I - Y21 c II = Id I or Ic I - Id I
Under the parity conditions Id I 2: 1 and either Id I - Ic I 2: 1
or I c 1 - Id I 2:. 1. It follows that Ic T + d 12 2:. 1 and thus A
is smallest for the identity transformation,
COROLLARY Let ¢ be any K-q.c, mapping of the
finite plane onto itself with K < V3. Then the vertices of any
equilateral triangle are mapped on the vertices of a triangle with
the same orientation.
Such a mapping can be approximated by piecewise affine
mappings. The triangle corres-
ponds to
l+iy'32
-1 + ivaand we know that the corresponding T = 2 ' The near-
t ' 'th I . -1 + i d h I'des pomt WI a rea p IS --2- , an t e non-euc 1 ean
Extremal Geometric Properties 61
*distance is log Y3. Hence K < y3 guarantees that Imp >0, and this means that the orientation is preserved.
The remarkable feature of this corollary is that it re
quires no normalization. The result is at the same time global
and local.
CHAPTER IV
BOUNDARY CORRESPONDENCE
A. The M-condition
We have shown that a q.c. mapping of a disk on itself
induces a topological mapping of the circumference. How regu
lar is this mapping? Can it be characterized by some simple
condition? The surprising thing is that it can.
Things become slightly simpler if we map the upper
halfplane on itself and assume that 00 corresponds to 00. The
boundary correspondence is then given by a continuous increas
ing real-valued function h(x) such that h(-oo) = - 00 and
h (+ 00) = + 00. What conditions must it satisfy?
We suppose first that there exists a K-q.c. mapping ¢
of the upper halfplane on itself with boundary values h (x ) .
It can immediately be extended by reflection to a K-q.c. mapping
of the whole plane, and we are therefore in a position to apply
the results of the precedin'g chapter. Namely, let e 1 < e3< e2
63
64 Quasiconformal Mappings
be real points which are mapped on e 1 ', e3 ', e2 '. If
p = p' =
and r, r' are the corresponding values on the imaginary axis
we have
K-1 1m r ~ 1m r' ~ Kim r
We shall use only the very simplest case where e 1, e~.
e2 , are equidistant points x - t, x, x + t and consequently p =
Yi, which corresponds to r = i. In this case we have that
Equivalently, this means that
or
(1) 1 - p UK) ~ p , ~ p UK)
Actually, what we prefer are bounds for
1 - p'-p
and from (1) we get
1 - p UK) h (x + t) - h (x) p UK)pUK) < h(x) -h(x-t} < I-p UK)
Even better, let us recall that p (r+ 1) = 1/p (r). For
this reason the lower bound can be written as p (1 + iK) - 1 and
by our product formula (Chapter III, B, (13)) we find
K 00 (1 -2nTTK ).8p(l+iK) - 1 = 16 e-TT II :;,.+~e _l_e-(2n-l)1TK
Boundary Correspondence
We have thus proved
) ( ) 1 h(x + t) - h (x ) ( )(2 MK- < h{x)-h(x-t) < MK
where
65
(3) () 1 rrK 00 ( 1- e-{2n-l)rrK ) 8MK =-e II
16 1 1 + e-2nrrK
We call (2) an M-condition. Obviously (3) gives the best possi
ble value, the upper bound
M(K) < 1~ err K
THEOREM 1. The boundary values of a K-q.c. mapping
satisfy the M(K) -condition (2).
It is important to study the consequences of an M-condi-
tion
(4) M-1 < h(x+t)-h(x)- h{x) - h{x + t)
< M
even quite apart from its importance for q.c. mappings. Let
H (M) denote the family of all such h. Observe that is is in
variant under linear transformations S: x ... ax + b both of the
dependent and the independent variables. In other words, if
h I: H(M) then Sl 0 h 0 S2 I: H(M). We shall let Ho(M)
denote the subset of functions h that are normalized by h (0)
= 0, h(1) 1.
For h I: Ho(M) we have immediately
(5) 1 < h (Y2) < MM+l M+l
66 Quasiconformal Mappings
and by induction this generalizes to
(6) 1 < <
This is true, with the inequality reversed, for negative n as
well. In other words we have for instance
which shows that the h f H0(M) are uniformly bounded on any
compact intervals (apply to h (x) and 1 - h (1-x)).
They are also equicontinuous. Indeed, for any fixed a
the functionh (a+ x) - h(a)h (a + 1) - h (a )
is normalized. Hence 0 s;. x s;. 1/2n gives
h(a+x)-h(a) < (h(a+l)-h(a))(MM)n+1
which proves the equicontinuity on compact sets. As a conse-
quence:
LEMMA 1. The space Ho(M) is compact (under uni
form convergence on compact sets).
Indeed, the limit function of a sequence must satisfy (4),
and this immediately makes it strictly increasing.
Actually, the compactness characterizes the H o(M) .
LEMMA 2. Let Ho be a set of normalized homeomor
phisms h which is compact and stable under composition with
linear mappings. Then Ho C Ho(M) for some M.
Boundary Correspondence 67
Proof. Set a = infh(-l) , (3 = suph(-l) for h f Ho'
There exists a sequence such that h (-1) .... a, and a subse-n
quence which converges to a homeomorphism. Therefore a >
-00, and the same reasoning gives (3 < O.
·For any h f Ho the mapping
k(x)_h(y+tx)-h(}) , t>O- h(y+ t} - My
is in Ho' Hence
<h (y- t) - h (y)
< {3ah (y + t) - h (y )
or1 <
h (y+ t) - h (y)< 1
a h(y) - h(y-t} -~
which is an M-condition.
We shall also need the following more specific informa-
tion:
LEMMA 3. If h f Ho(M) then
1
1 < f h (x) dx < MM+1 M+1
0
Proof. Let us set F (x) = sup h (x ), h f Ho (M) .
This is a curious function that seems very difficult to deter
mine explicitly. However, some estimates are easy to come by.
We have already proved that F <'/2) ::; M/(M + 1). Be-
causeh (tx)
hftT
6B Quasiconformal Mappings
we obtain, for x = Yz,h (1. )_2_ < F(Yz)h (t)
and hence
(7)
Similarly
for t > 0 .
h((l-t)x+ t) - h(t)
1 - h(t)
givesh(1+t) _ h(t)
21 - h(t)
For t < 1 this gives
h(l+t) < F(Yz) + (1 -F(Yz))h(t)2
and
(8) F(1+t) < F(Yz) + (l-F(Yz))F(t)2 -
Adding (7) and (8)
(9) F(~) + F( l~t.) < F(Yz) + F(t) .
Now1 2
f F(t)dt = Yz f F(~)dto 0
1
= Yz f (F (~) + F (~ + ~)) dt
o
1
< Yz F (Y2) + Yz ! F (t )dt
o
Boundary Correspondence 69
Hence1f F(t )dt :s F ('h)
o
and the assertion follows.
The opposite inequality follows on applying the result
to I-h(l-t).
Remark. From
M< M+l
the weaker inequalities
11
:s Jh dt :so
are immediate, and since they serve the same purpose, Lemma
3 is a luxury.
B. The Sufficiency of the M-condition.
We shall prove the converse:
THEOREM 2. Every mapping h which satisfies an M
condition is extendable to a K-q.c. mapping for a K that de
pends only on M.
12y
vex, y)
The proof is by an explicit construction. We shall in
deed set ¢(x, y) = u(x, y) + iv(x, y) wherey
u(x, y) = ~y J h(x+t)dt
-yy
f (h(x+t) -h(x-t))dt
o
(1)
70 Quasiconformal Mappings
It is clear that v (x, y) 2: 0 and tends to 0 for y .... O. More
over, u(x,O) = hex), as desired.
The formulas may be rewritten
X+Y
u = 1 f h (t )dt2y
(1) ,x~·y
x+y x
v 1 ( f h(t)dt - f h(t)dt)2y
x X-Y
and in this fonn it is evident that the partial derivatives exist
and are
Ux iy
(h(x+y) -h(x-y))
1 jX+Y hdt+ 21y (h(x+y) + h(x-y))-2i
X--Y
1- 2y2
X+Y
Jx
x
hdt - fX-Y
h dt) + iy
(h (x + y)
- h(x-y))
The following simplification is possible: if we replace
h (t) by hI(t) = h (at + b), a > 0, the M-condition remains
in force and ¢ (z) is replaced by ¢1(z) = ¢ (az + b). Thus
¢1 (i) = ¢ (ai + b), and since ai + b is arbitrary we need only
study the dilatation at the point i. Also, we are still free to
choose h(O) = 0, h(l) = 1.
The derivatives now become
Boundary Correspondence 71
U =x
U =y
'12(I-h (-1)),1
-'12 f h dt+ '12(1+h(-I)) ,
-1
v =x
v =y
'12 (1+ h (-1))1 0
-'12 f h dt - f h dt
o -1
The (small) dilatation is given by
d =\
( U - v ) + i (V + U y )x y x _
To simplify we are going to set
1
~ = 1- f h dt,
o
o
Tff3 = -h(-I) + f hdt
-1
ThusU x '12(1 + (3)
U y '!2 (~ - Tf(3)
giving
f3 -h(-I) ,
( > 0).
V x '12(1-{3)
v y '12(~+Tf{3)
d=«I-e') + f3(I-Tf)) + H(I+~) - f3 (I+Tf))
«1+e') + f3(I+~)) + H(I-e'} -f3(I-Tf))
1 + ~2 + f32 (1 + Tf2) - 2f3 (~+ Tf )
1+ ~2 + f32 (I+Tf2) + 2f3(~+Tf)
72 Quasiconformal Mappings
We have proved the estimates
.:s M, 1M+ 1 < (
1 < ...JLM+1 .:s." - M+1
(the last follows by symmetry).
This gives, for instance
1+d2
< M(M+ 1)1_d2
D < 2M(M+1)
< MM+1 '
As soon as we have this estimate we know that the
Jacobian is positive, from which it follows that the mapping ¢
is locally one to one.
It must further be shown that ¢ (z) .... 00 for z .... 00.
By (1) / we havex x+y
U1 (f hdt + f hdt)2y
x-y x
x+y x
v 2~ ( J hdt - J h dt )
x x-yx+y x
u2 + v22; [( f hdtY+ (f h dt)2 ]
x x-yy
If x :::: 0, u2 + v 2 1 ( J hdt Y> -2y20
0
If x .:s 0, u2 + v2 > 1 ( f hdt Y2?-y
Boundary Correspondence 73
and both tend to 00 for y ~ 00. When y is bounded it is
equally clear that u2 + v 2 ~ 00 with z.
Now ,= ¢ (z) defines the upper halfplane as a smooth
unlimited covering of itself. By the monodromy theorem it is a
homeomorphism.
1. Remark: Beurling and Ahlfors proved D < M2 . To
do so they had to introduce an extra parameter in the definition
of ¢.
2. Remark: It may be asked how regular a function
h (t) is that satisfies an M-condition. For a long time it was
believed that the boundary correspondence would always be ab
solutely continuous. But this is not so, for it is possible to con
struct functions h that satisfy the M-condition without being
absolutely continuous.
C. Quasi-isometry.
For conformal mappings of a halfplane onto itself non
euclidean distances are invariant. For q.c.-mappings, are they
quasi-invariant? Of course not. If noneuclidean distances are
multiplied by a bounded factor (in both directions) we shall say
that the mapping is quasi-isometric.
THEOREM 3. The mapping ¢ constructed in B 1S
quasi-isometric. Indeed, it satisfies
with a constant A that depends only on M.
74
(3)
Quasiconformal Mappings
It is sufficient to prove (2) infinitesimally; that is,
A-I ..hk.L < 1M..l < A..hk.Ly - v - y
Again, affine mappings of the z-plane are of no impor
tance, and it is therefore sufficient to consider the point (0, 1).
We have1 0
v (i ) Y:z f h dt - f h dt
o -1
and the estimates
M"2
are immediate (using Lemma 3).
This is to be combined with
and
l<,bzl2 = J/s [(l+e) + f32(1+r?) + 2f3(~+1])1
One finds that (2) holds with
A = 4M2(M+ 1)
We omit the details.
D. Quasiconformal Reflection.
Consider now a K-q.c. mapping <,b of the whole plane
onto itself. The real axis is mapped on a simple curve L that
goes to 00 in both directions. Is it possible to characterize L
through geometric properties?*
* For instance, we know already that L has zero area.
Boundary Correspondence 75
First some general remarks. Suppose that L divides
the plane into nand n* corresponding to the upper halfplane
H and the lower halfplane H*. Let j denote the reflection
z .... z that interchanges Hand H*. Then ep ° j ° ep-l is a
sense-reversing K2_q.c. mapping which interchanges n, n*
and keeps L pointwise fixed. We say that L admits a K2 _
q-c- reflection.
Conversely, suppose that L admits a K-q.c. reflection
w. Let [ be a conformal mapping from H to n. Define
H
in H,
in H*.IF = [F=wo[oj
2It is clear that F is K-q.c. So we see that L admits a re-
flection if and only if it is the image of a line under a q.c.
mapping of the whole plane. Moreover, we are free to choose
this mapping so that it is conformal in one of the half-planes.
We say that the conformal mapping [ admits a K~q.c. extension
to the whole plane.
. . * *We can also consIder the conformal mappmg [ from H
to n*. The mapping j 0/*-1 ° w ° [ is a quasiconformal map
ping of H on itself. Its restriction to the x-axis is h (x) =
[*-1 ° [, and we know that it must satisfy an M-condition. Ob
serve that L determines h uniquely except for linear transfor
mation (h(x) can be replaced by Ah(ax+b) + B).
(1)
76 Quasiconlormal Mappings
*
On the other hand, suppose that h is given and satis
fies an M-condition. We know that there exists a q.c. mapping
with these boundary values: we make it a sense-reversing map
*ping l from H to H . We cannot yet prove it, but there exists
a mapping ¢ of the whole plane upon itself which is conformal
*in H and such that ¢ 0 l 0 j is conformal in H . (This con-
dition determines J1 in the whole plane, and Theorem 3 of Chap
ter V will assert that we can determine ¢ when J1¢ is given.)
Thus ¢ maps the real axis on a line L which in turn deter
mines h.
How unique is L? Suppose L 1 and L 2 admit q.c. re
flections WI and w2' and denote the corresponding conformal
* *mappings by 11> 11 , 12, 12 . Assume that they determine the
same h = 1/-10 11 = 12*-1 0 12, The mapping
g = l/2 0 11-1
in °1 ,* I *-1 ...... *12 OlIn u 1 '
*is conformal in 0 1 U 01 and continuous on L1
. Is it confor-
mal in the whole plane? To prove that this is so we show that
g is quasiconformal, for we know that a q.c. mapping which is
conformal a.e. is conformal.
Introduce F1
and F2 as in (1). Form
G = F2
- 1 0 I * 0 I *-1 0 F2 1 1
in H*. It reduces to identity on the real axis, and we set G =z in H. Then G is q.c. Hence F2 0 G 0 F 1-1 is quasi
conformal. It reduces to 12* 0 11 *-1 in °1* and to 12 /0 11- 1
in °1 ; that is, to g.
*
Boundary Correspondence n
We conclude that g is conformal. Hence 12 differs
from 11 only by a linear transformation, and L is essentially
unique.
There are two main problems:
Problem 1. To characterize L by geometric
properties.
*Problem 2. To characterize f (and f).
We shall solve Problem 1. I don't know how to solve Problem 2.
The characterization should be in analytic properties of the in
variant f"j f'.
We begin by showing that a K-q.c. reflection retains
many characteristics of an ordinary reflection.
·z~o . i!
*We shall set ( == (U «() and suppose that (
¢C"Z), (0 = ¢(zo) when Zo is real.
All numerical functions of K. alone, not always the
same, will be denoted by C (K).
LEMMA 1.
< C(K)
78 Quasiconformal Mappings
Proof. We note that
pZo - Z
satisfies Ip I == 1. For any such p there is a corresponding
T situated on the lines Re T == ± %, 1m T 2: %.
We conclude that there is aT' corresponding to p
*('0 - ')/('0 - ,) that is within n.e. distance log K of these
lines. This means that T' lies in a W-shaped region indicated
in our figure.
We note that the points ± 1 where p = "" are shielded
from the W-region. Also, p .... 1 as 1m T .... "" and p .... -1 at
the points ± '12, provided that they are approached within an
angle. Therefore p' is bounded by a constant C (K) and this
proves the lemma.
Remark. It is not necessary to investigate the behavior
of p (T) at T = ± %, for the true region to which T' is restrict
ed does not reach these points.
Let 8 (,) be the shortest euclidean distance from 'to
L.
LEMMA 2.
C (K )-1 < C(K)
Boundary Correspondence
The proof is trivial.
79
*The regions nand n carry their own noneuclidean
metrics defined by
A Id~1Idz I
y
for ~ = /(z). The reflection w induces a K-q.c. mapping of
*H on H, and we know that it can be changed to a C (K) quasi-
isometric mapping. It follows that we can replace w by a re
flection w' such that (at points t w '(~))
C(K)-l Ald~1 s A*Idt I < C(K) A Id~1
But it is elementary to estimate A(~) in terms of 8(~).
"--Y'I \I ~o. I\ I, ./
..... - ... o w
For this purpose map n conform ally on Iw I < 1 with
w(~o) = O. Schwarz's lemma gives
But the noneuclidean line element at the origin is 21 dw I. SO
A(~o) = 2Iw'(~o)1 < _2_- 8(~o)
In the other direction Koebe's distortion theorem gives
'I. 1• Iw'(~o)1
128 (~o)
80 Quasiconformal Mappings
Combining the results with Lemma 2 we conclude
LEMMA 3. If there exists a K-q.c. reflection across L,
then there is also a C (K )-q. c. reflection which is differentiable
and changes euclidean lengths at most by a factor C (K) .
This is a surprising result, for a priori one would expect
the stretching to satisfy only a Holder con·dition.
I I
L
Now consider three. points on L such that ~3 lies be
tween ~1' ~2' Now P = (zl - z3)/(zl - z2) is between 0
and 1 which means that T lies on the imaginary axis. There*.
fore T lies in an angle
1 * -12 arc tan K - :s arg T :s 17 - 2 arc tan K
It can also be chosen so that IRe T*1ed to the following region
.,.-- ....",
o
*:s 1, so T is restrict-
Again it is obvious that Ip 1 has a maximum C (K) and
we have proved:
Boundary Correspondence 81
THEOREM 4. If ~1' ~3' ~2 are any three points on L
such that ~3 separates ~1' ~2 then
It is more symmetric to write
and in this form the best value of C (K) can be computed. It
corresponds to the point e ia and can be computed explicitly.
E. The Reverse Inequality.
We shall prove that the condition in the last theorem is
not only necessary but also sufficient. In other words:
THEOREM 5. A necessary and sufficient condition for
L to admit a q.c. reflection is the existence of a constant C
such that
(1) < C
for any three points on L such that ~ 3 is between ~1 and ~2'
More precisely, if K is given C depends only on K,
and if C is given there is a K-q.c. reflection where K depends
only on C.
82 Quasiconformal Mappings
Introduce notations by this figure:
lSI
Let Ak be the extremal distance * from ak to 13k in
0, and A/ the corresponding distance in 0*. Thus A1A2
* *= 1, A1 A2 = 1. Through the conformal mapping of 0,
let '1' '3' '2 correspond to x - t, x, x + t. This means that
Al = A2 = 1. Through the conformal mapping of 0*, '1' '3''2 correspond to h (x - t), h (x), h(x + t). If we can show
*that Al is bounded it follows at once that h satisfies an M-
condition, and hence that a q.c. reflection exists.
(2)
We show first that
c-2 e-2TT ::;
Al = 1 implies
I'2 -'11I '3 -'21
Indeed, it follows from (1) that the points of {32 are at distance
~ C-1 1'2 - '1 I from '2' while the points of a2 have dis
tance ::; C 1'3 - '2 I from '2· If the upper bound in (2) did
not hold, a2
and /32
would be separated by a circular annu
lus whose radii have the ratio e2TT• In such an annulus the ex-
* The extremal distance from a to {3 in E is the extremal length
of the family of arcs in E joining a to /3.
Boundary Correspondence 83
tremal distance between the circles is 1, and the comparison princi
ple for extremal lengths would yield ,\ 2 > 1, contrary to hypothesis.
This proves the upper bound, and we get the lower bound by inter
changing ~l and ~3 °
Consider points ~ ( a2
, ~/ ( f32
° By repeated application
of (1)
and with the help of (2) we conclude that the shortest distance between
a2
and f32 is :::: c4 e-2" 1~2 - ~31 . To simplify notations write
Because of (1), all points on ~ are within distance ~ from ~ .
Let r* be the family of all arcs in n* joining a2 and f320
Then
where p is any allowable function (recall Chapter I, D). We choose
p = 1 in the disk { I~ - ~21 ~ M1 + M2}, p = 0 outside that
.) *dISk. Then Ly(p ~ M2
for all curves y ( r , whether y stays
within the disk or not. We conclude that
'\2*~ 1 ( M2 )2
" M1 +M2
* *Since \ '\2
is proved.
1 , \ * has a finite upper bound, and the theorem
CHAPTER V
THE MAPPING THEOREM
A. Two Integral Operators.
Our aim is to prove the existence of q.c. mappings f with a
given complex dilatation Ill" In other words, we are looking for solu
tions of the Beltrami equation
(1) f-Z == !J.fz '
where Il is a measurable function and III I ~ k < 1 a.e. The so
lution f is to be topological, and f-Z' fz shall be locally integrable
distributional derivatives. We recall from Chapter II that they are then
also locally square integrable. It will turn out that they are in fact
locally LP for a p > 2.
The operator P acts on functions h f LP, P > 2, (with
respect to the whole plane) and it is defined by
(2) Ph«,) == _1 f f h(z)(~ - 1) dxdy" Z-s z
(All integrals are over the whole plane.)
LEMMA 1. Ph is continuous and satisfies a rmiforrn Holder
condition with exponent 1- 2/p.
85
86 Quasiconforma.l Mappings
The integral (2) is convergent because h £ LP and
t;/'(z(z-()) £ LP, 1/p + 1/q = 1. Indeed, 1 < q < 2, and for
such an exponent Iz(z-()I-q converges at 0, (;6 0) and 00.
The Holder inequality yields, for ( ;6 0,
IPh«()1 s 1;1 IIh lip II z(z:,) IIq
A change of variable shows that
and we find
(3)
with a constant Kp that depends only on p. «3) is trivially fulfilled
if (= 0.)
We apply this result to hI(z) = h(z+ (1) . Since
we obtain
which is the assertion of the lemma.
The Mapping Theorem
The second operator, T, is initially defined only for functions
h ( C02 (C2 with compact support), namely as the Cauchy princi-
pal value
(5)
We shall prove:
lim( .... 0
1
17
LEMMA 2. For h ( Co2, Th exists and is of class Cl.
Furthermore,
(6)
and
(7)
(Ph)z
(Ph)z
f f ITh 12
dxdy
h
Th
Proof. We begin by verifying (6) under the weaker assump
tion h ( COl. This is clearly enough to guarantee that
(Ph)~ = - ;; f f z~z?; dxdy
(8)
(Ph), = - ;; f f z~z?; dxdy .
Let y( be a circle wi th center , and radius (. By use of
Stokes' formula we find
88 Quasiconformal Mappings
117 II h-
z_z, dxdy 1 II hz
217i i="[ dz dZ
and
= lim£ .... 0
1217i J h dzz-'
1 I I zh~, dxdy = 1 I I dh dz17 217i ~
lim [- 1 I hdi" 1 f f ~dZd~ ]217i z-' + 217i£.... 0Iz~1 >£( -,)
y£
= Th(')
We have proved (6).
Observe that (8) can be written in the form
(9)P(h z )P(h)
h - h (0)
Th - Th(O) .
Under the assumption h £ C02 we may apply (6) to hz, and
by use of the second equation (9) we find
(10)(Th )z
(Th)z
These relations show that Th £ C 1, Ph £ C2 .
The Mapping Theorem 89
Because h has compact support it is immediate from the
definitions that Ph == 0(1) and Th == O(lz \-2) as z ~ c>o.
We have now sufficient information to justify all steps in the cal
culation
== ii f f Ph(Ph)zz dzdi"
which proves the isometry.
ii f f (Ph )hz dz di"
f f Ihl 2dxdy
q.e.d.
The functions of class Co2 are dense in L 2 . For this
reason the isometry permits us to extend the operator T to all
of L2 , by continuity. Unfortunately, we cannot extend P in
the same way, for the integral becomes meaningless when h is
only in L 2 , and even if we use principal values the difficulties
are discouraging.
The key for solving the difficulty lies in a lemma of
Zygmund and Calderon, to the effect that the isometry relation
(7) can be replaced by
(11)
for any p > 1, with the additional information that Cp ~ 1 for
p ~ 2. Naturally, this enables us to extend T to LP, and for
p > 2 the P-transform is well defined.
90 Quasiconformal Mappings
The proof of the Zygmund-Calderon inequality is given in
section D of this chapter. At present we prove:
LEMMA 3. For h ( LP, P > 2, the relations
(Ph)-Z h(12)
(Ph)z Th
hold in the distributional sense.
We have to show that
f !(Ph)¢-Z -! ! ¢h
(13)
! ! (Ph)¢z -! ! ¢Th
for all test functions ¢ ( Co 1. We know that the equations hold
when h ( C02 . Suppose that we approximate h by hn ( C0
2
in the LP-sense. The right hand members have the right limits
since 11Th - Thn II P $ CP II h - hn II p' On the left hand side we
know by Lemma 1 that
IP(h-hn)1 $ Kp Ilh-hnll p Iz I1-
2/
p
and since ¢ has compact support the result holds.
B. Solution of the Mapping Problem.
We are interested in solving the equation
(1)
where 111111"" $ k < 1. To begin with we treat the case where
11 has compact support so that f will be analytic at "".
The Mapping Theorem 91
We shall use a fixed exponent p > 2 such that kCp < 1.
THEOREM 1. If /l has compact support there exists a
unique solution I 01 (1) such that 1(0) = 0 and Iz - 1 ( LP.
Prool. We begin by proving the uniqueness. This proof
will also suggest the existence proof.
Let I be a solution. Then Iz = /lIz is of class LP,
and we can form PUz). The function
F = I-PU-)z
satisfies F z = 0 in the distributional sense. Therefore F
is analytic (Weyl's lemma). The condition I z - 1 ( LP implies
F' - 1 (LP, and this is possible only for F' = 1, F = z+ a.
The normalization at the origin gives a = 0, and we have
(2)
It follows that
(3)
If g is another solution we get
I z - gz = T[/lUz - g)l
and by Zygmund-Calderon we would get
and since k CP < 1 we must have Iz = gz a.e. Because of
the Beltrami equation we have also Iz = gz· Hence 1-g and
7 - i are both analytic. The difference must be a constant,
and the normalization shows that I = g.
92
(4)
Quasiconformal Mappings
To prove the existence we study the equation
h = T (/l h) + T /l .
The linear operator h ~ T (/l h) on LP has norm < k C < 1.- P
Therefore the series
(5)
converges in LP. It is obviously a solution of (4).
If h is given by (5) we find that
(6)
is the desired solution of the Beltrami equation. In the first
place /l (h + 1) ( LP (this is where we use the fact that /l
has compact support) so that P [/l (h + 1)1 is well defined and
continuous. Secondly, we obtain
(7)f-z /l(h+ 1)
T[/l(h+ 1)] + 1 h+l
and f z - 1 = h ( LP.
The function f will be called the normal solution of
(1).
We collect some estimates. From (4)
so
(8)
and by (7)
The Mapping Theorem 93
(9)
The Holder condition gives, from (3),
K(10) 1£((1) - £((2)1 s 1-~C II/lli p 1(1 -(211-2/p + 1(1 -(21
p
Let v be another Beltrami coefficient, still bounded by
k, and denote the corresponding normal solution by g. We ob
tain
f -g == T(lIf -vg )z z r z z
and hence
II f z -gz ll p s II T [v(fz- gz)lllp + lIT[(/l-v)fzll1p
< kCpllfz-gzllp + Cpll(/l-v)fzllp
We suppose that v .... /l a.e. (for instance through a se
quence) and that the supports are uniformly bounded. The fol
lowing conclusions can be made:
LEMMA 1. IIgz-fz lip .... 0 and g .... f, uniformly on
*compact sets.
More important, we want now to show that f has deriva
tives if ./l does. For this purpose we need first a slight gener
alization of Weyl's lemma:
* Since f-g is analytic at 00, the convergence is in fact uniform
in the entire plane.
94 Quasiconformal Mappings
LEMMA 2. If p and q are continuous and have locally
integrable distributional derivatives that satisfy Pz = qz'
then there exists a function f ( C 1 with fz p, f F = q.
It is sufficient to show that
f p dz + q dz 0
y
for any rectangle y. We use a smoothing operator. For ( > 0
define B/z) = 1/71(2 for [zl ::; (, B/z) = 0 for Izi > (.
The convolutions p * B( * B(, and g * B( * B(, are of
class C2 and
Therefore
and the result follows on letting ( and (' tend to O.
We apply the lemma to prove:
LEMMA 3. If 11 has a distributional derivative
o
p > 2,
then f ( C1 , and it is a topological mapping.
Proof. We try to determine ,\ so that the system
(11)
has a solution. The preceding lemma tells us that this will be
The Mapping Theorem
so if
95
(12)
or
,\-z (/1'\) z
(log '\)z
The equation
can be solved for q in LP, and we set
o = P (/1 q + /1z) + constant
in such a way that 0 ~ 0 for z ~ 00. Thus 0 is continuous
and
q
Hence ,\ = eO satisfies (12), and (11) can be solved with I
(C 1 . We may of course normalize by 1(0) = 0, and then I
is the normal solution since 0 ~ 0, ,\ ~ 1, I z ~ 1 at 00.
TheJacobian I/zI2-1/Z-12 = Cl-I/11 2)e20 is strictly
positive. Hence the mapping is locally one-one, and since f(z)
~ 00 for z ~ 00 it is a homeomorphism.
Remark. Doubts may be raised whether (eO) z = eO0 z
in the distributional sense. They are dissolved by remarking
that we can approximate 0 by smooth functions on in the sense
that on ~ 0 a.e. and (on) z ~ 0 z in LP (locally). Then for
any test function 1>,
96 Quasicon£urmal Mappings
q.e.d.
Under the hypothesis of Lemma 3 the inverse function
£-1 is again K-q.c. with a complex dilatation ii 1 = f1-1f
which satisfies lii1 0 £1 = 1111.
Let us estimate II iiI lip . We have
f fl'j11lPd~dTJ = f fIIlIP(I£zI2 -1£zI2)dxdY
and thus
If we apply the estimate (10) to the inverse function we
find
(13)
The Mapping Theorem
It is now obvious how to prove
97
THEOREM 2 For any Il with compact support and
IIIl to ~ k < 1 the normal solution 01 the Beltrami equation is
a q.c. homeomorphism with III = Il·
We can find a sequence of functions Iln £. C1 with
Iln .... Il a.e., Illnl ~ k and Iln = 0 outside a fixed circle.
The normal solutions In satisfy (13) with In in the place of
I and Iln in the place of Il· Because In.... I and II Iln \I p ....
1I1l11 p ' I satisfies (13), and hence I is one-one.
We recall the results from Chapter II. Because I is a
uniform limit of K-q.c. mappings In' it is itself K-q.c. As
such it has locally integrable partial derivatives, and these par
tial derivatives are also distributional derivatives. We know
further that Iz f 0 a.e. so that III = Iz/Iz is defined a.e.
and coincides with Il.
(Incidentally, the fact that I maps null sets on null
sets can be proven more easily than before. Let e be an open
set of finite measure. Then
mes In<e) = f <lln,zI2 -lln,z 12) dx dy
e
e
1 2/p(mes e) -
98 Quasiconformal Mappings
and since II f II is bounded we conclude that f is indeedn,z p
absolutely continuous in the sense of area.)
We shall now get rid of the assumption that /1 has com
pact support.
THEOREM 3. For any measurable /1 with 11/11100
< 1
there exists a unique normalized q. c. mapping f /1 with complex
dilatation /1 that leaves 0, 1, 00 fixed.
Proof. 1) If /1 has compact support we need only nor
malize f.
2) Suppose /1 = 0 in a neighborhood of O. Set
1 z2ii.(z) = /1 ('z) ~
z
Then ii has compact support. We claim that
Indeed,
f/1(z) = 1
f/1(11 z )
f/1(z)1 1
fii 01z)Z f ii(11z )2 7 Z
f!(z) 1 1 f!::. (IIz)Z fj1(l/z )2 _2 zz
Remark: The computation is legitimate because f ii is
differentiable a.e.
3) In the general case, set /1 = /11 + /12' /11 == 0
near 00, /12 = 0 near O. We want to find A so that
fA 0 /2 = f/1, fA == f/1 0 (1/12)-1 •
The Mapping Theorem
In Chapter I we showed that this is so for
and this ,\ has compact support. The problem is solved.
99
THEOREM 4. There exists a Il-conformal mapping of the upper
halfplane on itself with 0, 1, 00 as fixed points.
Extend the definition of Il by
;(z) = 1l(Z)
One verifies by the uniqueness that fll(i:) 71l(z). Therefore
the real axis is mapped on itself, and so is the upper halfplane.
SOROLLARY. Every q.c. mapping can be written as a
finite composite of q.c. mappings with dilatation arbitrarily close
to 1.
We may as well assume that f = fll is given as a q.c.
mapping of the whole plane. For any z, divide the noneuclidean
geodesic between 0 and Il (z) in n equal parts, the succes
sive end points being Ilk (z). Set
Then
(Ilk+1 - Ilk
1 - Ilk+1 Ilkf -1
o k
100 Quasiconformal Mappings
FIl/FIl(I) and FIl(I) ~ 1, the assertion
If fll is K-q.c. it is clear that gk = f k+1 0 fk-1 is
K l/n_q.c ., and
C. Dependence on Parameters.
In what follows fll will always denote the solution of
the Beltrami equation with fixpoints at 0, 1, 00. We shall need
the following lemma:
LEMMA. If k = 111111 00 ~ 0 then Ilf~ - 111 1,p ~ 0
for all p.
Note: II II R ,p means the p-norm over Iz I ~ R:
Suppose first that 11 has compact support, and let F 11
be the normal solution obtained in Theorem 1. We know that
h = F 11 - 1 is obtained fromz
h = T (11 h) + Til
and this implies Ilh lip ~ c 1111 lip ~ O. Here p is arbitrary,
for the condition k Cp < 1 is fulfilled as soon as k is suffi
ciently small.
Since f 11
follows for all 11 with compact support.
Let us write f(z) = 1/f ( ~ ). Then it is also true that
Iltll -11/ 1 ~ 0 when 11 has compact support. It is again suf-z ,p
ficient to prove the corresponding statement for the normal solu-
tion F 11. One has
The Mapping Theorem 101
_ 1 \P dxdy
Iz l4
For Iz I > R, F
The part ffl < Izl < R can be written
f f \z2(Fz(z) - 1) +
F (z)2 F (z)21<lzl<R
and tends to zero because IIF -l11 R .... O.z ,P
may be assumed analytic, and F (z) .... z uniformly.v
In the general situation we can write f = g 0 h where
J.Lh = J.Lt inside the unit disk and h is analytic outside. Then
J.Lh and J.L g are bounded by k and have compact support.
Sincev v v
f = (g 0 h)h + (g- 0 h)F = (g 0 h)hz z z z z z z
we get
Ilfz-1111,p < 1I[(iz -1) 0 h]hzlll,p + Ilhz - 1 11 1,p
For the first integral we obtain
f f l(gz-l) 0 hlPlhzl P dxdy
Izl<l
< _1_ f f Igz -liP Ih~ 0 h-1IP- 2 dx dy= 1-k2 _
hllzl<lI
~ 1_lk2 ( !! 1~_112Phllzl <11
. f f IhzI2P-4)~ 0 .
Izi < I
102 Quasiconformal Mappings
One of these integrals is taken over a region slightly bigger than
the unit disk, but this is of no importance. The lemma is proved.
It is now assumed that /l depends on a real or complex
parameter t and that
/l(z, t) = tv(z) + te(z, t}
where v and E f L"" and II E (z, t )11"" ... 0 for t ... O. We
shall show that f/l = f(z, t) has a t-derivative at t = O.
For any I~ I < 1 we can write
f(~) = 1 I f(z )dz217i z - ~
Iz 1=1- 1 II fz(z) dxdy
17 z - ~Izi < 1
(the Pompeiu formula). Replace z by 1/z in the line integral.
It becomes
1217i I
Izl = 1
f(1jz)dzz(l-z()
v
f(z )-1
1- z~dz
The convergence is guaranteed as soon as t is suffi-...ciently small to make K < 2. Indeed, the inverse of f satis-
fies a Holder condition with exponent 11K. For small Iz I wev
have thus, If(z)! > mlzl K and hence
IIzl=o
If(z)I-1 jzlldzl
11 - z~1
The Mapping Theorem 103
The constants can be recovered from the normalization
f(O) = 0, f(l) = 1. All told we get
(~ 2 z ~ z ) dx dy .1- z~ 1-z
f(~)
v
_1 f f ~z(z)17 f (z )2
Izl < 1
In the first integral, set
~ ~-1)- - + - dxdyz-l z
f- = Il fz = Il (fz-1) + Ilzv
~(z)and use a corresponding expression for f- withz(zl Z)2 Il (II z ) . Because
Hf -111 1 -->0' and ~ --> vz ,p t
we find
f(~) = lim HC) -Ct--> 0 t
_1 f f v(z)( z:~ ~ +~-1 ) dxdy
17 z-I -z-
Izl < 1
~ ranges over a com-
.~(~ -z2 1-z~
-~ f f v(l/z)
[z[ < 1
The convergence is clearly uniform when
~z )1-z dx dy .
pact subset of I~I < 1.
Finally, if 1/z is taken as integration variable in the
second integral, it transforms to the same integrand as the first,
104
and we have
where
Quasiconformal Mappings
i(I;) - kJJv(z)R(z, 1;) dxdy
R (z, !;) 1~ -
I;z-1
1;-1+ -- =
z
By considering f(rz), one verifies easily that this formula is
valid for all 1;, and that the convergence is uniform on compact
sets.
For arbitrary to we assume
in the same sense as above, and we consider
where
and
a~ Hz, t) = i 0 fila
= - 1 II ( v <to) ~S:) 0 (lllo)-1 R (z, fila (I;))dx dyIT I-lila 1
2 f Z
= - kII v (t 0' z )(I~Or R (lila (z) , fila (1;)) dx dy
The Mapping Theorem
which is the general perturbation formula.
We may sum up our results as follows:
THEOREM S. Suppose
wherev(t), Il(t), e(s, t) (Co, 111l(t)11 00 < 1,
and II e(s, t)1100 -+ 0 as s -+ O.
Then
f Il(t+s) (,) = f J1(t) (,) + ti (" t) + o(s)
uniformly on compact sets, where
105
1(" t)
If v (t) depends continuously on" t (in the L00 sense)
then we can prove, moreover, that (a/at) fez, t) is a continuous
function of t. It is enough to prove this for t = 0, so we have
to show that
By inversion, the integral over the plane can again be written as
the sum of two integrals over Iz I ::;. 1, and they can be treated
in a similar manner. We con~ider only the first part.
The important thing is that the improper integral con-
verges uniformly; that is,
!! Iff(z)12 IR(fIl(z), fll(')1 dxdy < E
\z-'i <0zl < 1
106 Quasiconformal Mappings
in a uniform manner. Indeed, this integral is comparable to
JJ IR(z, f J1 «())1 dxdy .
fJ1[z: Iz-<I <81The region of integration can be replaced by a disk with center
fJ1«() and radius < 28, say. The value of the integral is then
uniformly of order 0 (8).
For the remaining part of the plane (with a fixed 8) the
difference
tends rather trivially to 0, and it is also clear that
since II f J1 - 111 -+ 0 and the other factor is bounded.p •
D. The Calder~n-ZygmundInequality.
We are going to prove that the operator
Th (0 = lim _.1 f f h (z ) dx dyf -+ 0 17 (z _ ()2
Iz-(I >falready defined for h f C0
2, can be extended to LP, p 2: 2,
so that
(1)
We prove first a one-dimensional analogue, due to Riesz.
The proof is taken from Zygmund, Trigonometric Series, first
edition (Warsaw, 1935).
LEMMA. For
The Mapping Theorem
f ( Co 1 on the real line, set
107
Hf(~)
00
pr.v. 1 f J.i.!l dx .17 x-~
-xl
Proof. Set00
U + IV ~ f f(x) dx.. ~
, = ~ + i.", ." > 0
The imaginary part V is the Poisson integral, and it is immedi
ate that
V(~) = f(~)
The real part is00
u(~, .,,) 1 f x-t f(x) dx17
(x- ~)2 + .,,2-xl
00
1 f f(g+x) - f(g.·.,x) x 2dx
17 x x2 +.,,20
-+ Hf(~)
We observe now that
~ lul P
~lvlP
~ IF IP =
as .,,-+0.
p(p-l) lulp-2(u 2 + u 2)x y
p(p-l)lvI P- 2 (u} + u/)p2 IF IP- 2 (u} + u/)
It follows that
~ (IF IP - -p- lu IP)p-l p2(JFIp-2 - luI P- 2)(u} + u/)
> 0 .
108 QuasiconformaI Mappings
We apply Stokes' formula to a large semicircle
"'--,/ ,
I \L J
and find easily
00
f (IF «()I P - -p- lu«(W') d( ~ 0 .p-l
It is easy to see that the integral goes to zero for Tf .... 00.
Hence00 00
-""'l -""'l
for fixed Tf > O. We observe that
(J IF IP q'()2/P= IIu2+ v211p/2 < Il u2 1lp/2 + Il v2 1lp/2 .
Thus
Lettir.g Tf .... 0 we get the desired inequality.
The Mapping Theorem 109
We continue the proof of the Caldero'n-Zygmund inequal
ity (1), following the method in Vekua, Generalized Analytic
Functions, Pergamon Press, 1962.
We define the operator
dxdy
zlzl
again as a principal value. On setting zit can be written
re i() we see that
T* f(') = 1 f (~ 1 f('+re i()) - f('-re i())dr )e-i() d()2" r
o 0
This implies
00 H'+rei()) - H'-reie-, ~lIIT*fll < 17 max 1 fP 2 ()
17 r0
The norm on the right does not change if we replace , by 'ei(),
and then the integral becomes Hf()(') where f()(z) = f(ze i()).
The norm is of course a 2-dimensional norm. But it is clear
that our estimate of the I-dimensional norm can be used, for we
obtain
AP Ilf II PP () P
110 QuasiconformaI Mappings
so that, finally, II r*f lip :::; ~ Apllf lip'*We extend r to LP by continuity. The proof of (1)
will now be completed by showing that rf = - r*r*f for
f f C02 . This, of course, allows us to extend r to LP also.
We have ..1- _1_ = __1_ . If f f C 1 we obtainaz Iz I 2z Iz I 0
r*f(') - 1 IIHz+')..1- _1_ dxdy17 az Iz I
1 I I f (z+') _1_ dxdy17 z Iz I
(2)
1 ..1- 11Hz) dx dy17 a, Iz-~
~ ~ Ilf(Z) (lz~'1 - lh-) dxdy .
For any test function ¢ it follows that
This remains true for f f LP, because the integral on the
right is absolutely convergent (compare the proof for Pf).
This means that (2) is valid in the distributional sense.
The Mapping Theorem
We can now write
T*T*[(w)
111
(One should check the behavior for z = 0 and z = w. The
integral blows up logarithmically, and the boundary integral
over a small circle tends to zero.) We have to compute
...d... limaz R ~ ""
The differentiation and limit can be interchanged, for it is quite
evident that
aaz
tends to zero, uniformly on compact sets. Moreover, we can re
place our expression by
112 Quasiconformal Mappings
lim ~ f f 1 1 d~ dTf. J f f de dT/Jz Iz-w! I~-wl - Jz 1~llz-~
I~-w! <R. !~! <Rfor the difference, which is an integral over two slivers, tends
uniformly to zero.
By an obvious change of variable the first integral be-
comes
J~ f f!~I < Rllz-wl
R/lz- wi 217
J f f drdOJzoo -I=l_-'=r-'-ei-O1--
_% R. (z-w)lz-w!
and it is obvious that the limit is _ 17z-w'
cond limit is - 171 z .
We now obtain
Similarly, the se-
T*T*t(w) = ~ [1 f f t(z )(_1 - 1) dXdY]Jw 17 z-w Z
= -~ Pf(w) = -Tf(w)Jw
This completes the proof of (1).
The fact that C ~ 1 as p ~ 2 is a consequence ofp
the
The Mapping Theorem
Riesz-Thorin convexity theorem:
The best constant Cp
-+is such that log Cp
function of 1/p .
113
is a convex
Proof. We consider 1\1
P2 =1 both ~ 2.a1' a2
Assume
II Tfl11/al < C1l1fll1/al
II Tf Il 1/ < C211£111/a2 a
2
If a = (1- t )a1 + t a2 the contention is that
II Tf 11 1/ a :s; C11
-t C/ II f 11 1/ a <0 < t :s; 1).
Conjugate exponents will correspond to a and a' with
a + a' = 1. Similar meanings for a1
', a2
'. We note first that
I! Tf 11 1/ a = s~p f Tf· g dxdy
where g ranges over all functions of norm one in L 1/a' Be
cause simple functions (measurable functions with a finite number
of values) with compact support are dense in every LP it is no
restriction to assume that f and g are such functions.
For such fixed f, g set
1 = / Tf· g dx dy
The idea of the proof is to make f and g depend analytically
on a complex variable ,. 1 will be a particular value of an ana
lytic function cI> <'), and we shall be able to estimate its modu
lus by use of the maximum principle.
114 Quasiconformal Mappings
For any complex " define
ill1F(') If I· a _f
~' If IG(') Igi a -L
Ig Iwhere a(') = (1-')a 1 + 'a2 and a(')' = 1- a('). Ob
serve that , enters as a parameter: F (,) and G(,) are func
tions of z. We agree that F (,) = 0, G(,) = ° whenever
f=O, g=O. Note that F(t) = f, G(t) = g. Weset
¢(,) =
F (,) is itself a simple function I. Fj X j which makes
TF (,) = I. F. TX.. Similarly G(,) = I. G. X. *, say, and weII} }
find
<I> (,) = '"' F. G./!T X. . X.* dxdy .L.Jl} 1 }
We see at once that it is an exponential- polynomial: <I> «()
I. aje \' with real \. Therefore <I> (,) is bounded if ~ =
Re ( stays bounded.
Consider now the special cases ~ = ° and ~ = 1. For
~ = ° we have Re a(~) = a1 and hence
Ifl a/
a
Igl a1'1a '
It follows that
IIF (,) 11 1/ a1
II G (,) 11 1/ a '1
1 .
The Mapping Theorem
For simplicity we may assume that II f 11 11a
ly a normalization).
We now get
11S
1 (this is mere-
I¢(~)I ::; IITF(~)1I1/a IIG(~)1I1/a' < C 1 .1 1
A symmetric reasoning shows that
I¢(~)I ::; C2
on ~ == 1. We now conclude that
log I¢(~)I - (1-~) log C 1 - ~ log C2 < 0
on the boundary of the strip 0 ::; ~ ::; 1. Since the lefthand num
ber is subharmonic the inequality holds in the whole strip, and
for ~ == t we obtain the desired result.
CHAPTER VI
..TEICHMULLER SPACES
A. Preliminaries
Let S be a Riemann surface whose universal covering-S is conformallY isomorphic to the upper haIfplane H. The
-cover transformations of S over S are represented by linear
transformations of H upon itself which form a discontinuous
subgroup r of the group n of all such transformations. We
can write
S r \ H (orbits)
and the canonical mapping
77: H -+ r\His a complex analytic projection of H on S.
Conjugate subgroups represent conformally equivalent
Riemann surfaces. Indeed, if ro == Bor B o- 1, B o ( n then
z -+ BOz maps orbits of r on orbits of r 0 (for BoAz
(Bo ABo -I) Boz ). Therefore Bo determines a one-one con
formal mapping of So == r o\ H on S.
Conversely, let there be given a topological mapping
g: So -+ S .-
It can be lifted to a topological mapping g: So -+ S which
117
118
obviously satisfies
QuasiconformaI Mappings
17 0 g g 0 170
and we hav.e g
·,I-g-lSo --.;:;..g • S
If g is conformal, so is g
10
= B0
1 Bo-1 .
The classes of conform ally equivalent Riemann surfaces
correspond to classes of conjugate discontinuous subgroups of
n (without elliptic fixpoints).
But even if g is not conformal, it is still true that
A = g 0 Ao 0 g-1 ( 1
whenever Ao ( 1 0 , for
17 0 A 17 0 g 0 Ao 0 g-1g 0 170 0 Ao 0 g-1g 0 17
00 g-1
17
In other words, g defines an isomorphism () such that
() - --1AO = g 0 AO 0 g
It is not quite unique, for we may replace g by Bog 0 Bowhere B ( 1, Bo ( 1
0, This changes () into ()' with
Ao()' B 0 i 0 (Bo Ao Bo- 1) 0 g 0 B-1
which means that we compose () with inner automorphisms of
10
and 1. We say that () and ()' are equivalent isomorphisms.
Teichmiiller 5paces 119
LEMMA. 81 and 82
determine equivalent isomorphisms
01 and 02 if and only if they are homotopic.
Proof. If 8l' 82 are homotopic they can be deformed
into each other via 8 (t), say, which depends continuously on t.
We can then find g(t) so that it varies continuously with t,
and since
has values in a discrete set it must actually be constant.
Conversely, suppose 81' 82 determine. equivalent 01'
°2 , By changing i 1, i 2 we may suppose that 01 = 02' and
hence
Define g (t, z) as the point which divides the noneuclidean
line segment between i1
(t, z) and i2(t, z) in the ratio
t: (I-t).
Becausei
1(Az) Ail(z)
(A A 0)i
2(Az) Ag
2(z) 0
it follows thati(t, Az) Ai(t, z)
) -) 1Hence 8(t = 17 0 g(t 0170- is a mapping from 50 to 5,
and we have shown that 81 and 82 are homotopic.
Definition of T(50
) (the Teichmiiller space):
Consider all pairs (5, t) where 5 is a Riemann surface
and f is a sense-preserving q.c. mapping of 50 onto 5. We
120 Quasiconformal Mappings
say that (Sl' f 1) ...... (S2' f2) if f2 0 11- 1 is homotopic to a
conformal mapping of Sl on S2' The equivalence classes are
the points of T (So), and (SO' I) is called the initial point of
T (So),-
Every f determines a q.c. mapping I of H on itself,
and thereby an isomorphism () of ro ' Two isomorphisms corre
spond to the same Teichmiiller point if and only if they differ by
an inner automorphism of n.The space T (So) has a natural Teichmiiller metric: the
distance of (Sl' 11)' (S2' f2) is log K where K is the small
est maximal dilatation of a mapping homotopic to f2
0 f1
- 1
Let us compare T(So) and T(Sl)' Let g be a q.c.
mapping of So on Sl' The mapping
(S, f) .... (S, fog)
induces a mapping of T(Sl) onto T(So)' Indeed, if (S, f) '"
(S', f '), then (S, fog) '" (S', f' 0 g). This mapping is clear
ly isometric.
B. Beltrami Differentials.-
A q.c. mapping f: So .... S induces a mapping f of H
on itself which satisfies- -
(1) f 0 Ao = A 0 f
for A ':. Ao() Conversely, if f satisfies (1) it induces a map
ping f.
* For typographical simplicity both mappings will henceforth be de
noted by f.
Teichmiiller Spaces 121
From (1) we obtain
(A' 0 fH z
(A' 0 f)f-z
and thus the complex dilatation /If satisfies
/If = (/If 0 Ao)Ao 'I Ao '
or
(2)
A measurable and essentially bounded function /l which
satisfies (2) for all Ao (:ra is called a Beltrami differential
with respect to roo Another way to express the condition is to
say that
is invariant under r o '
Conversely, if /If does satisfy (2), then
/If 0 A = /Ifo
and it follows that f 0 Ao is an analytic function of f, or that
A = f 0 Ao 0 [I
is analytic, and hence a linear transformation.
The linear space of Beltrami differentials will be denoted
by B (r0) and its open unit ball with respect to the L 00 norm
is denoted by B1
<ro ) .
For every I! ( B 1 (ro ) we know that there exists a cor
responding f/l which maps H on itself. We normalize it so
that it leaves 0, 1, 00 fixed. It is then unique.
We set
122 Quasiconiormal Mapping<s
and write r 11 for the corresponding group, 011 for the isomor
phism. Since Oil represents a point in Teichmiiller space we
have actually defined a mapping
R1 era) .... T(Sa)
It is clearly continuous from L"" to the Teichmiiller metric.
There is an obvious equivalence relation: Ill"'" 112 if
and 0112 are equivalent isomorphisms. It is very diffi-
cult to recognize this equivalence by direct comparison of III
and 112. Because we cannot solve the global problem we shall
be content to solve the local problem for infinitesimal deforma
tions.
Before embarking on this road we shall discuss a differ
ent approach which has several advantages. The mapping ill
was obtained by extending 11 to the lower halfplane by symme
try. If instead we extend 11 to be identically zero in the lower
halfplane we get a new mapping that we shall call ill (again
normalized by fixpoints at 0, 1, ",,).
Clearly, ill gives a q.c. mapping of the upper halfplane
and a conformal mapping of the lower halfplane. The real axis
is mapped on a line L that permits a q.c. reflection.
J1 \ \ \
IIIIII \\\\\L
Teichmiiller Spaces 123
It is again true that
A = f 0 A 0 f -1Il Il a Il
is conformal, and hence a linear transformation (it is conformal
*in nand n , and it is q.c., hence conformal). We get a new
*group rll
which is discontinuous on nUn . We call it a
Fuchsoid group. We also get a surface r" n = S and a q.c.Il _ *
mapping So -> S as well as a conformal mapping So -> r \ n_ Ilwhere So has the conjugate complex structure of So .
Observe that f Il and fIl are defined for all Il ( L 00
with 111111 00 < 1 (Il ( B 1) even when the group ro reduces to
the trivial group. Our first result is
LEMMA 1. f Il = fV on the real axis if and only if
*fIl == fv on the real axis, and hence in H .
Proof 1). If f = f on the real axis if and only ifIl v
fIl (H) and fv (H) are the same and therefore
f 0 (l1l)-1 = f 0 (lV)-1 ,Il v
for both are normalized conformal mappings of H on the same
region.
2) Suppose fll = fV on the real axis. The mapping
h = (lV)-1 0 fll reduces to the identity on the real axis, so
h can be extended to a q.c. mapping of the whole plane by put
*ting h (z) = z in H . Consider the q.c. mapping A =f" 0 h 0 (I )-1. In f (H\ A = f 0 (I )-1 is conformal.
v Il Il v IlIn 9H),
124 Quasiconformal Mappings
is conformal. Thus A is a linear transformation, and the nor-
malization makes it the identity. This means fv fll
in H*.
q.e.d.
We shall now make the assumption that ro is of the
first kind, which means that it is not discontinuous at any point
on the real axis. It is then known that the orbits on the real
axis are dense. In particular the fixpoints are dense. Under
these conditions we can prove
LEMMA 2. III and 112 in B <ro) determine the same
Teichmiiller point if and only if f1 = fll2 on the real axis.
Proof. If fill = fll2 on the real axis, then A Il1 =
A Il2 on the real axis and hence identically. Therefore (/1 =(/2, and III and 112 determine the same Teichmiiller point.
Conversely, suppose (/1 and (J1l2 are equivalent iso
morphisms. This means there is a linear transformation S f nsuch that
A Il20 S = S 0 AIl1 for all A in ro .
We conclude that S maps the fixpoints of A Il1 on the
fixpoints of A Il2 (attractive on attractive). Since they corre
spond to each other we see that
S 0 fill = fll2
on the real axis. By the normalization, S must be the identity.
q.e.d.
COROLLARY. If ro is of the first kind, III and 112
determine the same Teichmiiller point if and only if
f f In H*.III 11 2
Teichmiiller Spaces 125
This means we may identify the Teichmiiller space T (So)
*with the space of conformal mappings of H of the form fll'
11 ( B1(ro) .
An even better characterization is by consideration ofthe
Schwarzian derivative
f ", (f ")2It ,zl = _11_ _ ~ L11 f' L. f'
11 11
Let us recall its properties with respect to composition. Con
sider
and let primes mean differentiation. We get
F '(z)
F"F'
F ",
IF, zl
For a better formulation, let us denote the Schwarzian by
[£1. The formula reads
[£ 0 g] = ([tJ 0 g )(g ')2 + [g]
There are two special cases. For f = A, a linear transforma-
tion,
[A 0 g] = [g]
and for g = A
[£ 0 A] = ([£] 0 A )(A') 2
126 Quasiconformal Mappings
On setting ¢J1 = [fJ11,
(¢ 0 A) A ' 2 [f 0
J1 J1[fJ11
We see that ¢J1 satisfies
(¢J1 0 A )(A ')2 = ¢J1
which makes it a quadratic differential. (¢ dz2 is invariant).
The following theorem is due to Nehari:
*LEMMA 3. If f is schlicht in the halfplane H, then
\[fll ~ ~ y-2
b bProof. Suppose that F (<:) - <: + 1 + 2 + '" is
~ 7schlicht for 1<:1 > 1. The integral ii II <:1 =r 'PdF measures
the area enclosed by the image of 1<: 1 = r, and is therefore po
sitive. One computes
F' = 1 - ~ + ...<:2
'PdF2~ f1<:1 =r
l- f (, + b1 + ... )( 1 - ~ - ... )d <:21 ;: <:2
= 17 ( r 2 - ~ _ ... )r 2
It follows that Ib 11 ~ 1. (More economically, Ib 112 + 21 b 212
+ ... + n Ibnl2 + ... ~ 1, which is Bieberbach's FHichensatz.)
Note that
F"= 2b 1
<:36b
F'" = _ _ 1
t
+ ...
gives [F 1
Teichmiiller Spaces
6b1- - +... and hence
~4
127
lim 1,4[Fll $ 6, -+ 00
Set ~ = Uz = (z-zo)/(z-zo), Zo xo+ iyo'
Yo < O. Consider F (,) = f(U-l ,). Then
[E] = ([F] 0 U)U,2
Here
-2iyU' = __0_
(z-zo)2
2iyoU ,.., --
z-zo
U,2 '" _ ~ U 4 (z -+ zo) .4 Yo
On going to the limit we find
and then
q.e.d.
In view of the lemma it is natural to define a norm on
the quadratic differentials by
II¢> II = sup I¢> (z)1 y2
'128 Quasiconformal Mappings
C. ~ is Open.
We have defined a mapping /1 .... ¢/1 from the unit ball
B 1(r) to the space Q(r) of quadratic differentials with finite
norm. The image of B 1(r) under this mapping will be denoted
by ~ (r). It is our aim to show that ~ (r) is an open sub
set of Q(r).
The question makes sense even in the case where r is
the trivial group consisting only of the identity. In this case
the spaces will be denoted by B l' ~, Q.
THEOREM. ~ is an open subset of Q.
Clearly, ~ consists of the Schwarzians [f] of functions
f which are schlicht holomorphic in the lower halfplane and have
a q.c. extension to the upper halfplane. We know already that
all ¢ in ~ satisfy II¢ II s 312 .
LEMMA 1. Every holomorphic ¢ with II¢ II < Y:z 1S m
~.
Proof. We choose two linearly independent solutions of
the differential equation
(1) TI" = - Y:z ¢ TI
which we may normalize by TI~ Tl2 - TI; Till. It is easy to
check that f = TI/Tl2 satisfies [[1 = ¢. Observe that the
solutions of (1) have at most simple zeros. Hence f has at most
simple poles, and at other points f' f O.
We want to show that f is schlicht and has a q.c. exten
sion to the upper halfplane. To construct the extension we con
sider
TeichmiiIIer Spaces 129
111(Z) + (Z-Z)11{(Z)
112(z) + (z - Z)11;(Z)*(z (. H )
We remark first that the numerator and denominator do not vanish
simultaneously (because 11{ 112 - 112 111 = 1). Therefore F is
defined everywhere, but could be 00.
(112 + (Z"- z) 11; )2
Y:z¢(z-z)2
F-z
Simple computations yield
1
(2)
and thus
= Y:z¢(z-z) 2
The assumption implies \Fz I ::; k \Fz I for some k < 1.
Hence F is q.c. but sense-reversing.
z ( H
= { f(z)F(z)
f (z)
The extension will be defined by
z (. H*(3)
It must be shown that f is a q.c. 1-1 mapping.
This is easy if ¢ is very regular. Let us assume, spe
cifically, that ¢ remains analytic on the real axis, and that it
has a zero of at least order 4 at 00. It is immediate that f and
F agree on the real axis, and it is also evident that ; is local
ly schlicht. The assumption at 00 means that there are solu
tions of (1) whose power series expansions at 00 begin with 1
and z respectively. We have thus
130
F( z) =
Quasiconformal Mappings
111 a1z + b1 + 0(\:1)
112 a2z + b2 + 0(1:1)1. This gives
a1z + ~ + oqz 1-1)
a2
z + b2
+ O(lz 1-1)
which is also the limit of f.
Now the schlichtness of f follows by the monodromy~
theorem. We may of course, compose f with a linear transfor-
mation to make it normalized.
To prove the general case we use approximation. Put
S z = (2nz-i)j(iz+2n). Then S H* CC H* and S z ~ Zn n nfor n ~ 00. Set cPn(z) = cP (Snz) Sn '(z)2. We have
y 2IcPn(z)1 = IcP(Snz )IISn '(z)1 2y 2 < IcP(Snz )I(ImSn(z))2
and hence IIcPn ll ~ IIcP II· Now cPn has all the regularity pro-~ ~ *
perties. We can find mappings f with [f 1 = -!.. in Handn n ~n
uniformly bounded dilatations. By compactness there exists a
subsequence which converges to a solution f of the originaln
problem.
If cPn ~ cP it is not hard to see that a normalized solu
tion of 11" = - J.h. cPn1l con verges to a normalized solution of
11" = - J.h. cP1I' Therefore, if we choose the same normalizations
- - *-we may conclude that fa = f in H and in H . Hence f can
be extended by continuity to the real axis and is a solution of the
problem with- 2
J1 = -2cPY
Teichmiiller Spaces 131
But if 1> l Q(r) one verifies that /1 l B (r ), and we
conclude
LEMMA 2. The origin of Q(r) is an interior point of
Suppose now that 1>0 l ~ and [fo1 = 1>0 where fo =* *f /1
0We assume that f0 maps H on n, H on n . Then
the boundary curve L of n admits a q.c. reflection A. Ac
cording to Lemma 3 of Chapter IV D, we can choose A so that
corresponding euclidean lengths have bounded ratio. This means
that A is C (K )-q.c. and
C(K)-1 < IAzl < C(K)
provided that fo is K-q.c.
If [f] 1> the composition rule for Schwarzians gives
1> - 1>0 = If, follo,2
The noneuclidean metric in n* is such that
p(Old(' =~-y
Therefore, 111> - 1>0 II S l shows that g
l[gl(OI S l p«()2
f 0 fo-1 satisfies
For sufficiently small l we have to show that g has a
q.c. extension.
We set t/J [g] and determine normalized solutions 71 1'
132 Quasiconformal Mappings
This time we construct
g«() = 171«()/172«(), ( ( 0*
i «()171«(*) + «(-(*) 17 1 '«(*)
(( 0 .172«(*) + «(-(*)172 '«(*)
(Recall that (* = A«(L) Computation gives
/1 ;( ()'h. «(- (*)2 t/J «(*) A ,«()
((01+ 'h. «(- (*)2 t/J «(*) A(()
* < Cp«(*)-1 .But IA(I < 1\::1 ~ C(K) and 1(-( I Hence
1/1;1 ~( . C(K) < 1(4) 1 - ( . C(K)
as soon as ( is small enough.A
It must again be shown that g is continuous and schlicht.
There is no difficulty if L is analytic and t/J is analytic on L
with a zero of order four at 00.
The general case again requires an approximation argu-
ment. Let f = f 0 S where S is as in the proof ofnOn' n
Lemma 1, and let L n be the image of the real axis under In'
Ln is an analytic curve that admits a K-q.c. reflection, and t/Jis analytic on Ln'
The Poincare density p of 0 * = f (H*) is > p,n n n -so that It/J I ~ (p2 implies It/J I ~ (pn 2 . Therefore we can
construct a sequence of normalized q.c. mappings i such thatA *[gn1 = t/J in On and /1; satisfies the inequality (4). A
A n A
subsequence of the gn tends to a q.c. limit g which is equal
Teichmiiller Spaces 133
to g in n*. This proves Theorem 1. Since /l; satisfies (4)
we conclude:
COROLLARY. For every sequence of ¢n ( ~ converg
¢o = [f/lo 1 ( ~, there exist /l n .... /lo such that
= ¢n'
The proof is by writing ¢ = ~ 0 f 01 .
We come now to the most delicate part:
THEOREM 2. ~ (r) is an open subset of Q(r) .
.Remark: This was first proved by Bers. The idea of the
proof that follows is due to Clifford Earle.
Given any /lo ( B, we construct a mapping
f3 0 : ~ .... ~
as follows: Given ¢ ( ~ there exists a /l ( B 1 such that
¢ = ¢/l' With this /l we determine A by
(5) fA = f/l 0 ([ /l0)-1
and hence ¢A = ¢A .1
It is evident that f30
is 1-1, and it carries
zero. Moreover, f30
is continuous. For if ¢n .... ¢
have just proved the existence of /In .... /l such that
The corresponding ¢A converge to ¢A'n
and set f3 0 (¢) = ¢A' It is unique, for if ¢ = ¢ , then f /l/l /l /l1
has the same boundary values as fl. Hence fA has the sameA
boundary values as f 1
134 Quasiconformal Mappings
LEMMA. (Earle). <P1l
( Q(f') if and only if for every
A (f' there exists a linear transformation B such that
fll 0 A 0 ([11)-1 = B on R.
Proof. <P1l
( Q([') is equivalent to [ill 0 A] = [ill]'
and this is true if and only if f11 0 A 0 f11-1 = t, a linear
*transformation, in nil .1) If B exists, then
C = f 0 A 0 f -1 = f 0 (f11)-1 0 B 0 f 11 0 ([ )-111 11 11 11*
on f 11 (R). The first expression is holomorphic in nil' the
second in nil' Hence C is a linear transformation. *2) If C exists, then
B = f 11 0 A 0 ([ 11)-1 = f 11 0 f -1 0 C 0 f 0 (f11)-111 11
on R. But the last expression is a conformal mapping of H on
itself, hence a linear transformation. The lemma is proved.
Assume now that 110 ( B 1(f'). We find that f3 0 maps
~ (f') on ~ (f'1l 0 ) , for
fA 0 Allo 0 (fA)-1 = fll 0 A 0 ([11)-1 = All
From the lemma we infer that Q(f') n ~ is mapped on
Q(f'1l0 ) n ~. The origin has a neighborhood N in Q(f'1l0 )
which is contained in ~ (f'1l0 ). Write N = Q(f'1l 0 ) n No =
Q(f'1l 0) n ~ n Nowhere No is a neighborhood in ~. Then
f3-1(N) = Q([') n ~ n f3-1(No) = Q(f') n f3-1(No)
which is a neighborhood of <P in Q(f'). From N C ~ (f'1l 0 )110
* Because it is quasiconformal and conformal a.e.
Teichmiiller Spaces
we get f3 -1 (N) c ~ (r) and this proves that ¢ has a/1 0
neighborhood in Q(r) which is contained in ~ (r) .
135
q.e.d.
Conclusion:
If the group r of cover transformations of H over So
is of the first kind, the Teichmuller space T (S0) is identified
with ~ (r), an open subset of Q(r). The Teichmiiller metric
defines the same topology as the norm in Q(r) .
Consider the case where So is a compact Riemann sur
face of genus g > 1. Then Q(r) has complex dimension
3g- 3. Choose a basis ¢t' ... , ¢3~-3' Every ¢ f Q(r) is a
linear combination
¢ = 71¢1 + ... + 73_-3 ¢3~-3
with complex coefficients. We find that ~ (r) can be identi
fied with a bounded open set in C3~-3.
We can show, moreover, that the parametrization by 7 =
(71 , ... , 73~-3) defines the Riemann surfaces of genus g as a
holomorphic family of Riemann surfaces.
According to Kodaira and Spencer a holomorphic family
may be described as follows:
There is given an (n+ I)-dimensional complex manifold
V and a holomorphic mapping 17: V .... M on an n-dimensional
complex manifold M. Each fiber 17-1(,), 7 f M is a Riemann
surface.
The complex structures of V and M are related in the
following way: There is an open covering lual of V so that
136 Quasiconiormal Mappings
for each a there is given a holomorphic homeomorphism ha:
Ua ~ C x M connected by ¢af3 = ha 0 hf3-1 (for intersect
int Ua' U(3)' For any T ( M the restriction of ¢af3 to
hf3 (Ua n Uf3 n 7T-1 (r)) shall be complex analytic (in fact, these
functions determine the complex structure of 7T-1(T).)
In our case, M will be the set ~ (r). Every T ( ~ (r)
*determines a ¢ = [i,), and ill is uniquely defined in H .
*Hence Oil' Oil' and the group r Il are determined by T. To
emphasize the dependence on T we change the notations to ¢T'
*0T' rT
, etc. Since iT is determined in H by the differential
equation [iT1 = ¢T' we know that iT depends holomorphically
on the parameter T. For A ( r, the corresponding AT ( r T
*is determined by the condition iT 0 A = AT 0 iT in H . This
means that AT depends holomorphically on T, a fact that will
be important.
The Riemann surfaces in our family will be S (T) = 0/rT
and V will be their union. Thus the points of V are orbits
r l with <: ( 0T and T ( ~ (r). The projection 7T: V ~ M is
defined so that 7T-1(r) = S(r).
Consider a point r T <:0 in V. We pick a fixed <:0 fromo
the orbit and d_etermine an open neighborhood N (<:0) such that
N is compact, contained in 0T • and does not meet its imageso
under r T' The neighborhood N (s <:0' TO) of r T <:0 will con-o 0
sist of all r T<: such that II ¢T - ¢T II < E and <: ( N (<:0) ._ 0
Here E shall be so small that N does not meet its images under
r T' This is possible because AT is near AT0 when ¢T is
Teichmiiller Spaces 137
near ¢T' As a consequence, there is only one (( N «(0) no
r T( and the mapping h: r l.... «(, T) is well-defined in
N (E, (0' TO) ,
The neighborhoods N (E, (0' TO) shall be a base for the
topology of V. We assert, in addition, that the parameter map
pings h: r l.... «(, T) defined on the basic neighborhoods make
17: V .... M into a holomorphic family, Indeed, if two basic neigh
borhoods U0 and U1 intersect, then in U0 n Ul' hO(rT() ==
«(, r) and h1(rT() == (AT (, T) for some AT ( r
T• Hence the
mapping h1 0 ho-1 is given by «(, r) .... (AT(, T), which we
know is holomorphic in both T and (. It is evident that the pa
rameter mappings define a complex structure on V which agrees
with the conformal structure of the surfaces S (T) and makes the
mapping 17: V .... M holomorphic.
D. The Infinitesimal Approach.
We continue with a direct investigation of f 11, All that
does not make use of the mappings fll
,
For any function F(p.) and any y ( L C>O we set
limt .... 0
when the limit exists, and we omit the argument 11 when the de
rivative is taken for 11 = O. We are assuming that t is real.
We have already derived the representation
i[vl«() == - ~ f f v(z)R(z, ()dxdy
138 Quasiconformal Mappings
where
R (z, ,) = 1 _ U - ~z-' z z-1
We apply the formula to the symmetric case: v(Z") = i/ (z) ,
and we write more explicitly
(1) i[v 1(') = - ; ff v(z)R(z, ')dxdy
H
-; f f i/(z)R(Z", ,) dxdy
H.It is evident that f[v] is linear in the real sense, but
not in the complex sense. To obtain a complex linear functional
we form
(2)
and we find
(3) lI>[v](') = - ~ ff i/(z)R(Z", ')dxdy
H
It is holomorphic for 'l H. Its third derivative is
17 ff i/(z) dxdy(Z" _,)4
H
If v l B (r) one verifies that ¢ is a quadratic differential
(¢ l Q(r)).
(4)
Fromfll(Az) = All fll(z)
with Il
(5)
t v we obtain after differentiation
i[v] 0 A = A[v] + A'f[v]
Teichmiiller Spaces 139
and hence Az
(the existence of A requires some, but not much, proof). Since
i [v lz = v differentiation of (5) gives
(v 0 A) A' = A- + A'vz
o because v ( B (i). We conclude that the. .A are analytic functions. Moreover, AIA' is real on the real
axis and can hence be extended by symmetry to the whole plane.
The explicit formula for i[v 1 shows that it is o(lz 12 ) at 00.
The apparent singularity of AIA' at A-loo is therefore re
movable, and there is at most a double pole at 00. We conclude
that
(6)
where PAis a second degree polynomial. From
(AlA
2)
(AlA
2)'
we deduce that
(AI 0 A2
) + (AI' 0 A2
) A2
(AI' 0 A2
) A2
'
(7)A'
2
We shall say that v is trivial, and we write v ( N (i) ,
if all A[v] = O. There is a whole slew of equivalent condi-
tions:
LEMMA 1. The following conditions are all equivalent.
a) A[v] = 0 for all A ( i, c) i[v] = 0 on R,
b) PA = 0 for all A ( i, d) cI> [v] == 0,
e) ¢[v1 == 0,
f) II v¢ dxdy = 0 for all ¢ ( Q(['). *i\H
140 Quasiconformal Mappings
Proof. a) < >b) by (6). c) > a) by (5). Con-
versely, if A '" 0 then since ;(0) = 0 it follows that ;(A 0)
= 0 for all A. .These points are dense on R (we are assuming
that the group is of the first kind) and hence, by continuity, f = 0
on R.
The definition of cI>, together with (5) and (6), gives
cI> ;, A - cI> = PA [v] + i PA [i v J
If cI> =0 0, it follows that PA [v] + i PA[i v] '" 0 on R. But
both polynomials are real on R, hence identically zero, so
d ) > b). Conversely, if f [v] = 0 on R, then cI> is pure-
ly imaginary on R and can be extended to be analytic in the
whole plane. Since cI> '" 0 (j z 12) it is a first degree polynomial.
But it vanishes at 0 and 1. Hence cI> =0 0 and c) > d).
Since cI> '" '" ¢, d) > e). Conversely, if ¢ = 0,
cI> is a polynomial and we reason as above to conclude cI> =0 o.Condition f) is the most important one. We prove it
only for the case that r\H is compact, and we represent it as
a compact fundamental polygon S with matched sides. If A =.. .0, then f 0 A = A'f. We know further that f z = v. By
Stokes' formula we find
f fv¢ dxdy
s- ii f f;z ¢ dz dz
s2~ f f ¢ dz
as
* For r\H non-compact, the condition shall hold for all ¢ f Q(r)
such that ffr\H 1¢ I dxdy < 00.
Teichmiiller Spaces 141
But the condition means that
f ¢ dz =
f/dz
fdz .
is invariant. Hence
is invariant, and the boundary integral vanishes.
To prove the opposite, consider a real ~ so that (3)
takes the form
~ -~ !!v (z ) R (z, ~) dx dy
H
We introduce a Poincare O-series
From the fact that IH IR (z, ~) Idx dy < 00 it is quite easy to
deduce that the series converges. We obtain now
~ = - ~ L: !f v(z)R(z, ~)dxdyA(S)
- ~ L: J!v(Az)R(Az, ~)IA '(z)1 2 dxdy
S
-~ L: Jf v(z)R(Az, ~)A'(z)2dxdy
= - ~ ff v !/J dx dy
which is zero if f) holds. Hence <I> is identically zero. The
lemma is proved.
We need the following lemma.
142
Then
(8)
Quasiconformal Mappings
LEMMA 2. Suppose ¢ is analytic in Hand
sup I¢ ly2 < 00 •
12 II p(z)y2 dxdy .17 (z _~)4
H
For the proof we note that
_ V. (z-z)2 = _ %[_1__ 2(z-C) +(z _~)4 ~z _~)2 (:z _~)3
_ 14~ [__1_ + z-( _ ~ (z-<:yld:Z (:z-~) (z_~)2 (z-~)3J
Assume first that ¢ is still analytic on R. Then integration
by parts gives
12 I I ¢ i dx dy = _ L17 (:z _ ~)4 217i
H RIt is easy to complete the proof by applying this formula to
¢(z+iE), E > 0, for the hypothesis guarantees absolute
convergence.
Compare formula (8) with (4). We have defined an anti
linear mapping
A: v ~ ¢[v1from B (r) to Q(r). On the other hand we may define
A*: ¢ ~ -¢ iwhich is a mapping from Q(r) to B (r). Lemma 2 tells us
*that AA is the identity.
By Lemma 1, e), v f N(r) if and only if Av = o.From AA* = I we conclude that
TeichmrlIIer Spaces 143
*v - A Av £ N (r) .
In other words, v is equivalent modN(r) to -¢;[vli. Of
*course this is the only A ¢ which is equivalent to v, for if
- ¢ y2 £ N (r) then fJhH 1¢ (z ) 12 y2 dx dy = 0, hence ¢ =
o.We conclude:
A establishes an isomorphism of B <r)/N (r) on Q(r) .
The inverse isomorphism of Q(i) on B(r)/N(r) is given by
A*.
A compact surface S, of genus g > Q, determines a
group i generated by linear transformations AI' ... , A2el
which satisfy
(9) ~ A2
Al
- l A2-1 ... A
2el-
lA
2elA
2el-
2-1 A
2el- l = I.
We say that lA l , ..• , A2el
l is a canonical set. If it belongs to
a surface, Al and A2 have four distinct fixpoints. By passing
to a conjugate subgroup we can make Al have fixpoints at 0,
00 and A2
to have a fixpoint at 1. When this is so we say that
the generating system is normalized.
Set
V set of normalized canonical systems,
T set of normalized canonical systems
that come from a surface of genus g.
It can be shown that V is a real analytic manifold of
dimension 6g - 6 . We are going to prove that T is an open sub
set of V, and that it carries a natural complex structure.
144 Quasiconformal Mappings
o all j
Let S determine r with normalized generators (A)
(A 1, ... , A2~)' We choose a basis v 1' ... , v3~-3 of
B (r)lN (r) and set
vCT) = T1V1 + .,. + T3~3 v3
and Tk = tk + itk'. For small T we obtain a system (A)V(T)
( T. The points of the manifold V near (A) can be expressed
by local parameters u1' ... , u6~6' and the mapping T ~ (A )V(T)
takes the form
uj = h j (tl' t 1 ', ... , t3~-3' t;g-3) .
We have to show
1) The h. are continuously differentiable,}
2) The Jacobian is f. 0 at T = O.
1) has already been proved. The coefficients of the A k
are differentiable functions of the uk' Therefore, if all Uk[v]
are 0, so are all A k [V] and hence all A[v]. Observe that
ah. ahj} = U. [Vk] - = u}.[ivk]
atk
} atk '
If the Jacobian at the origin were zero there would exist real
numbers (k' TJk such that
ah. ah j
~ (k at~ + TJk atk
'
This would mean
and hence
Hence ~ «(k + iTJk) v k ( N (r), and this is possible only if
Teichmiiller Spaces 145
A/1 .
all ~k' TJ k = o. We have proved that the Jacobian does not
vanish.
The proof shows that T is an open subset of V. It al
so follows that if 11/111 is small enough, then there exist unique
complex numbers T1(/1), , T3g-3(/1) such that
AT1 (/1)v1+ + T3/1-3 (/1)v3/1-3
Set /1 = tp and differentiate with respect to t for t = o.We obtain
A[~1[plv1 + ... + ~3~-3[p]v3g-31 A[p]
This implies
~1[plv1 + ... + ~3~-3[plv3~-3 - P £. N(r) .
Replace p by i p. We can then eliminate the term p to obtain
3~-3
L: (~k[P] + i ~k[ip])vk £. N(r)
1
from which it follows that
~k[i P1 = i ~k[P ]
In other words, the ~k are complex linear functionals,
which means that the Tk(/1) are differentiable in the complex
sense at /1 = o.From this we can prove that the coordinate mappings
(A/1) .... (T1(/1), ... , T3g-3(/1)) define a complex structure on T.
Indeed, we must show that on overlapping neighborhoods the
coordinates are analytic functions, and it suffices to show this
at the origin of a given coordinate system. Take /1 (T) =
l TjVj and /10 = /1(TO) near zero in B(r). Define A(r) in
146 Quasiconformal Mappings
B <r 11 0 ) by f 11 (r) = f'\ (t)
C in Chapter I, we have
11 .o f o. By the formulas of section
so that A depends analytically on T.
Now choose a basis A1
, ... , A3~-3 for B(r llo )/ N(r llo) . .;
Near (A 11 0) we have coordinate functions a 1(A), ... , a3~_3(A).
This means that for T near TO we can write uniquely
(AIl(r)) = (AIlO)A(r)) = (AIlO/a/A(r)) A)Because a/A) is complex analytic at A ". 0, we conclude )
that a i is a complex analytic function of T at TO' This is
exactly what we required.
.J
VAN NOSTRAND MATHEMATICAL STUDIES
VAN NOSTRAND MATHEMATICAL STUDIES are paperbacksfocusing on the living and growing aspects of mathematics. Theyare not reprints, but original publications. They are intended toprovide a setting for experimental, heuristic and informal writingin mathematics that may be research or may be exposition.
Under the editorship of Paul R. Halmos and Frederick W. Gehring,lecture notes, trial manuscripts, and other informal mathematicalstudies will be published in inexpensive, paperback format.
PAUL R. HALMOS graduated from the University of Illinois in1934. He took his master's degree in 1935 and his doctorate in 1938,and stayed on for another year to teach at the University. He thenspent three years at the Institute for Advanced Study, two of themas assistant to John von Neumann. He has returned to the Instituteon two subsequent occasions, in 1947-48 as a Guggenheim fellowand in 1957-58 on sabbatical leave. After teaching at SyracuseUniversity for three years he joined the faculty of the Universityof Chicago in 1946; he is now Professor of Mathematics at theUniversity of Michigan.
FREDERICK W. GEHRING attended The University of Michigan from 1943 to 1949, and received his doctorate from the University of Cambridge in 1952 after three years' study in England. Hespent the next three years as a Benjamin Pierce Instructor atHarvard University, and returned to The University of Michiganin 1955. From 1958-59 Professor Gehring was a Guggenheim andFulbright fellow at the University of Helsinki, Finland, and thefollowing year was an N.S.F. fellow at the Eidgenossische Technische Hochshule in Zurich, Switzerland. He is now Professor ofMathematics at the University of Michigan.
A VAN NOSTRAND MATHEMATICAL STUDY
under the general editorship of
PAUL R. HALMOS and FREDERICK W. GEHRING
The University of Michigan
•About this book:
In essentially self-contained exposition, this book presents the firsttreatment in English of a subject rapidly growing in its interest andimportance to mathematicians: German works in the same fieldoverlap only partly. Intended for advanced graduate study, thisvolume derives from lectures delivered by Professor Ahlfors atHarvard University during the Spring Term of 1964.
•The Author:
LARS V. AHLFORS is William Caspan Graustein Professor ofMathematics at Harvard University. As well as serving as chairmanof Harvard's department of mathematics from 1948 to 1950, theauthor has held professorships at the University of Helsingforsand the University of Zurich. A renowned mathematician, Professor Ahlfors was awarded the Field's medal for mathematicalresearch at the 1936 International Congress of Mathematicians; heis a member of numerous distinguished societies, including theNational Academy of Science. Professor Ahlfors has contributedmuch to the development of conformal mapping,. Riemann surfaces, and other branches of the theory of functions of a complexvariable: he is the author of Complex Analysis and, with L. Sario, ofRiemann Surfaces.
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