Conformal Geometry and Quasiregular Mappings

Lecture Notes in Mathematics Edited by A. Dold and B. Eckmann

1319 Matti Vuorinen

Conformal Geometry and Quasiregular Mappings I

Springer-Verla9 Berlin Heidelberg NewYork London Paris Tokyo

Author

Matti Vuorinen Department of Mathematics, University of Helsinki Hallitusk. 15, 0 0 1 0 0 Helsinki, Finland

Mathematics Subject Classification (1980): 3 0 C 6 0 ISBN 3-540-19342-1 Springer-Verlag Berlin Heidelberg N e w York ISBN 0-387-19342-1 Springer-Verlag N e w York Berlin Heidelberg

This work is subject to copyright. All rights are reserved, whether the whole or part of the material is concerned, specifically the rights of translation, reprinting, re-use of illustrations, recitation, broadcasting, reproduction on microfilms or in other ways, and storage in data banks. Duplication of this publication or parts thereof is only permitted under the provisions of the German Copyright Law of September 9, 1965, in its version of June 24, 1985, and a copyright fee must always be paid. Violations fall under the prosecution act of the German Copyright Law. © Springer-Verlag Berlin Heidelberg 1988 Printed in Germany Printing and binding: Druckhaus Beltz, Hemsbach/Bergstr. 2146/3140-543210

Contents Preface .................................................................

V

Introduction ..........................................................

VII

A s u r v e y of q u a s i r e g u l a r m a p p i n g s . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Notation and terminology ............................................ Chapter

I.

CONFORMAL

GEOMETRY

..................................

IX XVI 1

1.

M S b i u s t r a n s f o r m a t i o n s in n - s p a c e

2.

Hyperbolic geometry ...................................................

19

3.

Quasihypcrbolic geometry ..............................................

33

4.

Some covering problems ................................................

41

Chapter

II.

MODULUS

AND

.....................................

CAPACITY

..............................

1

48

5.

T h e m o d u l u s of a c u r v e f a m i l y . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

48

6.

T h e m o d u l u s as a set f u n c t i o n

.........................................

72

7.

T h e c a p a c i t y of a c o n d e n s e r . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

81

8.

Conformal invariants

Chapter

III.

.................................................

QUASIREGULAR

MAPPINGS

..........................

102 120

9.

T o p o l o g i c a l p r o p e r t i e s of d i s c r e t e o p e n m a p p i n g s . . . . . . . . . . . . . . . . . . . . . .

121

10.

S o m e p r o p e r t i e s of q u a s i r e g u l a r m a p p i n g s . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

127

11.

Distortion theory .....................................................

137

12.

Uniform continuity properties .........................................

152

13.

Normal quasiregular mappings ........................................

162

Chapter

IV.

BOUNDARY

BEHAVIOR

................................. ..........................

173

14.

S o m e p r o p e r t i e s of q u a s i c o n f o r m a l m a p p i n g s

15.

LindelSf-type theorems ...............................................

181

16.

Dirichlet-finite mappings

187

.............................................

174

Some open problems ..................................................

193

Bibliography ..........................................................

194

Index .................................................................

208

Preface

This book is based on my lectures on quasiregular mappings in the euclidean n space 1~n given at the University of Helsinki in 1986. It is assumed that the reader is familiar with basic real analysis or with some basic facts about quasiconformal mappings (an excellent reference is pp. 1-50 in J. V~is~l~'s book [V7]), but otherwise I have tried to make the text as self-contained and easily accessible as possible. For the reader's convenience and for the sake of easy reference I have included without proof most of those results from IV7] which will be exploited here. I have also included a brief review of those properties of MSbius transformations in R n which will be used throughout. In order to make the text more useful for students I have included nearly a hundred exercises, which are scattered throughout the book.

They are of varying

difficulty, with hints for solution provided for some. For specialists in the field I have included a list of open problems at the end of the book. The bibliography contains, besides references, additional items which axe closely related to the subject matter of this book. From its beginning twenty years ago the subject of quasiregulax mappings in n space has developed into an extensive mathematical theory having connections with PDE theory, calculus of variations, non-linear potential theory, and especially geometric function theory and quasiconformal mapping theory. Excellent contributions to this subject have been made, in particular, by the following five mathematicians: F. W. Gehring, O. Maxtio, Yu. G. Reshetnyak, S. Rickman, and J. V~is~l~. The subject matter of this book relies heavily on their work. I am indebted to them not only for their scientific contributions but also for the help and advice they have given me during the various stages of my work. It was O. Maxtio who suggested I start writing this book. The writing was made possible by a research fellowship of the Academy of Finland, which I held in 1979-85. A draft for the text was finished in the

VI fall of 1982 during my stay at the Mittag-Leffier Institute in Sweden. The following mathematicians have provided their generous help by checking various versions of the manuscript, pointing out errors, and contributing corrections: J. Heinonen, G. D. Anderson, and M. K. Vamanamurthy. Useful remarks were also made by J. Ferrand and P. J£rvi. At the final stage I have had the good fortune to work with J. Kankaanp££, who prepared the final version of the text using the TEX system of D. E. Knuth and improved the text in various ways. The previewer program for TEX written by A. Hohti was very helpful in the course of this project. The work of Kankaanp~£ was supported by a grant of the Academy of Finland. Hohti and O. Kanerva have provided their generous assistance in the use of the TEX system. Helsinki October 1987 Matti Vuorinen

Introduction Quasiconformal and quasiregular mappings in R '~ are natural generalizations of conformal and analytic functions of one complex variable, respectively. In the twodimensional case these mappings were introduced by H. Gr6tzsch [GR0] in 1928 and the higher-dimensional case was first studied by M. A. Lavrent'ev [LAV] in 1938. Far-reaching results were obtained also by O. Teichmfiller [TE] and L. V. Ahlfors [A1]. The systematic study of quasiconformal mappings in t t '~ was begun by F. W. Gehring [G1] and J. V~is£1~ [V1] in 1961, and the study of quasiregular mappings by Yu. G. Reshetnyak in 1966 [R1]. In a highly significant series of papers published in 1966-69 Reshetnyak proved the fundamental properties of quasiregular mappings by exploiting tools from differential geometry, non-linear PDE theory, and the theory of Sobolev spaces. In 1969-72 O. Martio, S. Rickman and J. V~.is£1~ ([MRV1]-[MRV3], [VS]) gave a second approach to the theory of quasiregular mappings which was based on some results of Reshetnyak, most notably on the fact that a non-constant quasiregular mapping is discrete and open. On the other hand, their approach made use of tools from the theory of quasiconformal mappings, such as curve families and moduli of curve families. The extremal length and modulus of a curve family were introduced by L. V. Ahlfors and A. Beurling in their celebrated paper [AB] on conformal invariants in 1950. A third approach was suggested by B. Bojarski and T. Iwaniec [BI2] in 1983. Their methods are real analytic in nature and largely independent of Reshetnyak's work. In this book a fourth approach is suggested, which is a ramification of the curve family method in [MRV1]-[MRV3] and in which conformal invariants play a central role. Each of the above three approaches yields a theory covering the whole spectrum of results of the theory of quasiregular mappings. So far the fourth approach of this book, introduced by the author in [VU10]-[VU13] has been applied mainly to distortion theory. This work has been continued in [AVV1], [AVV2], [FV], [LEVU], where some

VIII quantitative distortion theorems were discovered. These papers also include results which are sharp as the maximal dilatation K approaches 1. Perhaps surprisingly it also turned out in [AVV1] that to a considerable degree a distortion theory can be developed independently of the dimension n . In short, this fourth approach consists of the following. In a domain G in 1~n one studies two conformal invariants

),a(x, y)

and

#a(x, y)

associated with a pair of

points x and y in G . These invariants were apparently first introduced by J. Ferrand [LF2] in 1973 and I. S. G£1 [G.~L] in 1960, respectively. The systematic application of these invariants was begun by the author in a recent series of papers [VU10][VU13]. By their definitions, ~G(x, y) and ]~c(x, y) are solutions of certain extremal problems associated with the moduli of some curve families.

To derive distortion

theorems exploiting AC and ~ c we require two things: (a) the quasiinvariance of moduli of curve families under quasiconformal and quasiregular mappings ([MRV1]-[MRV3]), (b) quantitative estimates for )'G and /~G in terms of "geometric quantities". For a general domain G in l~ '~ these invariants have no explicit expression. In the particular case G -- B n such an expression is known for b o t h ~ c and /zc , and for G = 1~" \ {0} good two-sided estimates for the invariant AG will be obtained. We then generalize these results for a wider class of domains. In the two-dimensional case we can obtain the exact value of AR2\{0} (x, y) if we use the solution of a classical extremal problem of geometric function theory, the modulus problem of 0 . Teichmfller [KU, Oh. V]. This book is divided into four chapters. Chapter I deals with geometric preliminaries, including a discussion of MSbius transformations.

In Chapter II we study

certain conformal invariants and apply these results in Chapter III to obtain distortion theorems, the main theme of this book. The final part, Chapter IV, is a brief discussion of some b o u n d a r y properties of quasiconformal mappings.

A survey of quasiregular mappings The goal of this survey is to give the reader a brief overview of the theory of quasiconformal (qc) and quasiregular (qr) mappings and of some related topics. We shall also try to indicate the many ways in which the classical function theory of one complex variable (CFT) is related to quasiregular mapping theory (QRT) in R '~ as well as to point out some differences between C F T and QRT. This survey deals chiefly with results not discussed elsewhere in the book. For a general orientation the reader is urged to read some of the existing excellent surveys [A4], [L1], [L2], [BAM], [G4], [GS]-[G10], [I], and [V10], of which the first three deal with the two-dimensional case and the others the multidimensional case. Several open problems are listed in the surveys of A. Baernstein and J. Manfredi [BAM], F. W. Gehring [G9], and J. V~tis£1~. IV10]. 1. F o u n d a t i o n s .

In his pioneering papers [R1]-[R10], in which were laid the

foundations of QRT, Yu. G. Reshetnyak successfully combined the powerful analytic machinery of P D E ' s in the sense of Sobolev with some geometric ideas from CFT. Reshetnyak showed that the basic properties of qr mappings can be derived from the properties of the function ul(x) = log [ f ( z ) l , where f

is qr.

He proved that uf

satisfies a non-linear elliptic P D E which for n = 2 is linear and coincides with the Laplace equation. It follows from the work of J. Moser [MOS], F. John - L. Nirenberg, and J. Serrin [SE] that the solutions of this equation satisfy the Harnack inequality in

{ z: uf(z) > 0 }. Note that if f is analytic, then log If(z)[ has a similar role in CFT. Obviously only a part of C F T can be carried over to QRT: for instance power series expansions and the Riemann mapping theorem have no n-dimensional counterpart. 2.

Quasiconformal

balls.

By Riemann's mapping theorem a simply-con-

nected plane domain with more than one boundary point can be m a p p e d conformatly onto the unit disk B 2 . Liouville's theorem says that the only conformal mappings in R n , n > 3, are the Mhbius transformations. Thus Riemann's mapping theorem has

X no c o u n t e r p a r t in 1~~ when n > 3 : since MSbius transformations preserve spheres, the unit ball B '~ in R n can be m a p p e d conformally only onto another ball or a halfspace. A quasiconformal counterpart of the Riemann m a p p i n g t h e o r e m is also false: for n >_ 3 there are J o r d a n domains in R '~ h o m e o m o r p h i c to B n which cannot be m a p p e d quasiconformally onto B n although their complements can be so m a p p e d . Also, the unit ball B n , n _> 3 , can be :mapped quasiconformally onto a domain with non-accessible b o u n d a r y points, as shown by Gehring and V~is£1£ in [GV1]. This fact shows t h a t for each n > 3 the quasiconformal mappings in R n constitute a class of m a p p i n g s substantially larger t h a n the class of MSbius transformations. 3.

Topological properties.

A basic fact f r o m C F T is t h a t a n o n - c o n s t a n t

analytic function is discrete (i.e. point-inverses f - l ( y )

are discrete sets if f analytic)

and open (i.e. f A is open whenever f is analytic and A is open). By Reshetnyak's f u n d a m e n t a l work a similar result holds in QRT. Next let B /

denote the set of all

points where f fails to be a local h o m e o m o r p h i s m . In C F T it is a basic fact t h a t B / is a discrete set if f is n o n - c o n s t a n t and analytic. A topological difference between the cases n = 2 and n > 3 is t h a t B.f is never discrete if f is qr in R n , n _> 3 , and

Bf ~ 0. By a result of A. V. ChernavskiY dim B / = dim f B / < n - 2 if f : G --~ R n ( G a d o m a i n in R n ) is discrete and open ([CHE1], [CHE2], IV5]). Also the metric properties are different: if n ~- 2 and f is analytic, then cap B / =

0 , while if n > 3

and f is qr in R n , then either Bf = 0 or c a p B f > 0 (for the definition of the capacity see Section 7; see also [R10], [MR2], [$2]). By a result of S. Stoilow a qr m a p p i n g f of B 2 onto a domain D can be represented as f = g o h ,

where h is a qc m a p p i n g of B 2 onto itself and g is an

analytic function ([LV2]). Thus the powerful two-dimensional arsenal of C F T is applicable to the "analytic part" of f , greatly facilitating the s t u d y of two-dimensional qr mappings. No such result is known for the multidimensional case. A n o t h e r result which is known only for the dimension n = 2 is the powerful existence t h e o r e m for plane quasiconformal mappings (cf. [LV2]). In the multidimensional case there is no general existence t h e o r e m and all examples of qc and qr m a p p i n g s known to the author are based on direct constructions.

In the qc case several ex-

amples are given in [GV1]. In the qr case a basic m a p p i n g is the winding mapping, given in the cylindrical coordinates ( r , ~ , z )

by ( r , ~ , z ) ~

(r,k~,z),

k a positive

integer [MRV1]. An i m p o r t a n t example of a qr m a p p i n g is the so called Zorich m a p ping ([ZO1], [MSR1]) and its various generalizations due to Rickman (of. e.g. [Rill]).

XI Additional examples are given in [R12, pp. 27-32], [MSR2], and [MSR3]. One c a n a l s o construct new qc (qr) mappings by composing qc (qr) mappings. 4. Q u a s i c o n f o r m a l i t y v e r s u s L i p s e h i t z a n d H S l d e r m a p s . phism f: G ---* f G ,

A homeomor-

G C R '~ , is said to be K - q c if

(*)

M ( F ) / K < M ( f r ) _< K M ( F )

for all curve families r in G where M(F) is the modulus of F (see Section 5 below). This definition is somewhat implicit because the concept of modulus is rather complicated. To clarify the geometric consequences of (*) let us point out t h a t H(x,f)

= limsup: r--.o

"

If(x)

for all x E G , where d ( n , K )

f(z)l Iz-

f(Y)l :

l=r= lY- I } <

< oo depends only on n and K .

d(n,g) A well-known

property of conformal mappings can be expressed by stating that H ( x , f ) = 1 for g = 1 (while, unfortunately, d(n, K ) 74 1 as g --* 1 for n > 3 , cf. p. 193). A homeomorphism f: G ~ f G satisfying t~: - Y l / L < I](x) - fCY)l < Llx - Yl for all x, y E G , is called L-bilipschitz. It is easy to show t h a t L-bilipschitz maps are L ~('~-l)-qc. But the converse is false. The standard counterexample is the qc radial stretching x ~-* t x I ' ~ - l x ,

x E B n , a C (0, 1), which is not bilipschitz. All qc

mappings are, however, locally HSlder continuous; e.g., if f : B n --* B n is K - q c , then

for I 1, lYl < ½ If(x) -- f(Y)l <- A ( n , g ) where A ( n , K )

Ix - yl °' , ot = K U ( 1 - " ) ,

depends only on n and K . For details see Section 11 below.

Let B , OC, and )/ denote the classes of all bilipschitz, qc, and locally H61der continuous mappings.

By what was said above the inclusions B C •C

c )/ hold,

where the first inclusion is strict. Simple examples can be constructed to show t h a t also the second inclusion is strict. Many fundamental features of qc mappings are related to the strictness of the inclusion B C ~ C . For instance, one can construct qc mappings such t h a t the image of a segment is not even locally rectifiable and such that the Hausdorff dimension of a set is different from the Hausdorff dimension of its image ([GV2]). The HSlder continuity of qc mappings on the boundary of the domain of definition has been thoroughly investigated by R. N£kki and B. Palka in a series of papers (see e.g. [NP]).

XII 5.

LP-integrability.

A K - q c mapping has the property that its partial

derivatives are locally Ln-integrable.

Moreover, these partial derivatives are even

locally LP-integrable for some p = p ( n , K ) > n .

This was proved by B. Bojarski

for n = 2 and generalized to the multidimensional case by F. W. Gehring [Gh]. The m e t h o d of proof in [Gh], which makes use of so-called reverse Hhlder inequalities, has found several applications to the calculus of variations and to P D E theory ([GIA], [STR1], [STR2]). Some estimates dealing with the case K --* 1 were given by Yu. G. Reshetnyak in [R13] (see also [GUR]). In connection with qr mappings the integrability has also been discussed by n. nojarski and W. Iwaniec [BI2] and O. Martio [M2]. 6.

Stability theory.

The stability theory of K - q c and K - q r mappings in

R '~ in the sense of this book deals with the quantitative description of the behavior of these mappings when K --~ 1. Roughly speaking, the expectation is that the mapping should become more or less like a conformal mapping under this passage to the limit. By Liouville's classical theorem the two cases n _ 3 and n > 2 are substantially different, and we shall therefore consider them separately. Case A. n > 3. Liouville's classical theorem, which was mentioned above in con-

nection with quasiconformal balls, requires that the mappings be sufficiently smooth ( g3 is enough). By deep results of F. W. Gehring [G2] and Yu. G. Reshetnyak ([R3], R13]) the differentiability assumption can be replaced by the requirement that the mapping be 1-qc or even 1-qr. Recently a different proof was given by B. Bojarski and T. Iwaniec [BI1]. Next, as shown by Reshetnyak ([R3], [Rll], [R13]), one can show that as K --~ 1 any K - q r mapping must approach a Mhbius transformation. For the exact statement of these results the reader is referred to [R13]. The methods of [R13] involve normal family arguments. Unfortunately the "speed" with which the convergence to Mhbius transformations takes place as K --~ 1 is usually only qualitatively defined and no quantitative estimate for the "speed" in terms of K and n are known. Additional results have been proved by A. P. Kopylov [KO], J. Sarvas [$3], V. I. Semenov [SEM1], D. A. Wrotsenko [TR], and others. Case B. n _> 2. The paucity of such distortion theorems for K - q c or K - q r

mappings in R n , which are asymptotically sharp as K -~ 1 and provide quantitative distortion estimates, may be startling when compared to the rich qualitative theory described above in Case A. This state of affairs is due partly to the fact that to prove such results one needs to find sharp estimates for certain little-known special functions. Several results with explicit bounds dealing with the case K --~ 1 have

XIII

been proved by V. I. Semenov in several papers (e.g. [SEM1], [SEM2]). Some other distortion theorems of this kind together with associated estimates of special functions were developed in [VU10], [VUll], [AVV1]-[AVV3], [FV]. A survey including some two-dimensional results of this kind is given in [HELl. See also the important paper [AG] of S. Agard. 7. D i r i c h l e t i n t e g r a l m i n i m i z i n g p r o p e r t y .

Let G be a domain in R 2 and

v: G -~ R harmonic. For a domain D c G with D C G let S , , ( D ) = { u: G ~

R:

ulOD=

vlaD,

u • C2(G) }.

A well-known extremal property of the class of harmonic functions, the Dirichlet principle, states that they minimize the Dirichlet integral IT, pp. 9-14]. In the above notation this means that

/DlVvl

2em =

inf

[ lwl2em.

ueF, (D) JD

Analogous Dirichlet integral minimizing properties hold as well for the solutions of the non-linear elliptic PDE's which arise in connection with qr mappings. This important fact was proved by Yu. G. Reshetnyak [R5]. In [MIK3] V. M. Miklyukov continued this research and studied subsolutions of these PDE's. In a series of papers S. Granlund, P. Lindqvist, and O. Martio have considerably extended these results ([GLMll-[GLM3], [LI1], [LIM], [M6]). They have also found a unified approach to some function-theoretic parts of QRT including, in particular, the harmonic measure. See also [HMA]. Further results were obtained by J. Heinonen and T. Kilpel£inen. 8.

Value d i s t r i b u t i o n t h e o r y .

In 1967 V. A. Zorich [ZO1] asked whether

Picard's theorem holds for spatial qr mappings and whether the value distribution theory of Nevantinna [NE] has a counterpart in this context. These questions have been answered by S. Rickman in a series of papers [RI3]-[RIll], the main results being reviewed in [RI6] and [RI9]. Additional results appear in [MAWR] as well as in [BEll. An analogue of Pieard's theorem was published in [RI4]. One of the methods used in [RI4] is a two-constants theorem for qr mappings (analogous to the two-constants theorem of CFT [NED, which Rickman derives from an estimate for the solutions of certain non-linear elliptic PDE's due to V. G. Maz'ya [MAZ1]. An alternative proof which only makes use of curve family methods is given in [RIg].

XIV 9. Special classes of d o m a i n s .

The standard domain, in which most of the

CFT is developed, is the unit disk. During the past ten years an increasing number of papers have been published in which function-theory on a more general domain arises in a natural way. In the early 1960's two highly significant studies of this kind appeared in quite different contexts authored by L. V. Ahlfors and F. John, respectively. Ahlfors studied domains bounded by quasicircles, i. e. images of the usual circle under a qc mapping of R 2 , and found remarkable properties of these domains. In a paper related to elasticity properties of materials John introduced a class of domains, nowadays known as John domains. The importance of John domains was pointed out by Yu. G. Reshetnyak [Rll] in connection with injectivity studies of qr mappings. This direction of research was then continued by O. Martio and J. Sarvas [MS2], who also introduced the important class of uniform domains. Uniform domains have found applications in the study of extension operators of function spaces, e. g. in P. Jones' work ([J1], [J2]) as well as elsewhere ([GO], [GM1], [TR], IV12]). Other related classes of domains are QED domains IGM1] and ~-uniform domains ([VU10], [HVU D. The interrelation between some of these classes of domains has been studied by F. W. Gehring in [GS] and [G10], where also several characterizations of quasidisks are given. Important results dealing with function spaces and their extension to a larger domain have been proved by S. K. Vodop'yanov, V. M. Gol'dstein, and Yu. G. Reshetnyak in [VGR}, where additional references can be found. 10. C o n c l u d i n g r e m a r k s .

The above remarks cover only a part of the existing

QRT, and a wider overview can be obtained from the surveys of A. Baernstein and J. Manfredi [BAM] and F. W. Gehring [G9]. We shall conclude this survey by mentioning some directions of active research close to QRT. Recently qc and qr mappings have appeared in stochastic analysis in B. ~ksendal's work [OK1] and in the theory of manifolds (M. Gromov [GROM]). P. Pansu [PA] has studied quasiconformality in connection with Heisenberg groups, in which he has exploited among other methods the conformal invariant "~G of J. Ferrand [LF2]. Qc mappings also arise in a natural way in the study of BMO functions (H. M. ReimannT. Rychener fREIR], K. Astala-F. W. Gehring [ASTG], M. Zinsmeister [ZI]). In a series of papers V. M. Miklyukov [MIK4] has shown how the extremal length method can be used to study minimal surfaces. Extremely important are the partly topological results connecting geometric topology and quasiconformality, which were

XV proved by D. Sullivan, P. Tukia, J. V£is£1~, J. Luukkainen, and others.

Discrete

groups and quasiconformality have been studied in an important series of papers by P. Wukia ([WVl], [TU2]) and B. N. Apanasov, O. Martio and V. Srebro ([MSR1][MSR3]), F. W. Gehring and G. Martin [GMA]. Let us point out that we have confined ourselves here (and also elsewhere in this book) to the case of n-space, n > 2. For n = 2 the reader may consult the excellent surveys of O. Lehto ILl] and [L2] as well as his new book [L3]. The standard references for n = 2 are the books by L. V. Ahlfors [A2], H. P. Kfinzi [KI~I], and O. Lehto and K. I. Virtanen [LV2]. The variety of these results indicates the many ways in which qc and qr mappings arise in mathematics. Many fascinating connections between QRT and other parts of mathematics remain yet to be discovered.

Notation

and

terminology

T h e s t a n d a r d unit vectors in the euclidean space R n , n > 2, are denoted by el,...,en.

A point x in R n can be represented as a vector ( x l , . . . , x n )

s u m of vectors x = xIel + . . . + x n e n .

For x , y E R n the inner product is defined

by x . y = )"~.i~=1x i y i . T h e length (norm) of x E R " centered at x E R n with radius r > 0

or as a

is txl = ( x . x) 1/2. T h e ball

is B n ( x , r ) = { y E R n : l

sphere with the same center and radius is S n - l ( z , r )

x-y]

= { y E R n : Ix -


and the

Yl = r } , We

employ the abbreviations B n ( r ) = Bn(0, r) ,

B n = Bn(1) ,

S'~-l(r) = S ' ~ - ' ( 0 , r ) ,

S n-1 = S n - ' ( 1 ) .

T h e n - d i m e n s i o n a l volume of B n is denoted by ~n and the ( n - 1)-dimensional surface area of S n-1 by w , - 1 . and for x C R n \ { 0 }

For x, y E R n let [ x , y ] = { ( 1 - t ) x - i - t y : 0 < t

let [x,c~] = { s x : s :> 1 } u { c ~ } .

R n U (oo} is the o n e - p o i n t compactification of R n.

< 1}

T h e M h b i u s space ~ n =

T h e Mhbius space, equipped

with the spherical chordal distance q, is a metric space.

In addition to ( R '~, 1 I)

and ( R n, q) we shall require some other metric spaces such as the hyperbolic spaces ( B n , p B . ) and ( H n , P H . ) as well as (G, kG) where G c R n is a domain and k a is the quasihyperbolic metric on G . For a m e t r i c space ( X , d ) let B x ( y ,r) = { x E are n o n - e m p t y let d ( A , B ) = i n f { d ( x , y ) x, y E A } .

For x E X

:xC

X:d(x,y)

< r } . If A , B C Z

A, y E B } and d(A) = s u p { d ( x , y )

:

set d ( x , A ) = d ( { x } , A ) .

T h e set of natural numbers 0, 1 , 2 , . . . is denoted by N and the set of all integers by Z . T h e set of complex numbers is denoted by C . We often identify C = R 2 . For a set A in R '~ or R " the topological operations A (closure), OA (boundary), 1~'~ \ A (complement) are always taken with respect to R n . Thus the domain R n \ {0} has two b o u n d a r y points, 0 and o o , and the half-space It'* = { x E R " :

xn > 0 } has oo as a b o u n d a r y point. A domain is an open connected n o n - e m p t y set. A neighborhood of a point is a domain containing it. T h e notation f : D - - D ~ usually includes the assumption t h a t D and D ~ are domains in R n .

XVII Let G be an open set in R '~ . A m a p p i n g f : G -~ R m is differentiable at x E G if there exists a linear m a p p i n g f ' ( x ) : R ~ --+ R m , called the derivative of f at x , such t h a t

f(x + h) = f(x) + ff(x)h + lhle(x,h) where e(x, h) --* 0 as h --~ O. T h e Ja~=obian d e t e r m i n a n t of f at x is denoted by J r ( x ) . Assume next t h a t n = m and t h a t all the partial derivatives exist at x E G (thus f need not be differentiable at x ). In this case one defines the formal derivative of f -- ( f l , . - . , f,~) at x as the linear m a p defined by

(o:, (x), ""' ~x,~(x) o:, )

ff(x)ei : Vfi(x) = \O'~-xl

For an open set D C R '~ and for k E N ,

, i--- l,...,n.

Ck(D) denotes the set of all those

continuous real-valued functions of D whose partial derivatives of order p < k exist and are continuous. T h e n - d i m e n s i o n a l volume of the unit ball m n ( B n) is denoted by ~ln and the (n - 1 ) - d i m e n s i o n a l surface area of S n-1 by w ~ - i • T h e n

Wn_ 1 =

n~'~ n

and

rn/2

r(l + ½~) for all n = 2 , 3 , . . . where F stands for Euler's g a m m a function. For k = 1 , 2 , . . . we have by the well-known properties of the g a m m a function [AS, 6.1] 2,ff k 502k--1

----- ( k -

1)! ;

2k+lTr k ¢o2k = 1 . 3 . . - ( 2 k - 1) "

Algorithms suitable for numerical c o m p u t a t i o n of F(s) are given in [AS, Ch. 6] and in [ P F T V , Ch. 6]. We next give a list of the additional notation used. H ~ = l~

the Poincar~ half-space

1

P(a,t)

an (n - 1)-dimensional hyperplane

2

the group of MSbins transformations the group of orthogonal m a p p i n g s

3 3

the group of sense-preserving MSbius transformations n+l : x n + l = 0 }

x, f

a generic point of { x C R

d x ) , "2(x)

the stereographic projection

4, 6

4

XVIII

q(~, v)

the spherical (chordal)distance between x and y the antipodal (diametrically opposite) point

4, 5

5

Q(x,,')

the spherical ball

la, b,c, dl

the absolute (cross) ratio

a*

the image of a point a under an inversion in S n-1

To

a hyperbolic isometry with Ta(a) = 0

Lip(f) tz

the Lipschitz constant of f

p(x, v) J[*,v]

the hyperbolic distance between x and y

the geodesic segment joining x and y in R ~

D(x,M)

the hyperbolic ball with center x and radius M

J~(~,v)

a point-pair function (metric)

7 9

11

11

a spherical isometry with t~(x) = 0

14 20, 23 21

33

the quasihyperbolic ball with center x and radius M a point-pair function

p~(A,t)

22, 24

28

the quasihyperbolic distance between x and y

Dc(x,M)

10

39

the number of balls in a covering of the set A the locus of a path

35

46

49

t('~)

the length of a curve ~/

Mp(r), M(r)

the ( p - ) m o d u l u s of a curve family r

A(E, F; G)

the family of all closed non-constant curves joining E and F in G

49 49

51

A(E,F)

52

Cn

the constant in the spherical cap inequality the GrStzsch ring

59

65

RT,.(s)

the Teichmfiller ring

~.(~) = ~(~)

the capacity of Rv,n(s )

66

~.(~) = ~(~)

the capacity of R T , n ( s )

66

,(,)

a function related to the complete elliptic integrals

~K,.(')

a special function related to the Schwarz lemma

c(E)

a set function related to the modulus

65

74

67

68, 97

XIX

82

p-cap E, cap E

the (p-)capacity of a condenser

Aa(f)

the a-dimensional Hausdorff measure of F

88

the modulus of the GrStzsch ring

88

the modulus of the Teichmfiller ring the GrStzsch ring constant a point-pair function

AG(x,y)

86

88

102 103,

a conformal invariant (introduced by J. Ferrand)

118 the modulus (conformal) metric

103 106

a function related to an extremal problem

raG(z, y)

a point-pair invariant

116

~(y, f, D), ~(f, D)

the topological degree

121, 123

BI

the branch set of a mapping f

dim E

the topological dimension of a set E

J(G)

the collection of all relatively compact subdomains of a domain G

i(x,/)

123

123

the local (topological) index of f at x a normal neighborhood of x

N(I,A)

122

123

124

the maximal multiplicity of f in A

125

K(I), K o ( f ) , KI(I) the maximal, outer, and inner dilatations of f H(x,f)

the linear dilatation of a mapping f at x

A(K)

a special function related to the linear dilatation

C ( f ,b)

the cluster set of a mapping f at b

cap dens(E, 0)

the lower capacity density of E at 0

178

cap dens(E, 0)

the upper capacity density of E at 0

178

rad dens(E, 0)

the lower radial density of E at 0

178

rad dens(E, 0)

the upper radial density of E at 0

178

Dir(u)

the Dirichlet integral of u

187

174

128

134 136

Chapter I CONFORMAL GEOMETRY

This chapter is devoted to a s t u d y of some geometric quantities t h a t remain invariant under the action of the group of Mbbius transformations or under one of its subgroups. Examples of such subgroups are (1) translations, (2) orthogonal maps, (3) self-maps of R_~ = {x E R ~ : x~ > 0 } , and (4) spherical isometrics. The Mbbius invariance of the absolute (cross) ratio is of fundamental i m p o r t a n c e in such studies. The following three metric spaces will be central to our discussions:

(a) the

euclidean space R '~ , (b) the Poincar~ half-space R ~ = H ~ , and (c) the Mbbius space R ~ = R ~ U {oo}. Each of these metric spaces is endowed with its own natural metric t h a t is invariant under rigid motions of the space. In the particular case of R ~ , the invariant (hyperbolic) metric is often convenient in computations. This chapter is partly expository in character.

Some results, for instance vari-

ous well-known properties of Mbbius transformations in R ~ , are presented without proofs. For these results and further information on Mbbius transformations the reader is referred to C h a p t e r 3 in A. F. Beardon's book [BE] as well as to L. V. Ahlfors' lecture notes lAb].

1.

MSbius transformations in n - s p a c e

For x E R n and r > 0

let B"(x,r) ={z•R":lx-z

S~-'(x,r)

I
= { z • a N : Ix - zl = r }

denote the ball and sphere, respectively, centered at x with radius r . The abbreviations B'~(r) = B~(0, r ) , S ' ~ - l ( r ) = S ~ - 1 ( 0 , r ) , B '~ = B " ( 1 ) ,

S n - ' = S " - 1 ( 1 ) will

be used frequently. For t E R and a E R '~ \ {0} we denote P(a,t) = {xERn:x.a=t}U{oo}. Then P(a,t)

is a hyperplane in R '~ = R ~ u {0o} perpendicular to the vector a , at

distance t / l a I f r o m the origin. 1.1. D e f i n i t i o n .

Let D and D r be domains in R ~ and let f : D - - *

h o m e o m o r p h i s m . We call f

D ~ bea

conformal if (1) f E C 1 , (2) J r ( x ) # 0 for all x E D ,

and (3) I f ' ( x ) h ] --- I f ' ( x ) l l h ]

for all x E D and all h ~ R ~.

If D and D ' are

domains in ~ n , we call a h o m e o m o r p h i s m f: D --~ D r conformal if the restriction of f to D \ { o o , f - l ( o o ) } 1.2. E x a m p l e s .

is conformah Some basic examples of conformal mappings are the following

elementary transformations. (1) A reflection in P ( a , t ) : fl(x) = x - 2(x.a - t)]~2,

f1(c¢) = o o .

(2) An inversion (reflection) in S '~-1 (a, r ) : r 2(x

f 2 ( x ) ---- a +

- a)

F x - a ~ , f2(a) = oo, f2(oo) ---- a .

(3) A translation f s ( x ) = x + a , a E R '~ , f3(oo) = 0¢. (4) A stretching by a factor k > 0:

]'4(x) = k x , f4(oo) = oo.

(5) An orthogonal mapping, i.e. a linear m a p fs with Ifs(x) l : Ixl, fs(a~) : 0 0 .

1.3. R e m a r k .

T h e translation x ~-* x + a can be written as a composition of

reflections in P(a,O) and P ( a , ~lal 1 2 ) . The stretching x ~ k x ,

k :> 0 , can be written

as a composition of inversions in S ~ - 1(0, 1) and S '~- 1(0, V ~ ) . It can be proved, that an orthogonal m a p p i n g can be composed of at most n + 1 reflections in planes (see [BE, p. 23, T h e o r e m 3.1.3]). 1.4. E x e r c i s e .

Let f be an inversion in S ' ~ - l ( a , r )

t h a t f - 1 __ f and t h a t t x - a l l f ( x

)-a

t - - r 2 for all x E R

as defined in 1.2(2). Show n\{a}.

By considering

similar triangles show t h a t the following identity holds for x, y C R ~ \ (a} : r21x - y[ (1.5)

If(x)

-

f(y)l

-

Ix - ally

- al "

For x , y C R ~ \ {0} let p(x,y) = Ix -

1.8. E x e r c i s e .

yI2/(Ixllyl)

Applying

(1.5) show t h a t p(x, y) = p(f(x), f(y)) if f is a stretching or an inversion in S ' ~ - l ( r ) , r>0. 1.7. D e f i n i t i o n . A homeomorphism f : R n --+ R'~ is called a M6bius transfor-

mation if f = gl o -.. o gp where each gj is one of the elementary transformations in 1.2(1)-(5) and p is a positive integer. Equivalently (see 1.3) f is a MSbius transformation if f = hi o.. • ohm where each hj is a reflection in a sphere or in a hyperptane and rn is a positive integer. It follows from the inverse function theorem and the chain rule that the set of all conformal mappings of R n is a group. It is left as an easy exercise for the reader to show that the set of all Mhbius transformations constitutes a subgroup of the group of conformal mappings, and we denote it by ~ ( R

'~) or ~ M . Further, we shall write

~.M(D) = { f C DM(~n) : I D = D } for D C R'~. We denote by 0 (n) the set of all orthogonal maps in R n . A map f in ~)~ with f(oo) = oo is called a similarity transformation if If(x) - f(Y)I = eix - Yl for all x, y E R '~ where c is a positive number. 1.8. D e f i n i t i o n . Let D and D t be domains in R ~ . We call a C l - h o m e o m o r phism

f: D --* D'

sense-preserving (orientation-preserving) if Jr(x) > 0 for all

x E D\{oo, f-l(oo)}.

If Jr(x) < 0 for all x E D \ {oo, f - l ( o o ) }

then we call f

sense-reversing (orientation-reversing). One can show that reflection in a hyperplane or in a sphere is sense-reversing and hence the composition of an odd number of reflections. The composition of an even number of reflections is sense-preserving. For these results the reader is referred to [RR, pp. 137-145]. The set of all sense-preserving Mhbius transformations is denoted by ~ ( R ' ~ ) or N . A l s o w e l e t ~ ( D ) = { f c N : f D = D } 1.9.

Remark.

if D c R

n.

One can extend Definition 1.8 so as to make it applicable to a

wider class of mappings (including quasiregular mappings). This extended definition makes use of the topological degree of a mapping, which will be briefly discussed in Section 9.

It will be convenient to identify R~ with the subset { z E R '~ : x,~+~ = 0}t_){oo} of ~ + 1 .

The identification is given by the embedding

(1.10)

X ~-+ ~ = ( X l , . . . , X n , 0 )

;

X = (Xl,...

,Xn)

~ R n .

We are now going to describe a natural two-step way of extending a MSbius transformation of ~.'~ to a M5bius transformation of ~,~+1. First, if f in ~ j q ( ~ n ) is a reflection in P(a,t) or in S'~-l(a,r), let 7 be a reflection in P(~,t) or sn(~,r), respectively. Then if x C R '~ and y = f ( x ) , by 1.2(1)-(2) we get (1.11)

Xl,--.,

Xn, 0) ---- ( Y l , ' ' ' ,

Yn, 0) = f ( x )

By (1.11) we may regard 7 as an extension of f .

.

Note that 7 preserves the plane

x , + l = 0 and each of the half spaces x~+l > 0 and X~+l < 0. These facts follow from the formulae 1.2(1)-(2).

Second, if f is an arbitrary mapping in ~ M ( R ") it

has a representation f = fl o ... o fm where each fy is a reflection in a plane or a sphere. Then

= f l o • .. o fm is the extension of f , and it preserves the half spaces

x,~+l > 0, x,~+l < 0, and the plane xn+l = 0. In conclusion, every f in ~ j q ( ~ n ) has an extension 7 in ~ j q ( ~ + l ) .

It follows from [BE, p. 31, Theorem 3.2.4] that

such an extension f of f is unique. The mapping 7 is called the Poincard extension of f . In the sequel we shall write x, f instead of ~, f , respectively. Many properties of plane M6bius transformations hold for n-dimensional MSbius transformations as well. The fundamental property that spheres of R "

(which are

spheres or planes in R " , see Exercise 1.25 below) are preserved under MSbius transformations is proved in [BE, p. 28, Theorem 3.2.1]. m

1.12.

Stereographic

projection.

The stereographic projection

7I": I:{,n ----+

1 1 S~(~e=+ 1, 7) is defined by

(1.13)

x- e~+l R = ~(eo) rr(x) = e,~+l + t X _en+ll2 , x C ; = en+l •

Then ~r is the restriction to R ~

of the inversion in S ~ ( e ~ + l , 1 ) .

In fact, we can

identify rr with this inversion. Because f - 1 = f for every inversion f , it follows that 7r maps the "Riemann sphere" S~($e~+l,$)l 1 onto R ~ . The spherical (chordal) metric q in R n is defined by m

(1.14)

q ( x , y ) = IF(x) - ~ ( y ) l ; ~, Y < R ~ ,

where ~r is the stereographic projection (1.13). From the definition (1.13) and by (1.5) we obtain

12;-~l (1.15)

q(2;'Y) -- x / i + 12;I~ ~ 1

q(2;, oo) -

'

2; ~ ~ ~ y '

v ' l + 12;I2

en+l //11 ",,,

Diagram 1.1.

7r(X)

Formulae (1.13) and (1.14) visualized.

For x C R ~ \ {0} the antipodal (diametrically opposite) point ~ is defined by (1.16)

~"

2; -

-

12;I ~ and we set ~ = 0, O = c~ . Then, by (1.15), q(2;,~) = 1 and hence 7 r ( x ) , r ( 5 ) are indeed diametrically opposite points on the Riemann sphere. 1.17. E x e r c i s e . It follows from (1.15) that q(x, y) <_ min{ 1, Ix - y] } is always true. Applying (1.5) show that

q(2;'~) = q I~ ~'lul ~

lxtly---T

holds for 2;, y E R '~ \ {0}. Show also that

Ix - vl (l+[x[)(l+[yl)

212; - vl -< q(x,y) <_ ( 1 + I 2 ; [ ) ( l + [ y t )

for all x,y E R ~. Show that q(2;,y) > ½ [ x - y ]

2

-ll2;-yI for x, y E O B n

for 2;,y E B'~ and that q(2;,y) =

6

1.18.

:Exercise.

(1) For 0 < t < 1 let w(t) = t / v / 1 - t

2.

Show t h a t

q(O,w(t)el) --- t a n d t h a t t s

w(t) w-~

2t s

for 0 < s < t < ½v/-3. (2) Let x, y E B '~ with s = q(0, x ) , t = q(0, y ) . Show t h a t

q(x,v) _< , x / 1 - t2 + t x / ] - - ,2 < t + , . (3) Let x , y E R ~ \ {0} w i t h q(0, x) :> q(0, y ) .

Show t h a t the strict inequality

q(x, y) > q(O, x) - q(O, y) holds. (4) Show t h a t for x , y , z E B '~, x # z ,

1 Ix-yl q(x,y) < .,/~tx-yl V~ lx zl <- q(x,z----~ Ix z r 1.19. Definition.

Let ( X l , d l )

and (X2,d2) be metric spaces a n d let f: X1 -+

X2 be a h o m e o m o r p h i s m . We call f an isornetry if d 2 ( f ( x ) , f ( y ) )

x,y e Xl.

h m a p f in ~ M ( R " )

= d l ( X , y ) for all

is called a spherical isometry if q ( f ( x ) , f ( y ) )

=

q(x, y) for all x, y E R '~ . A similarity t r a n s f o r m a t i o n f is called a euclidean isometry if If(x) - f ( y ) [ = l z - y ]

for all x , y E R ~ .

O r t h o g o n a l m a p p i n g s and inversion in S n - 1 are e x a m p l e s of spherical isometries, while t h e t r a n s l a t i o n

x ~

x + el

and the stretching

x ~-* I x

are not spherical

isometries (see 1.54).

Reflection in a h y p e r p l a n e and t r a n s l a t i o n s are e x a m p l e s of

euclidean isometries. 1.20. Remark.

T h e inversion

zc2(X) = en+l +

2(~-cn+~) ix _ e,~+II 2

, x ~

R~

is also s o m e t i m e s called the stereographic projection. ~2(0)

=

- - e n + l , ~2(c~)

=

en+l and 7f2(S n ' - l )

=

S n-1

;

~(~)

=

e,~+l ,

It m a p s R ~ onto S ~ so t h a t .

F r o m (1.5) it follows t h a t the

spherical metric q in (1.14) can be defined in t e r m s of 7r2 as q(x, y) = 1[7r2 ( x ) - r 2 (Y) I, x, y C ~ , n .

We c a n identify 7r2 with the inversion in S n ( e n + l , x / 2 )

7;2 m a p s t h e half space xn+i < 0 onto B n+l 71"2(0) =

--en+l

•

• We see t h a t

in such a way t h a t 7 r 2 ( - e n + l ) = 0 ,

en+,

R"

--en + l

Diagram 1.2. 1.21. Balls in t h e s p h e r i c a l m e t r i c .

For x E A n and r E (0,1) we define

the spherical ball

Q(x,r) = {z E R'~: q(x,z) < r } .

(1.22)

Its boundary sphere is denoted by OQ(x, r) . From the Pythagorean theorem it follows that (cf. (1.16))

Q(x,r) ---- R'~ \ -Q(E, V/~ - r 2 ).

(1.23)

gn+l

~.

=

o

Q(=,,-)

Diagram 1.3. To gain insight into the geometry of spherical balls Q(x,r) it is convenient to study the image 7rQ(x,r) under the stereographic projection 7r (see the diagram). Indeed, by definition (1.14) we see that

~Q(=,r)

=

1 B~+'(:~(=),,) n S n (~e~+l,

~)

E i t h e r by this f o r m u l a or m o r e directly by the definition of the spherical metric (plus t h e fact t h a t Mhbius t r a n s f o r m a t i o n s preserve spheres) we see t h a t in the euclidean g e o m e t r y , Q(x, r) is a point set of one of the following three kinds (a)

an o p e n ball B " ( u , s ) ,

(b) the c o m p l e m e n t of B " ( v , t ) (c)

in R n

a half-space of R " .

Clearly, OQ(x, r) is either a sphere or a h y p e r p l a n e of R " . F o r m u l a (1.23) shows, in particular, t h a t 7cQ(x, 1 / x / ~ ) is a half-sphere of the R i e m a n n sphere Sn 1.24. Remark.

(~en+l,1 ~).1

In complex analysis, the spherical metric is often defined in the

following way. If "~ is a rectifiable curve in R n set

°(7) = a n d for x, y E R

n define

a(x, y) = i n f o ( ~ ) where ff runs t h r o u g h the collection of all rectifiable curves -f with y E ~/.

x E ff and

In a n a t u r a l way this definition is e x t e n d e d t h e n to all 5 ; y E R ~ .

It is

easy t o show t h a t a(x, y) is a metric on R ~ . M a k i n g use of (1.15) one c a n show t h a t

a(x, y) -= a(h(x), h(y)) if h is a spherical isometry. This spherical metric is equivalent to the metric q. In fact, the two relationships

a(x,y) = 2 a r c s i n q ( x , y ) , i< -

a(x,y) <~r ::4arctanl, q(x,y)

-

hold for all distinct x, y ~ R n . 1.25.

Exercise.

Q(0,1/v/2),

(1) Show t h a t

~ n \ B ~ __ Q(oo, 1 / v ~ ) ,

{ x E R n : xn > 0 } = Q(er~, 1 / V ~ ) ,

Q(O,r) -- B n ( r / v / 1 - r 2 ) ,

Q(O, t/v/-f + t2 ) . If t E ( 0 , 1 ) , x = tel , y OQ(y, 1 / x / 2 ) • (2) Show t h a t

Q(tel, 1 / v ~ )

1--t

l+tel,showthat

= Bn(uel, r) , where

and

B~ =

Bn(t) --

x, 5, e2, -e2 E

u =- 2 t / ( 1 - t 2) , r --

(1 ÷ t 2 ) / ( 1 - t 2 ) , p r o v i d e d t E ( 0 , 1 ) . Discuss also the case t E ( 1 , o o ) . (3) F i n d

q(Q(x,r))

and

q(OQ(x,r)), the spherical diameters of Q(x,r)

OQ(x,r), respectively, for x E R n ,

1/v

]

r c (0,1).

and

[Hint: Consider first the case r <

(4) Show that if z E R '~ , then there exists e E S n-1 such that e and - e are in OQ(z, 1 / V ~ ) • Conclusion: If Q(z, 1 / v ~ ) =

B~(a,r), then r 2 = 1 + lal 2 . Show 1 for all b E R n .

converselythat q ( B n ( b , ~ ) ) = 1.26. A b s o l u t e r a t i o .

For an ordered quadruple a, b, c, d of distinct points in

R~ we define the absolute (cross) ratio by

(1.27)

ta, b,c, dt -

q(a, c) q(b,d) q(a, b) q(c,d) "

It follows from (1.15) that for distinct a, b, c, d in R ~

la -- cl l b - dI bl Ic dI"

la, b,c, di-- la

One of the most important properties of MSbius transformations is that they preserve

absolute ratios, i.e. if f E f f ~ , then (1.28)

I f ( a ) , f(b), f(c), f(d) I = I a, b, c, d l

for all distinct a, b, c, d in R'~. As a m a t t e r of fact, the preservation of absolute ratios is a characteristic property of Mhbius transformations.

It is proved in [BE, p. 72,

T h e o r e m 3.2.7] that a mapping f : R ,~ --~ R n is a Mhbius transformation if and only if f preserves all absolute ratios. It follows from (1.28) t h a t l a, b, c, d I = A if and only if there exists an f in ~

(1.29)

with

f ( a ) = 0 , y(b) = e l , f ( d ) = ~ ,

By property (1.28), the absolute ratio is ~ ( R ratio we shall consider later some other ~ ( R For instance,

[a-bi/[a-c[

If(c)l = ~.

n)-invariant.

Besides the absolute

n) - or ~ ( D ) - i n v a r i a n t

and d(E)/d(O,E), E C R n \ { 0 } , a r e

quantities.

~(Rr~)-invariant

quantities. 1.30. R e m a r k .

The absolute ratio depends on the order of the points. In fact, 1

j0,el,~,~ I0, e l , ~ , x l 10,~,z,e,

J =-J~l = i 0 , ~ , e l , ~ l = !x- eli-

'

1 10,~,el,xf

I - ] ~ _ ell - 10, z , ~ , e l

' I"

A thorough discussion of the complex cross-ratio can be found in [BE, pp. 75-78].

10

1.31. E x e r c i s e .

Let f be in ~J~ such t h a t f ( 0 ) = 0 , f ( e l ) - - e l ,

and

f ( o c ) =- c¢. Show t h a t f is an orthogonal mapping. [Hint: Apply (1.28).] 1.32. E x e r c i s e .

For a , b , c , d E R'~ let

q(a, d) 2 q(b, c) 2 s(a, b, c, d) = q(a, b) q(b, d) q(a, c) q(c, d) ' sf(a, b, c, d) = s ( f ( a ) , f(b), f(c), f ( d ) ) . T h e n s is symmetric: s(a,b,c,d) = s(d,b,c,a) = s(b,a,d,c) = s(a,c,b,d) . Show t h a t s is ~ M ( ~ n ) _ i n v a r i a n t , i.e.

sf(a, b, c, d) = s(a, b, c, d)

(1.33) whenever a , b , c , d Ix - yl2/(Iz]ly]).

E R n and f E ~ M .

Applying (1.15) show t h a t s(O,x,y, oo) =

It should be noted t h a t the invariance p r o p e r t y in Exercise 1.6 is

a special case of (1.33). [Hint: Show t h a t s(a,b, c,d) is the product of two absolute ratios.] 1.34.

Automorphlsms

the m a p s in ~4(B'~).

of B n .

Assume t h a t

f

We shall give a canonical representation for is in M ( B '~) and t h a t

f(a) = 0 for some

a C B ~ . We denote a * - - iala , a • R , ~ \ {o}

(1.35)

and 0* = oo, oo* -- 0. Fix a • B '~ \ {0}. Let

aa(x) = a* + r2(x - a * ) * ,

(1.36)

r 2 = ]a] -2 - 1

be an inversion in the sphere S'~-1 (a*, r) orthogonal to S " - 1 .

S n-1

T h e n aa(a) = O,

S,,-l(a,,r)

Diagram 1.4.

11 Let p~ denote the reflection in the (n - 1)-dimensional plane P(a,O)

through the

origin and orthogonal to a and define a sense-preserving Mhbius t r a n s f o r m a t i o n by

T~ = p ~ o a ~ . Then, by (1.36), TaB '~ = B '~ , Ta(a) = 0 , and with ea -- a l i a I we have Ta(e~) = e~, T~(-e~,) -- - e a .

For a = 0 we set To = id, where id stands for the

identity m a p . T h e p r o o f of the following f u n d a m e n t a l fact can be found in [Ah, p. 21], [BE, p. 40, T h e o r e m 3.5.1]. 1.37. L e m m a .

If g E ~ ¢ q ( B n ) , then there is k E O(n) such that q = k o T a

where a -- g - ~(0) . 1.38. D e f i n i t i o n .

Let ( X l , d l ) ,

( X e , d 2 ) be metric spaces, let f: X1 --* X2 be

continuous and let L > 1. We say t h a t f is L - L i p s c h i t z if

d2 (f(x), f(y)) < L dl (x, y) for all x, y E X1 • T h e least constant L with this p r o p & t y is denoted by L i p ( f ) . If, in addition, f is a h o m e o m o r p h i s m and

dl(x,y)/L

< d2(f(x),f(y))

< Ldl(X,y)

for all x , y E X1 , we say t h a t f is L-bilipschitz or that f is an L - q u a s i i s o m e t r y . We call f a Lipsehitz (bilipschitz) m a p p i n g if it is L - L i p s c h i t z (resp. L-bilipschitz) for some L_> 1. If h E ~ 1.39.

and x E R ~

we sometimes write hx instead of h ( x ) .

The Lipschitz constant

a E B '~ \ {0}.

o f TaIB '~. Let Ta = Pa o a~ be as in 1.34,

Since Pa is a reflection in a plane and hence preserves euclidean

distances, it follows t h a t 1 T a x - T a y l = l e a x - a a y I .

As I a 1 - 1 - 1 < I z - a * l < l a l - l + l

for all z E B '~ , using (1.5) we get

lT~x - Tayl <_

(la1-2 -

1)Ix-yl - l + - l a I I x - y t 1 tat

1)lx-vl - 1-Iallx_v

1Tial

I

for all x , y E B ~ . Hence Lip(T~ IB ~ )

=supS ITox -Tovl Ix vl

: x, y E B n ,

x#y}

< -1+-

lal 1 -lal

12

In fact, 1 + lar

(1.40)

Lip(T~IB~) = Lip(T~IS~-X) -- 1 - - [ a I

as we see by a p p l y i n g (1.5) to a pair of points x , y E S '~-~ with [x - a*[ -- IY - a*l a n d t h e n letting Ix - a* I --+ la1-1 - 1. As T : 1 = T _ a , it follows f r o m (1.40) t h a t T a l B n is bilipschitz w i t h the c o n s t a n t 1.41. Exercise.

(1 + fat)/(1 - ta[).

(1) Show t h a t Ix - Yl tT~yi -

i ~ l i y - ~*1

for all x , y E B n . [Hint: A p p l y (1.5).] (2) Let r E ( 0 , 1 ) .

Find the "M6bius center" of the s e g m e n t

point a E (O, r e l ) such t h a t T a ( 0 ) = - T a ( r e l ) observe t h a t

ITa(a) - T~(O)I = I T a ( r e l ) - Ta(a)I

where Ta is as in 1.39. [Hint: First and hence, by the definition of T~,

a similar equality holds with aa in place of T a . lal = r / ( l + x / 1

[0, r e l ] , i.e. the

Next a p p l y (1.5) to aa to obtain

- r 2 ) . Note t h a t the point a can be found by a g e o m e t r i c construction,

as in the n e x t diagram.]

en

--e 1

el

--e n

D i a g r a m 1.5.

1.42.

Exercise.

(1) Show t h a t if p E (0,

~),

x~ -

(cos~,sin~)

V~ =

(cos p , - sin ~ ) t h e n there exists a Mbbius t r a n s f o r m a t i o n T~: B 2 ~ B 2 with T~el =

13 el, T~(-ex)=-el,

Ta(x~)=

e2 = - T a ( y ~ , ) ,

and

Lip(T~l]3 2) - - c o t

½~. ]Hint: By

1.39we see that a* = l~-fel and e2, x~, a must be collinear. Now T~ = t a n ( 1 7 r + l ~ ) ' Ild-la[ l_ia

•

COt

_I~,2 and the result follows from (1.40).]

(2) Let ~ e

( 0 , ½ r ) , F~ = { x e

S n-1 :xl = c o s ~ } ,

and let T~ e .M(B n) with

T~F~, = F~,. Show t h a t a = (cot ½~)2el. Assume next that 0 < a < / ~ < ½7r and TbF~---Fp.

Find b.

1.43. E x e r c i s e .

(1) Let 0 < s < 1. Applying (1.5) as in 1.39 show that

i -s

2

(1 + s2) 2 Ix -

Yl -< [Tax - T~,yI -<

1

1 _ ~ l x - Yl

for all a, x, y E ~ n (s) . (2) For a , x E B n with a • 0 and e~ = a l i a I show that

IT xl _< tTo(-l 1.44. S p h e r i c a l i s o m e t r i e s .

leo)l-

We are now going to study the action of spherical

isometries, proving a representation for them similar to that of ff~M(B n) in 1.37. By (1.40) and 1.37 we see that g E ~ J ~ ( B n+l) is a euclidean isometry iff g(0) = 0 . Next we we shall reformulate this fact for maps in f f ~ ( R '~) . Let p be the reflection in the hyperplane x n + l = 0 and f l the inversion in S n ( e n + l , v / - 2 ) , and set f = f l o p . Then f R ~ +1 = B '~+1 and f ( e n + l ) = 0 , q ( x , y ) = ½ 1 f ( x ) - f ( y ) ] now that h E ~ ( ~ n ) We see t h a t

p = foho

is given and that h E f f j q ( ~ n + l ) f - 1 E f f ~ ( B "+1)

for all x , y e R ' ~ . Assume is its Poincar~ extension.

is a euclidean isometry if and only if

h ( e n + l ) -- en+l • One can show that h is a spherical isometry iff h(en+l) = en+l

(see [BE, p. 42, T h e o r e m 3.6.1]). In particular, the inversion in S n - l ( a , r )

c R " is a

spherical isometry iff en+1 E S n ( ~ , r ) C R '~+1, i.e. iff (1.45)

r 2 = 1 ÷ latL

We recall (see (1.23), 1.25) that by virtue of (1.45) B n ( a , r) = Q ( z , 1 / V ~ ) for some z C R '~ . We define a spherical isometry tz in ~ (R ,~) which maps a given point z C R '~ to 0 as follows. For z = 0 let tz = id and for z = ~ is inversion in S n-1 and p is reflection in the ( n -

let tz = p o f ,

where f

1)-dimensional plane Xl -- 0.

For z e R ' ~ \ {0} let Sz be inversion in s n - l ( - z / I z l 2 , r ) ,

where r - - v / l +

Iz1-2

14 According to the criterion (1.45) the inversion s~ is a spherical isometry and it is easy to show t h a t s~(z) = O. Let p~ be reflection in the plane P ( z , 0 ) . (1.46)

tz : Pz

Defining

o sz

we see t h a t tz E ~ ( R '~) is a spherical isometry with tz(Z) = O. Hence tz(Q(z,r))

(1.47)

: Q(O,r) : B'~(r/V/-1 - r 2 ) ,

it~(x)l 2 _

q(x,z) 2

1 - q(~, ~)~

for all x , z C R ' ~ , r E (0, 1). In the above discussion we showed t h a t the inversion 8z is a spherical isometry by exploiting a result from [BE]. Next we shall show this by a direct c o m p u t a t i o n . 1.48. Lemma. Sn-I(-z/Izl

Proof.

2,

For a p o i n t z E R ~ \ {0} let s~ be the inversion in the sphere

~/1 + Izl-2). T h e n s~(z) = 0 and Sz is a spherical isometry.

It is easy to show t h a t s z ( z ) = 0. By (1.5) we obtain for x , y E R '~

(1+ Izl-=)lx-Yl

(1 + Izl-~)t~ -- Yl

where ~ = - z / I z l 2 as in (1.16). F u r t h e r by (1.5)

{~(x)l = I~.(~) -~(z)l

{~(y){

_

= (1-t-Izl-~)l~-zI {x-~llz-~l

Ix--z]

ly-z{ lzily-~i

S u b s t i t u t i n g these identities into (1.15) we obtain q(~(~),~(y))

= v~l + I ~ ( ~ ) j ~ v/1 + L~(y)L ~

(1.49)

(1 + Iz?)I~ - yt vqzPlx

-

~] ~ + Ix - zl ~ v/izl~iy - zi 2 + iv - zl ~

By the P y t h a g o r e a n t h e o r e m 17r(x) - r ( z ) [ 2 ,1, Dr(x) - 7r(z')l 2 = 1 or, equivalently, q ( x , z ) 2 ÷ q(z,~.') 2 = 1 or

(1 + Ix[2)(1 ,1, [z[ 2)

+

(1 "1" ]x]2)(1 ,1, lz[ -2)

=1,

15

e n Jr l.~_

/

,, . .

,,

....

o ....

~-

Diagram 1.6. This yields

1 + Ixl ~ -

(1.50)

1 + {y}~ =

Ix - z? ----~ 1 + Izl

+

Ix - ~i~ 1 + Izl -~ -

Ix - zl ~ + {z{~l x - ~'t ~ ~1 + I~{ 1 + Izl ~

ly - xl~ + {zl21y - ~'l ~ 1 + Iz{ 2 1 + lz{ 2

By substituting (1.50) into (1.49) we obtain

q(~z(x),~z(y)) = q(x, ~), showing t h a t sz preserves the spherical distance between points x, y in R '~ . It is left as an exercise for the reader to prove the case when x or y equals oo. 1.51. Lemma.

O

A MSbius transformation h is a spherical isometry if and only

if th(o) o h E O(n) . Proof. ~(B

Assume t h a t h E . q ~

is a spherical isometry.

T h e n f -- th(o) o h E

~) with f ( 0 ) = 0 , and hence f E O(n) by 1.37. T h e converse implication is

trivial.

ZI

In some questions it is useful to apply the following isometric decomposition of an inversion, which follows f r o m [BE, p. 31, T h e o r e m 3.2.4].

16

1.52.

Lemma.

Let a E R n , r > O, and let b E R '~ , u > O, be such that

B'~(a,r) = Q(b,u) . I f f is the inversion in S ' ~ - l ( a , r ) , then f = t~ I o f l o t b , where tb is the spherical isometry defined in (1.46) and f l is the inversion in S ~ - ~ ( u / 4 - f _ u2 ) = aQ(O,u) .

ert+l

o

',,,

b

Br~(a,r) ---- Q(b,u) Diagram 1.7. 1.53. Exercise.

Show t h a t B'~(a,r) and B n ( v ) , where r 2 < 1 ÷ la[ 2 , 2r

V

x/(1 + (lal + r)2)(1 + (]el- r) 2) "i 1 ÷ have equal spherical d i a m e t e r s .

Note t h a t

I~I = -

v < 1. Conclusion:

r=

T h e inversion f l

in

1.52 is in fact the inversion in a euclidean sphere with radius v and center 0. 1.54.

Lemma.

Each of the following Mhbius transformations is a bilipschitz

m a p p i n g in the spherical metric with the given constant: (1)

f ( x ) = k x , k >_ 1: L i p ( f ) = k .

(2)

T h e inversion in S ' ~ - l ( t ) , t E ( 0 , 1 ) :

(3)

T h e inversion in S n - l ( a , r ) , Lip(f)

(4)

f(x) =x+b:

Proof.

(~/(I

r 2

Lip(f) = t -2.

< 1 + lal2 :

+ (lal + r)2)(1-~- (!a I -r) 2) + 1 + lal 2 - r2"~ 2

\

]

2r

Lip(f)=l+½Ibl(lbl+~).

(1) Clearly fB'~(¼) = B " . ~;1B"(¼)

If 7r2 is the m a p in 1.20, t h e n

= s ~ n ~(-e~+l,

a n d ~r2B n = S_n = { x E S n : Xn+l < 0 } .

2/,/1

+ ks ) = A

17

gn+l

el

R'~

2/,/1 + k2

--Cn+l

Diagram 1.8. Hence 772ofo7721: S'~--+ S n maps A onto S_~ . Let a be the angle between

[0, en_F1 ]

and [e,~+l, ~ e l ] . Obviously tan a = ¼ and the Lipschitz constant of 772 o f o 77; 1 in the euclidean metric of R n+l (restricted to S '~ ) is the same as the Lipschitz constant of f in the spherical metric, L i p ( f ) . It follows from 1.42(1) that Lip(f) = k . (2) Since the proof is similar to the above proof, we indicate only the changes. First f maps S " - 1 ( t 2) onto S n-1 (and B~(t 2) onto R'~ \ B n). As above in the proof of part (1) we see that Lip(f) = t -2 . (3) The proof follows from 1.52, 1.53, and part (2). (4) Again the proof is similar to the one in (1). Observe first that g = 7r2 o f o ~r~-1 preserves the 2-dimensional plane containing e ~ + l , - e ~ + l g(en_F1 ) ---- e n + l ,

g(772(-b)) = -en+ 1 .

=

e.+l

41 + lbt Diagram 1.9.

and - b , and that

18 By 1.37 we see that g = k o T a ,

k E O(n+l),

Ta E f f ~ ( B n + l ) .

By elementary

geometry lal -- l / V / 1 + 41b1-2 , and hence (1.40) yields

1 + lal v/4 + IbI2 + Ib] L i p ( f ) = Lip(g) = - = l+½1bl(Ibl+~). 1 [a I V~4 + Ibl ~ Ibl -

1.55.

Exercise.

-

Let x, y E R " .

Show that q ( x , y ) -- t if and only if there

exists a spherical isometry h with Ih(x)l -- Ih(y)l = 1 and Ih(x) - h(y)l = 2 t . Prove t h a t the Lipschitz constant of Ta IB '~ in the euclidean metric is equal to the Lipschitz constant of Tal R n in the spherical metric.

1.56. Corollary. let s n - l ( a , r )

Let u E

(O, 1/v~],

let f

be the inversion in S"-1(a,r), and

: cgQ(b,u) for some b E r:~n . T h e n L i p ( f ) = u -2 - 1.

P r o o f . By 1.52 f and t b o f o t b g is the inversion in s " - l ( u / v / - f L i p ( f ) = u -2 - 1.

1 = g have equal Lipschitz constants. By 1.52

- u 2) = O Q ( O , u ) . Hence by 1.54(2) and 1.25(1),

[]

1.57. E x e r c i s e . Let x , y , w be three points in R n . Show that q(x,y)/c < q(x--w,y--w) where c = Lip(h) and h ( x ) = x 1.58.

<_ c q ( x , y )

w . [Hint: 1.54(4). 1

E x e r c i s e . (Continuation to 1.18(4) and 1.53.)

Assume that x , y , z E

B n ( a , r ) with x # z . Show that

1 tz- yl

q(z,y)

Iz- yl

c l z - ~J <- q(z,x----y <- c I ~ -

'

where c depends only on q(-B'~(a,r)). 1.59. E x e r c i s e .

Let f be the inversion in S n - l ( r ) ,

integer m = 2 , 3 , . . . show that there are f l , . . . , f m f=flo'"ofm

1.60.

Remark.

and

where r E (0, 1]. Given an

in ~ 4

with the two properties

Lip(f) = L i p ( f l ) . . . L i p ( f m ) .

As shown e.g. by the Riemann mapping theorem, the class

of conformal mappings in the plane is extremely large. According to a deep classical

19 theorem of Liouville the multidimensional case is radically different: If D is a domain in R n ,

n > 3 , and f : D

~

f D C R ~ is conformal, then f can be written in

the form f = gl D where g E ~34(tt'~).

This result was proved by Liouville for

C3-mappings, and a similar result under weaker hypotheses was obtained by F. W. Gehring in 1962 [G2, T h e o r e m 16] and by Yu. G. Reshetnyak in 1967 (see [R13] for more details). A new proof was recently given by B. Bojarski and T. Iwaniec [BI1]. Further generalizations of these results were obtained by Yu. G. Reshetnyak (see JR13] and the references therein). For additional references see [WI, p. 437]. 1.61. N o t e s .

A . F . Beardon [BE] has given a thorough account of the theory of

Mbbius transformations in ~ . " . See also L. V. Ahlfors [Ab] and J. B. Wilker [WI]. An illuminating representation of the stereographic projection and the spherical metric is contained in the classical book of D. Hilbert and S. Cohn-Vossen [HCV]. A thorough discussion of two-dimensional Mbbius transformations is contained in the books of C. Carath~odory [CA], L. R. Ford [FO], and H. Schwerdtfeger [SC]. The book of M. Berger [BER] contains numerous excellent illustrations related to the topic of this section.

2.

Hyperbolic geometry

Hyperbolic geometry can be developed in the context of two spaces or, as they are sometimes called, models. These two models of the hyperbolic space are the unit ball B " and the Poincar~ half-space H

=

= {

c It": x, > 0

).

These two models can be equipped with a hyperbolic metric p that is unique up to a multiplicative constant in either model. In either model the metric is normalized (by giving the element of length of the metric) in such a way that for all x, y E B '~ pHo ( h ( x ) , h ( y ) ) :

whenever h E ~ 34 and h B ~ = H '~ . Therefore both models are conformally compatible in the sense t h a t the two metric spaces (B '~, p) and ( H '~, p) can be identified. This compatibility is very convenient in computations because we may do a computation in that model in which it is easier, without loss of generality. In what follows we shall use the symbols R~_ and H '~ interchangeably.

20 For A c R ~ let A+ = {x E A : z~ > 0 } .

w:R~-~R+={xER:x>O} (2.1)

We define a weight function

by

w(x) = - -

1

x--(x,

,x~) CR ~

Zn

If ~/: [0,1) -~ R~_ is a continuous mapping such that ~/[0, 1) is a rectifiable curve with length s = £(q), then ~/ has a normal representation "7°: [0, s) --+ R~_ parametrized by arc length (see J. V£is~l~i [V7, p. 5]). The hyperbolic length of "/[0, 1) is defined by (2.2)

£h(~/[0,1)) =

/o

w(~t°(t))dt =

Idzt Xn

If A C R~_ is a (Lebesgue) measurable set we define the hyperbolic volume of A by (2.3)

rnh(A) = / A w(x)~dm(x)

where m stands for the n-dimensional Lebesgue measure and w is as in (2.1). If a, b E R~_ , then the hyperbolic distance between a and b is defined by

(2.4)

p(a,b) =

inf ~ h ( a ) =

c~CF~b

inf /~

c~@Fab

Xn

where F~b stands for the collection of all rectifiable curves in R~_ joining a and b. Sometimes the more complete notation PR~ (a, b) or p ~ (a, b) will be employed. The infimum in (2.4) is in fact attained: for given a,b E R +~ there exists a circular are L perpendicular to 0R~_ such that the closed subare J[a,b] of L with end points a and b satisfies

p(a,b) = gh(J[a,b]) = f 3 J[~,bl

(2.5)

Xn

J[a, bl

z[c, di I-I n

Diagram 2.1.

21

If a a n d b are l o c a t e d on a n o r m a l of 0 R ~ , t h e n

J[a,b] =

[a,b] = ( ( 1 - t ) a + t b :

0 < t < 1 } (cf. [BE, p. 134]). Because of the (hyperbolic) l e n g t h - m i n i m i z i n g p r o p e r t y (2.5), the arc

J[a,b]

geodesic segment

will be called the

joining a and b.

K n o w i n g the geodesics, we calculate the hyperbolic distance in two special cases. First, for r, s > 0 we o b t a i n

p(ren,sen)--:l~rd-dt/

(2.6)

= log r

.

Second, if ~p E (0, ½7r) we denote u v = (costa)el + ( s i n ~ ) e n

(2.7)

p(e,,

=

f

s ida ns

_ f

and calculate

s ida ns -logcot½~.

J[~,~.]

-

H~

~

/ sin a

a ¢ ~ ~ u ~

= (cosmic1 + (sinwle,,

0

Diagram 2. 2. We shall often m a k e use of the hyperbolic functions sh x = sinh x , ch x = cosh x , th x = t a n h x , c t h x = c o t h x a n d their inverse functions which are listed in 2.12. T h e above f o r m u l a e (2.6) a n d (2.7) are special cases of the general f o r m u l a (see [BE, p. 35])

(2.s)

Ix _ y]2 2x~yn

¢h p ( x , u) = 1 + - -

,

x, yEtt

'~ = R

n +.

p(x, y) is c o m p l e t e l y d(x, OH'~), y, -- d(y, OH'~), a n d

Note t h a t by this f o r m u l a the h y p e r b o l i c d i s t a n c e

determined

once the euclidean distances

Ix-Yl

x,

=

are

known. For a n o t h e r f o r m u l a t i o n of (2.8) let z, w E H n , let L be an arc of a circle p e r p e n d i c u l a r to O H ~ with

z,w E L

and let { z , , w , } = L n O H ~ , the points being

labelled so t h a t z , , z , w, w, occur in this order on L . T h e n (cf. [BE, p. 133, (7.26)]) (2.9)

p(z, w)

= log I z,, z, w, w , I"

22

p(z,w) = log t z . , z , w , w . I

i~ ~

Z,,

W~

aH"

Diagram 2.3. Note t h a t (2.6) is a special case of (2.9) when z, = 0 and w. = oo because

tO, z,w, °° I = Iwl/Iz{ for z,w E H '~ . The invariance of p is a p p a r e n t by (2.9) and (1.28): Given f in ~ N ( H ~) and x, y C H ~ , t h e n (2.1o)

p(x,y) = p ( f ( x ) , f ( y ) ) .

hyperbolic ball { x ~ I - I n :p(a,x) < M } is denoted by D(a,M). It is well known t h a t D(a,M) = B n ( z , r ) for some z and r (this also follows f r o m (2.10)!). This fact together with the observation t h a t Aten, (t/A)e,~ C OD(te,~,M) , A = eM (cf. (2.6)), yields For a E I - I n and M > 0

(2.11)

the

[ D(te~,M) = B~((tchM)e~,tshM) , Bn(ten,rt) C D(ten, M) C Bn(ten,Rt) , r = 1 - e -M , R = e M - - 1 .

~

D(te,~,M) = B~((tchM)e,~,tshM)

( e - M t ) e" ~ ~ t

I"I"

0

Diagram Z.4. The hyperbolic ball D(te~,M).

23

2.12. Remark.

T h e hyperbolic functions s h z ,

chx,

thx,

c t h x a n d their in-

verse functions arsh x , arch x , a r t h x , a r c t h x ( d e n o t e d b y s o m e a u t h o r s as sinh - 1 x , cosh - 1 x etc.) o c c u r often in w h a t follows. Recall t h a t

{

arshx=log(x+~+l),

x>O,

archx=log(x+v/~-l), x>_ 1 , + 1 arthx=½1og 0<x< 1, 1-x' X + 1 a r c t h x ---- ½ log x>l. ,

x--l'

For easy reference we record the following inequalities, whose proofs we leave as exercises: (2.13) (2.14)

l o g ( l + x) < a x s h x < 2log(1 + x) , x > O,

2log(1 +

1)) <_

< 21og(i +

• _> 1.

So far we have discussed only t h e h y p e r b o l i c g e o m e t r y of H a = R~_. N o w we are going to give t h e c o r r e s p o n d i n g formulae for B '~ . T h e weight f u n c t i o n w: B n -~ :R+ is now defined by (2.15)

2 w ( x ) = 1 - Ixl 2 ' x • B n ,

(cf. (2.1)). T h e hyperbolic distance between a a n d b in B '~ , d e n o t e d by p B ~ ( a , b ) =

p(a, b ) , is defined b y a f o r m u l a analogous to (2.5) ; the s a m e is t r u e a b o u t the hyperbolic volume of a m e a s u r a b l e set A c B n . For a , b • B ~ the geodesic segment

J[a,b]

joining a to b is an arc of a circle o r t h o g o n a l to S n-1 . In a limiting case the points a and b are l o c a t e d on a euclidean line t h r o u g h 0 .

Diagram 2.5.

24 In particular, J[O, tel]= [0,tell for 0 < t < 1 a n d we have t

(2.16)

p(O,tel) =

1 --'~-I 2 -

1 ~--s 2 - log ~---~ - 2 a r t h t .

[0,te~]

o

It follows f r o m (2.16) t h a t for s • ( - t , t )

p(sel,tel) = l o g

(2.17)

(l÷t l-s) 1~ ~. t~s

"

A c o u n t e r p a r t of (2.8) for B = is

(2.18)

Ix - y i ~ , x, y C B ~ sh2(½P(x'Y)) = (1 - I x 1 2 ) ( 1 - t y l 2) '

(cf. [BE, p. 40]). As in the case of H '~ , we see by (2.18) t h a t the hyperbolic distance

p(x, y) b e t w e e n x and y is completely d e t e r m i n e d by the euclidean quantities Ix - Yl, d(x, a B ' ~ ) , d(y, OB'~). Finally, we have also (2.19) where x , , y,

p(x, y) = log ] x,, x, y, y, I, are defined as in (2.9): If L is the circle o r t h o g o n a l to S ~-1

with

x , y • L , t h e n { x , , y , } = L r l S n--1 , the points being labelled so t h a t x , , x, y, y, o c c u r in this order on L . It follows f r o m (2.19) and (1.28) t h a t (2.20)

=

for all x,y • B '~ w h e n e v e r h is in ~ J ~ ( B ~ ) .

Finally, in view of (1.28), (2.9), and

(2.19) we have (2.21)

=

• B" ,

w h e n e v e r g is a MSbius t r a n s f o r m a t i o n with g B ~ = H ~ . It is well k n o w n t h a t the balls D ( z , M ) g e o m e t r y as well, i.e. D ( z , M ) = B'~(y,r)

of ( B n , p )

are balls in the euclidean

for some y E n n a n d r > 0 . M a k i n g use

of this fact, we shall find y a n d r . Let Lz be a euclidean line t h r o u g h 0 and z and { z l , z 2 } -- L z N O D ( z , M ) ,

Izll ~ Iz2[. We m a y assume t h a t z =fi 0 since w i t h obvious

changes the following a r g u m e n t works for z = 0 as well. Let e = z/Iz I and zl = se,

z2 = ue, u • (0,1) , s • ( - u , u ) .

It follows f r o m (2.17) t h a t

25

(1+ lzl l - s ) p ( z l , z ) = l o g ]---iz t 1 7 ~ (l+u

=M'

1-1zl)

p(z2,z) = l o g 1 - - u ' - - ~ l

=M"

Solving these for s and u and using the fact that

~

D ( z , M ) = B'~(½(zl + z2), ½1u - sl) one obtains the

following formulae (Exercise: Verify the computation.):

"~Diagram 2.6.

D ( x , M ) = B'~(y,r) ,

(2.22)

x(1 - t =)

ix12t2

(1 -]xI2)t r

,

1 -- [x---~

1

' t = th ~ M

and Bn(x,

(2.23) a=

a(1 -I<)) C D ( x , M )

t(1 + I<) l ÷ lxlt

' A-

C Bn(x,

t(1 + I~l) 1-1xlt

' t =

A(1 -I<)) , th ½M

We shall often need a special case of (2.22): (2.24)

D(O,M) =

Bn(th ~1 M ) .

A standard application of formula (2.24) is the following observation. Let Tz be in N(B ~) as defined in 1.34 with Tx(x) = 0. Fix x , y E B n and z E J[x,y] with p(z, x) = p(z, y) = ½P(x,y) . Then Tz(x) = - T z ( y )

(2.25)

and (2.24) yields

J" IT~(y)I = th ½p(z,y), ITs(x)]

th l p ( x , y ) .

T~

Diagram 2.7.

26 We next derive a few inequalities from (2.23). By studying the expression for the radius r in (2.23) we see t h a t

d( D ( z , M ) ) <_ d(-D(O,M) ) = 2 t h ~ M

(2,26)

- -

1

for all z E B '~ and all M > 0. This yields a s h a r p inequality between the euclidean and hyperbolic distances as follows. For given x, y E B '= choose z E

p(x,z) = p(z,y) = {p(x,y).

T h e n with M =

½P(x,y)

J[x,y] with

(2.26) yields the useful

inequality

(2.27)

Ix - vl -< 2th ¼p(~,v).

Equality holds here if x = - y .

Because t h A < A ,

(2.27) yields also the crude

estimate

(2.28)

t x - vt _< ½.(~, v)

for x, y E B '~. 2.29. E x e r c i s e .

Verify the following elementary relations.

(1) 1 - e - s < t h s <

1 - e -28 for s > 0 .

(2) If s > 0 , then th 2s

ths =

1 + ~/1 - th 2 2s Further, if u E [ 0 , 1 ]

and 2 s = a r t h u , t h e n

th~ -

u 1 + ~/i - ~

< ½(u + u 2 ) . -

(3) l o g t h 8 = - 2 a r t h e -28 , s > 0. X

(4) - - _ < l - e l+x (5) Show t h a t

-x_<x,for

x>-I

[AS, 4.2.32].

1 + thpx

1-thpx for p =- 1 , 2 , . . .

-

(1 + t h x ~ p ~l--thx )

and x > 0 .

2.30. E x e r c i s e .

Show t h a t for x, y E B ~ p(~,v)

_<

2t~-vl rain{ z - Ixl, 1 - Ivl } "

27

2.31.

Exercise.

Applying (2.23) show that if

D(x,M) = B'~(y,r),

then r

admits an estimate (1-[yl) b
(1-1Yl) B

where b and B depend only on M . Show also that the numbers a and A in (2.23) have lower and upper bounds depending only on M . In particular,

A/a

has an upper

bound depending only on M . 2.32. E x e r c i s e . Let x 0 E B '~, M > 0

and

v = min{ Iz - =01: p(=o,z) = M } ,

V = max{ l z - xol: p(xo, z) = M } .

V/v

Find an upper bound for 2.33.

Exercise.

by applying 1.43(1) .

Rewrite

(2.8) and

(2.19)

using the identity

2sh2A

=

ch 2A - 1. Given distinct points x and y in B ~ or H ~ one can express the Poincar~ distance

p(x, y)

in terms of the absolute ratio I x,, x, y, y,I by virtue of the formulae

(2.9) and (2.19) where x . and y, are the "end-points" of a geodesic segment containing x and y . Sometimes it will be convenient to express

p(x,y)

in a different way

without refering to the points x , and y, at all. Such an expression can be achieved by exploiting an extremal property of

p(x, y)

as we shall show in the next section (see

also Section 8). The formulae (2.8) and (2.18), which give explicit expressions for PH,~ (z, y) and PB- (x, y ) , respectively, are of fundamental importance for hyperbolic geometry. As a m a t t e r of fact, many formulae of this section can be derived directly from these formulae. For many applications it would be formally adequate to define the hyperbolic distance in terms of (2.8) and (2.18) without any reference to the geometric interpretation involving elements of lengths or the length-minimizing property of geodesics. These geometric notions and their invariance properties are, however, the reason why the hyperbolic metric is so useful and natural in many applications. The reader may show as an exercise that (2.23) follows from (2.18). The explicit

p(x,y) are somewhat complicated. Often it will be p(x, y) in terms of simple comparison functions. We now

expressions (2.8) and (2.18) for sufficient to give bounds for introduce such a function.

28 For an open set D in R n, D • R " , d e f i n e

(2.34)

j~ (x, y) : log (1 +

d(z) =d(z,OD) for z E D

Ix_-_y!

and

.~

min{d(x), d(y) } ]

for x , y E D . If A C D is non-empty define

JD(A) = sup{ jD(x,y) : x,y e A } .

(2.35)

An elementary (but lengthy) argument shows that JD(X,y) is a metric on D .

The following inequalities d(x) (1) jD(X,Y) > t log d(y) '

2.36. Lemma.

(2)

jD(x,y) < [logd(X) + l o g ( l + I x - Y l ) < 2jD(x,y) -

d(y)

d(~)

-

hold for all x, y E D. P r o o f . (1) The proof follows because d(y) < d(x) + Ix - Yt . (2) If d(x) < d(y), the proof is obvious in view of (2.34). If d(x) > d(y),

JD(~,~)

log\l{ +

I x - yl ~ < lo td(x) + d(x) J~__-y!) d(y) ] - g ~ d ~ d(y) d(y) ,1

d(x) =log~

+

log(1 + I~_~_yE~ d(y) ] < - 2jz)(x,y)

where in the last step the inequality in part (1) was applied. 2.37. E x e r c i s e .

'

[]

For an open set D C R ~ with D ¢ R n and for a non-empty

set A in D with d(A, OD) > 0 put

d(A) r D ( a ) - d(a, OD) Show that 1 ± log(1 + rD(A)) <: log(1 ÷ ~rD(A)) < jD(A) <: log(1 + rD(A))

2.38. E x e r c i s e .

Let G and G ~ be proper subdomains of R n with G~C G.

Show that jc(x,y) < jc,(x,y) for all x,y ~_ G'. For w E R ~ set R~o = R ~ \ { w } and define

ha(x,y ) = sup{jn~(x,y) : w e OG}

29 for all x,y E G . Show that if w E OG (2.39)

jG(x,y) = hG(x,y ) > lo g LY-wl x-w

; x, y E G .

Moreover, if d(x) <_d(y) and z e OG with Ix - z I = d(x) prove that Iv - zl

Ix- z---~-> }(expjA~,V)- 1) 2.40. E x e r c i s e .

(1) Let B--S'~-I(e,~,I) N { x E H '~ : x , ~ - > l } . Find

max{ PH- (en, X) : x e B} and min{ PH- (en, x) : x, e B } . (2) For an open set D C R n , D ~ R n , define

"3D(x,y)=log[(1 + tx--YI~ (1 + tx--YI~] d(x) ] d(y) ]J " Show that for all x, y E D

jD(x,y) <_ 7~,(x,v) _< 2j~(x,v). In the next lemma we show that JD yields simple two-sided estimates for PD both when D = B 2.41.

~ and when D = H

Lemma.

(1) jB,,(z,y)

_<

n.

pB~(z,y) <_ 4jB~(x,Y) For x,y E

B

~ •

(2) JHo(X,V) <--PHo(X,V) < 2jm(x,U) ~O. X,U ~ H " P r o o f . (1) By (2.19) Ix - vl ~ < k(min{1 I x - Yl ~2, sh2(½PB"(x'Y)) = (1 --Ix]2)(1 --lyI 2) --t2 -- ~l]-i - lyt}

and hence by (2.14)

pB.(x,y) ~ 4log(1 + t) < 4 j B . ( x , y ) • For the proof of the lower bound we may assume that Ix I > [Yl and Ix[ > 0. Let L be a euclidean line through 0 and x and fix y ' e "B'~(Ixl)NL such that Ix-y'i = I x - y t . Because lY'[-< [Yl it follows from (2.19) and (2.18) that (1 + Ixl

pB.(X,y) >_pB.(x,y')--> log

1-lxl+[x-yl)

l_txl-l+ixl-tx ~

>_jB.(z,y).

30 (2) Denote u -- 1 + ]x - yl2/(2xnyn).

,..(~,y) <-- 2log(1 + v ~ -

By (2.8) and (2.14) we get

1))= 21og(1 + xk:_ yl ~ < 2~;o(=,~). v f ~ ~ynn / --

For the proof of the lower bound we may assume that x,~ ~ yn and x = xnen • Let

y' = (x,~ + ix - yl)e,~. Because (y')n > Yn it follows from (2.8) and (2.6) that

...(~,y) > . . . ( = , y ' ) > log(i + I~:~l) =j.o(~,y).

[]

Xn

2.42. E x e r c i s e .

Solve 1.41(2) with the help of the hyperbolic metric. [Hint: [l+i~l~ 2 l+r Because of (2.17) the requirement that p(0, a) = ½P(0, re1) leads to ~ 1-t~1/ = 1 - , , i.e. t a l : r / ( l + v / 1 - r 2.43. E x e r c i s e .

2).] For an open set D in R ~, D ~ R

~D(x,y)----log(l+max{

,x--y[

~,let

[x--y[ 2 } )

Show that jD(x,y) ~ +D(X,y) ~_ 2JD(X,y ) . (See also 3.30.) 2.44. E x e r c i s e .

(1) Observe PH"

first

that, for t E (0, 1),

llh (ten, en) = PH, (ten, S n - l ( l~e n, -~jj

(cf. (2.8)). Making use of this observation and (2.11) show that 1 1 1 B n (~e~, ~) = [J D(te., log ~).

tc(0j) (2) For p > 0 and t :> 0 let A(t)

=

flH n ((0, tP),

(t,tv)).

Find the limits

limt--.0 A(t) and limt~o~ A(t) in the three cases p < 1, p = 1, and p > 1. 2.45. E x e r c i s e . The stereographic projection ~2 (see 1.20) provides a connection between the hyperbolic geometries (B~,p) and ( R ~_+ I ,p_) and the spherical geometry of (R'~, q). Verify

that

p(0, ael)

=

p - (772(0), 772(ael)) , a e (0, 1), by com-

puting the absolute ratios I - el, 0, ae 1, el ] and ] 772( - e l ) , 772(0), 7r2 (ael), ~r2(el) I (see (2.9) and (2.20)). Note that 2q(O, ael)

=

]~r2(0) - r 2 ( a e l ) [ . Let be1 be the orthogo-

nal projection of r2(ael) onto the xl-axis. Show that p(O, bel) = 2p(O, ael). [Hint: See the diagram 1.5 in 1.41(2).]

31

2.46.

Exercise.

where Sn(x,r)

( C o n t i n u a t i o n of 2.45.)

Show t h a t

7r2(aei) • S n N S n ( x , r )

is a s p h e r e o r t h o g o n a l to S ~ w i t h ael • S ' ~ ( z , r ) • Find x a n d r .

2.47. Exercise. that

Let x , y E B n and let T , E ~ ( B ") be as defined in 1.34. Show

Ix-yl _ 8 rT.yl = x/Ix - yf2 + (1 - rxl:)(1 - tyt ~) ~vq-4-~ '

w h e r e 82 = Ix - yl2/((1 - IxI2)(1 - lyl2)). [Hint: By (2.25) a n d (2.19) 1

ITzyl 2 -- thZ(½P(x,y)) = 1

82

ch2(½P(x,y)) - -------~ 1 + s "]

Next let z E J[x,y] be the hyperbolic m i d p o i n t of J[x, y] as in (2.25). Show t h a t

ITs< = r T z y t -

8

1 + v/1 - 8 ~ Ix - yl

V/Ix - yt 2 + (1 -

Ix12)(1

- lyl 2) + v/(1 - ]x12)(1 -

where s is as above. [Hint: Because t h A = t / ( l + v / 1

lyl 2)

'

- t 2 ) , t = t h 2 A , one can a p p l y

(2.25) and the above c o m p u t a t i o n . ] Moral: I n s t e a d of using these lengthy expressions for tTzyl a n d ITzyl involving euclidean distances it will often be m o r e convenient to use t h e equivalent f o r m u l a (2.25) involving the hyperbolic d i s t a n c e p(x, y ) . 2.48. Exercise.

Let x , y E R n a n d let t~ be a spherical i s o m e t r y as defined in

(1.46). Show t h a t

]t~y I =

Ix - Yl v / ( 1 + Iml=)(x + lyl 2) - Ix - yt =

[Hint: T h i s follows i m m e d i a t e l y f r o m (1.47) a n d (1.15).] Let a E [0, lrr] be such t h a t

s i n s = q(x,U).

Then ~ is the angle between the segments

[e=+l, t~xt = [e~+l, 01

a n d [e,~+l, t~y] at e,~+l (see (1.13).) Show t h a t the a b o v e f o r m u l a can be r e w r i t t e n as

It=yl = t a n ~ . Note the analogy with (2.25). 2.49. Exercise.

Show t h a t If(x) - f(y)l 2

(1 - I f ( = ) I =) (~ - I , ( y ) l =) for all f in ~ ( B

Ix - yl ~ ( 1 - [ x l 2 ) ( 1 - [ y l 2)

'~) a n d all x, y e B ~ . [Hint: A p p l y (2.19) a n d (2.21).]

32 2.50. E x e r c i s e .

Let 0 < t

< 1 and f E ~ M ( B n). Show that I x -- Yl

tf(~) - f(y)t <

1 -- t 2

for Ix[, [Yl - t . [Hint: Apply 2.49.] 2.51. R e m a r k .

The inequality (2.27) together with the formulae (2.25) and

2.47

yield for x, y E B ~ 2tx - yl 1~ - yl -< 2 t h l p ( ~ , y) = ~ / I x - yl 2 + b2 + b

where b = V/(1 - I x l 2 ) ( 1 - [ y [ 2 ) . 2.52. E x e r c i s e (Contributed by M. K. Vamanamurthy).

Starting with the iden-

tity (cf. 2.47) I~-ul th ½P(x,y)

=

V/I x _ Yl 2 +

(1 -"iX'[2)(1 - l y l 2)

for x, y E B '~ verify the following inequalities

(1)

I~ - yl < th l p C ~ , y ) 1 + I~llyl 1~1 - I~1 < th ½ p ( ~ , y )

(2) (3)

1 -

llx-

Yl < -

where Ixi' = V/1 - l x l

[xtly I -

<

t~ + Yl

-

1 + Ixllyl

I x - Yl < th 1 + IxIiyl + Ixi'lyt' -

<

I~ - yI

-

1 - ixllyl + J~l'lyl'

2 .

< I ~ - yl - 1 - Ixllyl

<

' '

lp(x,y)

Ix - ul

2(1 - m a x { l ~ t ~, i~f:})

'

Can you find similar inequalities for the spherical chordal

metric? [Hint: 2.48.] 2.53. N o t e s . The main source for this section is [BE] and the other references given at the end of Section 1. See also [T, pp. 508-514] and [RE].

33

3.

Quasihyperbolic geometry

In an a r b i t r a r y proper s u b d o m a i n D of R '~ one can define a metric, the quasihyperbolic metric of D , which shares some properties of the hyperbolic metric of B '~ or H '~ . We shall now give the definition of the quasihyperbolic metric and state without proof some of its basic properties which we require later on. The quasihyperbolic metric has been systematically developed and applied by F. W. Gehring and his collaborators. T h r o u g h o u t this section D will denote a proper s u b d o m a i n of R '~ . In D we define a weight function w: D --* R + by 1 w(x) = d(x, OD) ' x e D .

(3.1)

Using this weight function one defines the quasihyperbolic length £q('~) = £g('y) of a rectifiable curve -~ by a formula similar to (2.2). The quasihyperbolic distance between x and y in D is defined by /,

kD(x,y ) =

(3.2)

inf e g ( ~ ) = c~CP~y

where Fzv is as in (2.4).

inf

/w(x)]dx l

cz@P~ Jc~

'

It is clear t h a t k D is a metric on D .

It follows from

(3.2) t h a t k D is invariant under translations, stretchings, and orthogonal mappings. (As in (2.3) one can define the quasihyperbolic volume of a (Lebesgue) measurable set A c D , but we shall not make use of this notion.)

Given x , y E D there exists a

geodesic segment JD[X,y] of the metric k D joining x and y (cf. [GO]). However, very little is known a b o u t the structure of such geodesic segments

,]D IX, y]

when

D is given. Some regularity properties of geodesic segments have been obtained by G. M a r t i n [MA]. 3.3. 2PB,

Remarks.

Clearly, k H , ---- pH,~, and we see easily t h a t PB- --< 2 kB,, _<

(cf. (3.1),(2.15)). Hence, the geodesics of ( H ~ , k H . ) are those of ( H n , P H , ) ,

but it is a difficult task to find the geodesics of k D when D is given. The following m o n o t o n e p r o p e r t y of k D is clear: if D and D ~ are domains with D r c D and x , y E D ' , then k D,(x,y) > k D(x,y) .

34 In order to find some estimates for k D (x, y) we shall employ, as in the case of I-I n and B n , the metric JD defined in (2.34). The metric JD is indeed a natural choice for such a comparison function since both k D and JD are invariant under translations, stretchings and orthogonal mappings. A useful inequality is ([GP, L e m m a 2.1]) (3.4)

kD(X,y ) >_jD(x,y); x, y E D .

In combination with 2.36, (3.4) yields

(3.5)

k~(x, v) >

log dd(x) - ~ ' d(z) = d(z, OD) .

For easy reference we record Bernoulli's inequality

(3.6)

log(l+as) ~alog(l+s) 3.7. L e m m a .

; a ~ 1, s > 0 .

(1) I[ x ~ D , y E B , = B ~ ( x , d ( x ) ) , then

Ix-vl (2) I f s e (0, 1) and

Ix - yl 5 ~ d ( x ) , then 1

kD(x, v) -<- 1- -- - s

j~ (~, v).

P r o o f . (1) Select z E OB~ such that y E Ix, z].

Diagram 3.1. Because Ix, y] E F~ u , from 3.3 we obtain

/ d(w) Idwl ~- /

k D (x, y) ~ kB. (x, y) ~

i~,v]

d(x)

(

dC,*)

/

I'~ - z]

dt t

d(~)-J*-vl

Ix,v]

= log d(x) - Ix - vf = log 1 + ( = JR°\~z} (x, v) ) .

Idwl _ Ix-vl

T;=yl)

35 (2) For the proof of (2) we apply part (1), Bernoulli's inequality (3.6), and the definition of JD to obtain Ix -

1 --< i L s l ° g as desired.

(

1+

[x:y[~

1

d(x) ] <- 1----~s

jD(x,y )

O

We know by 2.41 and 3.3 that if D = B ~ , then an inequality similar to 3.7(2) holds for all x, y E D . For a general domain D this is not true, i.e. the ratio

A(D) = s u p

jD(x,Y) : x, y C D ,

x~y

may be infinite. For instance, A ( B : \ [0, el)) -- co. (For details, see 3.14.) 3.8.

D e f i n i t i o n . A domain G in R n , G :fi R '~, is called uniform, if there

exists a number A = A(G) > 1 such that kG(x,y ) < AjG(x,y ) for all x,y E G. By 2.41 the unit ball B '~ and the half-space H n = R ~

are uniform domains

with the constants 4 and 2, respectively. It follows from the definition that the class of uniform domains is invariant under translations, stretchings and orthogonal maps. It is not difficult to show that the image of a uniform domain under a bitipschitz mapping is again uniform. Next we shall study the quasihyperbolic balls DG(X, M) = { z E G: kG(X, z) < M } when x E G and M > 0. It follows from (3.5) that

e-Md(x) < d(z) < eMd(x) holds for z e -DG(x,M). Next, for z e B n(x, (1 - e-M)d(x))

we deduce by 3.7(1)

t h a t kG(Z,Z ) < M and for z e R '~ \ Bn(x, (eM - 1)d(x) ) we find by (3.4) and (3.5) that kG(X , z) > M . In conclusion, we have proved that

{ Bn(x, rd(x) ) c DG(X,M ) c Bn(x, Rd(x) ) , (3.9)

Da(x,/)

C { z E G : e-Md(x) <_d(z) <_ eMd(x)},

r--_l--e -M , R = e M - - 1 . For G = H n one can show that the numbers r and R are the best possible (see (2.11)). We shall write D(x,M) for DG(X,M ) if there is no danger of confusion.

36

MSbius transformations are hyperbolic isometrics.

T h a t is, if each one of the

domains G, G I is a ball or a half-space in R ~ and if f is a M5bius transformation m a p p i n g G onto G ~, then

pG( , y) =

(I(x), f(y))

for x , y E G (cf. (2.9), (2.19), (2.21)). Although the metric k c does not have this invariance p r o p e r t y it is not changed by more t h a n the factor 2 under MSbius transformations. (For a related MSbius invariant metric see J. Ferrand [FE].)

If G and G' are proper subdomains of R '~ and if f is a h~Sbius

3.10. L e m m a .

transformation of G onto G ~ , then 1

&c(x,y) < k c , ( f ( x ) , f ( y ) ) < 2 k c ( x , y )

for all x , y E G . A proof of L e m m a 3.10 was given by Gehring and Palka in [GP] where also a generalization to the case of quasiconformal mappings was obtained. This generalization will also be proved below in Section 12. 3.11. E x e r c i s e . T h e logarithmic spiral in R 2 has a p a r a m e t r i c representation r(w) = Ae B"~ in polar coordinates where A and B are constants and A > 0.

It

was shown by G. Martin and B. G. Osgood [MAO] t h a t the geodesic segments of k G , G = R n \ {0}, can be obtained as follows. Assume that x, y E G and t h a t the angle between the segments [0, x] and [0, y} satisfies 0 < ~ < 7r. T h e n the triple 0, x, y determines a 2-dimensional plane E and the geodesic segment of k c connecting x to y is a logarithmic spiral in E with equation

r(w) = l x I e x p ( ~ log lXl'~

lyt); 0_<~_<~.

Knowing this equation show (by integrating the element of length along this curve) that

(3.12)

kG(x,y) = V/ 2+log2 Ixl , a = R" \ {0},

holds for all z, y ~ a .

Making use of (3.12) study the set { z E G : k ~ ( e l , z) = t } .

Note the special case t = 7r.

37 3.13.

E x e r c i s e . Show that G = R n \ {0} is a uniform domain and that

k a ( x , y ) < A j G ( x , y ) for x , y • G where A2 = 1 +

~ 21.5 .

[Hint: Let x, y • G and let p • [0, ~r] be the angle between x and y. By a property I:~-vt and hence of the bisector of an angle in a triangle, sin 71~ -< /~l+Ivl Ix - vt

< 2arcsin I x l ~ IYl -

- vl

lxl + IYl

-< lo-g27rlog 1 + ~ l x l

-< log2 jG(X'Y)

"

By 2.36(2) and (3.12) we obtain the desired inequality.] 3.14. E x e r c i s e . Show that G = B 2 \ [0, el) is not a uniform domain. [Hint: For t •

(0,]!60) let xt = (1,t) and Yt = ( ¼ , - t ) , Y = { ( 0 , y ) : y > 0 } .

ka(xt,yt) > k G ( x t , Y ) = kH~(Xt, Y )

Show that

~ eye

when t --+ 0 (of. (2.7)), while j c ( x t , y t ) = log3 for all t • (0, ~60).1 3.15. E x e r c i s e . Let t • (0, 1). Show that

D a ( x , log(1 + t)) c B " ( x , td(x)) c D G ( x , log 1_..~) . 1--t [Hint: Apply (3.9). 1 3.16. E x e r c i s e . Suppose that there exists C > 1 such that for all x , y • G

kG(x,y ) <_ Csup{kR%{z}(X,y) : x • c~G} . Show that G is uniform. [Hint: Apply 3.13 and (2.39).] 3.17. E x e r c i s e . Let f: R " --+ R '~ be an L-bilipschitz mapping, that is

1x - y l / L <

If( x ) - f ( Y ) [

< nlx-Yl

for all x , y • R '~, and let G C R " be uniform. Show that f G Using the definitions show that

k G ( x , y ) / L 2 <_ k f G ( f ( x ) , f ( Y ) )

<_ L2kG(x,y) .

is uniform. [Hint:

38 Then deduce from (3.6) t h a t

j c ( x , y ) / L 2 <_ j : c ( f ( x ) , f(Y)) <_ L2jc(x, y) .]

3.18. E x e r c i s e .

Let G = R " \ {0} and f(x) = a2x/Ixl 2 for x e G , where

a > 0. Show that k G ( : ( ~ ) , : ( y ) ) = kG(z, y) for x , y E G . [Hint: Apply (3.12).] Show also that

jc(f(x),f(Y))

= jc(x,Y)

for x, y E G . [Hint: Apply (1.5).] Note that these assertions do not follow from 3.10. 3.19.

The symmetric ratio.

For distinct points a , b , c , d in R n define the

symmetric ratio by

(3.20)

s(a,b,c,d) = la, b,d, ct!a,c,d,b! .

Then by (1.27)

q(a,d) 2 q(b,c) 2 s(a, b, c, d) = q(a, b) q(b, d) q(a, c) q(c, d) '

(3.21)

which we recognize as the expression studied in 1.32. We recall that s is symmetric, i.e.

4 a , b,c,d) = s(a,c,b,d) = ~(d,b,c,a) = s ( b , a , d , ~ ) , and also ~ j ~ ( ~ n ) -invariant in the sense that (3.22) for all f in ~ ( R ' ~ ) . (3.23)

s.f(a, b, c, d) -- s(fa, fb, fc, fd) = s(a, b, c, d) Let D c R~ be an o p e n s e t with c a r d ( R ~ \ D ) >

SD(b,e) = sup{ l s(a,b,c,d) : a,d e c~D } .

It follows from (3.21) and (1.15) t h a t (3.24)

s(a,x,y,c~) =

Ix - yl 2 la - xlty - al

2 and define

39

3.25. The point-pair

invariant

sG .

We next list some i m m e d i a t e properties

of the f u n c t i o n s c ( b , c ) w h e n G c R n . (1)

sa(x,y ) = sc(y,x ) ,

(2)

~ I ~ ( f ( x ) , / ( y ) ) = ~G(x,~) for I c 9 ~ ( r ~ n) and ~,y ~ a ,

(3)

G' C G a n d x , y E G ~ imply s a , ( x , y ) > s a ( x , y ) ,

(4) for ~ e d

y~ a, ~G(~,Y)~0

and ~G(x,y)--~ o~ iff ~ - ~ Oa,

iff ~ - ~ y

s a ( x , y ) >_ (q(OG) q ( x , y ) ) 2 .

(5)

3.26. Lemma. Proof.

s B , (b, c) = ch

pB .

(b, c) - 1 for b, c E B '~ .

Because this equality is ~ N ( B ~ ) - i n v a r i a n t , r E ( 0 , 1 ) . T h e n r = th ¼ P ( x , y )

-re1 =-c,

we m a y assume t h a t

b =

by (2.25). It follows f r o m ( 3 . 2 0 ) t h a t

for a, d C S n - 1 we o b t a i n 4r2[a -- d[ 2

s(a,b,c,d)

4r2]a - d[ 2

= la _ b l l b _ dlla _ c l l c _ d l

= ia _ blla _ ctld _

blld _ cl

.

It is left as an exercise for the reader to show t h a t

'~-1} = 1-

min{ta-blla-cl:aeS

r2

and similarly for [d - blld - el. T h u s

s = sup{s(a,b,c,d):

a , d E S n - 1 ) <_

4r222

(1--r2) 2

This u p p e r e s t i m a t e is in fact a t t a i n e d if a = - e l = - d .

= { 4r ~2 \l--r 2] " Hence

s = 1 6 s h 2 ( l p ( b , c ) ) c h 2 (¼ p ( b , c ) ) = 4 s h 2 (½ p ( b , c ) ) = 2 ( c h p ( b , c ) a n d s B , ( b , c ) = c h p ( b , c ) - 1 as desired.

1)

[]

It is clear by (2.22) t h a t L e m m a 3.26 holds for the h a l f - s p a c e H n , too. Note also t h a t L e m m a 3.26 yields a f o r m u l a for p(b, c) involving the absolute ratio (3.27)

chp(b,c)=l+suP{½ta,

b,d, c I f a , c , d , b [

: a, d e S ~ - l } .

Recall t h a t a different f o r m u l a was given in (2.19). A n a d v a n t a g e of (3.27) over (2.19) is t h a t it generalizes to any d o m a i n G in R." with card(R. '~ \ G) > 2. For such a d o m a i n G define a f u n c t i o n P c by

(3.28) w h e n b, c E G .

chp~(b,c) = 1 + ~ ( b , c ) ,

4O 3.29. R e m a r k .

It is an interesting question whether the function Pc defined

in (3.28) is a metric. We shall discuss this below in Exercise 3.31. 3.30. E x e r c i s e .

1 Assume that a > 0 and define b by c h b = 1 + 7a

.

Show

that log (l + max {a, x/~ }) < b < l o g ( l ~

a+v/-a)

<21og(l+max{a,v/-a}) 3.31. E x e r c i s e .

.

Let G = R n \ {0} and s a as defined in (3.23). Explicitly, we

see by (3.24) that sG(~,y)

-

I ~ - yl ~ 21~lrvF ' ~' u ~ c .

Define Pc as in (3.28). Applying 3.30 show that

j a ( x , y ) < 2 p c ( x , y ) < 4jG(x,y) for x, y E G. 3.32. E x e r c i s e .

(1) Let D = R

' ~ \ { 0 } . Show that

1 + 2q(x,y) < expkD(X,y ) for x, y E D . (2) Let z e R ~ and G = R ~ \ { z } .

Show that

q(x,y) < ½ c ( e x p k G ( x , y ) --1) forall x, y e G , w h e r e

c = l + T I z1t ( I z l + ~ )

.

[Hint: Let D

~

R

n

\ { 0 } and

h(x) = x - z . Then by part (1) and 1.54 k c ( x , y) = % ( x

- z , y - z) > log(1 + 2 q ( x

-

z, y -

z))

> Iog(1 + 2q(x,y)/c) where c is as above.] Conclusion: If G is a proper subdomain of R '~ , then (cf. 3.3)

expkc(x,y ) > 1 + 2q(x,y)/A where A depends only on rain{ Izl : z E O G } . 3.33. E x e r c i s e . that f ( t ) / t

(1) Let f : [0, oe) --+ [0, oc) be increasing with f(0) -- 0 such

is decreasing on (0, c~). Show that f ( s + t) <_ f(s) + f(t) for s , t > 0.

41 (2) Let (X, d) be a metric space and let f be as in part (1). Show that

(X, fod)

is a metric space, too. (3) Let 0< a<

(X,d)

be a metric space and let

dl(x,y)

-- max{

d(x,y), d(x,y)~}

,

1. Show t h a t (X, dl) is a metric space, too.

(4) Give an example of a metric space (Y, d) such t h a t d z does not satisfy the triangle inequality for any fl > 1. 3.34. N o t e s .

T h e quasihyperbolic metric has been developed by F. W. Gehring

and his students.

Several interesting results can be found in [GP], [GOS], [MA],

[MAO]. Since 1978, when uniform domains were introduced by O. Martio and J. Sarvas [MS2], they have found many interesting applications, e.g. in P. Jones' works [J1], [J2] on extension operators of function spaces. An exposition of these results occurs in [GS], with several equivalent definitions of plane uniform domains. The above variant of the definition of a uniform domain is suggested by [GOS] and [VU10].

4.

Some covering problems

In this section we shall consider some geometric problems related to the hyperbolic or quasihyperbolic metric. A typical question, which we are going to answer, is the following. Let X be a compact set in B '~ and let iT be a covering of X by hyperbolic balls with fixed radii. The problem is to extract a subcovering iT1 of iT with X c [J 71 and to give a quantitative upper bound for card iT1 in terms of the parameters of the problem. 4.1.

(a, b, s ) - a d m i s s i b l e f a m i l i e s .

Let G be a proper subdomain of R n ,

a, b E G , and s E ( 0 , 1 ) . A family 7 = { B said to be

(a, b, s)-admissible

(4.2)

{

n(xi,ri)

:i--

1,...,p}

of balls in G is

if the following two conditions are satisfied:

(1)

__aeBn(xl'srl)'__

(2)

Bn(xi,srj)

A

_ bEBn(xp,srp), Bn(Xj+l,srj+l) 7£ O, j

---- 1 , . . . , p -

1.

We shall show that the smallest possible number of balls in an (a, b, s) -admissible family is roughly proportional to

kG(a,b ) , with

a constant of proportionality

T h e case G ----H '~ will be studied first. To this end note t h a t by (2.6) (4.3)

l+t

p(-Bn(x, txn))--log~-~-,

t G (0,1)

c(s).

42 for z = (3~1,... , Zn) E H n .

Let a, b E H n and s E (0,1).

4.4. L e m m a .

There is an (a, b, s) -admissible family containing at most 1 + p ( a, b) / log 1±8 1--s balls. (2) Every (a, b, s) -admissible family contains at least p(a, b)/ log ~ balls.

(1)

P r o o f . (1) Choose an integer p >_ 1 such that l+s l+s (p - 1) tog -1- - s <_ p(a,b) < plod 1 - s

(4.5) Select points y o = a ,

yj E J[a,b] such that P ( Y o , Y j ) = j l o g ~ , j= 1,...,p-I, and set yv = b. Let Bn(xj, sxjn) be chosen so that S'~-l(xj, sxj,~) is perpendicular to J[yy 1,Yj] and Yj-I, Yj E S n - l ( x j , s x j n ) ,

j = 1,...,p.

words, Bn(xj, sxjn) = D ( z j , M ) ,

and zj e J[Yj-I,Yj]

p(zj,yj-1) = p(zj,yj) = M ,

where 2M = l o g ~

1 _< j < p -

(In other and

1.) Here xjn is the n t h coordinate of

z j . In view of (4.3) and (4.5) the family { B " ( x j , x j ~ ) :

j = 1,...,p}

is the desired

(a, b, s) -admissible family. is (a,b,s)-admissible.

(2) Suppose that {B'~(zj,rj) : j = 1 , . . . , m }

By (4.3)

we get m

m

p(a,b) <_ E P ( - B n ( z j , j=l

srj)) <_ E p ( - B n ( z j , j=l

szjn)) = mlog 1 +____~s 1- s

from which the desired lower bound follows. Before formulating an analogue of Lemma 4.4 for B n we make a few observations about hyperbolic balls. By (2.23) D ( x , M ) = B'~(y,r) with r

( 1 + Ix])t

1--ty t

1+ Ixlt 2

E [th ~M, 1 th M ] ; t = th ½M ,

for all x E B n (see Exercise 2.31) and hence (4.6)

Bn(y, (th ½M)(1 - lY])) c D ( x , M ) c B'~(y, ( t h M ) ( 1 - ]y])) .

It follows from (4.6) that (4.7)

log 1 + s < p(B"(z, s(1 - l z t ) ) ) < 2log 1 +______~s 1 - s

-

--

1 - s

43

4.8. L e m m a . (1)

Let a, b E B n and s C ( 0 , 1 ) .

There is an (a, b, s) -admissible family containing at most 1 +p(a, b)/log 1--8 I+8

balls. (2) Every (a, b, s) -admissible family contains at least p(a, b)/(2 tog ~+~ 1 - - 8 1 balls. P r o o f . As in the proof of L e m m a 4.4 we cover j --- 1 , . . . , p }

B'~(yy, s ( 1 -

, p < 1 + 2-~p(a,b) tYYl)).

where

M

J[a,b]

by

is chosen so that

By (4.6) we may choose M = ½ 1 o g ~ _ : .

{-D(xj,M) -i)(xj,M)

: C

The proof of (1)

follows now as in L e m m a 4.4(1). The proof of (2) is similar to the proof of part (2) of L e m m a 4.4 except t h a t here we use the two-sided inequality (4.7) instead of the equality (4.3).

[]

In the next lemma we prove a counterpart of Lemma 4.8 for an arbitrary domain. 4.9. L e m m a .

Let G be a proper subdomain of t t '~ , a, b E G and O < s < 1.

(1) There is an (a, b, s)-admissible family containing at most 1 + k a (a, b)/dl (s)

balls, dl(s) = 2log(1 + s ) .

(2) Every (a,b,s)-admissible family contains at least d2(s) = 2 1 o g

1

1--8

balls,

'

P r o o f . (1) Fix a quasihyperbolic geodesic segment JG[a, b]. Choose points zj E

Jc[a,b], j = 1 , . . . , p

with ka(a, zi ) = M ,

kG(zj,zy+l ) = 2 M ,

J = 1,...,p-

2, zp = b, kc(zp_l,zp) < 2 M , where M > 0 will be chosen soon and 2 M ( p 1) < kc(a,b ) . We wish to choose M such that D a ( z j , M ) 1,...,p.

c B'~(zj, sd(zy) ), j =

In view of Exercise 3.15 it suffices to choose M such that log(1 + s) = M .

It is clear that the family {S'~(zy,d(zy)) : j = 1 , . . . , p } that p < l + k G ( a , b ) / d l ( S ) ,

is (a,b,s)-admissible and

dl = 2 1 o g ( l + s ) .

(2) It follows from Exercise 3.15 t h a t for all y E G 1

ka(B'~(y,sd(y))) <_ 2log _---L-1 ~ The proof follows from this inequality exactly in the same way as in part (2) of Lemma 4.4. We shall next give an immediate application of L e m m a 4.9 to positive functions satisfying the Harnack inequality.

44

4.10. Definiti on.

Let G be a proper s u b d o m a i n of R ~ and let u: G --~ R + tj

{0} be continuous. We say t h a t u satisfies the Harnack inequality in G if there exist numbers s E (0,1) and Cs > 1 (4.11)

such t h a t m a x u ( z ) < C8 m i n u ( z ) B~ -B~

holds true whenever B"(x,r) C G

and Bz -- Bn(z,sr).

The above definition does not require smoothness or any other regularity properties beyond continuity of u. It is well k n o w n that non-negative harmonic functions

satisfy (4.11) [GT, p. 161. 4.12.

Let u: G -~ R + u {0} satisfy the Harnack inequality in G.

Lemma.

Then u(x) <

, t-

kc (x, y)

for x , y E G where dl(s) -- 2 1 o g ( l + s ) . If G = t t ~ or G = B " , then we can replace t by p(x, y ) / l o g 1-t-~ 1--,~ Proof. i = 1,...,p}

Fix

'

x,y

E G

and an

(x, y, s) -admissible family

{ B '~(xi,ri)

:

with p < l + k G ( X , y ) / d l ( S ) (see 4.8). Let z i E B j N B j + I , By =

B'~(xj, s r j ) , j = 1 , . . . , p -

1. By (4.11) we get the desired inequality

t~(,T) < Cs'U(Zl) _< C2tt(z2) < - . - <

CPs--lU(Zp_I) <__CPu(y) .

0

Let u be as in L e m m a 4.12. It should be noticed t h a t , by virtue of 4.12, either u vanishes identically or u is strictly positive in G . 4.13. C o r o l l a r y .

Let f: (G, kG) -} (R, I l) be uniformly continuous

as

a

map-

ping between metric spaces. Then there is a number a such that If( x ) - f(Y)l <- 1 + a k c ( x , y ) for all x, y E G. Proof.

Because f is uniformly continuous, there exists a n u m b e r to such t h a t

If( x ) - f(Y)l -< 1 for x, y E G , k c ( x , y ) <_ to. If k c ( x , y ) > to we can exploit the m e t h o d of the proof of 4.9 to show t h a t If(x) - f(Y)l <- 1 + k G ( X , y ) / t o .

Hence we

m a y choose a = 1~to. T h e details are left as an easy exercise for the reader.

CI

45 The hyperbolic volume of a (Lebesgue) measurable set E in B n is defined by

rnh(E) = / E ( 2ndm(x) 1 : ~2-)n

(4.14)

(cf. Section 2). Let w,,-1 be the (n - 1)-dimensional area of S n-1 . Integration in polar coordinates yields (4.15)

mh(Bn(s)) = 2nWn-I

f

t n-1 2nw~_:( s ~-1 (l~---t2) ndt < n : - I \-~-s-- s /

"

0

The last inequality holds because

t'~-:dt s~_ 1 / ( T 7 t-~ '~ -< 0

0

/ <

dt (1 + t)'~(1 - t) '~

s~-'

dt n

sn-1/

(I ---t) 0

( l - - t ) 1-n n - I

0

Since t n - ' >_ 22-'~t for t E (½,1) we obtain

(4.16)

rnu(B~(s)) > 2nwn-122-n 1/2

tdt ,~ 22(1-'~)w,~_1 s)l_ n (i - t h ) > (1 n-1

1 n-1

for s C (½, 1). Finally, for x E B '~ and M > 0, by the invariance of rnh under the action of ~ N ( B ~) and by (2.24) and (4.15) we get

mh(D(x, M)) = mh(D(O, M)) = rnh(Bn(th ½M)) (4.17)

_<--2nwn-1 ( -n 1 1 -th½Mth ½/vl ]~n-I < --2nWn-leM(n-1) thn-l(½M)n-1

"

For what follows we shall need a lemma about coverings by families of euclidean balls [LA, p. 197, Lemma 3.2]. We shall give such a lemma here with a slightly more general formulation. Covering theorems of this sort are very useful in analysis. For a related result see [GU, Theorem 1.1]. 4.18. L e m m a .

Let (X,d) be any one of the metric spaces (R n, 11), (Bn,PB,,),

or ( I t ' ~ , p H . ) , and let B x ( z , r ) = { y C X :

d(y,z) < r } .

Let A be a non-empty

subset of X , J~ -- { B x ( z , r ( z ) ) : z E A} and suppose s u p ( r ( x ) : x E A} < c~. There exists a number c(n) depending only on n and a countable subfamily ~r1 C 3r such that (i) A C (.J ~rl and (2) each x E A belongs to at most c(n) elements of ~rl .

46

Let A c X ,

A~0,where

(X,d)

is a metric space, a n d f o r k

=

:

Because the space X is usually specified by the context, we write N o t e t h a t if A c X is n o n - e m p t y and c o m p a c t then P x ( A , t )

p(A, t) = Px(A, t).

< oo a n d { B x ( x j , t

) :

} , xj E A , is a covering of A .

4.20. Lemma. for X :

set

}.

c U j=l

j -- 1 , . . . , p x ( A , t )

t:>0

be any one oF t h e m e t r i c spaces in 4.18, let m x = m

Let (X,d)

a n , m X -~ m h i f X = B n or X = H n , and let A C X ,

c o m p a c t . T h e r e is a n u m b e r dl = 1 / r n x ( B x ( y ,

A =fi 0 , be

t)) d e p e n d i n g o n l y on n a n d t such

that

dlrrtx(A ) ~ p(Att) ~ c(~)dlmx( U Bx(z't)) zCA where c(n) is as in 4.18. T h e simple p r o o f of this l e m m a is based on a s t a n d a r d v o l u m e - c o m p a r i s o n argum e n t and on L e m m a 4.18, and left as an easy exercise for the reader. 4.21. Exercise.

Show t h a t m h ( U l z l < _ r D ( x , M ) ) <

m h is the hyperbolic measure of ( B ' ~ , p B , ) . I+R = e M l 1+- -~r

I--R

[Hint: Utzi<_ r D ( x , M )

(see (2.18)). Hence 1 - R > (1 - r)e - M --

4.22. ]Exercise.

1-n,where

= Bn(R)

where

a n d one m a y a p p l y (4.15).]

A p p l y 4.20 and 4.21 to show t h a t for r near 1

d3(1 - r) 1 - n < p ( B n ( r ) , M ) where d3 d e p e n d s only on n and M . 4.23. Exercise.

d2(n,M)(1-r)

<_ c(n) dl d~(1 - r) 1-n

[Hint: Apply also (4.16).]

For ~oG (0,½7r) let C(~o) = { z E R

At~ = C ( ~ ) A ( B n ( 1 ) \ ~ n ( } ) )

pHi(At

~ :z.e~

= Izlcos~}

and

, t > 1. Show t h a t

< log

where u 2 -- sin 2 ~ + ~t-1~2 ~t-G-fJ c°s2

1)1

J '

. [Hint: Consider the smallest euclidean ball B

c o n t a i n i n g A ~ , and find an u p p e r b o u n d for p ( B ) . ] 4.24. Remark.

For n = 2 , the hyperbolic a r e a o f D ( O , r ) is 47csh2(½r) ([BE,

p. 132, T h e o r e m 7.22]). N o t e t h a t for r --* 0 this is a p p r o x i m a t e l y ~rr 2 , the euclidean area of B 2 ( r ) .

47

4.25. E x e r c i s e .

In any ball B'~(x,r) in R " one can define a hyperbolic metric

Pr by making use of a formula similar to (2.20). equality

pr(x,x-i-ael) =

l o g ~1 - - a / r

for 0 -<- a < r

Generalizing (2.19), we have the "

Assume t h a t z E B ~

~

and p is the hyperbolic metric of B '~ , and let ~" be the hyperbolic metric of

D(z,M) with p(a, On(z,M)) > b > O. Find an u p p e r t e r m s of p(z, a) and b. [Hint: By the invariance of ~" and p z ----0 , whence D(z, M) --- B'~(th ½ i ) .] Let a E

4.26.

Exercise.

subset of G with

b o u n d for

'

D(z, M). ~(z,a) in

we m a y assume t h a t

Let G be a proper s u b d o m a i n of R n and F

d(F, cgG) >

M > 0

a connected

0 . Applying the covering l e m m a 4.18 show t h a t

[Hint: See [VUh, 2.18].] 4.27. N o t e s .

Chains of balls similar to those in L e m m a s 4.4 and 4.8, but often

without a quantitative u p p e r b o u n d for the n u m b e r of balls, are recurrent in analysis. With slightly different constants, 4.4 and 4.8 were given in [VUh], [VU6].

For 4.9

see [HVU]. Some formulae for the hyperbolic volume or area are given in [BE], [A5]. Instead of balls one could use cubes in L e m m a 4.18, see [GU, T h e o r e m 1.1].

Chapter II MODULUS AND CAPACITY

For n o n - e m p t y subsets E and F of R n let

/kEF be

the family of all curves

joining E and F in R ~ . For fixed F the modulus M(AEF ) of

AEF

is an outer

measure defined for c o m p a c t subsets E of R n \ F . T h e real n u m b e r M ( A E F ) gives q u a n t i t a t i v e information a b o u t the structure of the sets E and F as well as their position relative to each other.

Roughly speaking

M(AEF )

is small if E and F

are far a p a r t or if one of the sets E , F is "thin", while the modulus is large in the opposite case. If E and F are n o n - d e g e n e r a t e continua in R '~ , then M ( A E F ) and min{d(E),

d(F)}/d(E, F)

are simultaneously small or large. Because of its conformal

invariance, the modulus will be a most valuable tool in our subsequent studies in C h a p t e r III. We shall exploit the conformal invariance of the modulus and introduce in a s u b d o m a i n G of R n two conformal invariants A c ( x , y ) and

#~(x,y), x,y E G,

which describe the position of x and y with respect to each other and the b o u n d a r y of G . One m a y think of # a ( x , y )

as a conformally invariant "intrinsic metric" of G

while A c ( x , y ) is in a sense its dual quantity. The importance of # c

and Aa for

C h a p t e r III is based largely on the explicit estimates proved in this chapter as well as on the fact t h a t # a

and AG transform in a natural way under quasiconformal and

quasiregular mappings.

5.

The m o d u l u s of a curve family

For the sake of easy reference and for the reader's convenience we shall give in this section the basic properties of the modulus of a curve family. The proofs of several

49

w e l l - k n o w n results are o m i t t e d .

For the proofs of these results a n d for m o r e details

the reader is referred to original sources which we shall quote at the end of this section. Most of the material in Section 5 is based on C h a p t e r I of V£is£1/i's b o o k [V7].

A path in R n ( R '~) is a c o n t i n u o u s m a p p i n g 3 ' : A --* R n (resp. A c R

R n ) where

is an interval. If A ' C A is an interval, we call "~IA r a s u b p a t h of "I, T h e

p a t h ~/ is called closed (open) if A is closed (resp. open). (Note t h a t according to this definition, e.g. the p a t h qt: [0, 1] -* R n is closed a n d t h a t it is not required t h a t -~(0) = 3'(I) .) T h e locus (or trace) of a p a t h "1' is the set ~tA. T h e locus is also d e n o t e d by [3,[ or simply by 2 if there is no d a n g e r of confusion. We use the w o r d curve as a s y n o n y m for path. T h e length /~('~) of a curve ~/: A --+ R n is defined in the usual way, with the help of p o l y g o n a l a p p r o x i m a t i o n s and a passage t o the limit (see [V7, pp. 1-8]). T h e p a t h ~/: A -~ R '~ is called rectifiable if g('7) < co a n d locally rectifiable if each closed s u b p a t h of ~/ is rectifiable. If ~/: [a, b] --* R '~ is a rectifiable p a t h , t h e n "~ has a p a r a m e t r i z a t i o n by means of arc length, also called the normal representation of 7 .

T h e n o r m a l r e p r e s e n t a t i o n of q is d e n o t e d by ,~0: [0,~('~)] ~

R ~ . Making

use of the n o r m a l r e p r e s e n t a t i o n one defines the line integral over a rectifiable curve ~/. In a n a t u r a l way one t h e n extends the definition to locally rectifiable curves (for a t h o r o u g h discussion see IV7, pp. 1-15]). Let r be a family of curves in R '~ . By Y(F) we d e n o t e the family of admissible functions, i.e. n o n - n e g a t i v e B o r e l - m e a s u r a b l e functions p: R " -* R tA {co} such t h a t

~

pds ~ 1

for each locally rectifiable curve ~f in r .

For p > 1 the p-modulus of F is defined

by (5.1)

Mv(r ) =

where

inf f peT(r) J R

pPdm,

m s t a n d s for the n - d i m e n s i o n a l Lebesgue measure.

If 7 ( P ) = 0 , we set

Mp(F) = c o . T h e case 7(1") = 0 occurs only if there is a c o n s t a n t p a t h in I' because otherwise the c o n s t a n t f u n c t i o n co is in ~'(F). Usually p = n and we denote M~(lP) also by M(F) a n d call it the modulus of r .

If M(F) > 0 , the n u m b e r M(F) 1/(1-n) is

called the extremal length of F . We take the extremal length to be co if M(F) = 0. 5.2. L e m m a .

The p - m o d u l u s M v is an outer measure in the space of all curve

families in R '~ . That is,

50

(1)

Mp(¢)

(2)

r, c r= implies U , ( r , ) < Mp(r2) ,

= 0 ,

OO

OO

i=1

i=1

Let Yt and F2 be curve families in R " . We say t h a t F2 is minorized by r l and write F2 > £1 if every "/E F2 has a subcurve belonging to r l • 5.3. L e m m a .

F1 < F2 implies Mp(F1) _> Mp(F2).

The curve families F1, F 2 , . . . are called separate if there exist disjoint Borel sets El in R = such t h a t if q E Fi is locally rectifiable then f~ xids = 0 where Xi is the

characteristic function of R '~ \ Ei • 5.4. L e m m a .

If F 1 , F 2 , . . .

are separate and if F < Fi for all i , then

Mp(r) _> ~ Mp(r d . 5.5. L e m m a .

Let G be a Borel set in R '~ and F = { 3 : 3

isacurvein

G

with £('V) > r } . If r > 0 then

Mp(r) _<m ( C ) r -p Proof.

.

Because p = r1X c E Y(F) the proof follows from (5.1).

5.6. C o r o l l a r y .

[:3

If F is the family of non-constant curves in a Borel set G C R '~

with re(G) = O, then Mp(r) = O.

Proof.

If F j = { - ~ E F : £ ( 3 ) >

follows from 5.2(3) and 5.5.

~}, 3"=1,2,...,then

£=UFj

and the proof

[3

Curve families with zero p - m o d u l u s are sometimes called p-exceptional. We next give a general criterion for a curve family to be p-exceptional, which is a generalization of 5.6. 5.7. L e m m a .

A curve family F is p-exceptional if and only if there exists an

admissible function p E 7 ( F ) such that R pPdm < oo rt

for every locally rectifiable 3 E F .

and

f P ds = oo Jr

51

Proof. 1,2,...

k - l p E ~ ( I ' ) for every k =

If p satisfies the above conditions, then

and thus M p ( r ) <_ k -p JIt,~ pPdm

as k --* o o .

Hence F is p - e x c e p t i o n a l .

)0

Conversely, let Mp(F) = 0 a n d choose a

sequence Pk E 7(r) such t h a t f R . p ~ d r a < 4 - k , k = 1 , 2 , . . . .

Writing

{30

:

2

k=l

we infer t h a t f R . pPdrn < o o . On the other h a n d

Lpds>_L2k/Ppkds>2k/P for all k = 1 , 2 , . . .

i.e.f.~ p ds = oo for each locally rectifiable curve 7 in r .

CI

If r is a curve famity in R '~ and rr = { ' 1 ~ r : ~(7) < c o } ,

5.8. C o r o l l a r y .

then M ( r ) : M ( r ~ ) . Proof.

Set p(x) = 1 for Ixl < 2 and p(x) = 1/(ix llog Ixl)

for Ixl

2.

By

direct c o m p u t a t i o n R

02n--I

p'~drn = 2nan + ( n - 1)(log2) ~-1 < o o ,

where f~,~ is the n - d i m e n s i o n a l v o l u m e of B r~ and wn-1 is the ( n - 1 ) - d i m e n s i o n a l area of S ~-1 . Let r ~

= { 7 E I" : ~(7) = c o } .

t h a t f.~p ds = c~ for all 7 C I ' ~ .

In view of 5.7 it sumces t o show

If 3' is b o u n d e d , t h e n p(z) >_ a > 0 on

it is clear t h a t f,y p ds = c~. If 7 C roo is u n b o u n d e d we choose x E

171

and

171 ~ B ~ ( 2 ) .

It

follows t h a t

L

pds >

as desired.

~1 r l o g r

- oo

C3

For E , F, G C ~ n

we d e n o t e by A ( E , F ; G) the family of all closed n o n - c o n s t a n t

curves joining E a n d F in G . More precisely, a n o n - c o n s t a n t p a t h 7: belongs to A ( E , F ; G) iff (1) one of the end points 7(a),7(b) o t h e r to F , and (2) 7(t) C G for a < t < b.

[a,b]

~ R '~

belongs to E and the

52

5.9.

Remark.

If G = R n or R.'* we often denote A(E,F;G) by A ( E , F ) .

Curve families of this form are the most i m p o r t a n t for w h a t follows. The following If E = U j¢¢ = l Ej and CE(F) = Mp(A(E,F)) = cr(E), then cr(E) <_ ~ c r ( E j ) , see 5.2(3). More precisely if G C ~.'~ is a domain

subadditivity property is useful.

and F c G is fixed, then e~(E) = M p ( A ( E , F; G)) is an outer measure defined for E C G . In a sense which will be made precise later on, cE(F) describes the mutual size and location of E and F . Assume now t h a t D is an open set in ~ n

and t h a t

F C D . It follows from 5.2(2) t h a t

Mp(A(F, aD;D \ F)) < Mp(A(F, OD;D)) < Mp(A(F, OD)) . On the other hand, because

A(F, OD;D) < A(F, OD) and

A(F, OD;D\ F) <

A ( F , OD; D) , 5.3 yields (5.10)

Mp(A(F, 0 D ) ) = Mp(A(F, OD;D)) = Mp(A(F, O D ; D \ F ) ) .

As the relatively complicated definition (5.1) of the p - m o d u l u s suggests, it is usually a very difficult task to find Mp(F) when F is given. In fact, the real n u m b e r Mp(I') is known for very few curve families. If F has a simple structure, then one can sometimes compute Mp(r) in two steps. First, applying Hhlder's inequality and Fubini's t h e o r e m one proves a lower bound for fit- p'~dm when p is an admissible function.

Second, one shows t h a t this lower bound is attained by some particular

admissible function Pl • Making use of this m e t h o d one can c o m p u t e the modulus of a cylinder and of a spherical ring©(for details see V£is£1/~ IV7, pp. 20-23J). 5.11.

The cylinder.

Let E c { x C R ~ : x~ = 0} be a Borel set, h > 0,

F = E + he,~ and denote G--{xER"

: (xl,...,x,~_,,O) E E , O<x,~ < h } .

T h e n G is a cylinder with bases E and F and,

~

F

G

as shown in [V7] IMp( A ( E , F; G)) = m ~ - l ( E ) h ' - p

= re(G) h -p . Diagram 5.1.

E

53

B'~(b) \ B'~(a) and {ty : a < t < b}, y E Y. Then IV7]

5.12. T h e s p h e r i c a l r i n g . a Borel set in S n-1 . Let ~/y =

Let 0 < a < b, D =

(5.13)

; r={qy:yeY},

log

M(A(Sn-I(b),S'~-'(a); D))

(5.14)

let Y be

b~l-- . = w n - , (log a /

By (5.10) the formula (5.14) holds also if D is replaced by R n . Letting a -~ 0 we see by 5.3 that

M(A(S'~-'(b), {0}))

(5.15)

: 0.

It follows from (5.15), 5.2, and 5.3 (see the proof of 5.6 or [V7, p. 23]) that the family of all n o n - c o n s t a n t curves 3' passing through a prescribed point x0 C R '~ is n-exceptional. 5.16.

Remark.

The inequality of L e m m a 5.5 is sharp: if G is the cylin-

der in 5.11 with bases

E

is not usually the case.

and

F

then 5.5 holds as an equality.

However, this

Applying (5.14) we shall now give an example in which

L e m m a 5.5 gives a very crude estimate.

S n-1 , A(E,F;Gt), t

Let E =

(Bn(4) \ S ~ - 1 ( 2 ) ) U B ' ~ ( 2 e l , t ) , and Ft =

F = Sn-1(3) , e (0,½).

In this par-

ticular case, L e m m a 5.5 yields for M(Ft) an upper b o u n d independent of t .

At = A(S'~-l(2el,t),S'~-l(2ei,1);Gt).

Gt = Let

Because At < Ft , w e get by (5.14)

M(rt) < w.-1 (log 1~ ,-n -

i.e. M(Ft) ~ 0

\

t/

as t - - ~ 0 .

In conclusion, keeping E

and

F

~

B'~(2el't)

fixed and letting the domain Gt vary so that E, F c Gt and m(Gt) is constant, one can make

M(A(E,F;Gt))

arbitrarily small,

while this fact is not reflected in the form of the

Diagram 5.2.

upper b o u n d 5.5.

The most valuable property of the n - m o d u l u s is invariance under conformal mappings, which is the content of the next lemma. We first extend the definition (5.1) to curve families in R '~ when p = n . F be a curve family in R ~ .

If r

Let

contains a constant curve, we set M(F) = c o .

54

D e n o t e ro~ -- {,~ E I" : oo C " 7 } . M ( r ) = M(I' \ r o o ) .

If F does not contain a c o n s t a n t curve, we set

It follows from (5.15), 5.2, and 5.3 t h a t M(I'oo) = 0 . Hence for

curve families in R ~ , this e x t e n d e d definition of n - m o d u l u s coincides with (5.1). It s h o u l d be p o i n t e d out t h a t we have not included p - m o d u l u s , p ¢ n , in this e x t e n d e d definition (see 5.28). 5.17.

Let D and D I be domains in R~ and let f: D --* D I be

Lemma.

a conformal mapping. Then M ( f r ) = M(I') for each curve family r in D where

fr={fo~:~er}. It is easy to see t h a t 5.17 is false for the p - m o d u l u s ,

p ¢ n , even if f

is

t h e s t r e t c h i n g x H 2 x . Consider e.g. the curve family A = A ( E , F ; G) w h e r e E =

{zeB

n :zn =0},

F=Z+{hen},

h > 0,and

G={zCR

~ :x~+...+x~_ 2 1 <1,

0 < x,~ < 1 }. T h e n by 5.11 0 < M y ( A ) : rn(G)h -p 7~ m ( f G ) ( 2 h ) -p : M v ( f A ) : 2 ~ - P M v ( A ) for p e n . A n i m m e d i a t e application of the conformal invariance 5.17 is the following count e r p a r t of (5.14) in the hyperbolic and spherical geometries. 5.18. Corollary.

Let O < a < b

and x E B n a n d

A (a, b) = A ( OD(x, a), a D ( z , b); D(x, b) \ D(z, a ) ) . Then M ( A ( a , b ) ) = wn_lL(a,b) 1-n ;

(1)

If z E R n , 0 < r < s < 1, a n d

(2)

r(r,s)

th(b/2)

L(a,b) = log t h ( a / 2 ) "

= A(OQ(x,r), OQ(x,s)) then

M(r(r,s))=c~,_l log V ~ J J Proof.

Let T~ E M ( B n) be as defined in 1.34. T h e n T~(x) = 0 a n d we o b t a i n

by (2.25) (5.19)

T~D(x,c) -- D(O,c) = B n ( t h ½c), c > 0 .

55 Now 5.17 together with (5.19) and (5.14) yields M ( A ( a , b)) = M(T=A(a,

b)) = w,~_,L(a, b)1-,~

For the proof of (2), let t= C M ( R n) be a spherical isometry with t=(x) = 0, see (1.46). Then by (1.47) or 1.25(1) t x Q ( x , r ) = Q(O,r) = B n ( r / V / 1 - r 2 ) .

The proof of (2) follows from this equality, 5.17, and (5.14).

O

Next we shall discuss various symmetry properties of the modulus. If A C R~_ we denote by A* the symmetric image { ( x l , . . . , x n - 1 , - x , ~ )

E R ~ : (xl,...,x,~) • A}

of A in OR~_. The next three lemmas will be given without proofs. For proofs of 5.20, 5.21, 5.22 see [G6], [Z1], and [VU3], respectively. 5.20. L e m m a . Let E and F be disjoint compact sets in R~_ and let E* and F* be the symmetric images of E and F in OR~_. If F1 and F2 are the families of curves joining E to F in R~_ and E U E* to F U F* in R '~ , respectively, then

Mp(r2) = 2Mp(F1)

5.21.

.

L e m m a . f f (Fj) is an increasing sequence of curve families, i.e. r j c

F j + I , j = 1 , 2 , . . . , and p > 1, then

lim Mp(rS) = M p ( U r j ) .

j--*oo

Applying this lemma one can prove the following symmetry property of the modulus. 5.22. L e m m a . Let p > 1 and let E and F be subsets of R~_. Then Mp(A(E,F;R~))

~ 1Mp(A(E,F))

.

5.23. Corollary. Let E and F be sets in R'~ with q( E , F ) ~ a > O. Then M ( A ( E , F ) ) _~ c(n,a) < ~ .

56 Proof.

By the hypothesis there exists a ball Q ( z , r )

in R '~ \ ( E U F)

with

r < 1/v/2 and with the spherical diameter q(Q(z, r)) = a. By easy c o m p u t a t i o n (see 1.25(3)) q ( Q ( z , r ) ) = 2r~/i : r 2 and hence r > ~1 a .

Let ~" be the antipodal point

defined in (1.16) and t a spherical isometry with t(z~ = 0 as defined in (1.46). By (1.23)

t E , t F C Q(0, v/i - r 2 ) . Because t is a spherical isometry we obtain d(tE, tF) >_ q(tE, tF) = q ( E , F ) >_ a .

Next observe t h a t (see 1.25(1))

Q(O, ~c/1- r 2) = Bn(

r x / ~ - 1) = B .

These last two relations together with 5.22, 5.17, and 5.5 yield M(A(E,F))

as desired. 5.24.

for all j =

= M(t(A(E,F)))

= M ( A ( t E , tF)) <_ 2 M ( A ( t E , t F ; B ) )

[] Lemma.

Let F 1 , F 2 , . . .

1, 2 , . . . .

/ f p > 1, then

be separate curve families in R'~ with r j < F

Mv(r)'/(1--p)_>~ Mv(Fs)'/('-v) j=l

Proof.

Let { E j } be a family of disjoint Borel sets associated with the collection

{ r j } , let E = [.J E j , and let XE~ be the characteristic function of E j .

Fix pj E

5r(Fj) and. set a j = pjXE s . T h e n it is easy to see t h a t aj E 5r(Fj). Now choose a sequence (aj) so t h a t aj E [0, 1] and E aj = 1 and define a Borel function p by OO

P= E

OO

ajaj = E

j=l

ajpJXEj

"

j=l

We show t h a t P ~ Y ( P ) . Fix a locally rectifiable 3 c F and for each j a subcurve "~j E Fj . We obtain

3

-> E aj j

J

s

> E aj ; 1 j

57 Hence p E Y(F) and we obtain

pVdm =

NIp(r) <

pPdm =

aipiXE, .=

<--

E ap P; drn =S----1

"/

j

n

aSvPSp dm j

a~pP dm <_ E ajP

~

J

dm =

.

J

p~ dm .

J =1

Taking the infimum yields

My(r) < ~ ' ~ Pa s Nip(rj) .

(5.25)

j=l

We now apply this last inequality to prove the assertion.

Clearly we may assume

My(F) > 0 , which implies by 5.3 that NIv(Fj) > My(F) > 0. (If NIp(F) = 0 , the left side of the inequality is c~ and there is nothing to prove.) We may also assume that Mv(Fj) < co for all j , because

Mp(rs) 1/(1-p) = j=l

Mp(rs) 1/('-p) j=l

where }-~* refers to summation over terms with Mv(FS) < c~. Denote k

tk = ( ~

Mp(rs)l/O-P)) -1 , as = Nip(rs)l/(1-v/tk

j=l

for j =

1,...,k

and k - - 1 , 2 , . . ,

whence k

~a s =1. j=l

The above inequality (5.25) (with a s -- 0 for j > k + 1) gives k

k

j=l

j=l

Letting k --+ c~ yields the desired result.

[]

As an example of application we consider the following simple particular case of 5.24. Let

r1 =

1<

r 2 < ...

< rj

<

...

r = A ( s "-1, s n - l ( a ) ) ,

<

a and Fi = A ( S n - l ( r i ) , Sn-i(ri+l))

58 for i = 1, 2, . . . . It follows from (5.14) that in this particular case 5.24 yields (when p--n) oo

log a _> ~

log ri+ 1 = log ao

j=l

ri

where a0 = lim rj _< a . If we choose the sequence (rj) so that a0 = a , then equality holds. Hence 5.24 is sharp. For the proof of the next result the reader is referred to [MO, p. 82], [G1, pp. 514-515]. 5.26. L e m m a .

L e t s E (0,1) a n d

r l = zx( [0,se,l, s " - ' ; B " ) ,

r2 = zx( [0,sed,

T h e n M p ( r , ) = 2p-IMp(F2) for p > 1.

The next result will have interesting applications later on in this book. This result was conjectured by the author and a proof was supplied by F. W. Gehring ([VU10,

2.5s]). 5.27.

Lemma.

Let A1 = A([0, el], [t2el,cx~)) and A2 = A([0, e], [t2el,c~))

where e • S n - 1 a n d t > l . P r o o f . Denote

Then M(A2)<M(A1).

A l l = A([0, el], S n - ' ( t ) ) ,

A 1 2 ~- A 2 2 = A ( s n - l ( t ) ,

[t2el,oo))

.

A21 = A([0, e ] , S n - l ( t ) ) ,

Obviously

At ~ = A2 ~

M ( A , , ) = M(A2~) , M(A,2) : M(A22) . Let f

and

~ .

be the inversion in S n - l ( t ) . OO

Because

A12 --~ f A l l

we obtain by 5.17

M ( A , , ) = M ( f A , , ) = M(A,2 ) . Next, 5.24 yields

sn-'(t2) Sn

M(A2) 1/(1-n) _2>M(A21) 1/(1-n) q- M(A22) 1/(1-n)

D i a g r a m 5.3.

= 2M(All)l/(l-n) while the fact that A 1 is symmetric yields by 5.26

M(A,1) = 2 " - I M ( A , )

•

T h e desired inequality follows from the last two relations,

g

59 The family of all n o n - c o n s t a n t curves passing through a fixed point is n - e x c e p tional as was pointed out in the paragraph following (5.15). One can show that such a family is not p-exceptional if p > n (see [GOR, Chapter 3], [MAZ2]). We shall require this result in the following form, which is sometimes called the spherical cap inequality.

For this result we introduce first an extension of the definition (5.1) of

the p - m o d u l u s . r

Suppose that S is a euclidean sphere in R n with radius r and

is a family of curves in S .

We equip S with the restriction of the euclidean

metric of R '~ to S and with the ( n - 1)-dimensional Hausdorff measure run-1 with mn-l(S)

= w ~ - l r n-1 . Let ~ ( r )

be the set of all non-negative Borel-measurable

functions p: S --* R U {co} with f

pds > 1

for all locally rectifiable (with respect to the metric ds ) curves ~ in r and set

Ms(F):

inf

For ~ E (0,~r) let C(~) = { z e R " 5.28.

Lemma.

[ p'~drn,~_l.

oct(r) J s

: z.e,

Let S = S ~ - l ( r ) ,

>_ [ z ] c o s ~ ) .

~ E (0,~r], let K

be the spherical cap

S N C ( ~ ) , and let E and F be n o n - e m p t y subsets of K .

(1) Then Ms(A(E,F;K))

>> b_2_~ r

where b,~ is a positive number depending only on n . (2) If K = S , i.e. ~ = ~r , then b,~ may be replaced by c~ = 2r~b,~ in the above inequality. The proof of 5.28 (see [V7, 10.9]) is based on an application of Hblder's inequality and Fubini's theorem. A similar m e t h o d yields also the following improved form of 5.28 ([R12, p. 57, Lemma 3.1], [GV1, p. 20, L e m m a 3.8]). 5.29. L e m m a .

Assume that E ,

F , and K are as in 5.28(1). If ~ e (0, ½~r) ,

then Ms(A(E,F;K)) where d,~ depends only on n .

>_ d__~ ~r

60 5.30.

Remark.

T h r o u g h o u t the book we will denote by cn the number in

5.28(2). T h e number b,~ = 2-'~ca has the following expression bn = 2 1 - 2 n w n - 2 I l - n , (5.31) In =

i

~-12

1

b2 = 27c

2-,,

sin--1 t d t .

J0

Because -2t~<_ sin t _< t for 0 < t < ½~r , it follows from (5.31) that - 1 ) c15/ |( n - ' ) ' " \2/

< & < -

1 ) =" 2

for n _ > 2 . One can show that 2 n c n - + 0 when n - - ~ o c [AVV3]. By (5.1), any admissible function p yields an upper bound for Mp(F), that is Mp(F) < fR" PP din. The problem of finding lower bounds for Mp(F) is much more difficult because then we need a lower bound for fI~, pPdm for every admissible p. The next important lower bound for the modulus follows by integration from 5.28 and 5.29. 5.32. Lemma.

Let O < a < b

and let E , F be sets in R r' w i t h

E N S'~-l(t) ¢ 0 ¢ F A S'~-l(t) /'or t E ( a , b ) . Then

b M(A( E,F;Bn(b) \ Bn(a) )) >_cn l o g - . a

Equality holds if E = (ael,bel), F = ( - b e l , - a e l ) . 5.33.

C o r o l l a r y . //" E and F are non-degenerate continua with 0 E E n F

then M ( A ( E , F ) ) = ~ . P r o o f . Apply 5.32 with a fixed b such that Sn-l(b) ~ E ~ 0 ~ Sn-l(b) N F and let a - * 0 .

[]

We next give a typical application of L e m m a 5.32. Unlike 5.32 this application fails to give a sharp bound, but it yields adequate bounds in many cases (see e.g. Section 6). A sharp version of 5.34, which requires some information about spherical symmetrization, will be given in Section 7 (see 7.32 and 7.33).

61 5.34. L e m m a .

Let t > r > 0 and let E c B'~(r) be a connected set containing

at least two points. Then M ( A ( S ' ~ - I ( t ) , E ) ) > c,~ log P r o o f . Fix u, v E E

h(u) = - s e l

=-h(v).

2t + d(E) 2t - d( E) "

with l u - v I = d ( E ) = d and choose h E ~ N ( B n ( t ) )

with

By (2.27)

d(E) = [ u - v I < 2 t h ¼P(U,V) = 2 t h ¼P(h(u),h(v)) = 2 s , where p refers to the hyperbolic metric of B'~(t). Applying 5.32 to the annulus

Bn(tel,t+s)\-Bn(tel,t-s)

with E = h E

M(A(sn-I(t),E))

and F = S n - l ( t )

we obtain

= M ( A ( S n - I ( t ) hE)) > cn log t +_.~s ' -t--s > c,~ log 2t + d(E) 2t - d( E) "

We shall frequently apply the following lemma when proving lower bounds for the moduli of curve families. This lemma will be called the comparison principle for the modulus. In the applications of this lemma, the sets F3 and F4 will often be chosen to be non-degenerate continua (that is continua containing at least two distinct points) while the sets F1 and F2 will usually be very "small" sets when compared to F3 and F 4 . 5.35. L e m m a .

Let G be a domain in R n, let Fj c G , j = 1 , 2 , 3 , 4 , and let

Fij = A ( F I , F j ; G ) , 1 <_ i , j < 4. Then

M(r~2) _> 3-" min{ M(F13), M(r24),

inf M(A(t'/131,

I~=~1;a)) },

where the inl~mum is taken over att rectifiabIe curves "/13 C F13 and 324 E F24 • Proof.

By 5.2(1) we may assume that

Fj¢

p 6 F(F12) • If

(5.36)

pds >_ -5 1

fff 13

for every rectifiable "h3 C r13 or (5.37)

pds >_ -5 24

O , j = 1 , 2 , 3 , 4 . Fix

62 for every rectifiable "724 E F24, t h e n it follows f r o m 5.8 a n d (5.1) t h a t

JR["

(5.38)

p'~dm _> 3-~ min{ M(r,3), M(r=4) ) .

"713

F4

~,,t.

-

.. Fz

Diagram 5.4. If b o t h

(5.36) and (5.37) fail to hold we select rectifiable curves

"713 E F13

and

"724 E F24. Because p E 7(I'12) it follows t h a t

pds>_ 1 ")h 3 U o~ U " / ~ 4

for every locally rectifiable

a C A =

zx(l"7,s[,}"7 41;c)

Because b o t h (5.36) a n d

(5.37) fail to hold it follows f r o m the last inequality t h a t

pds> g for each locally rectifiable a E A . Hence (5.39)

/ R " p"dm > 3 - ' ~ M ( A ) > 5 - " inf M(A(I"713 [, t"724t; c))

where the i n f i m u m is t a k e n over all rectifiable curves "713 C F l s and q24 E F24. In every case either (5.38) or (5.39) holds, and the desired inequality follows. 5.40. Corollary.

~1

Let Fj c R '~ a n d r~y = A(Fi,Fj) , 1 < i , j < 4. Then

M(r~=) >_3-" min{ M(r,3), M(r=4), ~(r) } where r = min{ q(F,,F3), q(F2, F4) } and 6,(r) ----inf M ( A ( E , F ) ) Here the i n / ] m u m is taken over all continua E , q(F) > r.

.

F in R "

such that q(E) > r,

63 It is clear that 6n(0) -- 0 in 5.40. In fact, this follows from 5.18(2) if we choose r e (0, 1/V/2), set s -- x/~ - r 2 , and let r --* 0. We are going to show that 6,~(r) > 0 for r > 0. To this end the following corollary will be needed. 5.4:1.

C o r o l l a r y . I f x C R n , 0 < a < b < c o , and F1, F2 C B n ( x , a ) ,

r,j

F3 C R n \ B " ( x , b ) ,

= A ( F i , F y ) , then

(1)

M(F12) _> 3 -'~ min{ M(F13), M(F23), cn log b }

(2)

M(r,=)

a

>_ d(n,b/a)

min{ M(r,3), M(r=3) }.

P r o o f . We apply the comparison principle 5.35 with G = R n and F3 = F4 to get a lower bound for M(F12). It follows from 5.32 that the infimum in the lower bound of 5.35 is at least c, log b and thus (1) follows. For the proof of (2) we observe that by 5.3 and (5.14)

ma~{ M(rl~), M(r~s) } < A = ~._~ log ~ By part (1) we get 1 M(F12) _> 3-'~min{ M(I'13), M(F23), ~ (c,~log b ) m i n { M(F13), M(F2a) }}

> d(,~,b/a)min{ M(r,~), M(r~) } where d ( n , b / a ) = 3 - ' ~ m i n { 1 , 5.42.

Lemma.

~c,~log(b/a)}.

0

For n _~ 2 there are positive numbers d and D

with the

following properties.

(1) I f E , F C B n ( s ) are connected and d(E) > s t , d(F) > s t , then M ( A ( E , F ) ) _> d t . (2) I f E, F c R n

are connected and q(E) >_ t , q ( r ) > t , then

M ( A ( E , F ) ) > 6,(t) _ D t . P r o o f . (1) By 5.34 we obtain M (A(S n-1 (28), E)) > cn log 4s + ts > 2 48 - ts -

and similarly M ( A ( S n - I ( 2 s ) , F ) )

~

J&tc°gnl2t~

>_ ½c,~(logZ)t. Applying 5.41(1) with F1 = F ,

F2 = E , and Fs = S n - l ( 2 s ) and the above estimates we get M(F12) _> 3-n min{ ½er~(log2)t, c~log2 } > d t

64 where d = ½ . 3 - n c , ~ l o g 2 . (2) Observe first that both the first and last expressions in the asserted inequality remain invariant under spherical isometries (see 5.17). By performing a preliminary spherical isometry if necessary we may assume that - r e 1 E E , re1 E F , and r E [0, 1] (cf. 1.25(1)). Let E1

(El) be that component of E A B n ( 2 )

(of F N B ' ~ ( 2 ) , resp.)

which contains -re1 (re1). Then

d(E,) > q(E,) >_ min{ t, q(S'~-I,S'~-'(2)) } >_ t / v / - ~ , and likewise d(F1) > t/x/-l6.

The proof of (2) follows from (1) with D -- d/y'-l-6. O

By means of spherical symmetrization, which will be introduced in Section 7, one can give a different proof of 5.42(1) (see 7.38). 5.43.

E x e r c i s e . Let E

and

F

be non-degenerate continua in B '~. Find

in terms of n ,

a lower bound for

M(A(E,F;B~))

]Hint: Fix al E E ,

a2 E F with p(al,a2) = p ( E , F )

p(E),

p ( F ) , and p ( E , F ) .

and let x E g[al,a2]

be such

p(al,x) = ½P(E,F) . Let T~ E ~ ( B ~) be as defined in 1.34. By conformal

that

invariance 5.17 M ( A ( E , F ; B ' ~ ) ) = M(A(T~E,T~F;B'~)) . Now one can find a lower bound for the euclidean diameters

d(TxE),

d(T~,F) in

terms of p ( E ) ,

p ( F ) , and p ( E , F ) , see (2.23)-(2.25). After this apply 5.41 with

a = 1, b = 2,

F1 = T ~ E ,

F2 = T ~ F , and F3 = S'~-1(2). The desired result

follows now from a s y m m e t r y property of the modulus, see 5.22.] 5.44. :Exercise. For E C R

(5.45)

~

x E R '~ and 0 < r < : t

set

Mt(E,r,x) = M(A(S'~-I(x,t), E n Bn(x,r))) , M ( E , r , x ) = M2~(E,r,x) .

It follows from 5.3 that M t ( E , r , x ) < M s ( E , r , x )

for 0 < r < 8 <: t . Also a converse

inequality is true: (5.46)

M t ( E , r , x ) <_ M s ( E , r , x ) < a ( n , r , s , t ) M t ( E , r , x )

where a depends only on the parameters indicated. Prove (5.46) by applying 5.35 with F1 = Zn-B'~(x,r)

F2 = S n - l ( x , t )

F3 = S'~-l(x s)

F4 = s n - l ( x , r )

65

5.47.

Remark.

The m e t h o d described above fails to give the best possible

constant a in (5.46). The sharp result, due to Martio and Sarvas [MS1] yields the inequality

log(t/r) b== log(s/r)

Ms(E,r,x)
(5.48)

for 0 < r < s < t .

Here equality holds for E = B n ( x , r ) .

makes use of a radial quasiconformal m a p p i n g [V7, p. 49] of

The proof of (5.48)

Bn(x, s) \ Bn(x, r)

onto

Bn(x,t) \ Br~(x,r). 5.49.

The modulus

of a ring.

A domain D in R'~ is termed a

ring,

if

~ n \ D has exactly two components. If the components are Co and C1 we write

D = R(Co, C 1 ) .

The

(conformal) modulus modR(Co,C1)

(5.50)

:

of a ring

R(Co, C1)

( M ( A ( C ° ' C 1 ) ) ~ 1/(1-n) C.~n_ I

The

capacity

of

R(Co,CI)

is defined by

/

is M(A(C0, C , ) ) .

A ring is a special case of a condenser, which we shall define in Section 7. In the two-dimensional case the modulus of a ring R has the following geometric interpretation:

m o d R = t if and only if R can be m a p p e d conformally onto the annulus

{ z E R 2 : 1 < Izl <

et }.

Owing to this geometric interpretation the modulus of a

ring is often convenient to use in the two-dimensional case. In the multidimensional case there is no such geometric interpretation for the modulus of a ring because of the rigidity of the class of conformal mappings in R n , n > 3 (cf. 1.54). On the other hand there is also a geometric way of looking at the capacity of a particular ring, the so-called GrStzsch ring, which is applicable to all dimensions n k 2 (see (5.52) and (7.31)). For this reason we shall prefer the capacity to the modulus of a ring. 5.51.

The GrStzsch and Teichmiiller rings.

GrStzsch ring Ra,,~(s ) in TeichrnSller ring RT,n(s ) are

The c o m p l e m e n t a r y compo-

nents of the

R n are B '~ and [se1,oo], s > 1, while those

of the

l-el,0]

two special functions "/n(s), s > 1, and

and [sel,oO], s > 0.

r,~(s),

We shall need

s > 0, to designate the moduli of

the families of all those curves which connect the c o m p l e m e n t a r y c o m p o n e n t s of the Gr5tzsch and Teichmfiller rings in R '~ , respectively.

66

O0

cap R c , , , ( s ) = M ( r , )

--el

0

Sel

oo

cap RT,~(s ) = M ( A , ) = r,~(s)

= v,,(s)

Diagram 5.5.

(5.52)

,f *r,(8) = M(r,)

= -y(s),

[ r,~(s) = M(A.)

r(s).

T h e subscript n is o m i t t e d if there is no d a n g e r of confusion. We shall refer to these functions as the GrStzsch c a p a c i t y a n d the Teichmfiller capacity.

For s > 1, ~,~(s) = 2'~-lrn(S 2 - 1). The functions q,~ and T,~

5.53. Lemma.

are decreasing. Furthermore, l i m s ~ x + qn(s) = oo and lims~oo qn(s) = O. Proof. A(S n-',

Let

F1 = A([O, set] 1 , [Sel, 00]) ,

[sel,c~]).

r2

=

Z~([O, s1e l l , S n-1 ) , F3 =

It follows f r o m c o n f o r m a l invariance t h a t

M(r=) = u(ra)

=

3,,~(s) a n d f r o m 5.26 t h a t

~(s)

= 2"-'M(r,)

= 2"-~r(s 2 - 1).

For each fixed n _> 2 the functions qn and r~ are decreasing as follows easily f r o m 5.2(3). T h e limit values of qn follow f r o m 5.32 and (5.14).

O

For the sake of c o m p l e t e n e s s we set q,~(1) = T~(0) = oe and

~(oo) :

o.

5.54. Exercise. (1)

3'n(oe) =

Show t h a t

M(A([re,,set],[tel,uel]))

(2) M(A(S "-l,[sel,tel]))

=7

s)(t

(st-1

=q\~_s/,

5.55. Elliptic integrals and

u)

1 <s
' r<s
"y2(s) • T h e plane GrStzsch ring can be m a p p e d

onto an annulus by an elliptic function [BF 1. As shown in [HEll, [LV2, II.2] (5.56)

~2(s)-

2~r ,(1/4

67

for s > 1 w h e r e 77 K(V/1 - - r 2 ) ~1 it(r) - ~ ~-(~ , K ( r ) = -v [(1 - x2)(1 for 0 < r < 1. T h e function K ( r )

r2x~))-l/2dx

is called a c o m p l e t e elliptic integral of the first

kind and its values can be f o u n d in tables [AS], [BF]. T h e m o d u l u s # ( r ) satisfies the following t h r e e functional identities #(r)#(V/1-

r 2) = ¼7f2 ,

/1 -r\

1 2

tt(r)#(-~~r ) = ~

(5.57)

,

F r o m these one can derive several e s t i m a t e s for ~ ( r ) [LV2, p. 62]. By [LV2, p. 62] the following inequalities hold (5.58)

log -1 r

<

log 1 + 3v/1 - r 2 r

< #(r)

<: l o g -4 r

for 0 < r < 1. F r o m (5.58) it follows t h a t limr--.0+ #(r) = co whence, by virtue of the functional identities (5.57), l i m r ~ l - it(r) = 0 . For t h e sake of c o m p l e t e n e s s we set it(0) -- co a n d it(l) = 0 . By (5.56) a n d (5.57) we o b t a i n (5.59)

=

4

{s-l'~

,

>

1.

4-

J

.... j, 2

Diagram 5.6.

it(r) , 0 < r _< 1, and # ( l / r ) , r > 1 (from [AVV3]).

r

68

5.60. E x e r c i s e .

(:)

Verify the following identities 77 277 -

-

#(l/v/1

+ t)

~(

(V/I +

t

--

x/t) 2 ) '

T2(t) = 2v2(4[t + ~¢/t(1 + t)][1 + t + " v ~ ] )

(2)

1-r

5.61. E x e r c i s e .

,

2)

In the study of distortion theory of quasiconformal mappings

in Section 11 below the following special function will be useful 1

pg,,~(r) for 0 < r < 1, K > 0.

=

~Z:(K~/n(1/r))

(Note: Lemma 7.20 below shows that ~/~ is strictly

decreasing and hence that ~/~i exists.) Show that ~AB,n(r) ~-1 A,n(r) = ~l/A,n(r) and that

=

~A,,~(~B,~(r)) and

pK,2(r) = ~K(r) -= # - l ( - ~ # ( r ) ) . Verify also that (1)

~)2(r)-

(2)

~K(r) 2 -~ ~ 9 1 / K

2x/~ , l÷r :

1.

Exploiting (1) and (2) find ~ : / 2 ( r ) . Show also that (l-r)

~°:/K ~

(3)

( 2,/7

(4)

~K \ 1 _}_r / 5.62. E x e r c i s e .

1 - ~K(r) -- l + ~OK(r) ' _

1 _t_ P K ( r )

.

Verify the following identities for K, t > 0

(1)

~;~ ( ~ ( t ) / K ) =

(2)

"r2(t) =

r•'

(KT2(1/t)) ' 4 "

The above functional identities, e.g. (5.57) and 5.60(2), are restricted to the twodimensional case. For the multidimensional case n > 3 there is no explicit expression like (5.56) for ~n(s) or 7n(s) and no functional identities are known for "/n(S) or 7n(s) except the basic relationship 5.53. The well-known upper and lower estimates for "7,~(s) and Tn(s) will be given in Section 7. Next we shall show that for all dimensions n > 2 the Teichmfiller capacity ~-n(s) satisfies certain functional inequalities.

69 5.63. L e m m a .

The following functional inequalities hold:

(1)

T(s) <_ ~,(1 + 2s) = 2'~-lT(4s 2 + 4 s ) , s > 0 ,

(2)

T(S) <_2r(2s + 2sv/X + X/s) , s

>0

,

(3) ~-(4<~-(t)+T t-~ /,o<s 0 (4) T(U) < T ( --

u -F v - +

I

.

-

P r o o f . (1) Let F = A ( S n - X ( - ~ e l ,1i ) , [ s1e l , o o l ) .

Then by 5.53

M ( r ) = ~(1 + 2~) = 2~-~T(4s ~ + 4s)

while by 5.3 T(s) _ M(F) and the desired inequality follows. (2) We can map the Teichmfiller ring RT,n(s ) by a Mhbius transformation onto a ring in R n with complementary components I - e l , e l ] where b = 1 + 2s(1 + ~

and [be1, c~] U [-bel,c~]

1 / s ) . By a symmetry property of the modulus, Lemma

5.20, we obtain T(s) = 2M(A([0, el], [bel,ce]; {z C R " : zl > 0})) < 2 T ( b - 1 ) as desired. (3) Let F1 = A ( [ - e , , 0 ] , [ s e l , t e l ] ) ,

F2 = A ( [ - e l , 0 ] ,

[tel,oO]).

Then by

5.54(1) r(s) < M(FI U F2) < M(F1) + M(F2) = ~ _

_

t - s

/

+ fit)

(4) After a change of variables the second inequality in (4) follows from (3). The first inequality follows because ~',~ is decreasing (5.53). 5.64. C o r o l l a r y .

ZJ

T(s) _< 2T(VG ) _< 2'~-(s), s > 0.

P r o o f . The first and second inequality follow from 5.63(2) and 5.63(1), respectively.

ZI

5.65.

Remark.

Corollary 5.64 applied to ~'2 yields by 5.60(1) the following

two-sided inequality for the function # : tt(1/1~/i--~) < 2#(1/V/1 + v/t) <_ 4#(1/~, 1/~-'~) , t > 0 , which can also be derived from the identities (5.57). The second inequality in (5.58) can be derived from the lower bound in [LV2, p. 62] log (1 + lX/]--Z 4 r 2 )2 < # ( r ) . r

70

5.66.

Remark.

It follows f r o m (5.58) t h a t l o g ( l / r ) < # ( r ) < l o g ( 4 / r )

for all

r E (0, 1) w h e r e b o t h b o u n d s have the correct a s y m p t o t i c b e h a v i o r as r --~ 0 + . r -+ 1 -

For

the second inequality is very weak since #(r) -+ 0 as r --+ 1 - . An i m p r o v e d

t w o - s i d e d inequality for # ( r ) can be o b t a i n e d as follows. Exploiting (5.58) t o g e t h e r w i t h the functional identity (5.57) we o b t a i n 71-2

7[.2

4 l o g V/itr2

< #(r) -

71-2

4 # ( ~ )

<

4 log

1 \,1/]--~r'-

This inequality t o g e t h e r with (5.58) implies for r E (0, 1)

r

'

41og~

4

<#(r)

<min

log4, r

41og~

1

•

T h e a s y m p t o t i c b e h a v i o r in (5.67) is correct at b o t h ends r = 0 and r = 1. 5.68.

Exercise.

Let A , B , C , D

be distinct points on the unit circle S 1 in

the s t a t e d o r d e r and 2c~ and 2/3 the lengths of the arcs A B and C D , respectively. Find t h e least value of M ( A ( A B , CD)) . iHint:

IA- CLIP- D I = IA- BIIC- D t +

IB - CI]A - D I by P t o l e m y ' s t h e o r e m [CG, p. 42], [BER, 10.9.2].] 5.69.

Remark.

T h e function /~ has several interesting p r o p e r t i e s which are

given in [AVV3]. For instance the inequalities

ab

ab

hold for a, b E (0, 1) where a' -- v/1 - a 2 . It follows f r o m (5.57) t h a t the second and third inequalities hold as equalities for a = b. 5.70. Exercise.

Show t h a t for ~ > 0 a n d r E (1, 1 + 5) the inequality

r.(~) I/(l-n) >_ -7.(r) I / ( I - " ) + - / . ((I + s)/r) I/(i-n) holds w i t h equality if r = ~ 5.71. Notes.

+ 8.

Most results in this section are s t a n d a r d a n d are well represented

in the literature, e.g. in [V7]. T h e origin of some less s t a n d a r d results is indicated a b o v e in c o n n e c t i o n w i t h each result. on the results of this section.

Next we shall m a k e s o m e additional r e m a r k s

L e m m a s 5.7 and 5.24 are f r o m IF], 5.20, 5.21, and

71 5.22 from [G6], [Z1], [VU3]. The comparison principle 5.35 has its roots in [MRV2, 3.11], but under this name it was introduced and developed by R. N~kki IN2] and the author [VU8]. An account of the properties of the function #(r) is given in [LV2]. The inequalities in 5.63 and 5.64 are from [VU13]. For further properties of

rn(s)

the reader is referred to Section 7 and to [AVV3]. A highly interesting study of the complete elliptic integral K(r) and Gauss' arithmetic-geometric mean is contained in [BB]. 5.72. N o t e s .

The extremal length method of L. V. Ahlfors and A. Beurling

[AB] (1950) has its roots in the length-area method whose use is widespread throughout geometric function theory. Some historical comments about the origin of the length-area method are made by L. V. Ahlfors fA3, p. 50, 81] and by J. Jenkins in [JEll, IJE2]. According to Jenkins the first result of this type is due to H. Bohr in 1918 and slightly later results are due to W. Gross, G. Faber and R. Courant. In 1928 H. Grhtzsch [GRO] published his well-known work on quasiconformal mappings and in a subsequent series of papers developed his strip method, giving applications to a variety of problems. In his dissertation in 1955 J. Hersch [HEll established connections among harmonic measure, extremal length, and other conformal invariants. A survey of some function-theoretic applications of the extremal length is given by B. Rodin in [RO], which contains also a good bibliography of the subject. See also the bibliography in the book of G. V. Kuz'mina [KU]. In his book [O] M. Ohtsuka gives several function-theoretic applications of the extremal length. B. Fuglede IF] was the first to consider, in 1957, the p-modulus in the multidimensional case. He also considered the modulus of a surface family as well as the modulus of a system of measures (see also P. Mattila [MAT1]). Later these notions were developed mainly in connection with the theory of quasiconformal mappings (see O. Lehto and K. I. Virtanen [LV2], F. W. Gehring [G1], [G2], [G9] and J. V£is~l£ IV1], [V3], IV7], [V10]). The notion of p-capacity, which is closely connected with that of p-modulus (see Section 7) has been studied by many authors in the setup of nonlinear potential theory (see the references given at the end of Sections 6 and 7). The paper of C. Loewner [LO] is one of the first papers dealing with conformal capacity in space. See also the books of P. Caraman [C1, pp. 46-70] and A. V. Sychev [SY, pp. 26-35]. The book of Caraman contains a useful and very extensive bibliography.

72

6.

T h e m o d u l u s as a set f u n c t i o n

In this section we shall consider the problem of finding estimates for M (A (E, F ) ) when E and F are disjoint n o n - e m p t y sets in ~ n .

In view of the conformal invari-

ance of the n - m o d u l u s 5.17, one would like to find estimates which reflect this invariance property in the following way: The estimate should give the same lower/upper bound for M ( A ( E , F ) )

and M ( A ( h E , hF)) whenever h e ~ N ( R ' ~ ) .

In most es-

timates (see e.g. 5.42(1) or 6.1 below) this requirement is not completely met, the estimate remaining invariant only under the action of a subgroup of ~.M(~.n), e.g. under translations, stretchings, or spherical isometrics. Some aspects of this problem will be discussed in Sections 7 and 8. In the present section we shall prove the existence of a set function c(.) , defined in the class of all subsets of R ~ , with the following properties. 6.1.

Theorem.

For n > 2 there exist positive numbers d l , . • •, d4 and a set

function c(.) in R '~ such that

(1) (2)

c(E) = c(hE) whenever h: R'~

~ n is a spherical isometry and E

c(O) = O, A C B C I:U~ implies c(A) < c(B)

C

~n.

O<3

and c ( U y = I E j )

<

dl ~j°~=l c(Ej) if Ej c R Y .

(3)

If E C ~r~ is compact, then c(E) > 0 if and only if c a p E > 0 .

Moreover

c(R '~) < d2 < oo.

(4) (5)

c(E) >_ d3 q(E) if E C R'~ is connected and E ¢ 0. M ( A ( E , F ) ) _> d4min{ c(E), c(F) } , if E , F C R ~ .

Furthermore, for n ~_ 2 and t E (0, 1) there exists a positive number d5 such that (6)

M ( A ( E , F ) ) _~ d s m i n { c ( E ) , c ( F ) } whenever E , F c R'~ and q ( E , F ) >_ t .

It should be emphasized that the main interest in Theorem 6.1 lies in the inequalities (5) and (6). The condition cap E > 0 in 6.1(3) is not needed in this section and its definition will be postponed until Section 7.

73

We shall next give the reader some idea a b o u t the set function c(-). To this end define (see (5.45))

M t ( E , r , x ) = M(A( S n - t ( x , t ) , -Bn(x,r) A E; R'~)) , (6.2)

M ( E , r, x) = M2,(E, r, x)

whenever E c ~ n

xER '~,and 0

Moreover, let E - l = { z / l z t 2 : x E E }

and

a(E) = max{ M ( E , 1, 0), M ( E -1, 1, O) }

(6.3) m

for E C R '~ . It follows from the results of this section t h a t there are numbers ql and "~2 depending only on the dimension n such t h a t

(6.4)

?'la(E) < c(E) < "72a(E) . In w h a t follows we shall give a construction of the set function c(E). We remark

t h a t there m a y also be m a n y other methods of constructing

c(E):

it is clear by

(6.4) t h a t any m e t h o d which yields a set function differing f r o m a(E) by at most a multiplicative constant is adequate also for constructing c(E) . For the next l e m m a we recall t h a t the balls Q(x, r) of the metric space (R n, q) were defined in (1.22). 6.5. L e m m a .

Let r > 1 a n d t E (0,1) besuch that q ( S ' ~ - l , S ~ - l ( r ) ) > 2 t .

There is a n u m b e r b(t) depending only on n , r, and t such that the following holds.

If E c B '~ and G t = U=eE Q,(x, t), then M ( A ( E , OGt)) < b(t) M ( A ( E , S'~-l(r))) . Proof.

Let

F1 = E ,

F2 = S'~-l(r),

and

Fa = OGt = F4.

Because

q(F1,F3) >_ t and q(F2,F4) >_ t it follows f r o m the comparison principle 5.40 and 5.42(2) t h a t (6.6)

M(r12) _>3-nmin{ M(F13), M(F23), Dt}

where r i j = A(Fi, Fj).

We shall first find a lower bound for

M(r~3).

From the

choice of t it follows t h a t t < l / v / 2 and hence q(Q(z,t)) = 2t~/i - t 2 > t v ~

(see

74 1.25). It follows that F3 contains a continuum of euclidean diameter at least t V ~ . Hence we deduce by 5.9 and 5.34 that

2r + t v ~

M(r~3) > cn log - -

(6.7)

-

2r -

tv~

cnt log 2

> - --

r

:> --

Dt r

Here n is the number in 5.42(2) and (6.6). Since d(l'~I) > q(l~]) >- t and I'll c B~(r) for ~ ~ A ( F l , F 3 ; a , ) = A and M(zX) = M(ri3)

(of. (S.10)) we get by 5.~

rM(r~3) <_ U"i-~ • By (6.6) and (6.7) M(I',2) _> 3 -n min{ M(F,3), --Dt } r

_> 3 -"rain

M(r~),

n.r-+~M(r~3)

=

u(ri~),

b(t) = 3n/rain{ 1, ntn+l/(Ct,~rn+X)} . [] 6.8,

of c(E).

The construction

For E C R ~ ,

0 < r < t < 1 denote

(cf. (6.2)) (6.9)

f mr(E, r, x) = M(A(OQ(x, t), E n Q(x, r))) ,

/, m(E,x)

= ms(E, 1/v/2, x) ; s = ½~/3.

We define (see (1.16) and (1.23))

f c(E,x) = m a x ( m ( E , x ) , m(E,x~ },

(6.10)

/

c(E) = inf{ c(E, :~) : x c ~." }.

±

1~(~)

-

=(~1

=

1

½V~

1/v~

Diagram 6.1.

75 6.11. R e m a r k .

By 5.18(2)

lmt(E,r,x)

(6.12) If

[

F c-Q(x,r),

<

r

['t / l ~ r

2 ~]l-n

m ( E , x ) <_ m ( R ' ~ , x ) = w,~_,(logv/3) 1-

.

where r E ( 0 , 1 / v ~ ] , by (6.12) we obtain

(01 ) Hence c(F,x) -+ 0 as r -* O. Note that equality holds in (6.13) if F = Q ( x , r ) . Exploiting 1.18(1) one can simplify the upper bound in (6.12). 6.14. L e m m a .

There exists a positive number dl depending only on n such

that c(E,x) < dlc(E,y) for x, y E R '~ and E c R '~ . In particular, c(E) <_ c(E, x) <_ d i e ( E ) . P r o o f . Let (6.15)

U=

Q(x,

1/v~).

By 5.18(2) or by (6.12) we obtain

M(/~(aQ(x, l~v/3),OU)) -- M ( A ( O Q ( x , 1),OU)) : wn_i(log v/3) i - n -- a .

Fix x, y E R.'~. In what follows we shall assume that

c(E, x) = re(E, x) .

(6.16)

The other case c ( E , x ) = m ( E , ~ )

can be dealt with exactly in the same way; even

the constants will be the same in the other case. Let E~" = S n Q(x, 1/x/2) n Q(y, 1/x/2) , E~ = ( E \ E~) n Q ( x , 1 / v ~ ) . It follows from 5.9 and (6.16) that either

2M(A(OV, E;)) > c(E,x)

or

2 M ( A ( O V , E~)) > c ( E , x )

where V --- Q(x, l v / 3 ) . In the first case denote F1 = E{ , F: = OQ(y, l v/3 ) , F3 =

OV and F4 = O Q ( y , 1 / v ~ ) .

In the second case let F1 = E ~ , F2 =- OQ(y, l x / 3 ) ,

F3 -- O V , and F4 = OQ('~, 1 / v ~ ) .

In both cases (see 1.25(1) and (1.15))

rain{ q ( F l , F 3 ) , q(F2,F4) } >_ q(OV, OQ(x, 1/v/-2))

= q(S,_i(V/~), Sn_l) _ Vf3- 1

76 We obtain by 5.40, 5.42(2), and (6.15)

c(E,y) >_ M(F12) _> 3-'~ min{ M(F13), M(F24), D5 } > 3 - " min{ l c ( E , x ) , a, D6 } . Because c(E, x) < a by (6.12) and (6.15) we obtain from this inequality

c(E,y) > 3-~ min{ l c ( E , x ) , D5 } > d-~lc(E,x) ;

1 D~(logv/~,n-1/~n_l} d1-1 = 3-n min{ 3, which yields the desired bound. 6.17. L e m m a .

U]

If E C ~ n , then c(E)<w,~_,(log x/3 ) 1-,~ . x/2q(E)

P r o o f . Assume first that q(E) > l / x / 2 . In this case

c(S) < c(S,O)<

¢dn_l(1Og~/3) 1-n < ~n--l(lOg

-

-

-

~ v

)l-n q(E)

by (6.12). Assume next that q(E) < 1/x/2. In this case E C Q ( z , q ( E ) ) , and the proof follows from (6.13). 6.18. C o r o l l a r y .

z C E,

53

If E c R '~ is connected, then c(E) >_ d3 q(E) .

P r o o f . It follows from the definition (6.10) that c(E) = c(hE) whenever h is a spherical isometry. Hence both sides of the asserted inequality remain invariant under spherical isometries. By performing an auxiliary spherical isometry if necessary we may assume that 0 C E . Then E N B '~ has a connected component El with 0 E E1 and hence by (1.15) d(Ei) _> q(Ei) > min{ l / x / 2 , q(E)} _> q ( E ) / x / 2 . By 5.34 we obtain (see (6.2), (6.10), and 1.25(1)) 2 x / 3 + q(E)/x/2 > cnq(E)/x/6 c(E, O) >_ Mv~(Ex , 1, O) >_ cn log 2x/3-- q(E)/x/~ The proof with ds = c,J(dix/~)

follows now from 6.14.

[]

77

6.19. L e m m a . M(A(E,F)) > d4 min{c(E), c(F) }. P r o o f . Fix x E R n • Let z E {x, 5} with m(E,z) = c(E,x) and denote

F1 = EN-Q(z, 1/v/2), Fz = 0Q(z, ½~,/-3) • Let w C {x, 5} be such that m(F,w) -- c(F,x) and denote

F2 -- F n-Q(w,1/~,/2 ) , F4 = OQ(w, ½x/3 ) . We see that (cf. 1.25) min{ q(F1 F3), q(F2,F4) } > q(Sn-l(Vf3), S n-l) - V / 3 - 1 --

7

--¢5.

Set Fij -- A(Fi,Fj). It follows from the comparison principle 5.40 and 5.42(2) (see also 5.9) that M ( A ( E , F ) ) _> M(FI2) >_ 3-'~min{c(E,x), c(F,x), Dh} :> d4 min{ c(S, x), c(F, x) } > d4 rain{ c(E), c(F) } where d4 -- 3 -'~ min{ 1, D6(log V/3)~-1/w,_l} and the second last inequality follows from the fact that c(E,x), c(F,x) < W~_l(logx/3) '-'~ (cf. (6.12)). 6.20. L e m m a . Let E, F C R n be setswith q(E,F) > t > 0 .

Then

M ( A ( E , F ) ) < dhmin{ c(E), c(F) } where d5 depends only on n and t. P r o o f . Let E1 = E A Q ( 0 , I / v ~ ) ,

E2 = E \ E l ,

F1 -

FNQ(0,1/x/2),

F2 = E \ F1. Let F1 -- A ( E I , F 1 ) , F2 = A(E1,F2), F3 = A(E2, F1), and F4 = A(E2,F2). By 5.9 M ( A ( E , F ) ) _< 4max{ M ( F i ) : j = 1,2,3,4} . Without toss of generality we may assume that the maximum on the right side of this inequality is equal to M(F2) because in the other cases the proof will be similar. Let E~=U{Q(x,~t)

:xEE1 }, Ft=U{Q(x,~t):xEF2}.

78 If 7 C F2, then clearly J'YIn (6.21)

OE~ #

0 # j~tl c~ OF~ and hence by 5.3

M(A(EI,OEI)), M(A(F2,OF~)) }.

¼ M ( A ( E , F ) ) <_ M(r:) _< min{

We shall now find an upper bound for

M(A(E,,OE~)).

A simple calculation shows

that

q(Sn_l(v/~)

sn_l) _ V~- 1 ,

v~

1 >~

•

Since E1 c B n we get by 6.5

<_ M(A(E,,Sn-'(x/'3)))

M(A(E,,OE~))

b(t/8) = an/min{ 1,

Dtn+l/((8~)n+lan)}.

OFt))

A similar estimate holds for M(A(F2,

b(t/8) ,

as well. As a result we obtain in view of

(6.21), (6.10), and 6.14 M ( A ( E , F ) ) _< 4 b(t/8) min{ ~ 4 dl

6.22. C o r o l l a r y .

with d6 =

d2d5

[]

q ( E , F ) > t > 0, then M ( A ( E , F ) ) < d6.

c(E) < c(R n)

follows from 6.20.

c(F, 0) }

b(t/8) min{ c(E), c(F) }.

IfE, F c R n with

P r o o f . By(6.12) and 6.14

c(E,O),

= w n - l ( l o g v ~ ) 1-'~ =

d2.

The proof

[]

Recall that a different proof of 6.22 was given in 5.23. P r o o f o f T h e o r e m 6.1.

Part (1) is clear by the definition of c(-). Part (2)

follows from (6.10), 6.14, and 5.9: oo

o(3

c(U j=l

oo

E,., o) < j=l

oo

d, Z; j=l

•

j=l

The other assertions in (2) follow from 5.9. Tile proofs of (4), (5), and (6) were given in 6.18, 6.19, and 6.20, respectively. The proof of (3) follows from (5), (6), and the definition of a set with positive capacity, which will be given in Section 7 (see 7.12). [] 6.23. E x e r c i s e . Find a lower bound for

c(B'~(x,r)).

79

6.24. E x e r c i s e .

Applying (5.46) and the results of this section show t h a t (6.4)

holds. 6.25.

Exercise.

q(z,E) < t } .

Let E = {0}

U (U~-I sn-l(2-k)) and E(t) = ( z • R n :

Show t h a t M ( A ( E , OE(t))) > a t l - " l o g

I for small t where a de-

pends only on n . [Hint: Apply (5.14).] Conclusion: T h e function b(t) in 6.5 m u s t grow so fast t h a t b(t)t'~-l/log ~ 74 0 as t --~ O. From the proof of 6.5 it follows t h a t the rate of growth of b(t) is at most t - 1 - • , and the best rate of growth will be given in 6.27. An appendix

t o S e c t i o n 6.

In this a p p e n d i x we shall carry out some compu-

tations which we shall not need later on in this book b u t which m a y be of independent interest. We are now going to prove an improved form of L e m m a 6.5 and shall show t h a t the function b(t) in 6.5 can be chosen so t h a t its rate of growth is at m o s t t - ~ . It follows f r o m Exercise 6.25 t h a t the power - n

cannot be replaced by 1 - n (see

also 6.28). The following discussion is based on a Poincar~ inequality type result of Yu. G. Reshetnyak [R12, p. 60, Lernma 3.3], and the proof of L e m m a 6.27 below is also due to him. T h e author wishes to t h a n k Yu. G. Reshetnyak for contributing this result. For the p r o o f we need also some results from the early parts of Section 7. In particular, L e m m a 7.8 will be useful. (JR12, p. 60, L e m m a 3.3]). Let u be a function of class C ~ ( R '~)

6.26. L e m m a

such that u(x) = 0 for Ix[ _~ r > 0 . Then the inequality

f~ lul"dm<_(2r)'~fR IVul"dm holds. 6.27. L e m m a .

Let E be a compact set in B~(R)

and let E(t) = E + B'~(t)

for t > O. Then M(A(OE(t),E)) < a ( t ) M ( A ( 0 E ( 1 ) , E ) ) for t > o where a(t) = ~(1) rot t > 1 and ~(t) < ~ l t - " rot t ~ (0,1), and ~i

depends only on n and R . Proof.

Fix ~ > 0 .

with u ( x ) > 1 for x E E

fR

In view of (7.3)-7.s there exists a function u C C~°(E(1)) and

tVulndm <_e + M(A(OE(1),E)) r.

+cap(E(1),E).

80 There exists a constant bl depending only on n and for each t E (0, 1] a (real-valued) C ~ ( E ( 1 ) ) - f u n c t i o n ~t: E(1) --~ [0,1] with the properties (see e.g. [ST, p. 171])

(a) (b)

(x)=lforxeE, ~t(x) = 0 for x C E(1) \ E(t) ,

(c) IV (x)l <_bl/t The function

v(x) = u(x)~t(x) is admissible for the definition of c a p ( E ( t ) , E ) (see

7.2), and hence

M(A(OE(t), E)) = c a p ( E ( t ) , E) 4_/R-!VvI'~dm" Since ]Vv(x)I n ~

2'~(]Vu(x)i'z~t(x)n + ]~7~t(x)l'~lu(x)l n) we get by the properties

(a) and (c) of ~t

cap(E(t),E) <_

lw,

÷

lu( )r am

Moreover, by L e m m a 6.26 we obtain for t E (0, 1}

/R lu(x)i'~drn ~ 2n(R + l)n /R IVu(x)P din. Hence

£ .

~ a(t) / IVuI'~dm < a(t)(e + c a p ( E ( 1 ) , E ) ) JR a(t) = 2n(1 + 2nb?(R ÷ 1)'~t -'~)

cap(E(t),E)

n

for t C (0, 1]. For t _> 1 we define

a(t) = a(1). Because E > 0 is arbitrary the proof

follows from this last inequality in view of 7.8 and 5.3.

I:1

a(t) _< alt -~ of Lemma 6.27 provides the best possible integer power for the growth of a(t) . Next, we shall show that this rate t -'~ of growth for a(t) is in fact attained. We already know by Exercise 6.25 that the inequality

6.28.

Example.

We shall show that there exists a constant bl > 0 and for

arbitrarily small t E (O, 1) a set E = (6.29) where

Et in R ~ such that

M(E,t) = M(A(E, OE(t))) > b,t -'~ E(t) = E + B'~(t).

Let Q = [0,1] n-1 × { 0 } and let s E (0,1). It follows from 5.11 that (6.30)

M(Q,s)

~

.S 1 - n

.

81 •

Fix k > 4 and let Qj = Q + 2 - k j e , ~ , t E (2-k-2,2 -k-l)

j = 0,...,2 k

2k

Set Ek = U j = o Q j "

For

we obtain by 5.4 and (6.30) M ( E k , t ) >_ ( 2 k + 1)t 1-~ _> l t - n .

In conclusion, we have proved (6.29) with bl _- ~1 • 6.31.

Remarks.

Modulus estimates in the spherical metric a p p e a r in [LV2,

1.6.5], [V7, Section 12], [SR1], [MRV2, L e m m a 3.11], and in IN2]. This section is taken from [VU8]. H. Renggli [REN] and W. P. Ziemer [Z2] have also constructed some set functions related to moduli of curve families.

7.

The capacity of a condenser

In the present section we shall introduce, as a special case of curve families and their moduli, the notion of a condenser and its capacity, and we shall examine various properties of condensers.

An i m p o r t a n t property of the capacity of a condenser is

t h a t it decreases under a special geometric transformation called symmetrization. Of the several kinds of s y m m e t r i z a t i o n discussed in the literature (see e.g. [PSI, [G1], [$1], jR12, p. 74]) we shall consider only spherical symmetrization.

An immediate

consequence of the a b o v e - m e n t i o n e d monotoneity is the fact t h a t condensers obtained as a result of spherical s y m m e t r i z a t i o n are of extremal character - - their capacities yield lower bounds for the capacities of a wide class of condensers in R '~ . T h e extremal condensers of GrStzsch and Teichmiiller are of particular importance, and the wellknown estimates for the capacities of these condensers are given in this section. One of the m a i n themes of this section is the relationship of the capacity of a condenser to its geometric structure. T h e hyperbolic and quasihyperbolic geometries are useful instruments in the study of this interrelation in Sections 7 and 8. In this context the hyperbolic and quasihyperbolic geometries are useful for proving estimates for the capacity only of ring domains with n o n - d e g e n e r a t e c o m p l e m e n t a r y components. 7.1. Tj:R n ~

Definition.

For j = 1 , . . . , n

let R~ = { x E R '~ : x j = 0}

R~ be the orthogonal projection T j x = x - x ] e j .

and let

Let D c R ~ be an

open set and u: D --~ R a continuous function. T h e function u is called absolutely continuous on lines, abbreviated as A C L , if for every cube Q with Q c D , the

82 set Aj C TjD C R~ of all points z E TjQ such t h a t the function t ~-~u(z+tej),

z+te I E Q, is not absolutely continuous as a function of a single variable [HS, p. 282], satisfies rnn-l(Aj) = 0 for all j = 1, . . . . n . By well-known properties of absolutely continuous functions of a single variable the derivative exists almost everywhere and is B o r e l - m e a s u r a b l e (see [HS, p. 285], [V7, pp. 87-89]. From this fact and f r o m Fubini's t h e o r e m it follows t h a t an ACL function u: D --+ R has partial derivatives with respect to every variable X l , . . . ,xn a.e. (with respect to n - d i m e n s i o n a l Lebesgue measure) in D .

We say t h a t an ACL function

u: D --+ R is ACL p , p > 1, if ¢gu(x)/Oxi E LP(K), j = 1 , . . . , n , whenever g C D is c o m p a c t . A v e c t o r - v a l u e d function is said to be ACL ( ACL p ) if and only if each coordinate function is in this class. 7.2. D e f i n i t i o n .

Let A C R n be open and let C C A be compact. T h e pair

E -- (A, C) is called a condenser. Its p-capacity is defined by (7.3)

p-cap E = inf f JR

tVulPdm, n

where the infimum is taken over the family of all n o n - n e g a t i v e ACL p functions u with c o m p a c t support in A such t h a t u(x) _> 1 for x E C . Here v

(z) =

{ au (x) '" "'

)

A function u with these properties is called an admissible function. It follows f r o m (7.3) t h a t p-cap E is invariant under translations and orthogonal maps. W i t h o u t alteration of the real n u m b e r p-cap E , one can take the infimum in (7.3) over several other classes of functions as can be shown by approximation. For instance one m a y take functions u E C°~(A) with c o m p a c t s u p p o r t in A and u(x) > 1 for x E C (see [MRV1]). T h e following monotone property of condensers is a consequence of the definition. If ( A , C ) and (A',C') are condensers with A' c A and C C C ' , then (7.4)

p-cap (A', C') _> p-cap (A, C ) .

T h e p - c a p a c i t y of (A, C) reflects the metric s t r u c t u r e of the pair

C , R '~ \ A

as we

shall see later on. If p = n we denote n - c a p (A, C) simply by cap(A, C) and call it the capacity or conformal capacity of the condenser (A, C ) .

83

An A C L v function u: D --* R

TM

where D c R n is open, is said to be abso-

lutely continuous on the rectifiable curve a: [a,b] -~ D iff f o a ° : [0,~(a)] --* R m is absolutely continuous as a function of one variable. We shall m a k e use of the following result of B. Fuglede IF], IV7, 28.1, 28.2]. 7.5.

Let D be an open set in R n and let f: D -+ R

Lemma.

TM

be A C L v .

Then the family of all locally rectifiable paths in D having a closed subpath on which f is not absolutely continuous, is p-exceptional. 7.6. L e m m a .

Let G be a domain in R n , let u: G--~ R be an A C L p function,

-co < a < b < co, andlet

A, B C G be n o n - e m p t y sets such that u(x) < a for

x E A and u(x) > b for x E B . Then Mv(A(A,B;G)) Proof.

<_ (b - a) - v / a

l~7u]Vdrn "

Define an A C L v function v: G ~ R by

-

b-a

,

xEG.

T h e n v(y) > 1 for y E B and v(y) ~ 0 for y C A .

Let A = {-~ E A ( A , B ; G )

:

"y is rectifiable} and A~ = { ~/E A : v is not absolutely continuous on a closed s u b p a t h of "~ } . Fix "~ E A \ A~ w i t h the n o r m a l r e p r e s e n t a t i o n

~/0: [0, c] --~ G , c = ~('~), and with

70(0) e A , "~°(c) e B . T h e n .~o has a Lipschitz c o n s t a n t

1 and I('~°)'(t)l = 1 a.e.

in [0, c] (see [V7, 2.4]). By [V7, 1.3] we get 1 < v('7°(c))

(7.7) _<

-

-

v('~'°(0)) <

~0CI(v o "f°)'(t) ! dt

/o

:

IVvtd

.

Since v is A C L v , 1Vvl is a Borel function a n d thus IVvl E ~r(A \ A~) in view of (7.7). By 7.5 we o b t a i n

Mp( (A, . ; G))

I vlPdm = (b a)

I ulPd

[]

84

Let (A, C) be a condenser for which A is bounded. It follows from 7.6 that

Mp(A(C, OA;A)) <
>_1}

and

OAC

{zER~:u(z)

<_0}.

Thus

Mp(A(C, OA;A)) <_p-cap (A,C)

.

Also the converse inequality holds true according to the following result of W. P. Ziemer [Z1], but the proof is longer and will be omitted. 7.8. T h e o r e m .

p-capE =

7.9. R e m a r k .

bounded

//" E = (A, C) is a

condenser in R '~ ,

then

Mp( A(C, OA; A)) .

By 5.9 the curve family on the right side of Theorem 7.8 may be

replaced by some other families as well. We shall need 7.8 mainly in the case p = n . We now show that 7.8 holds also for unbounded condensers if p = n . Let (A, C) be an u n b o u n d e d condenser, let z E C and r > 1 +

d(C),

and

Ar = A n B'~(z, r2).

By

the monotone property of the capacity, by 7.8, 5.9, and 5.14 we obtain c a p ( A , C ) _ cap(AN,C) =

M(A(C, OAr;Ar))

<_ M(a(C, OA; A))

+ M(Zx(C,S"-'(z, r2)))

< M (A(C, OA; A)) + wn-, (log r) 1-n Letting r --, oo shows t h a t c a p ( A , C ) < M ( A ( C , OA; A)). The converse inequality follows from 7.6 and 7.7. In conclusion, we have proved that the equality (7.10)

cap(A, C) =

M(A(C, OA; A))

holds whenever ( A , C ) is a condenser in R '~ , whether A is bounded or not. We now extend the definition of a condenser to R'~. Assume that C c ~ n

is

compact and that there exists an open set A c R '~ with A ~ R '~ and C c A . Then we say that (A, C) is a condenser in R n and define its ( n - ) c a p a c i t y by (7.10). In view of 7.8 this extended definition is compatible with the definition (7.3) in case A C R '~ . (We shall not need the p-capacity, p ¢ n , of a condenser in ~.'~ .)

85 Now let ( A , E ) be a condenser in R n or ~ n .

It follows from (7.10) and the

conformal invariance of the modulus 5.17 that c a p ( A , E )

is a conformal invariant.

Likewise, by virtue of (7.10), many properties of cap(A, E) may be derived directly from the properties of the modulus. In particular, we shall often make use of Remark 5.9 specialized to condensers. If p ~ n , then p - c a p ( A , E) is not invariant under conformal mappings, while n - cap(A, E) has this invariance property. The corresponding property of the modulus immediately yields this conclusion. 7.11. L e m m a .

Let (A, F) be a condenser with A c B n . Then there are posi-

tive numbers al depending only on n , and a2 depending only on n and d(F, OA) such that alc(F) <_ c a p ( A , F ) < a=c(F) . P r o o f . By the proof of 6.19 and 7.8 c a p ( A , F ) _> d4 min{ c(R'~ \ A, 0), c(F,O)} . Since A C B '~, c(P,.'~ \ A, O) = Wn_l(lOgx/3) 1-n = c(R'~,0) _> c(F, 0) (cf. (6.10), (6.12)) and thus we obtain the desired lower bound cap(A,F) > d4c(F). For the upper bound let t = q ( F , R '~ \ A ) . since F c A c B

Then by 1.17 t > ½ d ( F , R ' ~ \ A )

n. By the proof of 6.20 and 6.14 c a p ( A , F ) < 4b(t/8) m i n ( c ( F , O ) , c(R ~ \ A, 0) } <_ 4 dl b(d(F, O A ) / 1 6 ) c ( F ) ,

which yields the desired upper bound. (Note that a slightly better upper bound can be derived from 6.27.) 7.12. denoted

D e f i n i t i o n . A compact set capE

cap(A,E) =0.

E in R '~ is said to be of capacity zero,

= 0 , if there exists a bounded open set A c o m p a c t set E C R . n ,

A with

E C A and

E : f i ~ n , is said to be of capacity zero if

E can be mapped by a MSbius transformation onto a bounded set of capacity zero. Otherwise E is said to be of positive capacity, and we write cap E > 0.

86

7.13. R e m a r k s .

By conformal invariance the second p a r t of the above definition

is independent of the choice of M6bius transformation. We next show t h a t the first p a r t of the definition is independent of the choice of b o u n d e d open set A with A D E . Indeed, if Aj D E , j = 1 , 2 , are b o t h bounded, say Aj c B'~(R) , j = 1 , 2 , then by 6.1(4) d2 = c ( R n) > c ( R n \ a j )

>d3/v/-~R2;

j=1,2.

This inequality together with 6.1(5),(6) yields for j = 1,2 d4 min{ c(E), d3/V/2 + R 2 } <_ c a p ( A j , E ) < ds min { c(E), d2 } where

d2,d3, d4 are positive numbers depending only on n and where d5 depends

also on q(E, (0A1) U ( 0 A 2 ) ) . In other words, cap(Aj,E) = 0 if and only if c(E) = 0. Hence the condition cap E = 0 is independent of the choice of open bounded set A with E C A (see also JR2, L e m m a 2]). This a r g u m e n t also shows t h a t in the above definition of cap E = 0 , E C R n compact, one can replace the b o u n d e d set A by a ball Bn(r) with r >_ d(O,E) + 2d(E) , say. It should be observed that we have only defined the conditions cap E = 0 and

cap E > 0 for a compact set E and t h a t in the latter case the "capacity" of E will not be specified as a real number. In view of 7.9 and (5.15) countable c o m p a c t sets are examples of sets of capacity zero. The following t h e o r e m shows t h a t sets of capacity zero are always very thin JR12, p. 72]. The definition of the Hausdorff dimension and the a - d i m e n s i o n a l Hausdorff measure can be found in [FA] and [MAT2]. 7.14. Lemma. Suppose that F is a compact set in R n of capacity zero. Then

for every a > O, the a-dimensional Hausdorff measure Am(F) of F is zero. In particular, int F -- O, and F is totally disconnected. 7.15.

Remarks.

(1) In the dimension n = 2 the logarithmic capacity is

often used in complex analysis. H. Wallin [W1] has proved t h a t a c o m p a c t set is of logarithmic capacity zero if and only if it is of capacity zero in the sense of the above definition ( n = 2). He has also constructed a c o m p a c t C a n t o r - t y p e set E in R '~ of positive capacity (in the sense of 7.12) with A s ( E ) = 0 for all a > 0. See also V. G. M a z ' y a - V . P. Khavin [MK]. (2) Various sufficient or necessary conditions for capacity zero can be found in the literature [MK], [W2], [R12, p. 71], [R7], [MV], IV9].

87

7.16.

The spherical symmetrization.

If x0 E R '~ , E c R '~ and if L is a

ray f r o m Xo to c o , t h e n the spherical symmetrization E* of E in L is defined as follows: (1)

xoEE*

iff x o C E ,

(2)

coEE*

iff c o E E ,

(3) for r E (0, c o ) , E* N S'~-l(xo, r) ~ 0 iff S n S'~-l(xo, r) ~ ~, in which case E* MS=-l(xo,r)

is a closed spherical cap centered on L with the same

ra=_l measure as E A S'~-l(xo,r) .

OO

Diagram 7.1. Let ( A , C ) be a condenser and x0 E 1~'~ . Denote by C* and B the spherical s y m m e t r i z a t i o n s of C and R ~ \ A in two opposite rays L1 and L2 e m a n a t i n g from Xo, and let A* = R " \ B . T h e n it is easy to verify t h a t (A*, C*) is a condenser [S1]. An i m p o r t a n t p r o p e r t y of spherical s y m m e t r i z a t i o n is given in the following theorem [GI], IS1]. 7.17. T h e o r e m .

/ f (A, C) is a condenser, then for p > 1 p-cap (A, C) >_ p-cap (A*, C* ) .

C,, O0

Diagram 7.2.

88

This inequality is sharp in the sense t h a t there is equality if (A*, C*) = (A, C) (e.g. Xo = 0 ,

C = [0,e,],

A = Bn(2)

t h a t the m i n o r a n t p - c a p ( A * , C * )

and Li

is the positive x l - a x i s ) .

Note

in 7.17 depends on the choice of the center of

s y m m e t r i z a t i o n , the point x0, in an essential way. For instance, if n > 3, Ej = { x e S'~-1(2 - / )

: x3 = 0 } ,

E = {0} U ( [ . J j = I E j )

and if E*

is the spherical

s y m m e t r i z a t i o n of E in the positive x l - a x i s (in which case x0 = 0 ) , t h e n E* = {0,

~el, ~el,...} I I

and clearly c a p ( B ~, E*) = 0 . It is left as an exercise for the reader

to find a spherical s y m m e t r i z a t i o n with center ~ 0 which provides a strictly positive m i n o r a n t for c a p ( B n, E) . 7.18. T h e G r S t z s c h a n d T e i c h m i i l l e r r i n g s .

Let us recall the GrStzsch and

Teichmiiller rings Rc,n(S ) and RT,n($ ) which were introduced in Section 5. T h e y can also be understood as condensers in the following way:

Rc,.(~) = (R" \ {te, : t > ~), ~ n ) ,

~ ~ (1, ~ ) ,

R~,n(~) = (R" \ {t~,: t > ~}, I - e l , 0 ] ) , ~ ~ (0,oo). We define functions

•

= On

and

g¢ = ~ n

modRa,r~(s ) = l o g ¢ ( s )

by

and

m o d R T , n ( s ) = log ~ ( s ) . In other words (cf. (5.52)) J~ cap Ra,n(s ) ----(Mn_ 1 (log O(s)) ' - n = "Tn(s) ,

(7.19)

[ cap R r , n (s)

7.20.

Lemma.

= ~_

1 (log

k~(s) ) 1-n = "rn(s) •

The function O(t)/t is increasing for t > 1 and gJ(t - 1) =

O(x/'t) 2 for t > 1. Moreover, the functions "~,~ and rn are strictly decreasing. Proof.

For the first p a r t fix 1 < s < t , let R - - Rc,~(t ) and let R t and R "

be the two rings into which R is split by the sphere {x[ = t / s .

By 5.24 and 5.14 we

obtain logO(t) = m o d R > m o d R t + m o d R " = log(t/s) + log O(s) whence

¢(t)/t >_ ¢ ( s ) / s

as desired.

It follows, in particular, t h a t

¢

and

~

are

strictly increasing and hence by (7.19) "/n and T,~ are strictly decreasing. The asserted identity is the functional identity 5.53 rewritten. By 7.20 the function

C]

logO(t) - logt is increasing and therefore has a limit as

t --~ oo. We define a n u m b e r ,kn by (7.21)

l o g a n = ~_,oo(.limlogO(t) - logt) .

89

This number is sometimes called the Grhtzsch (ring) constant. Only for n = 2 is the exact value of the Grhtzsch constant known, ),2 = 4 [LV2, p. 61, (2.10)]. Various estimates for )`n, n >_ 3, are given in [G1, p. 518], [C1, pp. 239-241], JAN2]. For instance it is known t h a t )`n C [4,2e'~-1),

)`n-<4exp

-s

ds

; a(n,s)=

-i,

(s2+1~

~l/n --* e as n --* oo. These technical results will not be proved here. Some

and that ,,,~

of them are summarized in the next lemma. 7.22. L e m m a .

F o r each n ~_ 2 t h e r e e x i s t s a n u m b e r

A,~ ~ [4, 2e n - l ) , As = 4 ,

s u c h that

(1)

t ~_ ¢ ( t ) ~_ Ant , t > 1 ,

(2)

t + 1 <__

Furthermore,

,,n

<

(t + 1 ) ,

t > 1

--+ e as n --~ oo and, in p a r t i c u l a r ,

)`,~ -+ oo as n -+ o o .

P r o o f . The bounds 4 _< An <_ 2e '~-I are given in [G1], JAN2]. The lower bound in (1) follows from the fact that the boundary components of R c , n ( t ) are separated by the annulus A = B'~(t) \ B n with m o d A = logt and from 5.3. The upper bound in (1) follows from 7.20 and the definition (7.21) of A,~ above. Inequality (2) follows from (1) and 7.20, and the last assertion is proved in [AN2].

O

Because of the functional identity 5.53 the properties of T can be derived from those of ff and conversely. A simple argument similar to 5.63(3) shows that the strictly decreasing function T is continuous on (0, oo). In what follows we may use these simple properties without notice. The following fundamental difference between dimensions n = 2 and n > 3 should be observed: for n > 3 no explicit expression like (5.56) is known for ~,~(s). It is an interesting open problem to find such a formula also for the multidimensional cause. The Gr6tzsch and Teichmfiller condensers have some important extremal properties which are connected with the spherical symmetrization.

In what follows we

shall often require a lower bound for the capacity of a ring domain in terms of the Teichmfiller capacity T,~(S) which follows from the spherical symmetrization lemma 7.17. For this reason various estimates for ~[n(s) and 7,~(s) will be very useful - -

90 in fact they wilt be necessary for our later work in the multidimensional case n _> 3 when no exact formulae for rn(s) or %~(s) are known. Before giving these estimates we shall discuss qualitatively the behavior of rn(s) and 3%(s) • First we note that by (5.14) and 5.32 the limit values of "/n(s) and rn(S)

are "Yn(S) = c ~ , (7.23)

lim+rn(s) = o o , 8

lim "In(s) = O, lim rn(s) = O. 8 "-*00

For convenience we set 3%(00) = 0 = rn(oo) and %~(1) = o0 = r,~(0) . L e m m a 7.22 yields the inequalities wn-1 (log AnS) 1-" <__"Tn(S) _< W , - l ( l O g s ) 1-n ,

(7.24)

wr~-I (log(A2s)) 1 - " <_ r , ( s - 1) <_ w , _ l ( l o g s ) 1 - " for s > 1. In passing we shall show how this upper bound for "),=(s) can be slightly improved. First, fix s > 1 and choose h • ~34(B '~) with hi0, ~el] 1 = I--eel,eel], a>0.

Then a=s-V/~-I

by 2.42 and b y c o n f o r m a l i n v a r i a n c e 5.17, 5 . 3 , a n d

5.14

ff~(s) : M (A(S '~-', [0, {eli)) : M (A(S ''-1,

(7.25]

<_ W n - l (log(s + ~ ) ) l - n

[-ael,ael]))

< w n - l ( l o g s ) 1-n •

We note that (7.25) yields a slightly better upper bound for ~n(s) than (7.24). Note also that by combining the first inequality in (7.24) with (7.25) and letting s --+ oo yields An _> 2 for all n > 2. An even better upper bound for "7,~(s) will be given in L e m m a 7.26. Each of the bounds for "yn(s) in (7.24) is asymptotically sharp as s tends to o0, but not of the correct order as s tends to 1, as can be seen from 7.26 below. The following theorem due to G. D. Anderson [AN1] yields inequalities t h a t are asymptotically sharp as s tends to 1. 7.26. T h e o r e m . (1)

For s E (1,oo) and n > 2

"~,~(s) _< Wn_llZ(1/,S) 1-n < 6dn_l(lOg(s + 3 ~2V/~'~1)) l - n ,

,-1 Moreover, if s • (0,oo) and a = 1 + 2 ( 1 + v ' T + s ) / s ,

,o,(,+1) then

.

91 (3)

c n l o g a < r,~(s) < c,~tt(1/a) < cnlog(4a)

and (1 + 1/x/-s) 2 < a < (1 + 2/V/-S) 2 hold true. Furthermore, when n = 2, the first inequality in (1), the second inequality in (2), and the second inequality in (3) hold as identities. P r o o f . (1) The proof of the inequality "Tn(s) < w,~-l#(1/s) 1-n will be omitted (see [AN1]). The second inequality follows from (5.58). (2) & (3) It is left as an easy exercise for the reader to verify that (2) and (3) are equivalent, that is, one can be derived from the other in view of (5.53). Hence it suffices to prove (2). Here we shall only prove the lower bound in (2); the proof of the upper bound will be omitted (see JAN2]). Let h be an inversion in S n - l ( - e n , v/2) which maps B '~ onto H ~ and 0 to e~. By (2.22) h preserves hyperbolic distances. Because "~n(s) = M (A([0, ~e~], S n - 1 ) ) and pB.(0, lge~) = l o g ~

we see by (2.6) and ( 2 . 1 7 ) t h a t h(½e,~) = ~8-1 e~. By

conformal invariance, 5.26, and 5.32 we obtain

~,~(s) = M ( A ( I St 8- +' l en, e,], 0 H ~ ) )

8-, e-, e",], - 8s-+', e " ' - e 4 ) ) -- 2~'- 1M (A ( [ ~-;5 > 2n--lcn log s + 1 -

s - l '

which is the desired lower bound. The proof for the bounds for a is elementary.

The assertions concerning the case n = 2 follow from (5.56), (5.57), (5.59), 5.60(i), and 5.30.

CI

It should be observed

that 7.26(1) yields a slightly better explicit bound

(7.25). We shall next summarize the preceding inequalities for ~=(s) . Let Ul(S ) = Ojn_lU(1/S) 1-n ,

2" l c o , ( s -

\s+l/

!3 '

vl(s) = wn- l (log Ans) 1-'~ , v2(s) = 2'~-1cn log s + i s--1 By (7.24) and 7.26

(7.27)

max{ vl(s), v2(s) } _< q=(s) <_ min{ ul(s), u 2 ( s ) } .

than

92 For n = 2 the right side of (7.27) is sharp. In fact, it follows f r o m (5.56) (or (5.59)) t h a t for n = 2 the right side holds as equality for all s > 1. One can rewrite (7.27) for

r,~(s) using

the functional identity 5.53.

4,

......... ::::::::::::::::::::::::::::::::::::::::::

2"

2"

l

" ......................

0,0

r

i

1

i

1

~

i

i

~

0.~.

0.2

0.3

0.4

0.5

0.6

0.7

O,B

0.9

Diagram 7.3.

%

r

1,O

T h e graph of ~/s(1/r) , 0 < r < 1, lies in the s h a d e d region.

Lower bounds:

f(r)

l+r = 4c31og 1--~-r'

U p p e r bounds:

F(r)

= 4c3#

7.28. R e m a r k s .

( l~- r )

47r log2(9.9002/r ) ()~3 < 9.9002).

g(r) = ,

G(r)-

( # 4~r (r)) 2

(from •[AVV31) .

(1) T h e last inequalities in 7.26(2) and (3) can be improved

in view of (5.67). (2) T h e inequality 7.26(3) can also be written as follows c

(1)

log(l+t)

7.29. Hyperbolic

metric and capacity.

, t>0

As in Section 2 we let

J[x,y]

denote

the geodesic segment of the hyperbolic metric joining x to y , x, y ~ B '~ . It is clear by conformal invariance t h a t c a p ( B n,

J[x, y])

= c a p ( B ~,

TxJ[x, y])

93 where Tz is as defined in 1.34. We obtain by (2.25) and (7.25) (7.30)

cap(B",alx,

1

y])=~ln(th½p(x,y))

1

< W.-l(-logthTp(,y))

X

1--n

.

Next by (7.30), 7.26(2), and (5.58) we get

(7.31)

2"-1c. p(~, y) _< cap(B ", J[~, y]) < 2~-I~..(e-~(~,~)) < 2~-le.(p(x,y) + log4)

For large values of p ( x , y )

(7.31) is quite accurate. For small p ( x , y )

one obtains

better inequalities than (7.31) by combining 7.26(1) and (7.30). 7.32. L e m m a .

Let x, y E B n and let E c B '~ be a c o n t i n u u m w i t h x, y E E .

Then 1

cap(B~,E) > cap(n", JE~,~I) -- ~ ( t h ½p(~,y)) " P r o o L Let Tz be as in 1.34 and let * denote spherical symmetrization in the positive xl-axis.

Then the center of symmetrization is the origin and by 7.17 we

obtain cap(B '~, E) = cap(B '~, T ~ ( E ) ) >_ cap(B '~, ( T , ( E ) ) * ) . By (2.25) we see that [0, (th ½P(x,y))el] C ( T x ( E ) ) * and the proof follows from (7.30). [] 7.33. :Exercise. Show that 5.34 follows from 7.32. [Hint: We may assume that t = 1 in 5.34. Apply (7.31) and 2.41(1).] The next result gives a very useful lower bound for the capacity of a ring domain. 7.34. L e m m a .

Let R = R(E,F)

be a ring in R ~ and let a, b E E , c, oc E F

be distinct points. T h e n

capR = M ( ~ ( E , F ) ) >

(la-cl~ ~7-c-~l.

Here e q u a l i t y holds for S = [ - e l , O ] , a = 0, b = - e l ,

F = [sel,oo),

c = sel,

d=oo.

P r o o L Observe first that the right side remains invariant under the similarity transformation f ( x ) = ( x -

a ) / l a - b I . Then If(c)[ = [ a - c[/]a - b] and f ( a ) = O.

94

d.~- oo

% -Ibtel

a = 0

C ~

O0

Sel

D i a g r a m 7. 4 .

T h e spherical s y m m e t r i z a t i o n s of f E

and f F

in the negative and positive x l - a x i s ,

respectively, c o n t a i n the c o m p l e m e n t a r y c o m p o n e n t s of R T,,~(In-hi) " T h u s by 7.17

Ita-cl~ '1.35. L e m m a . be d i s t i n c t p o i n t s .

L e t R -- R ( E , F )

be a r i n g in R '~ a n d let a, b E E , c, d E F

Then

capn > T(la, b,c, dt) Here equality holds if b = slel , a = s2el ,

c

~

83ei

,

d

~

s4e

1 ,

and

s l < s2 <

83 ~ S 4 .

Proof.

By (1.29) we m a y assume t h a t

a = 0,

b = el,

d = oo, and

[c I =

I a, b, c, d I • T h e p r o o f follows now f r o m 7.34. T h e assertion c o n c e r n i n g the equality follows f r o m 5.54(1). 7.36.

[3

Corollary.

Let

R = R(E,F)

be a r i n g a n d let a , b E E ,

c, d E F

be

d i s t i n c t p o i n t s in R '~ . T h e n

capR

> T(

t - s

-

where S~---

Ia - b

[a - b[ I+[ a-c I+lc-d

I '

t-~

la - bl + Ia la-b I+[a-c]+l

c[ c-d

I "

H e r e e q u a l i t y h o l d s for E = [O, SCl] , a = O, b = s e l , F = [ t e t , e t ] , c -- t e l , and O<s
d = el,

95 P r o o f . Because the points are finite, (1.15) and 7.35 yield c a p R > ~ ( ~ a - c' [b - d[

-

btlc

dl)

The desired inequality follows from this and the fact that Ib-dl _< Ib-al+ l a - c l + l c - d l • [] The statement concerning equality follows from 5.54(1). 7.37. Corollary. /f R = R(E, F) is a ring, then (1)

capR > T (

(2)

capR > v \ q ( S ) q ( F ) ]

--

q(S

;q()) F

~ 4q( E, F) ~

'

"

P r o o f . (1)Choose a,b E E and c,d E F so that q(a,b)~-q(E)

and q(c,d)=

q(F). Then q(a, c) q(b, d) q(a, b) q(c, d)

1 q(E) q(F)

and (1) follows from 7.35 because v is decreasing. (2) Choose a E E , such that

c E F such that q ( a , c ) - : q ( E , F )

q(a,b) >_ ½ q(E)

,

and choose b E E ,

dEF

q(c,d) >_ ½ q(F) .

With this choice of a, b, c, d the proof follows from 7.35.

[]

7.38. L e m m a . Let E and F be disjoint continua in R '~ with d(E), d(F) > O.

Then M ( A ( E , F ) ) _~ r(4m 2 + 4 m ) _> c,~ log(1 + 1/m) where rn -- d ( E , F ) / m i n { d ( E ) ,

d(F)} and c,~ is as in 5.32.

P r o o f . Fix a E E , c E F with ] a - c ] = d(E,F)

and b E E , d E F with

ta - b I = ½ d(E) and tc - d I -- ½ d(F), respectively. By 7.35 we obtain r(I a - cl Ib - d I~ > v( la - c I (Ia - _ b i ~ - I a - c I + I c - d I ) ) M ( A ( E , F ) ) _>

la bl Ic dll

Here

u --

bl :

2 d(E, F) (d(E) + 2d(E, F) + d(F) ) ~_ 2m ~r 4m 2 ÷ 2 m , d(E) d(F)

and the first inequality follows. The second one follows from 7.26(3).

[]

T(U)

96 7.39. C o r o l l a r y . Let E and F be disjoint continua in R ~ with 0 < d(E) <<

d(F). Then M(ZX(E,F)) >- 2 ' - " ~ / d (e---~ E,r)~ ) P r o o f . The proof follows from 7.38 and 5.63. 7.40. E x e r c i s e .

[:71

(1) Show that if R = R ( E , F) is a ring in R'~, then

/

c a p R > q(E)cn log~l +

q(F) q(E,F)]

[Hint: Apply 7.37(2), 7.26(3), and (3.6).] (2) Derive 5.42(2) from 7.37(1). We are going to generalize the formula (7.31), which relates the hyperbolic distance p(x,y)

and the capacity of the condenser (B '~,g[x,y]) in a simple fashion.

Now we shall discuss instead of this particular condenser a general ring R ( E , F) and the hyperbolic distance will be replaced by the function

Ix - yl j c ( x , y ) - - - l o g ( 1 + min{-d(~): ~/(y)}) which was introduced in (2.34). 7.41. L e m m a .

is a ring with o c ~ E U F ,

ff R = R ( E , F )

then

cap R > cn rain{ j R , , \ E ( F ) , j R . \ F ( E ) } .

If c~ E F , then c a p R _> c n j R , , \ r ( E ) . P r o o f . The proof follows immediately from 7.38, 7.34, and the definition of

j~(A) (see (2.35) and 2.37).

O

Applying this lemma with E = S '~-1 , F = J[x,y], x,y E B '~ we obtain in view of 2.41(1)

cap(Bn,j[x,y]) >_ cnjB,~(J[x,y]) = C n j g . ( x , y ) >

¼c. p(~,~).

Hence 7.41 implies (7.31) with a slightly different constant. Thus we may regard 7.41 as a generalization of (7.31).

97 7.42. R e m a r k . sets E and F .

L e m m a 7.41 has a converse which is valid even for disconnected

Indeed one can show t h a t for a given integer n _> 2 there exists a

h o m e o m o r p h i s m h,: [0, co) -+ [0, o0) with the following properties. If E and F are c o m p a c t disjoint sets in R " , then M(A(E,F))

T=min{j~.\E(F),jR.\f(E)}.


We shall outline a proof for this estimate. Clearly we m a y assume that 0 <

d(E) <_

d(F). Set t = d ( E , F ) . Then 5.5 yields

+-o/,t/) < while for

I+

d(E) < t we obtain by (5.14)

<- .._,(log

t

hi--

d-~]

"

These two inequalities together imply the desired bound. 7.43. A s p e c i a l f u n c t i o n .

The functions ff~ and r , as well as their inverses

and various combinations of these will occur often in C h a p t e r III. Of particular importance is the function ~ g : [0, 1] -+ [0, 1], which will occur in the quasiregular version of the Schwarz l e m m a as well as in its m a n y applications. This function is defined as follows. For 0 < r < 1 and K > 0 we define a special function 1

~K(r) = ~/zl(g.~n(1/r) ) = ~g,n(r)

(7.44) and set ~ K ( 0 ) ---- 0 ,

~ K ( 1 ) ---- 1.

It is easy to see t h a t

~ K : [0,1] -~ [0,1] is a

h o m e o m o r p h i s m . Next we shall derive some explicit estimates for ~ g "

Recall first

t h a t by (7.24)

< for s > 1.

<

1-"

It is left as an exercise for the reader to derive from this the following

inequality (7.45)

tc*/)~, ~ " ~ ; l ( K ' / n ( t ) ) ~ ~ t a

for all t > 1 and K > 0 , w h e r e (7.46)

a=K

1/(1-~). From (7.45) it follows t h a t

raA~ ~ <-- ~ K (r) _< Anr a

holds for all K > 0 and r E (0,1). It is easy to see t h a t 0 < A _< B < co implies ~ A ( r ) <

~B(r). In particular,

~,/g(r) <_ r ---- ~,(r) <_ ~pg(r) for K _> 1. We next improve the u p p e r bound in (7.46) for K >_ 1. T h e resulting explicit bound is sharp for K = 1.

98 7.47. T h e o r e m .

(1) (2)

For n >_ 2 , K >

1, and O < r < 1

~K(r) _~ A1-c~r ~ , a = K 1~(l-n) ~ l / K ( r ) ~_ A l - ~ r ~ , fl = g l / ( n - l ) P r o o f . ( 1 ) L e t K _> 1, r E (0,1), and M ( r ) =

~oK(r ) we have M ( r ' ) = a M ( r )

modRa,,~(1/r ) . Setting r ' =

and r' _~ r. These facts follow from the definitions

(7.10) and (7.44). From the proof of 7.20 it follows that M ( r ) + log r is a decreasing function on (0,1), so that M ( r ) + logr > M ( r ' ) + l o g r ' . Let An _> 4 be as in (7.21). Since log An _> M ( r ) + log r by (7.21) we obtain Art

0 _< l o g -

r

- M ( r ) <_ l o g ~ - M ( r ' )

and, further, because 0 < a < 1 alogA'Zr - a M ( r )

~ log-~-M(r').

This inequality yields r' _< A

for r E (0,1). Because this holds for r = 0

r

and r = 1 , t o o , we have completed the

proof of (1). (2) The proof of (2) is similar. 7.48.

Exercise.

O

Applying (7.24) and 7.26(1) show that A,~ > 4. Derive from

(7.24) and 7.26 also some inequalities between the constants c,~ and wn-1 • [Hint: Note that by (5.57) # ( 1 / V ~ ) -- ~r.] 1 0)n_ 1

Find also lower bounds for A,~ in terms of

and ca.

7.49.

Exercise.

Show that

m o d R c , , ~ ( 1 / r ) and a = K U ( t - , ~ ) .

~K,~(r)

= MZl(aMn(r))

where M,~(r) =

For the proof of (7.46) the crude upper bound

"/,(8) < W,_l(lOgs) 1-" was used. Derive improved versions of (7.46) by using the two upper bounds in 7.26(1). than 7.47.)

(Note: The resulting inequality will yet be weaker

99 It is clear t h a t r c" < r 1 / K , a = K 1 ~ ( l - n )

,

for K > 1 and 0 < r < 1. This fact

t o g e t h e r w i t h 7.47 a n d t h e n e x t l e m m a , shows t h a t WK(r) < c ( K ) r 1/K for K > 1, w h e r e c ( g ) d e p e n d s only on K a n d where c ( K ) - ~ 1 as K--~ 1. 7.50.

Lemma.

F o r n _7 2 , K >_ 1, a n d a = K 1~(l-n) -- l i f t

the following

two inequalities hold:

(1)

A1 - a ~ 2 1 - a K

(2)

A 1-~ > 2 1 - ~ K -fl ~ 2 1 - K K - K .

Proof.

< 21-1/KK.

(1) It follows f r o m 7.25 t h a t (1 - a) l o g A n <

From 1-e -z_<x,

(1 - a ) ( n -

1) + (1 - a) l o g 2 .

x _> 0 , one can deduce t h a t (1 - a ) ( n - 1) ---- (1 - K ' l ( 1 - n ) ) ( n - 1) _~ l o g K .

B e c a u s e 1 - a < 1 - 1 / K we conclude t h a t

(1-a) logAn
A -a = Ap-1)

> (21-

K)-a = 21-aK-

>_ 2 ' - K K -K .

Next we shall prove a " d i m e n s i o n - c a n c e l l a t i o n " p r o p e r t y of the function

~gK, n ,

K > 0 , by finding d i m e n s i o n - f r e e m i n o r a n t and m a j o r a n t functions. 7.51. Lemma.

For K > 0 and 0 < r < 1 there exist positive numbers al a n d

a2 in (0,1) such that a I ~_ 99K,n(r) ~ a2 for all n >_ 2. In particular, al a n d a2

are independent of n . Proof. (7.52)

By 7.26(2) we have Al°gS-+lsl-<3(s)-
A=2n-lc'~'

for s > 1. Because "r is strictly decreasing we o b t a i n f r o m this

< 1 + lz-I(t/A) 1 - #-l(t/A)

100

and t

../- 1 (t) > cth 2---A " These two inequalities together with (7.52) yield for r E (0, 1) 1

1-r

_~-1

(1))

1 +/z-l(KlogT) 1 # - l ( K l o g T)

b2

where T = (1 + r ) / ( 1 - r ) . Both bounds are independent of n . In view of (7.44) we m a y choose ax = l/b2 and a2 = l/b1 .

[]

For n = 2 , K > 0 , t > 0 , let aK(t) = T ; I ( T 2 ( t ) / K ) .

7.53. E x e r c i s e .

Show

that (7.54)

a K(t) - 1 - A 2 ' A = ~K,2

Let t K = 2 ( 4 K ) - K .

Then for K_> 1 and t E (0, tK]

_<

•

I

(see 7.50). Conclude t h a t for K > 1 and t C (0, tK]

O~K(t) <-- 3--4"1 6 1 _ l / K ( 1 _ ~ ) l / K Next, applying 5.61(2) and 7.47 show t h a t for K > 1 and t > 0

aK(t ) <_ 1 6 g - U K ( 1 + t)K--1/Kt 1/K .

7.55. E x e r c i s e .

Let G be a uniform domain in R n with connected b o u n d a r y

(recall 3.8). Show t h a t if E is a connected subset of G , then M ( A ( E , OG)) > c kG(E ) where c is a constant. 7.56. E x e r c i s e .

(1) Applying the functional identities of # , one can write the

constant b2 in the proof of 7.51 also in other ways. Show t h a t 9T2

1 a2 -- b2

_

_l(

rogr)

(2) Find an improved form of 7.50 by replacing the inequality 1 - e - ~ < x in the proof of 7.50 by the b e t t e r inequality 1 - e -~ < th x (cf. 2.29(1)).

101 7.57. : R e m a r k .

7n(S) has several interesting properties

The Grhtzsch capacity

which are studied in [AVV3]. It is shown there that ~/n(1/th(a + b)) < "Tn(1/th a) + "~n(1/th b) holds for all a, b > 0. Several inequalities involving the function

99K,n(r) can also be

found in [AVV1], [AVV2], and [AVV3]. One of these is 2r ~ r ~

-

(l+r')

a+(1-r') ~

<_

;

rI

~ / 1 - r ~ a = K U(1-n)

which holds for all K > 1, r E [0,1], and n > 2. 7.58. E x e r c i s e .

Let M ~ ( r ) , r E ( 0 , 1 ) , n > 2, be as in 7.49. T h e n M2(r) =

~ ( r ) . Show that the relationship

(a)

Mn(r)M,~(V/1-r 2 ) =c

for all r E (0, 1) is equivalent to

(b)

[~K,n(r)]2+ [~l/K,n(V/1--r2)]2= 1

for all r E (0,1) and all K > 0. Recall that both (a) and (b) hold for n = 2 by (5.57) and 5.61. Next, applying 7.26 show that (a) is false for n > 3 and, therefore, also (b) is false for n > 3. 7.59. A n o p e n p r o b l e m .

Let E and F be disjoint compact sets in H ~ and

let F* denote the reflection of F in 0 H n . Consider two curve families F = A ( E , F) and F* =

"A(E,F*). It seems natural to conjecture that M(F) :> M(F*) ([BBH,

p.501, 7.57]).

The validity of this conjecture can be verified in certain particular

cases, e.g. when E and F are balls. In particular, the conjecture holds true when n : 2, as F. W. Gehring and N. Suita have independently shown to the author. Some applications of this fact are given in [LEVU]. 7.60. N o t e s .

The m e t h o d of symmetrization has found many applications in

geometry (see [BER, 9.13]) and in various branches of analysis, e.g. in the study of isoperimetric inequalities (see [PS], [BA], [HE2]), and in real analysis. O. Teichmfiller [TE] applied these ideas to geometric function theory and proved a special case of L e m m a 7.17 above. Other function-theoretic applications are given in [HA2].

102 In R 3 the eonformal capacity was studied by C. Loewner [LO], who applied his result to quasiconformal mappings. Many results of this section are connected with the fundamental results of F. W. GeAring [G1], [G2]. A multidimensional version of Teichmiilter's work on symmetrization is contained in [G1] and [S1]. See also [PS]. The literature dealing with p-capacity is vast: the reader is referred to [MK], [FR], [GOR], [MAZ2], and [STR2], [W2], as well as to the bibliographies of these works. One of the main goals of this section is to find estimates for M (A (E, F ) ) in terms of geometric quantities such as

d(F) } d(Z, F)

mAn{ d(E),

For 7.34-7.37 see [G1] and [(]7]. For 7.41 and 7.42 see [VU10] and [VU13]. The natural setup and motivation for 7.47 is the Schwarz lemma [HP], [WA], [SH], [MRV2], which we shall study in Section 11. For n = 2 Theorem 7.47 is due to O. Hfibner [HU] and the same m e t h o d appears also in [LV2, p. 64] and, in the n-dimensional context, in [AVV1]. For a different proof ( n = 2 ) see P. P. BelinskiY [BEL, p. 15]. Also 7.50 and 7.51 were proved in [AVV1]. For 7.38 see [VU10], [VU13], and [GM1]. From the vast literature dealing with condensers in the plane we mention [B], [KL], [KU], and IT, CA. III].

8.

Conformal invariants

In the preceding sections we have studied some properties of the conformal invariant M ( A ( E , F; G ) ) . In this section we shall introduce two other conformal invariants, the modulus metric

~a(x,y)

and its "dual" quantity

AG(X,y),

where G is a domain

in R ~ and x , y E G . The modulus metric # ¢ is functionally related to the hyperbolic m e t r i c

PG

if G = B " , while in the general case # a reflects the "capacitary

geometry" of G in a delicate fashion. The dual quantity Ac(x ,y) is also functionally related to

PG

if G = B ~ . For a wide class of domains in R ~ , the so-called

QED-domains, we shall find two-sided estimates for

AG(X,y)

Ix - yl

r (x, y) = min{

Oh), d(y, Oh) }

in terms of

103

8.1. T h e c o n f o r m a l

invariants

Ac a n d

#c.

If G is a p r o p e r s u b d o m a i n

of R '~ , t h e n for x , y E G w i t h x ~ y we define (8.2)

Ac(x,y ) =

inf M(A(Cz,Cy;G))

C~ ,Cy

where Cz = "/z[0, 1) and ~z: [0, 1) ~ G is a curve such t h a t z E [Yzt a n d ~z(t) --~ OG w h e n t --* 1, z = x, y . It follows f r o m 5.17 t h a t m a p p i n g s of C .

AG is invariant u n d e r conformal

T h a t is, Afc(f(x),f(y)) = A G ( x , y ) , if f: G -~ f G is c o n f o r m a l

a n d x, y E G are distinct. 8.3.

Remark.

If c a r d ( R " \ G) = 1, t h e n A c ( x , y ) - oo by 5.33. T h e r e f o r e

Aa is of interest only in case c a r d ( R '~ \ G) > 2. For c a r d ( R " \ C) > 2 a n d x, y E G , x ¢ y , t h e r e are c o n t i n u a

C,

and

Cy as in (8.2) with

C~ ~ Cy = 0 and thus

M ( A ( C z , Cy; G)) <: oe by 5.23. T h u s , if card(l~ n \ G) > 2 , w e m a y a s s u m e t h a t the i n f i m u m in (8.2) is t a k e n over c o n t i n u a Cz a n d Cy with C z Q Cy = 0.

Diagram 8.1. For a p r o p e r s u b d o m a i n G of ~ n (8.4)

a n d for all x , y E C define

#c(x, y) = inf M ( A ( C z y , OG; G)) Cz9

w h e r e t h e i n f i m u m is t a k e n over all c o n t i n u a Cxy such t h a t C~y = q[0, 1] a n d ^/ is a curve w i t h ~(0) = x and ~(1) = y . It is clear t h a t # c

is also a c o n f o r m a l invariant

in the s a m e sense as AG . It is left as an easy exercise for the reader to verify t h a t # c is a metric if c a p 0 G > 0. [Hint: A p p l y 5.9 and 6.1.] If c a p 0 G > 0 , we call # c

modulus metric or conformal metric of G .

the

104 8.5. R e m a r k .

Let D be a subdomain of G . It follows from 5.9 and (5.10)

that

#a(a,b) <_ #D(a,b)

tinct

a,b E D .

for all a,b E D and

)~c(a,b) > )~D(a,b)

for all dis-

In what follows we are interested only in the non-trivial case

c a r d ( R n \ G) > 2. Moreover, by performing an auxiliary Mbbius transformation, we may and shall assume that oo E ~ n \ G throughout this section. Hence G will have at least one finite b o u n d a r y point. In a general domain G , the values of ,~a(x, y) and p c ( z , y) cannot be expressed in terms of well-known simple functions. For G = B ~ they can be given in terms of

p(x, y) and the capacity of the Teichmiiller condenser. 8.6. T h e o r e m .

The following identities hold for all distinct x, y E B '~ : 1

1

#n"(x'y)=2'~-lr(sh2½P(x,y))

(1)

(2)

y) =

gr

=

(th-~p(x,y))'

(sh ½p( , y))

P r o o f . (1) The proof of part (1) follows directly from 7.27, (7.32), and 5.53. (2) Because the assertion is 0 N ( B ~ ) - i n v a r i a n t , we may assume that x = r q = -y

and r = t h ( t p ( x , y ) )

(see (2.25)). By a s y m m e t r y property 5.20 of the modulus

and by 5.54(1) we obtain AB.(x,y) < M(A([-el,-rel],

rel, - r e t ]

..... 1 M ( A ( [ - 1

:

½T(

[rel,e,]; B n ) )

4r=

[re,, rel]; 1

Rn))

1 :T

Hence it will suffice to prove the inequality " > " Let C z , Cy be as in (8.2) and 0 < e < ½(1 - I x [ ) . subsets E ,

F of Cx, Cy with x E E ,

Let Z 8 = E U h E ,

F 8 = FUhF

Choose compact connected

y E F and d ( E , S n - l ) = d ( F , S '~-~) = e.

where h(x) = x/Ixl 2. By 8.3 we may assume that

C ~ C l C y = 0 and hence at most one of the sets E and F can contain 0. We may assume 0 ~ F , hence F s is compact. Let S y m ( F ~) denote the set obtained from F s by spherical symmetrization in the positive Xl-axis and let S y m ( E s) be the set obtained from E 8 by spherical symmetrization in the negative x l - a x i s . By 7.17, 7.8, and 5.9 cap( R ~ \ E 8 , F 8 ) > cap( R '~ \ S y m ( E ~ ) , S y m ( F ~) )

> M(A([-le,,-rq] _

1 , [re,, re11)) - 2M(A(YI,Y2) )

105 where Y1 = [ - r le1' -re1] and Y2 -- [(1 - e)el, (1 - e)-lel] . This inequality together with 5.54(1) yields c a p ( R '~ \ E 8, F *) > r(sh 2 ½P(x,y)) - 6(e) where

6(e)

--~ 0

M(A(Cz,Cu;Bn))

as

e --* 0.

Letting

e --, 0 and applying 5.20 yields

> l r ( s h 2 l p ( x , y ) ) . Since Cz and Cy were arbitrary sets with

the stated properties, the desired inequality ),B,(x,y) > l r ( s h 2 ½P(x,y))

follows. O

From 7.26(3) we obtain the following inequality for x, y E B '~

8.7. R e m a r k .

(exercise) l r ( s h 2 l p ( x , y ) ) >_ -Cn logth l p ( x , y )

Here the identities 2ch 2A = l + e h 2 A

and sh2A = 2 c h A s h A

were applied (see

also 2.29(3)). Recall that sh 2

~p(x,y)

tx - ut 2 = (1 - Ixl~)(1 - lyl ~)

by (2.19). Similarly, by 7.26(3) we obtain also ½~(sh ~ ½p(x,y))

< '

, 4 - "ic'~t~(th2(¼P(x'Y))) < 2c~1°g th2 ¼P(x,y)

= Cn log

8.8. L e m m a .

2 th ¼P(x,y) "

Let G be a proper subdomain of R '~ , x E G , d(x) --- d(x, OG) ,

B~ -- S ' ~ ( x , d ( x ) ) , let y E B~ with y ~ x , and let r -- I x - y l / d ( x ) . following two inequalities hold: (1)

)~G(x,y) :> ~B~(X,Y) ~- 1T ~

(2)

/~a(x,Y) --< I~B.(x,Y) = "7

l

:> CnlOg--r ' _< Wn--1 log r

P r o o f . (1) By 8.5, 8.6(2), and 8.7 we obtain Aa(x,y ) >)~B.(x,y )= iv ( -

---- cn log

2

\1

1 + X/I--- r 2 r

r:

"~ > - c ,

-- r 2 }

logth l ( 2 a r t h r )

-

g

1 > ca log r

(2) The desired inequalities follow from 8.5 and (7.24).

O

Then the

106 8.9. T h e f u n c t i o n p ( x ) . (8.10)

For x E Rn \ {0, e l } , n _> 2, define p ( x ) = inf M ( A ( E , F)) E,F

where the infimum is taken over all pairs of continua E and F in R'~ with the properties 0, el C E and x,c~ C F . 8.11. L e m m a .

The i n e q u a l i t y p(x) > max{ ~(IxI), ~ ( I x - ell)}

holds for all x E R ~ \ { 0 , e l } .

E q u a l i t y hotds i f x -- s e l a n d s < O or s >

P r o o f . The proof follows directly from 7.17.

1.

[]

The main result of this section is the following theorem. 8.12. T h e o r e m .

For I x - eli _< Ix], x e R " \ {0, el}

p(x)_<2~-(Ix-~11) when I~:+~,l > 2 , (2) p(x) _<4T([z-el[) when Ixl _> 1, (1)

(3) p(x) < 2~+'~(Ix- ~ l ) The proof of Theorem 8.12 will be divided into several parts. Due to symmetry properties of the above definition (8.10) (that is axial symmetry in the x l - a x i s and symmetry in the ( n - 1)-dimensional plane xl = ½ ) it is clear that the values of p ( x ) are determined by its values in the set (8.13)

D, = ( (Xl,0, .

.,0, .x , ). : x, > ~-, ' x,,_> o } \ {e~}.

All the upper bounds (1)-(3) in Theorem 8.12 are based on Lemma 5.27 and on the functional inequalities of T(S) in Lemma 5.63. 8.14. L e m m a .

I f x C R '~ \ B n ( - 2 e l , 3 ) , t h e n

4(1:,:1- 1) > min{ Ix - ~,1, Ix - ~,I ~

P r o o f . Write x = x + 2 e l - 2 e l

and x - e l

}.

=x+2el-3el.

of cosines Ixl 2 = 1x + 2e,t 2 + 4 - 4(x + 2el) . e , , Ix-

~,12 : Ix + 2~,1 = + 9 - 6 ( x + 2el) "el •

Then by the law

107 From this we obtain 3I=1 ~ - 2[= - e~ t ~ = tx + 2e~ i ~ - 6 ~ 9 - 6 = 3.

Hence ixI > (1 + :2I x - e:12) :/2 so that ~[X -- el[ 2

]~1- 1 _> 1+

Case A. I x -

ell

V 2/

1 + ~]x-

el [2

(__ 1. Then Ixl-

1 _>

~lx - - e,I- 2

1+

> &Ix--

~

t~ .

+g

Case B. ] x - e l ] : > 1. Then I x l - 1 :>

-32 1 x - e l l l X - eli 1+

v

1 + g2 l x - e l l 2

> -

21x-ell 1+

:> ¼tX_el] ,

+ S2

since t H t/(1 ÷ V/1 + 2t2) is increasing on (0, oc) . The proof follows from the above inequalities. 8.15. L e m m a .

(1)

Let E :

[0, el] and F :

[x,(x:)] for x E R " \ B

'~ . Then

p(x) < M ( A ( E , F ) ) _< ~(Ixl- 1).

If x E R n \ B n ( - 2 e l , 3 ) , then

(2)

p(x) < M ( A ( E , F ) ) <_ 2 r ( I x - ell)P r o o f . The inequality (1) follows from 5.27. It follows from 5.63(2) that ~(u) _<

2T(2U + 2X/~ ) < 2T(2X/~ ) and hence r(¼s 2) < 2r(s). From 5.63(2) it also follows that ~-(¼s) < 27(s). In conclusion, for s > 0 the following inequality holds

~(lmin{~,~2}) < 2~(~) The proof of the inequality (2) follows from (1), the above inequality, and from Lemma 8.14.

O

108 8.16. E x e r c i s e .

1 i7r let xt = q 4 - t ( ( c o s a ) q + ( s i n a ) e , ~ ) ,

For 0_< a <

t >0.

For fixed a and arbitrary t > 0 show that p(:~,) _< ~(t cos ~ ) .

8.17. P r o o f o f T h e o r e m 8.12(1).

Let Y = { x E S n - 1 ( - e l , 2 ) : x l = ½}.

Note that d ( e l , Y ) = v ~ . It suffices to prove the result for x E D1 \ B ' ~ ( - e l , 2 ) •

Case A. Choose

Ix- ell <_v~. x E sn--l(--el,2

) N D, w i t h

t'~ - e l t

= tx - e l i .

1

I

D1

S n - - 1 ( e l , IX

2.

ozo 1

- 2 1

rxo - zol = ~ - 4 s i n

(~Z)

lz0 - x l = ½ ÷ 2sinfl

Diagram 8.2. Then 1 5 - el I = 4 sin ½fl where fl is the acute angle between the segments I - q , q} and [ - e l , ~ ] . Let xo = ~i ( e l - en). Then ] 5 - Xo[~ = (2sinfl + ½)2 + (½ _ 4sin21~fl)2 = - 21 + 12 s i n 2 ½fl 4- 2 sin fl -- ½(1 4- A) where A = 2 4 s i n 2 ½ f l + 4 s i n f l . Hence I ~ - x01 Ix0 - q j

1 =

A

1 + v~+ A

109 It is left as an exercise for the reader to show that A

> le - el I = 4 s i . ½8

1+ ~/I+A holds for all x in D, \ B ' ~ ( - e l , 2 ) .

-

Let E1 = [xo,ell , E2 = [0, xo], and

F = { Xo + t(~ - Xo) : t _> 1 }. By 5.27 and the last two inequalities we obtain M ( A ( E j , F ) ) < r ( j-~ -_xol -

1) < r ( l ~ - e t l ) .

I~0-ell

-

Because 1 7 - ell = I x - e l l , we obtain by 5.27 and 5.9

p(x) <_ M(A(E1 U E 2 , F ) ) < 2~(1~- ell) as desired. Note that the condition Ix - e l i _< V~ was used only for the construction of 7 .

Case B. I x - e l l >_ V/-~. It is easy to see that in this case for x E D1 I~1- 1

v~-

Ix - el I

v~

1 :>- 1 -4

and hence by 8.14(1)

p(x) <_f(lx I - i) <_ r(¼1x - ell). Because T(S) <_ 4r(4s) by 5.63(2) we obtain the desired inequality also in Case B. [] 8.18. E x e r c i s e . Show that A/(1 + v/1 + A) > 4sin ½8 in the above proof.

For x E D I \ B n ( - 1 7 ( e l + 3 e , ~ / t a n a ) , 3 / ( 2 t a n a ) ) 0 < a < l~r, the following inequality holds 8.19. T h e o r e m .

and

p(x) <_4 r(2(sin a)lx - ell) • P r o o f . Let xo = ~1 ( e l -

en/tan a ) .

{xo + t ( x - xo): t > 1 }, Fj = A ( E j , F ) ,

Let E1 = [xo,ell , E2 = [0, Xol , r = j = 1,2. It follows from 8.15(1) that

M(Fj) < r ( Ix = x ° I -

~,lxo-

ell

1)

110 for j = 1, 2. Because of the choice of x , 8.15 yields T ( I x - xol

1) < 2 v ( I x - e l l

I~o- ~I

-

t~o

"~ _-- 27((2sinc~)i x _ ell)

~iJ

-

These inequalities together with 5.9 yield the desired b o u n d

p(x) <_M ( A ( E 1 U E 2 , F ) ) 8.20. a

=

~1

Proof of Theorem

and x0

~1( e l

:

--

< 4v((2sina)lx-

8.12(2).

ell) • []

We m a y assume t h a t x E D1 \ B ~. Let

en/tan a ) . As in the proof of T h e o r e m 8.19 we get

p(x) ~ 4r(~/~lx- ell) < 47(Ix- ell) as desired. 8.21.

[] Proof

of Theorem

8.12(3).

We m a y assume t h a t x E D1 • If x,~ k

1V~(1 -- X l ) , t h e n L e m m a 8.19 with ~ = 7r/6 yields O0

(8.22)

p ( x ) <~ 4 T(IX -- 611),

If x . < 1V/3(1 - x l ) , choose E E D1 with 51 : Xl and 5,~ : l x / 3 ( 1 - - X l ) . 1 Let Xo = ~(61

-

-

%/3en) ,

E = [0, x0] U [x0,el], and

F:[x,~]U{xo+t(~-Xo):

t > 1}.

Diagram 8.3. By (8.22) and 5.53

p(x) <_ M(A(E,F)) < 4 r ( [ ~ - ell) + M(A([x,~],E)) _< 4T(Ix-- e,I) + M(F) _~4r(lx - ell) +~(2) = 4 T ( l x - ell) + 2"-1T(3) --< 2 m a x { 4 , 2 ' ~ - l } T(I x -- ell)

< 2~+1~(Ix - ell) where Y = A ( [ x , ~ ] , S'~-l(~,r~-ell)).

[]

111 For x E R n \ {0} we denote by r~ a similarity map with rx(0) = 0 and

rz(x) -- e l . Then it is easy to see that lr~(y)-ell = Ix-yl/Ixl. It follows immediately from the definitions (8.2) and (8.10) that (8.23)

AR,\{o} (x, y) ----min{

p(rz (y)), p(ry(x)) }.

Next we deduce the following two-sided inequality for Art-\{o} (x, y) .

For distinct x, y E R '~ \ {0} the folIowing inequaIity holds

8.24. T h e o r e m .

1 <_ ~ R o \ ( 0 } ( x , y ) / T ( l x - y I / m i n { I x t , tyt}) < 4 . P r o o f . We may assume that Ixl _< lYl. Denote G -- R " \ {0}. We prove first the lower bound. By (8.23) and 8.11 we get

Ac(x,y) >_min{ r(Ir~(y) - exl), r(Iru(x) - eli) } = ~ ( m ~ x ( Ix - y l / I x l , Ix -

yf/lul

}) = T(I~ - y l / I x l ) ,

which is the desired lower bound. For the proof of the upper bound let V be the (n - 1)-dimensional plane orthogonalto

[0, x] at ½x and let H 0 , Hz be the components of R ~ \ V ,

x E H~.

Consider two cases.

Case Z. y E H~. Because lYl -> Ixl it follows from 8.12(2) that

:~c(.~,y) <- p(r~(y)) <_ 4T(l~ Case

- yl/Izl) •

B. yEHo. Let E, =[0,½x] , E2 = [ ½ x , x] , and F = {1~ + t(y - ½~) :

t > 1 }, r i -- A ( E i , F ) ,

j = 1,2. Then by 5.27 M ( r i ) < r(21Y~1½xl

1)

for j = 1,2. Because IY - ½xl :> ½V/31Yl and IYt -> Ixl we obtain

~a(x,y)<_

M(rl)+

M(F2)<

2r (v~I~ I

1)

From 5.63 it follows that r(s) _< 2 r(4s) and hence the above inequalities imply

,~G(x,y) < 4 r ( 2 ( V / 3 - - 1 ) ~ ) as desired.

O

_<4 r ( ~ )

112 The above results provide us with some efficient estimates for ) ' c , which we now give. 8.25.

Corollary.

Let G be a p r o p e r subdomain of R '~, x and y distinct

points in G and m ( x , y ) = min{d(x), d(y)}. Then

_< inf

4T(f

zEaG

Proof.

-yl/m(x,y)).

The first inequality follows from 8.5. For the second one fix z0 E OG

with m ( x , y) = d({x, y}, {zo}) • Applying 8.24 to R n \ (z0} yields the desired result. O We next show t h a t 8.25 fails to be sharp for a J o r d a n domain G in R " . For t e (0, 1) consider the family Gt = B n ( - e l , 1)U B n (el, 1)U B n (t) of J o r d a n domains. T h e n by 8.25 ~a,(-e,,e,)

< 4T(2)

for all t E (0, ~) . But this is far from sharp because in fact A c , ( - - e , , e , ) <_ M ( A ( [ - 2 e , , - e l ] , / l\l--n __~ W n _ i ~ l o g ~ ) as t --~ 0.

[el,2e,];Gt)) ' 0

However, for a wide class of domains, which we shall now consider, the

u p p e r b o u n d in 8.25 is essentially best possible. 8.26.

QED domains.

A closed set E in ~ n

is called a c-quasiextrernal dis-

tance set or c-QED exceptional set or c-QED set, c E (0,1], if for each pair of

disjoint

continua F1, F2 c R n \ E M ( A ( F , , F 2 ; R '~ \ E)) > c M ( A ( F 1 , F 2 ) ) .

(8.27)

If G is a domain in R,'~ such t h a t R ' ~ \ G domain.

If e = 1 then the set E

is a c-QED set, then we call G a c-QED

is called a null set for eztremal distances or a

NED set. 8.28. E x a m p l e s .

(1) The unit ball B n is a I _ Q E D set by [GM1] or by the

above L e m m a 5.22. (2) If E is a compact set of capacity zero, then E is a 1 - Q E D set. For instance all isolated sets axe 1 - Q E D sets. T h e class of all 1 - Q E D sets contains all closed sets in R '~ of vanishing ( n (3) B 2 \ [ 0 , e l )

1)-dimensional Uausdorff measure (see IV3], [GM1]).

is not a c-QED set for any e > 0 .

113 8.29. T h e o r e m .

Let G be a c-QED domain in R n. Then

Ac(x,y) _> ~ ( s ~ +2s) _> 2~-"~ ~(s) where s = Ix - Yl/ min{d(x), d(y) ) . P r o o f . Let C , and Cy be connected sets as in (8.2) with x E C , and y C Cy. Let r l -- A(Cx, Cy;G)

and F2 = A ( C , , C y ) .

We may assume that d(x, OG) <

d(y, OG). Fix u e C~ and v e Cy with I x - u I = d(x, OG) and I Y - v l = d(y, OG) >_d(x, OG). Because l u - v I < l u - x ] + ] x - y ] + l y - v l we obtain by 7.26 and 5.63(1) M(F,) _> CM(F2) >

_> C T

c T ( I x - Yl lu - vl Ix ulty vl)

I~-

~l l y -

~1 I~ ~---~-

> ~ ~(~ + 2s) > ~ ~(4~ ~ + 4~) > 2 ~ - ~ ~(~) as desired.

C3

It should be noted that the lower bound of 8.29 is very close to that of 8.24 ; in fact it differs only by a multiplicative constant. In the next few theorems we shall give some estimates for the conformal metric t t ~ . 8.30. L e m m a .

Let G b e a p r o p e r s u b d o m a i n

of R n , s E ( 0 , 1 ) , x, y E G. If

kc(x,y) < 21og(1 + ~), then (1)

~G(x,y) < -

~(t h ( % ( x , y1) / ( 1 - ~ ) ) J )

Moreover, there exist positive numbers bl and b2 depending only on n such that

,G(x,y) _< b,kc(x,y) + b~

(2) for aI1 x, y E G.

P r o o f . ( 1 ) C h o o s e a quasihyperbotic geodesic segment Jc[x,y] connecting x to y and let z E JG[x,y] with k a ( x , y ) = 2 k c ( x , z ) = 2kG(y,z ) • Then by (3.4)

Ix - zt

j~(x,z) = tog(1 + min(d-(~):-d(z)}) -< kc(x'z) <- log(1 + s)

114 and hence

x C B'~(z, sd(z)). Let Bz = B'~(z,d(z)). By 3.7(1), (3.6), and (3.4) we

obtain

Ix - zJ

kt~(x,z ) <_log(1 + d(z)---ix'-z I) <- log 1 <--log

-

(

1-s

1+

Ix - z I ~ d(z)

]-

(1

Ix - zl + ( 1 - s)d(z)')

1 <--ja(x,z)

1

<

1-s

1-s

-

kG(x,z) .

Because of the symmetric choice of the point z, we get a similar upper bound also for

kB~ (z, y) . Hence %. (~, y) < %. (~, z) + % . (z, ~) 1 < --(kG(x 1-s

1 - -,z) + kG(z,y)) -1-s

-

Denote by PB, the hyperbolic metric of

k~ (z, y ) .

Bz (see 4.25). Now by 3.3 and the above

results we get

p~. (z, y) < 2 %. (z, y) < - --

2

k~(~, y)

- - 1 - - s

and hence by 8.5, 8.6(1), and 5.53 1 1

1

= ~/(th where p

= PB.

t h ( k c ( x , y)/(1

•

e Ja[x,y} with xl = x , xv+l = y and ka(xj,xj+l )=21og(l+s) for j = l , . . . , p - 1 and ka(xv,xp+l ) < 2 1 o g ( l + s ) and p < 1 + ka(x , y)/(2 log(1 + s)) (cf. the proof of Lemma 4.9(1)). Then by part (1) (2)

Choose points x l , . . . , x p + l

P

jml

where

b2 --- ~/(1/th[(21og(1 + s))/(1 - s ) ] ) .

b~/(2 Iog(1 -t- s)) follows.

The desired result with

bl

:

[]

It should be observed that Lemma 8.30(2) is a generalization of the upper bound in (7.31) to the case of an arbitrary domain. The lower bound in (7.31) will next be generalized to the case of domains with connected boundary.

115 8.31. L e m m a .

Let G be a domain in R n such that OG is connected. Then

for all a, b E G , a ¢ b , (1)

#G(a,b) >_ r(4m 2 + 4 m ) _> cnjG(a,b)

where c,~ is the constant in 5.28 and m --- min{d(a), d ( b ) } / t a - b I . lf, in addition, G is uniform, then (2)

IzG(a, b) > B kG(a , b)

for all a,b E G . P r o o f . Statement (1) follows from 7.38 and (8.4), while (2) follows from (1) and 3.8.

IS]

The above results in 8.29 and 8.31 are invariant under similarities but not under ~(Rn).

This is an aesthetic flaw; since AG and /zG are conformal invariants one

would naturally expect conformally invarlant results. Next we proceed to give bounds for AG and #G in terms of conformally invariant majorant/minorant functions. For distinct a, b, c, d in R '~ let (8.32)

m ( a , b , c , d ) = max{ [a,b,d, c l , [a,c,d,b] } .

If G C R '~ is a domain with card(R n \ G) > 2 then let (8.33)

mG(b,c ) ----s u p ( m ( a , b , c , d ) : a,d e c3G ) .

It is clear that m is symmetric, that is, (8.34) and also ~ ( R (8.35)

re(a, b, c, a) = re(a, c, b, a) = re(b, a, d, c) '~) -invariant, that is, my(a, b, c, d) = m ( f a, fb, f c, f d) = m(a, b, c, d)

for all f E ~¢M(R n) (cf. (1.28)). For x, y C R n \ { a )

( a E l ~ n)

Ix - yl

(8.36)

m(a, x, y, oo) = min{ Ix - a I, lY - al } "

It follows from (8.36) that (8.37) for all x, y E G

3"G(X,y) = log(1 + m G ( x , y ) ) , G = R n \ (a} , where JG is as in (2.34).

116

8.38. T h e p o i n t - p a i r

invariant

invariant s y m m e t r i c function

rnG

m c . Next let us consider the conformally

for an a r b i t r a r y domain

G

c

R~

with

c a r d ( R n \ G) > 2. The following properties are immediate:

(1) G1 c c~ ~ d ~, y e Cl ~ (2) For a fixed y e G ,

"~c~(~,Y) > "~G: (z,y).

m c . ( x , y ) -+ 0 iff x ~

y and m G ( x , y ) --+ CC iff

x~OG.

(3) m~(~,u) >_ q(aC)q(~,u)

(4)

r n c ( x , y ) <_ q(aG) q ( z , y ) / q ( ( z , y } , a G ) 2 .

8.39. L e m m a . Proof.

p(b,c) = l o g ( l + rnB.(b,c))

for b,c E B ~ .

By ~ M ( B n ) - i n v a r i a n c e we m a y assume b = - t e a

=-c.

Then p(b,c)=

2log[(1 + r ) / ( 1 - r)] or, equivalently, r = th ¼P(b,c). For all a , d e OB ~

m ( a , b , c , d ) <_

21b - c I (l-r) 2

4r (l-r) 2

m(-~l,--rel,rel,el)

and hence rnB. (b, c) =

4 th ¼P(b, c) = e p(b'c) - 1. (1 - th ¼P(b,c)) 2

T h e similarities between re(a, b, c, d) and s(a,b, c, d) (see (3.21)) should be observed. In fact one could use s(a, b, c, d) instead of re(a, b, c, d) and prove analogous estimates. Let a and d be distinct points in ~ n 8.40.

and D = ~ n \ { a , d } .

1 < AD(b,c)/T(mD(b,c))

Theorem.

<_ 4

for distinct

b,c C D =

R ~ \ {a, d } P r o o f . The proof follows readily from (8.35) and 8.24. 8.41.

Corollary.

[]

L e t D c R "~ be a c-QED domain with c a r d ( R '~ \ D) > 2.

T h e n for distinct x, y E D

2 ~ - ~ ( m ) _< ~ ( m : + 2m) <_ ~.(~,y) _< 4~(m) where m ---- rnD(X , y ) . Proof.

The proof follows from 8.29, 8.25, and 8.40.

[]

117

8.42. Exercise.

Let b,c E B n • Show t h a t

p(b,c) < log(1 + p(b, c) > log (1 +

(min{1 -Ibl, 1

J ' 2 Ib - cl

m i n { 1 - IbI,1 - I c l } (I + m i n { l -

Ibl, 1 - I c I } ) ] "

Let d E S n-1 . Show t h a t

p(b, c) >_ log (1 + 8.43.

Remarks.

21b-el Ib -- JI ;

J

For n -- 2 an explicit expression for the function p(x) can

be deduced from [KU, T h e o r e m 5.2, p. 192]. This explicit expression is a real n u m b e r determined by certain elliptic integrals with a complex argument. Because of this fairly complicated definition it is difficult to see how the exact value of p(x) changes with x or, say, w i t h the angles a between [0,x] and [0, eli and ~ between [0, eli and [ e l , x ] , respectively. In [VU13, 4.3] it was conjectured t h a t for n = 2 the constant 4 in 8.24 can be replaced by a smaller one, c = 1.1712 . . . .

#(1/v/3)/#(1/x/~),

which would be sharp as shown in [VU13, 4.3]. A weaker version of this conjectured two-dimensional result with c 2 in place of c was established in [LEVU]. Also some bounds for AB2\{0} (x, y) were found in [LEVU]. 8.44. E x e r c i s e .

Mori's ring RM,2(a, fl ) in R 2 has two c o m p l e m e n t a r y com-

ponents C 1 = { t e l : t _ > 0} and Co = { ( c o s ~ , s i n ~ ) E R 2 : ~ r - a

< ~ < 7r+fl},

0 < a _< ~ < r . Find an expression for cap RM, 2(a,~) by m a p p i n g R 2 \ C1 conformally onto I-I2 . (For n = 2 p((½, y)) can be

expressed in

t e r m s of the capacity of

Mori's ring, see [KU, T h e o r e m 5.2, p. 192].) 8.45.

[AVV3].

Remark.

One can show t h a t

AB.(x,y) 1/(l-'~) is a metric on B "

It is t e m p t i n g to conjecture t h a t for all proper s u b d o m a i n s

G of R ~,

Ac(x,y) 1/(1-'~) is a metric. Even the particular case n = 2, G = R 2 \ {0}, is open. As shown in [LF2] Aa(x,y) -1/n is always a metric. Next we shall find an u p p e r bound for the function aK, "(t) defined as (8.46)

aK,,~(t ) = rgl(r,~(t)/K) ,

t > O, K > O.

It is easy to show using the basic functional identity 5.53 t h a t

aK,n(t ) = 1 -B2B 2 ;

B=991/K,n(1/1VFf--'~) "

118 For n = 2 we can go one step further using the identity 5.61(2) and obtaining (7.54) as a result. Further from (7.54) one can easily deduce that ~K,2(t) has a majorant of the form A t 1/I~ , A constant as we have pointed out earlier. Although the multidimensional analogue of 5.61(2) is false (recall 7.58), we nevertheless can find a similar majorant for ag,•(t ) valid for all dimensions n _> 2. 8.47. T h e o r e m .

For n > 2 , K > 1, and t E (0, 22-3K) the following inequal-

ity holds T Z I ( v n ( t ) / K ) ~_ 4 3 - 1 / K t l / K .

Proof.

Let

and b = log0 + 2(1 +

x --

By 7.26(3)

we obtain c.b _< ~.(t) -- K ~ . ( x ) _< c . K , ( 1 + 2(1 - ~ - 4 - ~ ) / x ) and further x < 4#-1(b/K) - (1 - # - l ( b / K ) ) 2 " The inequality l o g ( l / r ) < #(r) < log(4/r) (cf. (5.58)) shows that e - u < I~-l(u) < 4e - u for u > 0. Therefore i ~ - l ( b / K ) < 1 for t E (0,22-3K) and also x <_ 4 3 ( t _ ~ 2 ( l t ~ ) )

holds for t E (0,22-3K) as desired. 8.48.

Notes.

1/K < 4 3 - 1 / K t l / K

[]

This section is taken from [VU10] and [VU13]. NED sets in

the complex plane were introduced by L. V. Ahlfors and A. Beurling [AB]. J. V£is£1£ IV3] studied NED sets in n - s p a c e and finally F. W. Gehring and O. Martio [GM1] introduced QED sets.

See also V. V. Aseev and A. V. Syehev [ASY] as well as

J. V£is£I£ [V12]. The conformal metric ~ c has been studied by I. S. G£1 [G.~L] and T. Kuusalo [K1]. Its dual invariant AC was introduced by J. Ferrand [LF2]. For n -- 2 the function p(x) is closely connected to a modulus problem of O. Teichmiiller and the shape of the extremal ring for p(x) has been thoroughly examined (see G. V. K u z ' m i n a [KU, p. 192, T h e o r e m 5.2]). 8.49. A n a p p e n d i x t o S e c t i o n 8.

We shall give here an alternative proof for

T h e o r e m 8.12(1) which is slightly simpler than the proof given in 8.17. This proof is due to M. K. Vamanamurthy. First a lemma is needed.

119 8.50. Lemma.

I f t > 0 and s = t 2 / ( 1 + V/i-+ t 2 ) ,

then the inequality

28(1 + X/1 + 1/8 ) > tv/'2 holds. P r o o f . The assertion is equivalent to v~t2 1 + ~

(

1+

W/

1 + x/-i-÷ t 2 ) t2 _>t,

1+

or to v~(t+V/t 2+1+

l~--+--fl) > 1.

1 + VII + t 2

This is equivalent to f(t) =

v/2(t+~~/l+ I ~ T - ~ ' ) >_ 1. 1+ v/l+t 2

But here the left side f(t) >

V ~ ¢/1 + t 2

v / l + v/l + t 2

v~u

- - -

> 1

v/l + u s -

since u / x / l ÷ u 2 is increasing on [1,oo) and u = ~ / l + t 2_> 1. 8.51. A s e c o n d p r o o f for T h e o r e m 8.12(1). have x = x + e l - e l

and x - e l

=x+el-2el.

[]

For x e D1 \ B n ( - e x , 2 )

we

Hence

12:12 - 12: ÷ ~xl 2 ÷ 1 - 2(x + el) .el ,

12:-ell ~= [ x + e 1 1 2 + 4 - 4 ( x ÷ e l ) ' e l . These

inequalities yield

212:12 - 1 2 :

- ~,12

=

[2: -t- ell 2 -- 2 _> 2.

Thus

&12: - ell 2 = t 2 and hence 2 12:1- 1

-

-

12:1~ 1 > 12:1 ÷ 1 -

½[2: - e l i 2

_

t2 --8.

-

1 + v/l+ ½[2:- e112

1 + V/1Wt 2

By 5.63(2) and 8.50 T(12:I- 1) < 2 T(2,(1 + v ' l + 1 / 8 ) ) < 2 T ( t v ~ )

as desired.

[]

= 2 T(12: - ell)

12:12 - 1 _>

Chapter III QUASIREGULAR MAPPINGS

The s t u d y of quasiconformal and quasiregular mappings in this and the following chapter will be based on the transformation formulae for the moduli of curve families under these mappings.

In most cases it will be enough to make use of these

t r a n s f o r m a t i o n formulae specialized to the conformal invariants /z~ and AG . These special cases of the general t r a n s f o r m a t i o n formulae are convenient to use because they together with the results of Section 8 provide immediate insight into some relevant geometric quantities. In the case of the conformal (pseudo)metric

#c

the t r a n s f o r m a t i o n formula

f: G ~ f G C R n is a Lipschitz m a p p i n g between the (pseudo)metric spaces (G, #G) and (fG, tZfG). From this result and a similar re-

reads: a quasiregular m a p p i n g

sult for the conformal invariant ,ka we derive several distortion and growth theorems for quasiregular mappings. To this end we shall make use of some results from C h a p t e r II t h a t will enable

I~a(x,y ) and AG(x,y ) . Except for the special case G = B " formulae for I~G(X,y) and Aa(x , y) are unknown, but one can us to find simple estimates for the functions

give u p p e r and lower bounds for t h e m in terms of Iz -

Yl

rG(X'Y) = min{d(x), d(y)} '

d(x) = d ( x , 0 e )

for a wide class of domains G (see 3.8 and 8.26). When G -- B '~ the t r a n s f o r m a t i o n formulae for # c and AG yield two variants of the Schwarz l e m m a (see 11.2 and 11.22, respectively). A central t h e m e of this chapter is a circle of ideas centered in the Schwarz l e m m a and its various generalizations, including a s t u d y of uniform continuity properties of qr mappings. In particular, we shall also discuss some properties of normal quasiregular mappings.

121

9.

Topological properties of discrete open mappings

In this section we shall survey some topological properties of discrete open mappings. A thorough discussion of this topic, including the definition of the degree of a mapping, requires machinery from algebraic topology (see [RR]). In this section no proofs will be given. 9.1. D e f i n i t i o n . The set T '~ consists of all triples (y, f, D ) , where f : G --* R n is a continuous mapping, G C R ~ is a domain, D is a domain with D c G and y E ~n \ fOD. 9.2.

Lemma.

There exists a unique function /z : T n ~

Z , the topological

degree, such that (1)

V ~-~ it(V, f , D )

is a constant in each component of R= \ f O D .

(2)

I i t ( y , f , D ) I = 1 if Y C f D

(3)

it(v, id, D) = 1 if V E D and id is the identity mapping.

and l i D is one-to-one.

(4) Let ( y , f , D ) E T "~ and D 1 , . . . ,Dk

be disjoint domains such that k ( y , f , Di) E T n and f - l ( y ) M D c U i = i D i . Then k

it(y, f , D) = ~

it(y, f , D i ) .

i=1

(5)

Let ( y , f , D ) , ( y , g , D ) E T ~ be such that there exists a homotopy ht: D --+ R n , t E [0, 1], with ho = l i D ,

h 1 =

g l D , and (y, h t , D ) e T ~ for a11

t E [0,1]. Then l t ( y , f , D ) = i t ( y , g , D ) . 9.3. L e m r n a .

(1) If ( y , f , D ) E T ~ and y ~ f D , then i t ( y , f , D ) = 0 .

(2) If f is a c o n s t a n t c, then # ( y , f , D ) = 0 for all y # c. (3) If f: D ~ R '~ is differentiable at xo E D and J f ( x o ) = d e t f ' ( x o ) # O, then there exists a neighborhood U of xo such that (y, f, U) E T n and #(y, f, U) = sign J f ( x o ) for y E f U .

122

It follows from 9.3(3) t h a t if f

is a reflection in the plane x,~ -- 0 , then

/z(y, f , B '~) -- - 1 for y E B n . We next extend the definition 1.7 of a sense-preserving C 1- h o m e o m o r p h i s m . 9.4.

Definition.

A m a p p i n g f: G -~ R n is called sense-preserving (orien-

tation-preserving) if /z(y, f , D) > 0 whenever D is a domain with D C G and y E fD\fOD.

If # ( y , f , D )

< 0 for all such y and D , then f is called sense-

reversing (orientation-reversing). Reflection in a plane and inversion in a sphere are sense-reversing mappings ([RR,

pp 137145]) 9.5. L e m m a .

If f

Let f: G-+ R~ and g: f G - ~ ~ n be mappings and set h = g o f

.

and g are both sense-preserving or both sense-reversing, then h is sense-

preserving. If one of the maps f and g is sense-reversing and the second is sensepreserving, then h is sense-reversing. 9.6. R e m a r k s .

The approach to the degree theory in [RR] is based on algebraic

topology. An alternative approach can be based on Sard's t h e o r e m and on approximation of continuous functions by C °°-functions, for which the degree #(y, f, D) can be defined as the s u m of the signs of the Jacob ians, evaluated at the points of D n f - 1 (y). See [DE], [HEI], [R12]. 9.7.

Lemma.

Let ( y , f , D )

and ( y , g , D ) E T ~ be such that flOD = glbD

a n d eo ~ f D U g D . Then #(y, f , D) = #(y, g, D ) . For 9.5 see [V4] and for 9.7 see [RR, pp. 129-130]. T h e a s s u m p t i o n oo ~ f D U g D in 9.7 cannot be dropped, as the example D = B '~ , f = id , and g an inversion in S ~-1 , shows. T h e branch set B f of a m a p p i n g f: G -+ R.'~ is defined to be the set of all points x C G such t h a t f is not a local h o m e o m o r p h i s m at z . is a closed subset of G . We call f open, light if f - l ( y )

It is easily seen t h a t B f

open if f A is open in R '~ whenever A C G. is

is totally disconnected for all y E f G , and discrete if f - l ( y )

is isolated for all y E f G . The next l e m m a is a f u n d a m e n t a l property of discrete open mappings (see A. V. Chernavski~ [CHE1], [CHE2] and J. V£is£1£ [VS]).

123

9.8.

Let f: G --* R '~ be discrete open. Then d i m B f -: dim f B f

Lemma.

dim f - I f B/<_ n - 2 , where dim refers to the topologlcal dimension. 9.9. :Remarks.

It is clear t h a t G \ B f is open, and f r o m 9.8 and a well-known

n o n - s e p a r a t i o n p r o p e r t y of sets of dimension <_ n - 2 (see [HW, p. 98]) it follows t h a t G \ B I is a domain. Stoilow's theorem (see [LV2]) implies t h a t B f consists of isolated points for n -- 2 . In the multidimensional case n :> 3 B f never contains isolated points, as one can show by applying some properties of covering mappings ( m o n o d r o m y theorem). Let G C R n be a domain. We denote by J(G) the collection of all subdomains D of G with D c G . 9.10. Defin ition.

Let f : G --~ R n be discrete. Fix x E G and a neighborhood

V E J(G) of x such t h a t {x} -- U A f - l ( f ( x ) ) . The n u m b e r ~ ( f ( x ) , f , U ) is denoted by i(x, f) and called the local (topological) index of f at x . (Exercise. Making use of 9.2(4) show t h a t i(x, f) is independent of the neighborhood of x and has the required properties.) Now let f : G -+ :Rn be discrete open.

It follows from 9.8 t h a t

G \ By is

connected. Hence i ( x , f ) has a constant value, either +1 or - 1 , in G \ By. In the first case f is sense-preserving, and in the second case sense-reversing. In b o t h cases we have by 9.2(4) if D E J ( G ) ,

y E f D \ f O D , and D A f - l ( y ) = { x l , . . . , x k } k

(9.11)

it(y, f, D) = ~

i(xj, f ) .

j=l

A domain D C J(G) is said to be a normal domain of f : G--~ R n if fOD =

O f D . A normal neighborhood of x is a normal domain D such that D A f - l ( f ( x ) ) =

(x). It follows from 9.2(1) t h a t #(y, f, D) is a constant if D is a normal domain of f and y E f D .

This constant is denoted # ( f , D ) .

f, yE fD,and

f-l(y)__{xl,...,xk}.

Let D be a normal domain of

It follows from (9.10) t h a t k

i(xj,f).

.(:, D) -j=l

9.12. Exercis e . always true.

If f : G - - * R

n is open and D E J ( G ) , t h e n

O f D c fOD is

124

f open

Diagram 9.1.

In classical function theory (see [BU, p. 84},

[WHel) the

local topological index

is usually called the winding number of a point. We shall next list several topological results about discrete open mappings without proofs. The proofs of Lemmas 9.13-9.15 are given in [MRV1]. 9.13. L e m m a .

Suppose that f: G ~ R n is open, that U c R n is a domain,

and that D is a component of f - l U

such that D E J ( G ) . Then D is a normal

domain, f D = U , and U E J ( y a ) .

If f: G -~ R '~ , x E G , and r > 0, then the z-component of f - l B ' ~ ( f ( x ) , r )

is

denoted by U ( x , f , r) . 9.14. L e m m a .

Suppose that f: G --~ R n is a discrete and open mapping. Then

l i m r - . o d ( U ( x , f , r ) ) = 0 for every x E G . If U ( x , f , r ) E J ( G ) , then U ( x , f , r )

is a

normal domain and f U ( x , f , r) = B n ( f ( x ) , r) e J ( f G ) . Furthermore, for evezy point x E G there is a positive number az such that the following conditions are satisfied for O < r < az :

(1)

U(x, f , r )

(2)

U(x,f,r) = U(x,f, az)M f-lB'~(f(x),r).

(3)

O U ( x , f , r ) -- V ( x , f , Crz) N f - l S n - l ( f ( x ) , r )

is a normal neighborhood of x .

(4) ~'~ \ v(x, f, r) (5) ~ \ ~(x, f, r)

if r < g z .

i~ ~onnected. is connected.

(6) If 0 < r < s <_ a z , then U ( x , f , r ) C U ( x , f , s ) ,

and U ( x , f , s ) \ - U ( x , f , r )

a ring, i.e. its complement has exactly two components.

is

125

If f : G - ~ R

n, ACR

n and yER

n,denote

g ( y , f , A ) = card(A A f-l(y)) , N ( f , A ) = sup{ N ( y , f , A )

: y C R 'z } ,

N(f) = N(f,G) . Here N(y, f, A) is called the multiplicity of y in A and N ( f , A) the maximal mul-

tiplicity of f in A . 9.15.

Suppose that f: G --* R n is sense-preserving, discrete, and

Lemma.

open. (1)

If D E J ( G ) , then N ( y , f , D )

<_ # ( y , f , D )

N ( y , f , D ) = t t ( y , f , D ) for y E R ' ~ \ f A ,

for a11 y E R ~ \ f O D and

A=aDU(DMBI).

(2)

If D is a normal domain, Y ( f , D ) = # ( f , D ) .

(3)

If A C G is compact, N ( f , A ) < 0o.

(4)

Every point x E G has a neighborhood V such that if U is a neighborhood of x and if V C V , then N ( f , V ) = i ( x , f ) .

(5)

x e BS i ~ i(x, f ) >_ 2 .

It follows f r o m 9.15(4) t h a t the local index i(x, f) of a s e n s e - p r e s e r v i n g discrete o p e n m a p p i n g f can be defined in terms of the m a x i m a l multiplicity of f as follows

i(x, f ) = lim N ( f , B" (x, r ) ) .

(9.16)

r ---*0

A trivial e x a m p l e is the f u n c t i o n g: B 2 --~ B 2 , g(z) = z 2 with i(O,g) = 2. 9.17. Remark.

Let f : G -* R '~ be continuous, A j C R '~ , j = 1 , 2 , . . . .

Then

one c a n show t h a t N ( y , f , U A y ) _< ~

N ( f , UAj) < E If A is a Borel set in G , then N(y, f , A ) 9.18.

An open

problem.

g(y,f, Aj),

N ( f , Aj) .

is m e a s u r a b l e (cf. [RR, pp. 216-219]).

Let f : G --* R n be discrete open, x0 E G ,

t E

(0, d(xo,OG)) , and assume t h a t f S ~ - l ( x o , t ) = O f B ~ ( x o , t ) , t h a t is, B~(xo,t) is a n o r m a l domain.

A s s u m e , further, t h a t

B S N S'~-l(xo,t) = 0 a n d n > 3.

Is it

t r u e t h a t f]B'~(xo,t) is o n e - t o - o n e ? For n = 2 we have the obvious c o u n t e r e x a m p l e g: B 2 --~ B 2 , g(z) -- z 2 . This p r o b l e m is given in [BBH, p. 503, 7.66].

126

9.19. Path

lifting.

Let f : G -+ R'~ a n d let 13: [a,b) ~ A n be a p a t h and let

x0 E G be such t h a t f ( x o ) - - 13(a). A p a t h

a: [a,c) -4 G is said to b e d

maximal

lifting of 13 starting at xo if: (1)

~(a) = xo-

(2)

foo~=131[a,e ) .

(3)

If e < c' < b, t h e n there does not exist a p a t h c~= c~'l[a,e ) a n d f o ~ '

c~': [a,c') -4 G such t h a t

=131[a,c').

If 13: [a,b) .4 R'~ is a p a t h and if C c ~.'~, we write 13(t) -+ C as t -4 b if the spherical distance q(13(t), C) --+ 0 as t -4 b. 9.20. Lamina.

Suppose that f: G -4 R n is light a n d open, that xo E G , a n d

that 13: [a, b) -4 R n is a path such that 13(a) = f ( x o ) and such that either limt--+b 13(t) exists or 13(t) . 4 0 f G

as t -4 b. Then 13 h a s a m a x i m a / 1 i f t i n g c~: [a, e) -~ G starting

at too. If a(t) -4 xx E G as t - + c, then c

Iimt-+bfl(t) • Otherwise

b and f(xl)=

a(t) -4 OG as t -4 c. If f is discrete a n d if the local index i ( a ( t ) , f ) is constant for t E {a, c), then a is the only m a x i m a / l i f t i n g of 13 starting at xo • This l e m m a is proved in [MRV3, 3.12]. It follows f r o m the l e m m a , in particular, t h a t a locally h o m e o m o r p h i c m a p p i n g has a unique m a x i m a l lifting s t a r t i n g at a point. 9.21.

Remarks.

In the sequel L e m m a 9.20 will be applied in the following

situation. Let f : G -4 R ~ be n o n - c o n s t a n t qr, x0 E G , a n d let 13: [0, 1] -4 R = be a p a t h w i t h ]~(0) = f ( x o ) and 13(1) E O f G . T h e n 9.20 shows t h a t 13 has a m a x i m a l lifting c~: [0, c) --~ G s t a r t i n g at x0 with a ( t ) - 4 0 G A mapping whenever

K

as t -4 c.

f : G -4 R n is called proper if f - l K

is a c o m p a c t subset of G

is a c o m p a c t subset of f G , a n d closed if f C

is a (relatively) closed

s u b s e t of f G w h e n e v e r C is a (relatively) closed subset of G . 9.22. Lemma.

Let f: G -+ R n be discrete open. Then the following conditions

are equivalent: (1)

f is proper.

(2)

f is closed.

(3)

N(f,G) =p
a n d for all y E f G

k p = ~ i(xj,f), j=l

{ X l , . . . ,.Tk}

:

f-l(y).

127 For the proof of 9.22 see [V5], [MSR1], [VU1], and the references in these papers. As the simple example z ~-* z 2 shows, a maximal lifting of a p a t h starting at a branch point need not be unique. The next lemma is a quantitative statement of this fact. For a proof see [RI2]. 9.23. N(f,G)

Lemma.

L e t f: G -~ R ~ be discrete, open, and closed.

< c¢ a n d let fl: [a,b) --* f G

D e n o t e p --

be a path. T h e n there exist p a t h s ai: [a,b) --~

G , 1 < j < p , for which (1)

foaj=-/~,

(2)

c a r d { j : a j ( t ) = x } = [ i ( x , f ) [ for x E f-l[/~[ and t E [a,b) ,

(3)

v

= f - 1 I#I

9.24. R e m a r k s .

•

It follows easily from the definitions t h a t an open continuous

mapping f: G -~ R '~ obeys the m a x i m u m principle, i.e. if D E J ( G )

then

max If(z) l = mDax If(z) l • OD For further results concerning with discrete and open mappings see [CH] and [TY].

10.

Some properties of quasiregular mappings

In the present section we study some fundamental properties of quasiregular mappings. According to deep results of Yu. G. Reshetnyak [R2], [R12], a non-constant quasiregular mapping is discrete, open, and differentiabte a.e., and it satisfies Lusin's condition (N) [HS, p. 288].

By definition, condition (N) holds if and only if sets

of measure zero are m a p p e d onto sets of measure zero. The proofs are beyond the scope of this book. Applying these results one can prove the transformation formulae, the so-called K o - and Kl-inequalities, for the moduli of curve families under quasiregular mappings.

Also these important results are stated without proof. Of

these the K o - i n e q u a l i t y is due to O. Martio, S. Rickman, and J. VZis~l£ [MRV1], while the K l - i n e q u a l i t y was proved by E. A. PoletskiY [P1] and in an improved form by J. V£is~.l£ [VS]. A simplified proof of PoletskiY's result was given by M. Pesonen [BE2]. 10.1.

Quasiregular

mappings.

Let G C R '~ be a domain.

A mapping

f : G --~ R '~ is said to be quasiregular (qr) if f is ACL '~ and if there exists a constant

128 K >_ 1 such t h a t

(lO.2)

If(x)[ ~ <

a.e. in G .

KJ:(x),

If'(x)l = Ihl=l max

lf'(x)h I

Here i f ( x ) denotes the formal derivative of f at x (cf. Notation and

terminology). The smallest K > 1 for which this inequality is true is called the outer

dilatation of f and denoted by K o (f) . If f is quasiregular, then the smallest K > 1 for which the inequality (10.3)

Jr(x) < K l ( f ' ( x ) ) n '

l ( f ' ( x ) ) = min I f ' ( x ) h l , Ihr=s

holds a.e. in G is called the inner dilatation of f

and denoted by K x ( f ) .

The

maximal dilatation of f is the number K ( f ) = max{ K I ( f ) , K o ( f ) }. If K ( f ) <_ K ,

f is said to be

K-quasiregular

( K - q r ) . If f is not quasiregular, we set K o ( f ) =

K l ( f ) = K ( f ) = oo. It follows from linear algebra (see [V7, p. 44] and JR12, p. 22]) that

(lO.4)

K o ( f ) < K I ( f ) n-x , K i ( f ) <_ K o ( f ) '~-1

hold. Moreover, these inequalities are best possible. 10.5. L e m m a .

Let f: G --~ R n be a non-constant qr mapping. Then

(1)

f is sense-preserving, discrete, and open,

(2)

f is differentiable a.e.,

(3)

f

satisfies condition (N), i.e. if A C G

and

m(A) = O, then also

r e ( f A) = O . For proofs of these results, see [R2], JR4], JR121. Next we extend the definition of a qr mapping. 10.6. Q u a s i m e r o m o r p h i c

mappings.

Let G c R~ be a domain. A mapping

f : G ~ R ~ is called quasimeromorphie (qm) if either f G = {co} or the set E = f-l(c~)

is discrete and f l -- f [ G \

(E U { ~ } )

is qr.

We set g ( f )

= K(fl),

K o ( f ) = K o ( f l ) , and g 1 ( f ) = g x ( f l ) . 10.7. Q u a s i c o n f o r m a l m a p p i n g s .

If f is a homeomorphism satisfying (10.2)

and (10.3) with [J:(x)l in place of J r ( x ) , then f is called quasiconformal (qc).

129 10.8. R e m a r k s .

For n -- 2 and K -- 1 the class of K - q r maps coincides with

the class of analytic functions. By 10.7 a qc mapping may be sense-reversing, while a qr mapping in the sense of (10.2) is always sense-preserving. Reshetnyak replaces

Jr(z)

by

Igf(x)l

In his book [R12]

in the definition of a qr mapping and hence

qr maps in the sense of JR12] may be sense-reversing. This is largely a question of technical convention, since by topology (see L e m m a 9.8) each discrete open mapping is either sense-preserving or sense-reversing. 10.9. C u r v e f a m i l i e s a n d q u a s i c o n f o r m a l m a p p i n g s .

We now give an al-

ternative definition of a quasiconformal mapping. Let G, G ~ be domains in R '~ and let f : G --~ G I be a homeomorphism. Then f is (10.10)

K-quasiconformal

if

M ( F ) / K < M ( f r ) < g M(F)

for every curve family I' in G . Moreover, the dilatations of f are defined as

Kl(f)

M(fr) = sup

M(F)

K°(f)

'

M(F) = sup M(fr) '

where the suprema are taken over all curve families r in G such that M(F) and M ( f r ) are not simultaneously 0 or co. Thus

(2o.11)

M(r)/go(f) < M(fr) _< Ki(f) M(r)

for every curve family 1~ in G . The equivalence of the two definitions 10.7 and (20.10) of a qc mapping is proved in IV7] and also in [C1, pp. 81-110]. The next example shows that (I0.11) does not generalize directly to the case of qr mappings. 10.12. r =

Examples.

(s1,sl(1/e))

(1) Let

K(fk)

= z k, k e N \ { 0 } ,

z e C = R 2, and

By M(r) = 2~ ,

Moreover,

fk(z)

M(fkr) < 2r/log(e k) = 2 r / k .

= 1 because fk is analytic. If k _> 2, we see that the left inequality

of (10.11) fails to hold for (non-univalent) analytic functions.

f(z) = e x p z , F = A(A0, A1). Then fY c A(SI(e),S 1) and M ( f F ) < 2r/loge = whereas M ( F ) - - c ~ by 5.11 or by 5.33 and 5.17. Since K ( f ) = (2) Let Aj = { ( x , y )

E

R 2 : x = j},

j = 0,1,

z E R 2, and 2 r by (5.14), 1 also in this

example, we see that the left inequality of (10.11) fails to hold for analytic functions. A f o r t i o r i , it fails to hold for qr mappings.

130 By inserting a multiplicity factor in the left side of (10.11) one obtains the K o inequality for quasiregular mappings ([MRV1]).

Suppose that f: G -~ tt~ is a quasiregular mapping and that A is a Borel set in G such that N(f, A) < co. If F is a family of paths in A, 10.13.

Theorem.

M(r) _
L(x, f ) = limsup If(x + h) - f(x)I h-~0

Ihl

for x C G . Thus L ( x , f ) = If'(x)l whenever f is differentiable at x . It is easy to see that x ~-~ L(x, f) is a Borel function. Suppose that a E 7 ( f F ) . p(x)=

Define p: R '~ -~ R U {co} by setting

la(f(x))L(x,f)

ifxEA;

to

otherwise. Let F0 be the family of all rectifiable paths ~/E I' such that f is absolutely continuous on ~/. By Lemma 7.5 M(ro) = M(F). From the formula concerning change of variables in integrals it follows that

f p d s > _ ~ crds>_io~ for all ~ C Fo. Thus p E Jr(Fo). A more detailed proof is given in [MRV1]. Hence we obtain M(r)

=

M(ro) _<.Zn"dm =/, o(S(x))" L(x,.f)"dm(x) <- K ° ( f ) /A e(f(x))n J(x, f) drn(x) .

Since f is A C L " ,

J(x,f) is integrable over every domain D E J(G). Thus the

transformation formula in [RR, Theorem 3, p. 364] yields

f

J A rhD

:) d (x) = f

dR

o(v)- ::(y, :,A D)d (v)

<_N(f, A)/I~- a"dm . The theorem cited above is formulated in [RR] for finite-valued functions, but we may apply it to m i n ( k , a n) and then let k -~ co. Since D E J(G) is arbitrary, we obtain

#

M(F) < Y ( f , A ) K o ( f ) JR~ andre. Since this holds for every a E F ( f F ) , the theorem follows.

[~

131 The right side of (10.11) holds for qr mappings, too, as the following theorem shows. We shall mainly need the special case rn = 1 of this result. T h e proof is omitted ([V8]). 10.14.

Theorem.

Suppose that f : G --* I[ n is a n o n - c o n s t a n t qr mapping,

that F is a path family in G , that F' is a path family in R "

and that m is a

positive integer such that the following condition is satisfied: There is a set E o C G of measure zero such that for every path /3: I -* R " in F' there are paths ax , . . . , am in F with f oai C /3 for all i and such that for every x E G \ Eo and t E I , ai(t) = x for at m o s t one i . Then

M(F') _< K I ( f ) M ( F ) . m

In this result it is not required that f F = Y'. applications f F < F ' . in f D ,

As a m a t t e r of fact, in many

If D is a normal domain of f , if F' is a family of paths

and if F is the family of all paths a in D such that f o a c

condition in 10.14 is satisfied with m = N ( f , D ) ,

F ' , then the

E0 = By by 9.22 and 10.16(1)

below. Due to the connection (7.10) between the conformal capacity and the modulus of a curve family, one can formulate the K o - and K I-inequalities for condensers as well. If f : G --~ R " is discrete open and (A,C) is a condenser in G such that A is a normal domain of f , then (A,C) is called a normal condenser. Also the next result is from [V8]. 10.15.

Theorem.

Suppose that f : G --+ R ~ is a n o n - c o n s t a n t qr mapping.

Then

(1)

cap(fA, fC) <

KI(f) cap(A,C) - M ( f , C)

for all condensers (A,C) in G where M ( f , C) =

inf

yEfC

~_~ i(x, f ) z e c n f - ~(~)

and

(2)

cap(A, C) <_ K o ( f ) N ( f , A) c a p ( f A, f C )

for all normal condensers (A, C) in G .

132 In the next theorem we list some basic properties of quasiregular mappings. 10.16. T h e o r e m .

Let f: G --+ R ~ be a non-constant qr mapping. Then

(1)

m ( B f ) = r n ( f B f ) = O.

(2)

Jr(x) > 0

a.e. in G.

(3) If g : G ' - - + R "

isaqrmappingwith

fGcG'

,then

g o(fog) <

K o ( f ) K o ( g ) and Kl(f o g) < K,(f)Kt(g). 10.17. R e m a r k s .

Part (3) of 10.16 follows immediately from 10.15. Part (1) of

10.16 can be much improved, see [R10], [MR2], [$2]. In most of our later applications of the K o - and K x-inequalities, one may appeal to the following particular cases, which are the transformation formulae for #G and

AG . 10.18.

Theorem.

(1)

If f: G -+ R n is a non-constant qr mapping, then

# f G ( f ( a ) , f ( b ) ) <_K,(f) tta(a,b) ; a, bE G .

In particular, f: (G,#G) --+ ( f G , # I G )

iS Lipschitz continuous.

If N ( f , G ) < oo,

then

(2)

AG(a,b) ~_ K o ( f ) N ( f , G ) Afa(f(a), f(b) )

for all a, b E G with f(a) # f(b) . Proof.

(1) F i x

~, b e O and a c u r v e

,~: [0, 11 --~ a

f f

Diagram 10.1.

such that

~(0) = a,

133 a(1) = b, and denote r ' = zx( (fo~)[0,1], o : a ) .

Let r be the family of all maximal

liftings of the elements of Y' starting at ]a]. T h a t is, q E I" iff there exists /3 in Y' such t h a t 2 is a maximal lifting of /3 starting at a point of lat. Then fY < F ' ; by the definition (8.3) of the conformal invariant # a and by 5.3 and 10.14,

urn(f (a), f(b)) <_M(r') <__M(fF) < KI(f) M(F). Because I]3] n OfG # 0 for all /3 E Y', it follows from 9.20 that t~/] n OG # 0 for all ~/E F . Then by 5.2(2)

M(r) <_M(A(Ia],OC;C)). The proof now follows from this and the preceding inequality since a is an arbitrary curve in G with a ( 0 ) = a , a ( 1 ) = b (see (8.4)). (2) Let /3j: [0,1) --+ f G be paths such that /3j(t) ~ OfG, j = 1 , 2 , as t ~ 1,

f(a) = 131(0), f(b) = /32(0) and I/3,1 0 1/321A f C = 0. Let q,f: [0, c/) -+ G be a maximal lifting of /3j, j = 1 , 2 , with "Ta(0) = a , 3'2(0) = b. Since /3j(t) --+ Of G as t --~ 1 it follows from 9.20 that %-(t) --+ OG as t --+ c j , j = 1 , 2 . Let F = A(Iql t, 13,21;G) . By (8.2) and 10.13

)~c(a,b) <_ M(r) < Ko(f) N(f, G) M(fr) . Because t3i: [0,1) --+ f G ,

j = 1 , 2 , were arbitrary curves satisfying the conditions

mentioned above and because f F c A(lflll, ]/32[; G ) , the proof follows from the last inequality, (8.2), and 5.2(2).

[]

10.19. C o r o l l a r y . f f f : G ~ G ~ = f G is a qc mapping, then

(1) (2)

#c(a, b)/Ko(f)
hold for all distinct a, b E G. P r o o f . The right inequalities were proved in 10.18. Because K o ( f -1) = Kx(f ) ,

K1(f -1) = K o ( f ) , the left inequalities also follow from 10.18. According to 10.18, each qr mapping f: G ~

[]

fG is a Lipschitz mapping of

the (pseudo)metric space (G, # c ) onto (fG, # I t ) • We shall employ the inequalities of Section 8 for M ( A ( E , F ) ) ,

which enable us to give a geometric meaning to this

134

general result in m a n y interesting cases and to replace the metric space (G,/~c) by other metric spaces. Depending on the context, one may wish to replace ( G , # ~ ) by some less abstract space such as ( B n , p ) , If (X, dx) , (Y, dy)

(G, kc) , ( G , j c ) or even ( R n , I I).

are (pseudo)metric spaces and f : (Z, dx)

-+ (Y, dy)

is

continuous, then (10.20)

ws(t ) = s u p { d y ( f ( x ) , f ( y ) )

: d x ( x , y ) < t } , t > O,

is called the modulus of continuity of f .

This definition clearly depends on the

metrics d X and dy . If confusion seems possible we shall specify the metrics.

It

is increasing and t h a t wi(t) -+ 0 as t --~ 0 iff

is clear t h a t wi: (0, co) -+ ( 0 , ~ ]

f: (X, dx) -+ (Y, dy) is uniformly continuous. A t h e o r e m t h a t yields an u p p e r bound for the modulus of continuity is often called a distortion theorem. One can derive numerous distortion results for qc and qr mappings directly from 10.18, 10.19, and the estimates of Section 8. Examples of such results will be given in Section 11. We shall next give an application of 10.18 which yields a b o u n d for the

linear dilatation H(x, f) defined by L(x,f,r) H ( x , f ) = limsuPr_~0 l ( x , f , r ) ' (10.21)

L ( x , f , r ) = m a x { i f ( x ) - f(z)] : ]x - z I = r } , 0 < r < d(x, OG) , l ( x , f , r ) = min{ If(z) - f(x) l : I x - z 1 = r } ,

whenever f : G --+ R '~ is continuous and x C G .

10.22.

Theorem.

If f: G --+ R n is a non-constant qr mapping and x E G,

then

H(x,f) <_ c(n, Ko(f) Proof.

i(x, f) ) < oo .

x = 0 = f(x).

We m a y assume t h a t

Let a 0 be as in 9.14,

U =

U(O,f, ao) , and choose t > 0 such t h a t B=(3t) C U . For each r E (0, t] choose xr, yr C S n - l ( r )

with

If(xr)I = L(0, f , r ) ,

]f(Yr)] = l ( O , f , r ) .

Let Ar be the

y r - c o m p o n e n t of f - l [ o , f ( y r ) ] and Br the x r - c o m p o n e n t of f - t [ f ( x r ) , c ~ ) . 0EAT

and B ~ N b U # O

by 9.20. Denote F~ = A ( A r , B ~ ; U ) . By 5.9, 5.3, and 5.14

we obtain

(lO.23)

M(r,) +

Then

(log 3t] r

/

_>

M(A(A,, U 71B,.;

.

135

Next, 7.17 and 5.54(1) yield (10.24)

(3t + r~ . I-re1,-3tell)) = r \ \ ]

M ( A ( A r , U N B r ; R n ) ) _~ M ( A ( [0, rel],

If If(Xr)l > [f(Yr)l t h e n by 5.27 we obtain

M(fY.) _
(10.25)

This inequality holds trivially if ]f(xr) l =

1)

.

If(Yr)l. By 10.13, 9.10, and 9.15

M(r.) _< Ko(f) i(x,f) M(fr.). We combine the latter inequality with (10.23) and (10.24) and let r --~ 0 . As a result we o b t a i n r(1) ~_ Ko(f)

i(O, f) T ( H ( 0 , f ) -- 1) , (

T(1)

H(O,f) <_ 1 + r - l k K o ( f ) i(O,f) ) as desired.

O

10.26. Corollary.

ff f : R n --. R ~ is a

qc mapping with f ( 0 ) = 0 , then

If(x)I ~_ c(n, Ko(f))lf(y)l rot Ixl = lyt Proof.

T h e proof is similar to t h a t of 10.22; in fact, it is slightly simpler.

10.27.

Remark.

[:3

Making use of the functional identity in 5.53 one can write

the constant in 10.22 also as follows ~(x/~)

This equality together with 7.51 and 10.22 shows that the linear dilatation H(x, f) of a K-quasiregular mapping f has an upper bound depending only on Ki(x, f) . In particular,this upper bound is independent of n. 10.28. Exercise.

Let f: R n --~ R n be a K - q c mapping with f(0) = 0 and

let m = rain{ If(x)] : Ixl = 1 } , M = max{ If(x)] : Ix] = 1 }. W i t h o u t appealing to 10.22 or 10.26 show t h a t

M/rn < d(n, K) . [Hint: Let F t = A ( S n - l ( m ) ,

Sn-l(i)).

Because Sn-l(")

S "-1 N f-IS'~-I(M), 7.34 yields a lower b o u n d for M ( F ) ,

r

f-lsn-l(m) = I - I t ' .]

~ 0

136 10.29. R e m a r k .

A. Mori [MOR2] proved that the linear dilatation of a K - q c

mapping of a plane domain has an upper bound

e ~'K

.

His result was extended to the

multidimensional case by F. W. Gehring [G2] who found the upper bound d(n,K) = exp(2(

~/(V/~))1/(n--1))

Kwh-1

for the linear dilatation of a K - q c mapping of a domain G in R ~ . The bound in 10.22 and 10.27 yields a better bound c(n, K ) with c ( n , K ) = [2-1 (~/(V~)/K)] 2 _< d ( n , K ) / l O .

For these facts see [VU11] and [AVV1]. With a different (larger) constant 10.22 was proved by Yu. G. Reshetnyak [R10] and O. Martio, S. Rickman, and J. V£is£1~i [MRV1]. 10.30.

Exercise.

Show that d(2, K ) = e '~K . Next using 7.26(1) show that

c(2, K) < d(2, K ) / 1 0 . Applying 7.47 and 7.50 find a dimension-independent upper bound for c(n, K ) . 10.31. R e m a r k .

The sharp upper bound A(K) for the linear dilatation of a

K - q c mapping of R 2 onto R 2 was found by O. Lehto, K. I. Virtanen, and J. V~iis~il£ [LVV]. For further results of this type see [HEL]. It can be shown that A(K) = c(2, K ) - 1 = (see [LVV] and 5.61(2)) and that A(K) ,-~ l e ~ : K for large values of K .

It can be

shown (cf. [AVV3]) that e ~r(K-1) <_ A(K) < e ~ ( K - 1 / K )

for K > I . 10.32. N o t e s .

A thorough study of the K o - and Kz-inequalities is contained

in [RI12]. Theorem 10.18 and Corollary 10.19 are from [VU10], Theorem 10.22 from [VUll]. 10.33. R e m a r k .

Quasiregular mappings have important normal family prop-

erties, which were established by Yu. G. Reshetnyak [Rb] (for a simple proof see P. Lindqvist [LI1]). These properties will not be used in this book.

137

11.

Distortion theory

In the present section we shall put into effective use the transformation formulae 10.18(1) and (2) for the conformal invariants Ac

and /zc .

Most results of this

section are of the following general type: we combine the transformation formulae in 10.18 with some particular estimates for Ac and /z~ proved in Chapter II and as a result obtain distortion theorems. Besides the fundamental distortion theorems, the qr variant of the Schwarz lemma, and the Hhlder continuity, we prove several additional special distortion theorems. 11.1.

Theorem.

Let E c R ~ be a compact set of positive capacity and let

f: B r~ ---+R r~ \ E be a K - q m mapping. Then <- a K

<_ b K (_togth¼P(x,y))l-,~

for distinct x, y E B '~ where a and b depend only on n. P r o o f . It follows from 6.1 and 8.5 that

#:B- (f(x), f(y)) > d4 rain{ c(E), q(f(x), f(y)) } > d4 q(f(x),f(y)) min( d3, c(E) } . Because E

is of positive capacity we deduce from 6.1 that

1 _> c(E)/c(R '~) >_

c(E)/d2 > 0 , and therefore

#IB', (f(x), f(y)) > d4 c(E) q(f(x), f(y)) rain{ d3/d2 , 1 } . The proof follows now from 10.18(1), 8.6(1), and (7.30).

O

It follows from 11.1 and the monotone property 6.1(2) of the set function c(E) that for fixed K and #B,~(x,y), the distance q(f(x),f(y))

decreases if the set E

becomes larger. In other words, the larger the set omitted by the mapping f , the less f can oscillate as a mapping between metric spaces f : ( B ' ~ , # B . ) --+ ( R ~ , q ) . Later on we shall encounter a similar phenomenon with other metric spaces in place of ( B n , / z B , ) and ( R n, q). The next result is a counterpart of the Schwarz lemma for qr mappings. consider here the function ~ K = ~ r , ~ introduced in (7.44).

We

138 11.2.

Theorem.

Let f: B n --* R '~ be a non-constant K - q r mapping with

f B n C B '~ and let a = K x ( f ) 1/(1-'~) . Then

(1)

th½P(f(x),f(y)) <~K(th½P(x,y))

(2)

<_ A ~ - ~ ( t h ½ P ( x , y ) ) ~ ,

p ( f ( x ) , f ( y ) ) < g x ( f ) ( p ( x , y) + log 4 ) ,

hold for ali x, y E B '~ , where An is the constant in (7.21).

P r o o f . Fix x, y E B n . Because f B n C B ~ it follows from 8.5, 8.6, and (7.32) that # ~ ° (i(x), i ( y ) ) > #Bo (/(x), f(y)) = ~ ( 1 / t h b) where b-- ½ p ( f ( x ) , f ( y ) ) .

Similarly, by 10.18(1) and 8.6,

,i~o (y(x),/(~)) <_KI(/) ~ o (~, y) = Kx(i) ~(1/ th a) 1 where a = -~p(x, y ) . These inequalities together with 7.47(1) imply (1). For the proof

of (2) we note that by (7.31) and 10.18(1) Ap(f(x),f(y))

< ~ ( 1 / t h b ) ~ K x ( f ) A ( p ( x , y ) + log4)

where A ---- 2 n - i o n . Hence we have proved also (2). 11.3. C o r o l l a r y .

~3

Let f: B ~ --~ B n be a K - q r mapping with f(O) = 0 and let

a = K x ( f ) 1/(1-'~) . Then

(1)

IfCx)l _< ~,~(Ixl) ~ A~-~I~I ':~ ~ 2 1 - ~ / K K I x I ~/K ,

(2)

IfCx)I <- a+---~'a-1 a---- (41--~l+txl) gz(y),

for a11 x E B n .

P r o o f . Apply (2.17) and 11.2 with y = 0 and recall that A~n- ~ <_ 2 ~ - I / K K by 7.51.

O

The following invariance properties of 11.1 and 11.2 should be noted. The inequality of 11.1 yields the same upper bounds for

q(:(x),/(y))

and q((h o f ogl)(x), (h o :

o~x)(y)),

139

while the second one yields the same u p p e r bounds for

fl(f(X),f(y)) a,nd P((gl 0 f o g2)(X), (91 0 f 0 92)(X)) whenever 91, 92

E .M(Bn)

and h is a sense-preserving spherical isometry.

It should also be observed t h a t the explicit estimate 11.3(1) is sharp if K -~ 1. In 11.2 we assumed t h a t f B " C B "

and proved t h a t f : ( B ~ , p ) --~ ( B ~ , p ) is

uniformly continuous with a quantitative bound for its modulus of continuity. If, in addition, B n \ f B '~ ~: 0 , one would expect a b e t t e r result t h a n 11.2. For instance, one could hope to replace the target space ( B ~ , p )

in 11.2 by ( f B ' ~ , k f B , ) .

In

the particular case of Mhbius transformations this indeed is possible by 3.9 (later on we shall prove t h a t this is possible also for quasiconformal mappings). Now we are going to show t h a t for qr mappings and even for b o u n d e d analytic functions such an expectation is futile. 11.4.

Example.

Let g: B 2 --* B 2 \ {0} = gB 2 be the exponential function

g(z) = exp[~_--zy~, ~z+l~ z E B 2 . We shall show t h a t g: (B2,p) -+ (gB2,kgB~) fails to be uniformly continuous. To this end, let xj = (eY - 1)/(eJ + 1), j -- 1 , 2 , . . . .

It follows

from (2.17) t h a t p(O, xi) = j and thus p ( x j , x j + l ) = 1. Let Y = B 2 \ {0}. Since

g(xj) = e x p ( - e J ) we get by (3.4) and (2.34)

ky

_>jy

= log [1

+ (expe j+t) (exp(-e j)

~-log[l+exp(e j+l-e

y)-l]

=e j+l-e

In conclusion, g: (B2,P) --* (Y, k y )

as j -+ oo.

- exp(-ei+l))]

j--~ oo

cannot be uniformly continuous,

because p(xy,xy+l) -~ 1. In this example 0 ( g B 2) consists of a point component {0} and the unit circle 0 B 2 . We now show t h a t if each b o u n d a r y component of the image domain is n o n degenerate, then the situation will be different, at least under an additional condition. Later on we shall show that this additional condition, which requires t h a t the image domain be uniform, can in fact be removed, and t h a t the exponential function in 11.3 is in a sense an extremal case. 11.5.

Theorem.

Let f: B n --~ R ~ be a non-constant qr mapping, let E C

t t '~ \ f B ~ be a non-degenerate continuum such that c~ E E , and let G = R ~ \ E be a domain.

140

(1)

Then f: ( B n , p ) -~ (G,ja) is uniformly continuous.

(2)

I f G is uniform, then f: (B n, p) ~ (G, kc) is uniformly continuous.

Proof.

(1) T h e proof follows the same general p a t t e r n as the one in 11.2. T h e

particular estimates needed for the present case are supplied by 7.41, 8.6, and (7.30). (2) T h e proof follows from (1) and the definition 3.8 of a uniform domain. 11.6.

Lemma.

(:3

Let G and G t be proper subdomains of R n , where G is

uniform and G ~ has connected complement ~ n \ Gl. If f: G ~ R n is a qr mapping with f G C G ~, then for all x,y E G jc,(f(x),f(y)) < aljG(x,y) ÷ as, where al,a2 are positive numbers depending only on n , K i ( f ) , and the constant

in the definition of a uniform domain. Proof.

By 8.31, 10.18(1), 8.30(2), and 3.8 we obtain

c~ Jc, ( f ( x ) , f ( y ) ) < #G, ( f ( z ) , f ( y ) ) <_ g i ( f ) #G(x, y)

_< Kx(f)(bl < Kx(f)blAjc(z,y) +K1(f)b

11.7. E x e r c i s e .

.

Show t h a t the hypothesis t h a t R ' ~ \ G ~ be connected cannot be

removed from 11.6 if n = 2 and G --- B '~ . [Hint: Show t h a t the exponential function in 11.4 provides a counterexample in the present case, too. Recall t h a t p ~ JB- by 2.41(1).] 11.8. E x e r c i s e .

Observe first t h a t 11.2(1) and (2) hold also for a qr m a p p i n g

f : B '~ --~ H '~ . Show t h a t

If(z)l _< 22~1f(0)1 ~ ]

, ~-- K , ( f ) ,

for a qr m a p p i n g f : B '~ --~ H " when x E B n . [Hint: Apply 11.2(2) for a qr m a p p i n g of B = into t t ~ and the inequality pH=(x,y) > Itog(Ixl/tYl)l, x , y E H ~, required inequality then follows from 2.41(2) and (2.39).]

The

141

11.9. Exercise.

Show t h a t if f : B n -+ B '~ is K - q r , t h e n for all x E B ~ 1 -IS(~)l > 2-2K( 1 -IS(0)l)

(l--j~),, 1-Ixl

K

[Hint: O b s e r v e t h a t by 2.36(1) a n d 2.41(1)

1 -lyl p(x, y) > J,3. (x, y) > log - 1 -Ixl for all x, y E B ~ . Now a p p l y 11.2(2) a n d (2.17).] i1.10.

Theorem.

Suppose that f : G --~ R '~ is a b o u n d e d qr m a p p i n g and that

F is a c o m p a c t subset o f G . Let a = K i ( f ) 1/(I-~) and C -- A ~ - a d ( f G ) / d ( F ,

OG) ~

where An is as in (7.21). T h e n f satisfies the HSlder condition

(11.11) for x E F , Proof.

If(x) - f(y)] < C Ix - y]~ yCG. Set r -- d(F, O G ) . S u p p o s e first t h a t

I x - Yl < r . Define g: B '~ -÷ B n

by g(z) = f ( x + rz) - f ( x )

d(fC) T h e n g(0) = 0 a n d K i ( g ) = K z ( f ) Setting z = ( y - x ) / r

by 10.16(3). B y 11.2 we get Ig(z)l <_ A~-alzl ~ .

we o b t a i n (11.11). Next a s s u m e t h a t I x - y l

> r . Since A,~ _> 1

(in fact, An > 4 , see 7.22) we have

]f(x) - f ( Y ) l <~ d ( f G ) < r - ~ d ( f G )

11.12. Theorem.

Ix - yl ~ < C I x - yl~ .

Let f : B '~ -+ B n be a K - q r m a p p i n g into B ~ . T h e n If(x) -f(Y)l-<

b~ ( t h ½ P ( x , y ) )

for all x , y e B " , w h e r e b g ( s ) = 2 ~ K , ~ ( S ) / (1 + V/1 -- ~ , ~ ( s ) if f

0

) . T h e result is sharp

is a rotation fixing the origin and x = - y . Proof.

If we let t ' =

½ p ( f ( x ) , f ( y ) ) , it follows from (2.27) a n d 2.29(2) t h a t

I f ( x ) - f ( y ) [ _< 2 t h ½t' =

2tht' 1 + ~/1 - t h 2 t r

T h e desired inequality follows now f r o m 11.2(1). assertion follows f r o m the one in (2.27).

O

Since ~ l , n ( r )

= r , the s h a r p n e s s

142

11.18. C o r o l l a r y .

Under t h e a s s u m p t i o n s o f 11.12 l f ( x ) _ f(y)[ _< ~K,,~(a) + ~2K,n(a)

where a = t h ½ P ( x , y )

for all x, y E B ~ .

P r o o f . The proof follows from 11.12 and the inequality 2 1 + v/1 -

11.14. T h e o r e m . a(r)


O<x
C]

x2 -

For n > 2, r E (0, 1), and K E [1, c~) there e x i s t s a n u m b e r

w i t h limr--.0 a(r) = 1 such t h a t i f f : B ~ ~ B '~ is a K - q r

mapping into B ~ ,

then I f ( x ) - f ( Y ) I <-- a(r) A ~ - C ~ I x - yl a

for all x , y E B " ( r )

w h e r e a = K 1/(1-~) . w

P r o o f . Let r E (0,1) and x , y E B n ( r ) . Then th ½ P ( x , y ) <_ th ½ P ( - r e l , r e l ) :

2r

1+

r2

by 2.47. By the inequality in the proof of 11.13, by 11.12, 11.2(1), and 2.47 we obtain

If(x)

-

f(y)l ~ bK (th ½p(x,y)) 1 ÷ V/1 - ~ , n ( 2 r / ( 1 < -

÷ r2))

Aln-a [1 + ~ K , n ( 2 r / ( 1 ÷ r2))] Ix -- y[a [Ix - yl ~ + (1 -ixl~)(1

- tyI:)] "/~

We may choose 2r

r 2) - a

The following result is a generalization of Liouville's theorem concerning the growth of entire analytic functions. 11.15. T h e o r e m .

S u p p o s e t h a t f : R ~ --+ R '~ is a qr m a p p i n g a n d t h a t

lim~--.oo I x ] - a t f ( x ) l = 0 w h e r e a = K i ( f ) I / ( 1 - ~ ) . T h e n f is a c o n s t a n t .

143 P r o o f . We can write lf(x)l _< Ixl~,(Ixl) where e(R) -+ 0 as R --+ oo. and choose R > Ixl. Applying (11.11) for G - - B " ( R )

x E R"

and F = {0} we

obtain If(x) - f(0)l < Clxl ~ where C -- ~ln-ad(f, O G ) - a d ( f G ) < 2 ~ - a e ( R ) .

If(x)-

f(0)l < 2Aln-a~(R)]xl a . Letting R - + ¢x~ yields f ( x ) - - f ( 0 ) .

constant.

Thus

Hence f is a

[]

11.16.

Remarks.

The exponent

a in 11.2(1) and 11.15 is best possible.

f ( x ) -- x[xl ~ - 1 ,

As to 11.2(1), the function f : B " --~ B ~,

g(f)

Fix

x E B n,

gl(f ) =

= a 1 - " , is a desired example (see IV7, 16.2] for the calculation of K i ( f ) ).

T h e same function, as a mapping of R "

limz-+oolxl-~lf(x)l

= 0

onto

R '~ , shows that the condition

in 11.15 cannot be replaced by the requirement that

]x[-~[f(x)] be bounded.

For ~ e (0,1~) let c ( ~ ) = {z e R " = z . e . ~ Izlcos~}.

We next

formulation of 11.2 for maps into a cone or into an infinite cylinder. 11.17. T h e o r e m .

Let f: B n -* R n be a non-constant qr mapping.

(1) If ~o e (O, ½r) and f B " c C ( ~ ) , then for alI x e B ~

If(x)l _< If(o)14 ~(,1---:~) where a depends only on n and K t ( f ) . (2) If f B '~ C { x E R n : x 2 + . . . + x 2~_I<1}, then forall x , y ~ B '~

lY(x)l < lf(Y)l + A K , ( f )

(p(x,y) + log4)

where A is a positive constant depending only on n . P r o o f . (1) By 5.29, 10.18(1), 8.6(1), (7.31), and 7.26(2) we obtain

d_&log If(x)l If(O) l

-< # f n - ( f ( x ) , f(O)) _< K l ( f ) # B - ( z , O)

< K t ( f ) 2 n _ l c n l o g ( 4 1 + [xl~ -

l_--Z-~

T h e proof of (1) with a = 2 ~ - X c ~ K i ( f ) / d n

]

"

follows.

(2) Assume first that If(x)] > If(Y)[ + 1. From 5.29 we deduce that

[f(x)l

.IBo(I(z),I(y))

_> d,~

~(r)-----; -> 2d" (l/(x)l,r -I/(Y)I[f(y)]+l

1).

give a

144 Here ~o(r) E (0, 17r) is such that

n : x ~ + . . . + x ~ _2,

s~-l(r) A{xeR

rp(r)

for r > 1, i.e. p ( r ) = arcsin(1/r) and

< 1}

= Sn-I

(r) nC(p(r))

< ~1 r . By 10.18(1) and (7.31) we obtain

as in the proof of (1) If(x)l _< If(y)f + 1 +

TKl(f)(p(x,y ) + log4)

-< If(y)] + AKI(f)(P(x,Y) + log4) where T =

2'~-2c,,7r/d,,

If(y) i + 1 as well, the proof of (2) is complete. 11.18. R e m a r k .

Since equality holds for If(x)l -

and A = T + 1 / l o g 4 .

For small values of

O

p(x, y)

one can improve 11.17 by applying

7.26(1) instead of 7.26(2). Recall also 7.28(1). 11.19. T h e o r e m .

Let f: B " ~ B n

be a qr mapping with

N ( f , B '~) -- N < oe.

Then lp(f(x),f(y))

th

< 2 (th

hola~ ~o~ aU ~ , y ~ B ~ w h e r e ~ = 1 / ( N K o ( f ) )

all

x C B

lp(x,y))~

. F , ~ t h e r m o ~ e , Z f(O) = O, t h e n ~o~

'~ _< (

If(x)[ 1 + Xf~--i)(x)l 2

Proof. We may assume that (11.20)

)~B~(x,y )

=

Ixl

2\1 + ~

~,~ /

f(x) ~ f(y). It fo]]owsfrom 8.6(2) and 8.7 that

1 (sh2½P(x,y)) ~T

> -c~logth¼P(x,y)

Because f B n C B '~ , it follows from 8.5, 8.6(2), and 8.7 that (11.21)

AfB~ (f(x),

f(y)) ~

AB~ (f(x),f(y))

< cn log

2 th

¼p(f(x), f(y))

The proof now follows from (11.20), (11.21), and 10.18(2). If f(0) -- 0, the assertion follows from the above inequality and (2.17), 2.29(2).

[]

11.22. E x e r c i s e . Observe first that the proof of 11.19 yields the inequality

7._1 ( r(sh 2 a) sh 2 b < where a =

½P(x,y)

and b =

\NK

½p(f(x), f(y)).

o(f)]

Next assume, in addition, that f(0) = 0

and N = 1. Exploiting the functional identity 5.53 and the definition (7.45) show that the above inequality with y -- 0 yields

If(~)l ~ _< 1 -p~/K,r~(v/i - - Ixl ~ ) for all x E B n . (Compare this to the Schwarz lemma 11.3.)

145 11.23.

E x e r c i s e . Assume that f : B ~ --+ B '~ is K - q c with f(0) = 0 and

f B ~ = B '~ . Show that

If(z)l 2 _< min{ ~oK,,~(Ixl),l--~O:lK,,,(Vl'7--1*l 2 2 If(x)] 2 _> max{ ~02/K,n(iXl) , 1 -- ~02 K,,(vq

=)

},

I~1~) } -

[Hint: Apply 11.22 and 11.3 also to f - 1 .] Recall that in the case n = 2 we have ~ 2 , 2 ( r ) = 1 - 9 9 12/ ~ : , 2 ( x / 1 - r =) for all K > 0 and 0 < r < 1 by 5.61(2) while the analogous relation fails to hold for n _> 3 by 7.58. 11.24.

Theorem.

Let f : B '~ -+ R ~ \ { 0 }

be a q r m a p p i n g w i t h

N(f,B ") _<

p < ca. Then for x, y C B n

If(x)l ~ If(y)l (1 ÷

T -I

(Ar(sh 2 ½P(x,y))))

,

where A = l l ( 2 p K o ( f ) ) . P r o o f . If If(x)t < If(Y)] there is nothing to prove. Hence we may assume that [f(x)t > [f(Y)l- By 5.27 and 8.5 we obtain

ASB,<(f(x),f(y)) < Ac(S(x),f(y)) <

T( If(x)l

-

IS(y)l

<_ M ( A ( [0, f(y)], [ f ( x ) , c a ) ) )

1)

where G = R n \ {0}. Next, by 8.6(2)

~ B - ( z , ~ ) = ½ ~(sh 2 ': p ( * , y ) ) and by 10.18(2) AB.

(x, y) ~ p K o (f) AfB= ( f ( x ) , f ( y ) )

The desired bound follows from these relations.

.

0

We require the following important theorem of Martio, Rickman, and Vgis£I/i. [MRV3, 2.3] on locally homeomorphic qr maps of B n , n _> 3.

The proof of this

theorem makes use of an ingenious method of V. A. Zorich [ZO1]. The proof will be omitted. A similar result for qm mappings was proved by Martio and Srebro [MSR4]. 11.25. T h e o r e m .

For n >_ 3 and K >_ 1 there exists a number ¢ = ~b(n, K ) E

(0, 1) such that every locally homeomorphic K - q r mapping f: B ~ -+ R ~ is injective

in B'~(x, ( 1 - 1 x l ) ¢ ) for al! x E B n .

146 11.26.

E x e r c i s e . Applying (2.23) show that D ( x , M )

where T = ( 2 t h ½ M ) / ( 1 -

th½/),

tx] < 1.

B'~(z, ( 1 - Izl)¢) where IzI < 1, ¢ E (0, 1), i 11.27. T h e o r e m .

c B'~(x, T(1 - I x l ) )

Conversely show that D ( z , i )

c

= 2arth(¢/(2 + ¢)).

Let f : B r~ -+ Rr~\{0} be a locally homeomorphic qr mapping

and n > 3. Then

where C and a are positive numbers depending only on n and K ( f ) . P r o o f . Let ¢ = ~b(n,K(f)) be as in 11.25 and define gz(z) = fz(x+z(1-]xl)~b) for z c B

'~ and x C B " . Then gx is injective and K - q c in B '~ by 11.25.

We are going to show first that [f(x)l satisfies the Harnack inequality (4.11) in B '~ with s E (0, 71¢ ] and

(11.28)

C s = I + r-I(Ar(16/9)) , A = 1/(2Ko(f) ) .

To this end let B'~(z,r) C B ~ and xl,x2 E B n ( z , s r ) , s e (0, ½~b]. By 11.24 we obtain

I f ( x 1 ) _ Igz(m)l < l + r - l ( A r ( s h 2½p(yl,y2))) IS( 2) Igz(y )l where yj = ( x j - z ) / ( ( 1 - ] z [ ) ¢ ) E B n and A = 1 / ( 2 K o ( f ) ) . Because lyJl <- 71 for j = 1,2 it follows from (2.17) that

2B = P(Yl,Y2) <- 2 l o g 3 , and hence sh 2 B _< 1 6 / 9 . We have thereby proved (11.28). By virtue of (11.28) and 4.12 we obtain

If(m)l ~ C~÷tlf(0)I where C~ is as in (11.28) and t = (log 1+_¢__~,,, l_)~))/log~,

s = ½¢.

We have thus

proved the desired inequality with C = Cs and a = (tog C8)/log 22_+-~. (Recall the relationship (10.4) between g o ( f ) 11.29.

and K ( I ) . )

0

E x e r c i s e . A counterexample to show that 11.25 is false for n = 2 is

easily found. Denote fj(z) = e x p ( j z ) , j = 2 , 3 , . . . ,

z E B 2 . By considering the

family {fj} we see that 11.25 is false for n = 2. Find a counterexample to show that 11.27, too, fails to hold for n = 2.

147 Next we shall study the behavior of the function Ix - yt r a ( x ' Y ) = min{d(x), d(y)}

under quasiconformal mappings. 11.30. T h e o r e m .

Let G and G' be proper subdomains of R '~ and f: R '~ --~

R n a K - q c homeomorphism such that f G = G'. Then for a11 x, y E G

_< P r o o f . We may assume that d ( f ( x ) , O G ' ) < d ( f ( y ) , O G ' ) .

Fix z' E aG' such

that If(x) - z'] = d ( f ( x ) , O G ' ) and z E OG such that f ( z ) = z ' . Then by 10.18(2) A D ( x , y ) <_ K A D , ( I ( x ) , f ( y ) ) , where D = R n \ {z} and D ' = f D = R '~ \ { z ' } . By 8.24 ~D(x,y) > r ( Ix -- y]~ > 7(rG(x,y)) -

t]x-z]/-

x)--f(Y)l~ ) = 4 T(rG,(f(x),f(y)) ) A D , ( f ( x ) , f ( y ) ) < 4 r \ ( i f (~--]-Ox)-----~ The desired result follows immediately from the above inequalities. Applying T h e o r e m 11.30 with G = R '~ \ {0} yields the following result. 11.31. C o r o l l a r y . Let f: R '~ --+ R '~ be a K - q c mapping with f(O) = O. Then ~or x, ~ ~ R '~ \ { 0 )

if(x)-f(y)l

< r_l(

min{ll(~)l t.r@t) ,

1

~"(

-

Ix--yl

)

mi~l.isl} )

"

Next we shall prove a result where f need not be defined on the whole space R n as it was in 11.30 and 11.31. 11.32.

Theorem.

c-QED,andlet

Let G be a proper subdomain of R '~, suppose that G is

f: G--~ f G

be K - q c with f G c R '~ . Then


C

T:rG(x,y)))

148

Proof.

B y 8.29 we o b t a i n

AG(x,y) >>_cr(r 2 + 2r) _> 2 1 - n c r ( r ) w h e r e r = r c ( x , y ) . N e x t b y 8.25 we g e t

)~:G(f(x), f ( y ) ) < 4 r ( r s G ( f ( x ) , f ( y ) ) ) . T h e d e s i r e d i n e q u a l i t y now follows easily. 11.33.

Example.

We shall now show t h a t t h e

necessary. Let G = B 2\[0,el) map f(z)=

O

a n d let f : G - - ~

c-QED

c o n d i t i o n in 11.32 is

B~_ = B ~ N H 2 b e t h e c o n f o r m a t

~/'z, z e G . L e t x j = ( 1 / 2 , 1 / j ) , yj = ( 1 / 2 , - 1 / j ) ,

j = 4,5, ....

Then

r c ( x j , y j ) = 2 , w h i l e it is e a s y t o see t h a t rfG(f(xj) , f(yj)) as j

-+ c o .

In p a r t i c u l a r ,

rfc(f(xj),f(yj)

) oo

) has no u p p e r b o u n d in t e r m s of

r c ( x j , y y ) . O n e c a n show t h a t G is n o t a c - Q E D d o m a i n for a n y c > 0 . T h e f u n c t i o n rG(x ,y) is i n v a r i a n t u n d e r s i m i l a r i t i e s a n d , a c c o r d i n g l y , t h e s a m e is t r u e a b o u t 11.30 a n d 11.32. N e x t we shall give s o m e N ( R '~) - i n v a r i a n t r e s u l t s . 11.34. andlet

Theorem.

L e t D c R ~ be a c - Q E D d o m a i n with c a r d ( R '~ \ D ) > 2

f: D --* f D c R ~ be K - q c . T h e n for x , y ~ D

~nfD(f(x),f(Y)) ~. T-I(2n~IK TO"~'D(Y',Y))) where wtD is as in (8.33). Proof.

T h e p r o o f follows f r o m 8.41 a n d 10.18(2).

11.35.

Theorem.

Let f: B ~ --* R ~

53

be a K-quasimeromorphic mapping, tet

a, d E R "~ \ f B '~ be distinct and suppose that N ( f , B ~) < p < o o . Then q(a,d)q(f(x),f(y)) q(a,f(x))q(f(y),d) for all x, y E B '~ .

-

i---

lyl

))

149

Proof.

By 8.6(2)

AB,(x,Y) for distinct x, y E B

'~. Let D - - R

½ T ,1((

_]xl2)(l_ty] 2)

~\{a,d}.

By 8.5 )~I~" - < ) ' D a n d thus by 8.40

we o b t a i n

~ . o (I(~), f(y)) _< ~ (I(~), f(y)) _<4 T(m~(f(~), f(y)) <- 4 T(I a, f(x),d, f(Y) I) • T h e desired conclusion follows f r o m the above inequalities and 10.18(2).

[]

th½P(x,y)= Ix-yl/v/[x-yl2+4xnyn x, y ~ H '~ . Show t h a t if f : H '~ --~ H '~ is K - q r with f(en) = e, t h e n 11.36.

Exercise.

Show t h a t

+

for all x E H " .

[Hint:

-

Apply 11.2(1).] A s s u m e

for

4x.

next t h a t f : H " --+ H "

is K - q c and

f(e~) = e ~ . Show t h a t

If(x)-e.I:

T-1 1 ~(l~

[Hint: Find first an expression for sh ½P(x,y) and t h e n a p p l y 11.22.] 11.37.

Exercise.

(1) Let f : B n --* B n be K - q r a n d c~ = K 1/(1-n) . From

the p r o o f of 11.14 derive the inequality

If(x) - f ( Y ) I ~ (2)~,~)l-ap(x,Y) ~ for all x, y E B

".

(2) Next e x t e n d this inequality to a K - q m m a p f: B n --+ Q ( z , r )

where Q(z,r)

is a ball in the spherical m e t r i c as defined in (1.22). Show t h a t

q(f(x), f ( y ) ) < (2A~) 1 - " p(x, y) ~ c(r) for all x , y e B n w h e r e c(r) : r / v / 1 - r 2 . (3) Find a f o r m of 11.14 where the m a j o r a n t is i n d e p e n d e n t of n . (4) Let f : B n --* R n be a K - q r m a p p i n g with Bf = 0. Show t h a t for n___ 3 there exists a n u m b e r d(n, K) such t h a t for all r E (0, 1)

N ( f , B ~ ( r ) ) < d(n,K)(1 - r ) 1-'~ . [Hint: 4.22 and 11.25.]

150 11.38.

Exercise.

numbers T > 0

Let f : B " ~

and A > 0

R"

be K - q r and assume that there are

such that p ( x , y ) < T

that

implies I f ( x ) - f ( y ) l < _ A .

Show

A,-~ [th ½P(=,Y)]~ L ~

ts(=)-s(y)l_< A .

for all x, y E B '~ with p(x, y) <_ T where a = K 1/(1-") . Next combine this inequality with 4.13 to obtain a bound valid for all x , y C B n . Next we shall survey some distortion theorems for quasiconformal mappings, which will not be proved in this book. In 1956 the following theorem of A. Mori [MOR1] appeared. 11.39.

Theorem.

Let f : B2 --~ B2 be a K - q c mapping w i t h f(O) = 0 and

f B 2 = B 2 . Then If( x ) - f ( Y ) l <-

161x- Yl 1 / K

for all x, y E B 2 . F u r t h e r m o r e , the n u m b e r

16 cannot be replaced by any smaller

absolute constant.

It has been conjectured that the best constant in place of 16 is 1 6 1 - 1 / K

([LV2,

p. 68]). In 1985 H. Qu [Q] proved that the constant 16 ~(1-1/K) will do (cf. [SEM2, p. 205]). In [FV] R. Fehlmann and M. Vuorinen proved the following theorem. 11.40. fB"

Theorem.

L e t f: B " ~ B '~ be a K - q c

m a p p i n g w i t h f(0) = 0 and

= B '~ . Then If(=) - f(~)J <_ M I ( . , K )

i= - ~l" , ~ = K I/(I-") ,

for all x,y 6 B n , where the n u m b e r M I ( n , K )

has the following three properties

(I) M l ( n , K )

--+ 1 as K--+ I, uniformly in n.

(2) M I ( n , K )

remains bounded for fixed K

(3)

and varying ~%.

M,(., K ) <_ 3~. for all K >_ 1.

We remark that a multidimensional generalization of 11.39 (essentially part (3) of (11.40)) follows if one extends Mori's original argument to R " .

This fact was

observed by B. V. Shabat in 1960 tSH] (see also F. W. Gehring [G2, p. 387] and K. Ikoma [IK]). The point of 11.40 is that a quantitative constant is given which satisfies the property (1). See also G. D. Anderson and M. K. Vamanamurthy [AV]. Some related results are given by R. N£kki and B. Palka [NP] as well as by F. W. Gehring and O. Martio [GM2].

151 11.41. R e m a r k .

It is an open problem whether the constant M1 (n, K) in 11.40

can be chosen so that it remains bounded when both n -* oo and K --* oo. The following theorem was proved by P. Tukia and J. Vgis/il/i ([TV], [Vll]) and in its present improved dimension-independent form by G. D. Anderson, M. K. Vamanamurthy, and M. Vuorinen [AVV1]. 11.42.

Theorem.

For K >_ 1 and s E (0, 1) there exists a h o m e o m o r p h i s m

77: [0, c~) --~ [0, c~) with ,7(0) -- 0 and with the following properties. If f: B n -+ R '~ , n > 2 , is a K - q c m a p p i n g into R n and x , y , z E B n(s) with x ~ z , then I f ( x ) - f(y)[ < rlflx-yl'~. . If(x)- f(z)I \ix_ z I,/ •

11.43. E x e r c i s e . Show that the inequalities 7(1) _> 1 and

If( x ) - f(Y)l > 1//~(I x- zl~ If( x ) - f(z) l -

kl x-

Yl "/

for all distinct x , y , z E B'~(s) follow from 11.42. Show also that 7(1) yields a bound for the linear dilatation of the mapping f . 11.44. A n o p e n p r o b l e m . For K>__ 1, n _ 2 , a n d

~k,,~(r)

r E (0,1) let

= ~ : ( r ) = sup{ If(x)l : f c QCK(Bn), f(0) ----0, Ix[ < r}

where QCK(B '~) = { f : B n --~ f B '~ I f i s K - q c a n d

f B '~ C B'~}.

As shown in

[LV2, p. 64]

(11.45)

~D~,2(r ) = ~OK,2(r ) ~ 4 I - I / K

for each r E (0,1) and K > I . (11.46)

r IlK

By 11.3(1)

• ',, 1 - ~oK,n(r ) <_ pK,~(r) <: :~,~

C~

r O~ , a ---- K 1/(1-'~)

"~

for n _ 2, K _> 1, r E (0,1). i . V. Sychev [SY, p. 89] has conjectured that (11.47)

~*K,n(r) ~ 4 i - a r a

for all n _~ 2 and K > 1. Because A2 = 4, (11.47) agrees with (11.45) for n = 2. In [AVVd] it is shown that ~*K,n ~ ~ g , n for n :> 3. It follows from 11.19 and 11.23 that ~*K,n(r) ~ 4 r 1/K ,

(11.48)

/

<

1 -

-

152 From (11.48) and (11.46) it follows, as shown in [AVV2], that

p*K,n(r) < 41-1/K:r 1/K

(11.49)

holds for all n _> 2, K _> 1, r E (0,1). Note that the right hand side of (11.49) is bounded when K -~ oc.

Recall that An -* co as n -~ oc by 7.22 and that

A~-c' < 2 1 - U K K by 7.51. Note that Sychev's conjecture (11.47) still remains open. 11.50. N o t e s .

Distortion theorems for qc and qr mappings have been proved

by many authors (see the bibliography of [C1]). The Hhlder continuity of plane qc mappings was proved by L. V. Ahlfors [A1], and the Schwarz lemma by J. Hersch and A. Pfluger [HEP] and P. P. Belinski~ (see the references in [BEE, p. 13]). For n = 2 the explicit bound 41-1/K in 11.3(1) was found by C.-F. Wang [WA] with the aid of a parametric method, and a simplified proof was given by O. H~bner [H/J]. See also O. Lehto and K. I. Virtanen [LV2, p. 65, (3.6)] as well as P. P. BelinskiY [BEL, p. 15, formula (16')]. The n-dimensional form of the proof in [H()} and [LV2] was given by G. D. Anderson, M. K. Vamanamurthy and M. Vuorinen [AVV11. The Hhlder continuity of qr mappings in R '~ was proved by E. D. Callender [CALl, F. W. Gehring [G2], Yu. G. Reshetnyak [R1], JR12, pp. 36-38]. A spatial form of the Schwarz lemma was found by B. V. Shabat [SIt] and O. Martio, S. Rickman, and J. Vgisglg [MRV2]. Most of these bounds depend essentially on n , with bounds that approach oc as n -+ o0. Dimension-free bounds (such as 11.3(1)) were given in [AVV1] and [AVV2]. For 11.27 see [VU10] and [MIll. Both 11.34 and 11.35 were proved in [VU13]. For results similar to 11.34 and 11.35 see [G7, p. 233] and [RI2].

12.

Uniform continuity properties

The present section is devoted to the study of uniform continuity properties of a qr mapping f: G -~ f G as a mapping between the metric spaces (G, kc) and

(fG, k i t ) and to the study of its restrictions flD : (D, kD) ~

(fD, kfD )

whenever D is a subdomain of G . We shall consider the modulus of continuity of f (cf. (10.20))

wl(t ) = sup{ k f c ( f ( x ) , f ( y ) ) : kc(x,y ) ~_ t } .

153 If f is a Mhbius transformation, then WflD(t ) < 2t for all domains D in G by 3.10. In this section we shall prove an analogous result for qc maps. The situation for non-homeomorphic qr mappings is entirely different, as Example 11.3 in the preceding section shows. However, under a natural additional condition, one can prove a positive result even for non-homeomorphic mappings. This additional condition is a necessary and sufficient condition for a qr mapping f: (G, kG) --+ ( f G , kfG ) to be uniformly continuous. The condition requires that the function df defined by

x,

~ dr(x) = d ( f ( x ) , O G ' )

satisfy the Harnack inequality (4.11) in G . Applying some results of Section 8 we shall show that this Harnack condition is satisfied if f is qc or if f is qr and, in addition, N ( f , G ) < c~. Furthermore, under mild restrictions on Of G the Harnack condition holds independently of N ( f , G ) .

For instance, it is sufficient to require

Of G to be connected. We shall first prove some preliminary results. 12.1.

Lemma.

Let R > O, u , v C R '~ \ B'~(R), u ¢ v , and let F be a

continuum with u,v E F . Then M (A(F, S n - l ( R ) ) ) > "~(1 + a(u,v)) where

a(u,v)

=

2 m i n { Ivl(lul - R ) , lul(Ivl - R) } R tu - "1

P r o o f . Let h(x) = Rx/Ixl 2 , txt > R . Then h ( R " \ B ~ ( R ) ) -- B '~ . By (1.5)

lh(u) - h(v)l - lu - ~l R . lu11,1 This together with the definition (2.34) yields j~o (h(u), h(v)) = log(1 + 2/a(u, v ) ) . By conformal invariance 5.17, 7.32, and 2.41(1) M(A(F, S ' ~ - I ( R ) ) ) : M ( A ( h ( F ) , S'~-1)) > ~/(th

>_ ~ t h ( ½ J B . ( h ( u ) , h ( v ) ) )

"

Because th(½ log(1 + s)) = s/(2 + s), we obtain

M(A(F, S~-I(R))) _> -~(1 + a(u,v)) as desired.

~3

1

½p(h(ul,h(~)))

154

Let f: G --~ R n be a qr mapping, let G and f G

Lemma.

12.2.

be proper

s u b d o m a i n s of R n , x E G , # C (0, ½), and let z E O f G with dy(x) =

If(x)-

z I = d(f(x),cgfG) .

A s s u m e that Ix - Yl < ½ d(x) implies If(Y) - zl ~ 0 d r ( x ) .

Then the inequality

If(x) - f(Y)l < A dr(x) - " / - l ( K ~ / ( d ( x ) / ( 2 1 x - Yl)) ) - A - 1 1 d(x) , where K = K i ( f ) holds for Ix - y[ < -~

Let Bz = B n ( x , ½d(x)).

Proof.

and A = 2(0 -1 - 1).

We m a y assume t h a t f ( x ) ~ f ( y ) .

By the

m o n o t o n e property 8.5 of # c , 10.18(1), and 8.8(2),

d(x)

_< g ~ ( 2

I¥-yl) where K = K i ( f ) .

'

Next apply 8.5 and 12.1 with R -- O d l ( x ) to get .:c(f(x),f(y))

> "~(1 + a)

where a

~-~

2min{ If(Y) - z l ( I f ( x ) - zl - R ) , If(x) - z[(If(y ) - z I - R) }

R If(x) - f(Y)! Since If(Y) - zl ~- If(x) - f(Y)t + If(x) - zl and R = Odf(x)

we

obtain

~:(~) ,f(~--_ ](y)[) .

a<2(0-'-l)(l+

This inequality together with the above ones yields

d:(x)

~'(1 + 2(0-I- 1)(1 + i f ( ~ ) - - - ~ ( y ) i ) ) < The desired inequality is now easily obtained from this. 12.3.

Corollary.

Kq(2~(X)yl)

.

O

Under the assumptions of 12.2, there exists a n u m b e r to

(0, ½) depending only on K i ( f )

and 0 such that Ix - Yt <- to d(x) implies

ff(x) - :(y)I <_½.

,~:(x)

155

Proof.

D e n o t e K = g i ( f ) . As in (7.44) let 1 ff_l(g?(1/r))

~K(r)

, r e (0, 1 ) .

We can now rewrite t h e inequality of 12.2 as

i f ( x ) - f(Y)l <

dS(x )

A~K(2lx--

yl/d(Q)

= U

- 1 - (1 + A ) ~ K ( 2 l x - yl/d(x))

where A = 2(0 -1 - 1) . Hence it suffices to require B = g1 , in other words

(3A + 1) ~K(21~- yl/d(~)) : (60 -1 - 5)~K (2Ix- yI/d(x)) = 1. In order to find a n u m b e r to i n d e p e n d e n t of the dimension we recall t h a t by 7.47(1) a n d 7.50

99K, n(t) <_ 2 1 - 1 / K K t 1/K holds for K >

1, n_>2,and

t E ( 0 , 1 ) . Hence it suffices t o choose to so t h a t

(60 -1 - 5) 2 1 - 1 / K K ( 2 t o ) U K = 1 or, equivalently, to -- (12K0 -1 - 1 0 K ) - K • Because 0 C (0, ½) we see t h a t (12.4)

(12-~)

-

<_to<_

(5)

Hence to d e p e n d s only on 0 and K as desired.

[]

After these auxiliary results we now prove the m a i n result of this section. 12.5. Theorem.

For K > 1 and $ E ( 0 , 1 )

there exists a n u m b e r c with the

following property. Let G and G ~ be p r o p e r s u b d o m a i n s of R n and let f: G ~ R '~ be a n o n - c o n s t a n t qr m a p p i n g with f G C G ~ satisfying the Harnack condition d ( f ( x ) , OG') >_ 0 d ( f ( y ) , OG') for all x , y E C with ix - Yl <- l d(x) • If K i ( f )

= K

and a

= K 1/(l-n)

k~,(I(~),I(~)) _< cmax{ k~(x,y)", kG(x,Y) for all x , y E G .

}

,

then

156 O u t l i n e of p r o o f .

The proof will be carried out in two steps.

step we choose a number t E (0,½), t = t( K, O) , such that I x - y l

If(x) - f(Y)I < ½ d(f(x),OG')

In the first

<- td(x) implies

whenever x C G . Moreover, we prove the theorem for

I x - y] <_ t d(x). In the second step we assume that Ix - Yl >- t d(x) and prove the theorem in this case by exploiting quasihyperbolic geodesics as in Lemma 4.9. P r o o f o f 12.5.

Fix x,y E G with y E B'~(x, ½d(x)) = B~. Choose Zo E OG'

such that I f ( x ) - z o l -- d(f(x), 8G') = d ( f ( x ) ) . From the Harnack condition it follows that f maps Bz into R ~ \ B~(zo, 0 d(f(x)) ). Let t = t(K, O) be the number given by 12.3. Because 0 E (0, ±) 2 we obtain by (12.4)

(12.6)


Then

I f ( x ) - f(Y)l < 1 d(f(x)) - "~

(12.7) for I x - Yl <- t d(x).

CaseA.

l x - y l <-td(x). Let B1 = B ' ~ ( x , td(x)) and B2 = B ~ ( f ( x ) , ½ d ( f ( x ) ) .

For y E B1 we obtain by 8.8(2)

(td(x) .~,

(x,y)

= ~'.lx

- yl:

"

Observe that f B 1 C B2 by (12.7). Hence the monotone property 8.5 of #G together with 8.8(2) yield

.:,,: ( S ( x ) , f ( y ) ) > ,Bo (:(x), /(Y)) = ~ ( -

d(f(x))

21:~:7(y)t)

"

Because

#:B, ( f ( x ) , f ( y ) )

< K ttB2(x,y )

by 10.18(1), the above relations yield

(12.8)

2If(x) - f ( y ) l < d(f(x))

(l~-yl~

1

- ~/-'(K'~(td(x)/lx - Yl) ) = ~ K \ t d(x) ]

where we have used the function ~K introduced in (7.44). Because Ix - Yl <- t d(x), also (12.7) holds, and hence by 3.7(1)

kG,(f(x), f(Y)) <_

[

If(x)- f(y)t

~

l°gd + ~(f(x~-17(Y) -s(y)lJ -<

2If(x)- f(y)I

4f(x))

157 This inequality together with (12.8), 7.47, and 7.50 yields

k a,(f(x), f(y)) < 21-1/KK(

(12.9)

-

Ix -- Yl'~a \t--'~(x)]

1/(1-~). It follows from 3.7(1) and (12.6) that

where a = K

1 7 I ~ l o g ~l - t - - < log 6 < 6 "

tx - Yl k G ( x , y ) < log(l-{- d ( x ) ( 1 - t ) ) Therefore we have by (3.4) for Ix - y] < t d(x) Ix

(12.10)

d(~)

Y_____<[ e x p ([k c ( x\ , y ) ) -

,

,

- 1<

kc(x,y)

-

1-1/6

-

5 6

kc(x,y ) .

Here the second inequality follows from the well-known fact that [AS, 4.2.33]

ea--1 <

a

1-a

for a < 1. By (12.9), (12.10), and (12.6) we obtain

k~, (f(~), I(u)) <_ cA k~(~, y)~

(12.11) where

CA : 2 X - U K K ( 5 / 6 ) c ' ( 1 2 K / O ) Kc" <_ 2 1 - U K K ( 1 2 K / O ) K

(12.12)

Case B.

Ix - Yl > t d ( x ) .

.

Let Ja[x,y] = J be a geodesic segment of the quasi-

hyperbolic metric k c . Choose points X l , . . . , X p + l on J as follows. Let xl = x and assume that the points x l , . . . , x j p = j,

have been chosen. If y C B n ( x j , t d ( x j ) )

we set

xp+l = y and the process of choosing points ends. Otherwise we choose Xj+l

to be the last point of J on S n - ~ ( x j , t d ( x j ) )

when we traverse from x to y along

the geodesic segment J . It follows from (3.4) that

kc(xj,~j+l) _>log(1 + t) for 1 < j _< p - 1. By the length-minimizing property of the geodesic J p--1

( p - 1)log(1 - t) <_ E

kG(xj'xi+l)

~- k c ( x l ' x P + l )

= kc(x'Y)

j=I

and hence p < 1 + k c ( x , y ) / l o g ( 1 + t ) . By the definition of the number t (see (12.6)) we see by 3.7(1) that

kG, (f(x~), f(xj+,)) < log 2

158 for all j = 1 , . . . , p .

Therefore by the triangle inequality for k G, and by (3.4) we

obtain the desired inequality k G , ( f ( x ) , f ( y ) ) < plog2 < (log2) (1 + -

-

ka(x,Y) log(1

)

+ t) ]

kc(x,y) < (log 4) log(1 + t) ' because Ix - Yl -> t d(x) in the Case B. By (12.6) the constant admits the following upper bound log4 (12K/0) K log4 3 cB -- log(1 + t) -< 71og(8/7) <-- (12K/O)K "

(12.13)

Finally, by (12.12) and (12.13) we see that in both Cases A and B we can choose (12.14)

c -- m a x { c a , CB } < 3 . 2 - 1 / K K ( 1 2 K / O ) K .

12.15. E x e r c i s e .

[]

In the above computations we applied the fact that in view

of (12.6), t < 1/7. However, it was required in 12.5 that 0 < 71 and hence t < 1/14 by (12.6). Using this fact improve the constant c in (12.14). 12.16.

C o r o l l a r y . Let f: G ~ R '~ be a non-constant qr mapping such that

f G C G ' . Then

Y: (a, ka)

(a',kG,)

is uniformly continuous if and only if the Harnack condition of 12.5 holds. P r o o f . By 12.5 it will be enough to prove that uniform continuity implies the Harnack condition. Assume that f is uniformly continuous in the above sense. Hence there exists a number D such that k c ( x ,y) <_ log(3/2) implies k G,(I(x), f ( y ) ) <_ D.

It follows from (3.4) that I x - Yl < ½d(x) for k a ( x ,y) < log(3/2). Hence for

Ix - Yl <- l d ( x ) we obtain by (3.5)

log d(f(y)) t -< kc,(f(x),f(y)) < D where d ( f ( x ) ) = d ( f ( x ) , O G ' ) . Thus the Harnack condition of 12.5 is fulfilled with 0 =e

-D

.

We next show that p-to-one qr mappings satisfy the Harnack condition of 12.5.

159 12.17. T h e o r e m .

Let G and G I b e p r o p e r s u b d o m a i n s o f R '~ and let f: G--+

R ~ beaqrmappingwith

fGCG'

and N ( f , G ) < p < ~ .

Then forall x, y E G

with I~- yl < ½d(z) d ( f ( x ) ) ~_ [1 + r - ' ( A r ( 1 / 2 4 ) ) ] d(fCy)) where d ( f ( x ) ) = d ( f ( x ) , O G ' ) and A = 1 / ( 8 p K o ( f ) ) . P r o o f . We may assume that d ( f ( x ) ) > d ( f ( y ) ) . Because If(x) - f(Y) I d(f(x)) min{ dCf(x)), d ( f ( y ) ) } > d ( f ( y ) ) - 1 by the triangle inequality, Corollary 8.25 yields

~c' (f(x),f(y)) <

:d(f(x)) d(f(y))

4r\

1).

It follows from 8.6(2) and (2.17) that

~B- (0, z) _> ~1 z(sh2(½ log ~)) > t I"(1/24) for all ]z[ < ½. Denote Bz = B n ( x , ½d(x)). Then

~ ( ~ , y ) > ~.~(x,y)> ½ ~(1/24) by 8.5 and the above inequality. The desired inequality follows now from 10.18(2). O 12.18. E x e r c i s e .

Applying the functional identity -¢(t) : 2'~-lT(t 2 - 1) of 5.53

show that l+r-l(Mr(t))

= [W-I(Mw( 1 ~ ) ) ]

2

for all M > 0 and t > 0. Next show that the constant in 12.17 has an upper bound in terms of P K o ( f ) . 12.19.

[Hint: Apply 7.51.]

Corollary.

Let f: G - + f G

be a qc mapping where G and f G are

proper subdomains of It n . Then k/G ( f ( x ) , f ( y ) ) < c max{ ka(x, y ) a , k c ( x , y) } holds for all x , y E G where ~ = K r ( f ) 1~(l-n) and c depends only on K o ( f ) .

160 P r o o f . By 12.17 and 12.18 the Harnack condition of 12.5 holds with a dimensionfree constant 00 • The proof follows now from 12.5.

O

12.20. C o r o l l a r y . Let f: G - + f G be a K - q c mapping, where G and f G are

proper subdomains of R '~ . Then

kfG (f(x), f(y)) ~ c I max{ ka(x, y)1/K, ],gG(Z,y) } holds for all x, y E G where C1 depends only on K . P r o o f . Because K > K o ( f ) with K o ( f )

and because the constant c of 12.19 increases

we can make c independent of K o ( f ) by replacing K o ( f )

with K .

This yields a new constant cl depending only on K with Cl _> c . Because a =

K i ( f ) 1/(1-n) >_ 1 / K we obtain max{ k c ( x , y ) a, k c ( x , y ) } <_ k c ( x , y ) 1/K for k c ( x , y ) < 1 and max{ k c ( x , y)~, k c ( x , y) } = k c ( z , y) for k c (x, y) _> 1. The desired dimension-free inequality follows.

Z]

It follows from Example 11.4 that Corollary 12.19 does not hold for qr mappings and not even for analytic functions. However, if Of G satisfies some additional conditions, then 12.19 can be generalized to qr mappings. Next we shall prove such a result when Of C is connected. 12.21. T h e o r e m .

Let f: G -+ R n be a non-constant qr mapping and let Of G

be a continuum containing at least two distinct points. Then k~c ( f ( x ) , f ( y ) ) < c~ max{ k c ( x , y) ~, k c ( x , y ) }

for all x , y C C where c2 depends only on n and K z ( f ) . P r o o f . Let x , y E G with I x - y[ <_ ½d(x). By 10.18(1) and 8.8(2) we obtain

ic(f(x), f(y)) < KI(f) uG(x, y) _< KI(f) q(2). Further, in view of (3.5) and 8.31(1)

cn l o g d ( f ( x ) )

-< # f G ( f ( x ) , f ( y ) ) .

161

F r o m these inequalities it follows t h a t d ( f ( x ) ) = d ( f ( x ) , O f G )

satisfies the H a r n a c k

condition of 12.5 w i t h 0 ----e x p ( - K , ( f ) Hence the p r o o f follows f r o m 12.5.

~(2)/cn) .

[]

It should be o b s e r v e d t h a t T h e o r e m 12.21 is applicable to qr m a p p i n g s also when

N ( f , G) -- oo. 12.22.

Let f: G ~ R '~ be a non-constant K - q r mapping and let

Theorem.

D e J ( G ) (for n o t a t i o n see 9.10). Then kiD (S(x), S(Y)) -< c(D) m a x { k D (x, y)•, k D (x, y) } holds for all x , y E D where a = g 1/(i-n) and c(D) depends only on K N ( f , D ) . Proof.

It follows f r o m 12.17 and 12.18 t h a t

d(f(x),OfD)

nack condition of 12.5 in D with a d i m e n s i o n - f r e e c o n s t a n t

K N ( f , D) . T h e p r o o f follows now f r o m 12.5. 12.23.

h(z)

~--Z p ,

Example. p

=

For n = 2 and

2,3,...,

z e C.

satisfies the Har-

0 d e p e n d i n g only on

[]

K = 1 consider the analytic functions

T h e points a v = ½ and bp = ½exp[(zc/p)i] are

m a p p e d by fp onto a vt = 2 - p a n d b'v = --2 - p , respectively. L e t D = B 2 \ { 0 } =

f p D . By (3.4) kzp.(a

,b

) >

' by) ' = log 3 :> 1 , (av,

kv (av, bp) <_ 7r/p

,

where the last inequality follows by integration along the circular arc { z C C : z = 12 e x p ( t i ) , 0 < t < ~r/p } (see the definition (3.2) of the quasihyperbolic metric). By T h e o r e m 12.22 1 < log3 _< c(D)max{(Tr/p)

I, 7 r / p }

:

c(D)zc/p

a n d hence

c(D) >_ p / r >_ N ( f p , D ) / ~ r . In p a r t i c u l a r , we see t h a t c(D) --~ cc as N ( f , D) ~ co in 12.22. 12.24.

Corollary.

Let f: B n --* Y ,

Y = R n \ {0}, be a qr mapping with

N ( f , B '~) < o o . Then f : (B '~,p)

, (V, k y )

162

is uniformly continuous. In particular, f : (Bn, p)

~ (R,~, q)

is uniformly continuous. P r o o f . Theorem 11.24 shows that the Harnack condition of 12.5 is fulfilled and hence the first assertion follows from 12.5. The second assertion follows from the first one (see 3.31).

[=]

12.25. E x e r c i s e . Show that 12.20 yields a bound for the linear dilatation of a K - q c mapping. [Hint: Apply 12.20 to G \ {x}, x E G .] 12.26. R e m a r k .

(1) Let ~ denote the least constant with which 12.19 holds.

As shown in [AVV2] the following inequalities hold

l+

log2

(g)

_<

_< 2 K

8K ,

where A(K) is as in 10.31 and K = K ( f ) . (2) The condition in 12.21 that Of G be a non-degenerate continuum can be replaced by the requirement that Of G be sufficiently thick at each of its points in a sense involving n-capacity. See [VU12]. (3) This section is taken from [VU10]. Corollary 12.19 is due to F. W. Gehring and B. G. Osgood [GOS].

13.

Normal quasiregular mappings

The properties of bounded analytic functions of the unit disc have been studied extensively in classical function theory. In their fundamental paper [LV1] of 1957 O. Lehto and K. I. Virtanen proved that many boundary properties of bounded analytic functions, or more generally of meromorphic functions omitting at least three distinct values in the extended complex plane, have natural generalizations to a wider subclass of meromorphic functions, namely the normal meromorphic functions. This class of functions is very convenient to study because of its invariance properties. The notion of a normal meromorphic function also provides a natural setup for the study of the Schwarz lemma and the Schottky theorem as well as their many ramifications.

163

For a bibliography of normal m e r o m o r p h i c functions the reader is referred to A. J. Lohwater's survey [LOH] (see also [PO]). The goal of this section is the s t u d y of some growth properties of normal quasiregular mappings.

In the case of m e r o m o r p h i c functions there are several equivalent

characterizations of normal functions, of which we mention here only three: namely one based on the s t u d y of normal families, one based on the notion of the spherical derivative, and finally one making use of uniform continuity between a p p r o p r i a t e metric spaces. Of these the last one seems to be the most natural definition in the present context, since we are interested not only in knowing whether a function is normal but also in estimating its modulus of continuity. 13.1. D e f i n i t i o n .

A continuous m a p p i n g f: B n --+ ~ n

is said to be normal if

wy(t) -+ 0 as t --+ 0 where wf(t) :sup{q(f(x),f(y))

Then q ( f ( x ) , f ( y ) ) definition.

< wf(p(x,y))

: x, y E B '~ and p ( x , y ) <_t} .

holds for all x , y C B " by virtue of the above

In other words, f : B = --* R ~ is normal if and only if f

is uniformly

continuous as a m a p p i n g between metric spaces f : (B n, p) ---+ ( R n, q). 13.2. R e m a r k s .

(1) Since the hyperbolic metric (2.21) is a conformal invariant

it follows t h a t f is normal if and only if f o g this case w f = Wyog.

is normal for each g E ~M(Bn ) .

If f : B '~ --+ R.'~ is normal and

In

h E ~ ( R ' ~ ) , then so is

h o f and Whof < Lip(h) w f where Lip(h) is the Lipschitz constant in the spherical metric. In particular, Whof = w f if h is a spherical isometry. It follows t h a t in most cases we m a y assume t h a t f(0) -- 0 , by considering tf(o) o f

in place of f , where

tf(o) is the spherical isometry defined in (1.46). As we shall see in 13.4, from these invariance properties and from the Schwarz l e m m a 11.2 it follows t h a t w f ( t ) < c t ~ , a -- K 1/(1-'~) , if f : B n --~ ~ n

is K - q m and normal.

(2) T h e above definition of normality extends immediately to the case of functions defined in an a r b i t r a r y proper s u b d o m a i n G of R ~ , if we use k G in place of p . It should be observed t h a t if G is a multiply connected plane domain then this definition of a normal function is not the same as the definition in [LV1]. 13.3.

Exercise.

Assume t h a t f : B " -+ R'~ is normal and fx E ~ ( R '~) with

f l ( x ) = xA-a. Show t h a t ~dflof(t ) < [1A-½1aI(laI+v/4 + l a l 2 ) ] w / ( t ) . [Hint: 1.54(4).]

164

Assume f l ( x ) = a + r2(x - a ) / l x -

Find an u p p e r bound for Wf~o/(t ) . [Hint:

a[ 2 .

1.54(2).] By virtue of the results in the preceding sections we see t h a t the class of normal qm m a p p i n g s is wide. Several examples will be given in 13.7. The following result m a y be viewed as a generalization of the Schwarz l e m m a to the context of normal functions. 13.4.

Theorem.

Let f: B ~ --~ R'~

be a normal q m m a p p i n g and let M =

1

s u p { t : w f ( t ) = 7 }" T h e n

q(f(x),f(y)) < a(.,K)(th-lfll(~z~Y)) c~ o~= Ki(f) I/(l-n) th 7 M

-

for aU x , y ~ B ~ where a ( n , K )

'

= max{l, ;~

'

/x/3}

and , ~

is the Grhtzsch

constant (7.24).

Proof.

Since a ( n , K )

B ~ with p ( x , y ) < M . may assume t h a t

> 1, the assertion is trivial if p ( x , y ) > M .

Fix x , y E

In view of the conformal invariance of the right side, we

x = 0 and y = ( t h l p ( x , y ) ) e l

(see (2.25)).

Because the left

side is invariant under spherical isometries we may also assume f(0) = 0. fD(O,M)

Hence

C Q(0, 1) = B,z(1/v/~) by 1.25(1). Denote hi : B n ---* D ( O , M ) , h i ( x ) = x t h g M1 , h2 : B n ( v ~ )

~

B " , h2(x) = x V ~ .

T h e n g = h2 o f o hi: B n --* B ~ is qr with K l ( g ) = K i ( f ) , g o ( g )

= Ko(f),

g(0) = 0 and thus by 11.3(1) and 1.17 v/3q(f(Y),f(O))

<- V ~ I f ( Y ) - f(O)t <_ l g ( y / t h ½M) - g(O)l

(th½p(x,y)

-< "k~-a\

th½M

/

where a = K x ( f ) . This is the desired inequality. 13.5.

Corollary.

A q m m a p p i n g f: B '~ --~ R '~ is n o r m a l if and only if there

are n u m b e r s ~ ~ (O, 11 and Z > 0 such that ~ : ( t ) < Z t ~ rot all t C If f : B '~ ~ R

'~ is continuous, then the set Et = { z E B

(0,~).

'~ : I f ( z ) t = t } ,

t >0,

is called the t-level set of I f l . We are next going to give a geometric characterization

165

of a normal qr mapping which requires that the oscillation of the mapping "near" a level set is bounded. It should be observed that the hypothesis S ~-1 N f B '~ # 0 in the following theorem is merely a technical normalization: if it fails to hold, then f omits a ball of R~ of spherical diameter = 1 and hence f will be normal by virtue of 11.1. 13.6. T h e o r e m . Let f: B n --~ R n be a non-constant qm mapping with S ~-1A

f B n ¢ 0 and let E = { z E B n : If(z)] < 1 }. Then the following conditions are equivalent: (1)

f is normal.

(2)

There exists a positive number T such that If(z)[ _< e whenever z E B ~ \ E and p(z,E) <_ T .

P r o o f . Since the implication (1) =~ (2) is obvious, only (2) =~ (1) remains to be proved. Fix x , y E B ~ with p(x,y) < 1 T . We consider two cases.

Case 1. p(x, E) < ½T. In this case flD(x, ½T) omits the set F1 = R= \ B'~(e) by the hypothesis of the theorem and

e(F1) > C ( F l ' ° ° ) - t°n-1 (log(ev/'3)) 1-n : d l dl dl by (6.13) and 6.14.

Case 2. p(x,E) > ½T. In this case f l D ( x , ½T) omits I3" (i.e. f D ( x , ½ T ) n B '~ = 0 ) and

c(Bn, 00) c(Bn)

->

dl

--

¢°n-l(logvT~)l-n dl

a2 .

In both cases we apply 10.18(1) to f I D ( x , ½T), and we obtain by (2.24) and

s.s(2) / th I T #D(=,T/2) (X, y) = "t ~ th ~-¢~, y ) ) " Because dl < d2 we obtain by 6.1 in both Cases 1 and 2

/Zfo(~,T/2)(f(x),f(y)) >_ flmin{ d l , d3 q ( f ( x ) , f ( y ) ) } > flmin{ d l , d3} q(f(x), f ( y ) ) . This together with the previous inequality, 7.26(1), and 10.18(1) shows that f is normal.

C3

166 13.7.

Examples.

We now list some sufficient conditions for a qr mapping

f: B '~ --~ R ~ to be normal: (1)

c(~t n \ f B '~) > 0 (see 11.1).

In particular, an injective qr mapping of

B n (i.e. qc mapping) is normal, because c ( b f B n) > 0 by 14.6(1) and 6.1. Likewise, bounded qr maps are normal. (2)

f B '~ C G , where G is a proper subdomain of R n and d y : B " --~ R + , dr(x) -- d ( f ( x ) , OG) satisfies the Harnack inequality (see 12.5 and 3.31).

(3)

f : (B n,p) ~ (R n, 1 l) is uniformly continuous (see also 16.12).

The above sufficient condition 13.7(1) for a qr mapping to be normal may be much refined. As the following important theorem of S. Rickman [RI10] shows it suffices to assume that c a r d ( R n \ f B n) exceeds a sufficiently large finite number p(n, K ) depending only on n and K . The next result is a qr variant of the Schottky theorem, which has a fundamental role in classical complex analysis IT, p. 268], [A3, p. 19], [NE, p. 62]. Some applications of this result are given in [VU14]. 13.8.

T h e o r e m ([RI10]). For n > 3, K _> 1 there exists p = p ( n , K ) such

that every K - q m mapping f : B n - - ~ R ' ~ \ { a l , . . . , a p } ,

where ai # ai for i # j , is

normal. Moreover, if oc q~ f B n , then log + [f(x)l < Co ( - l o g s 0 + log + ]f(O)l) (1 - Ix[) -C 1 where log+t = l o g m a x { 1 , t } , so = -~ min{ q(ai,aj) : 1 <_ i , j <_ p , i ~ j } , and C

and Co are constants depending only on n , K , and so. We are now going to prove that every normal qr mapping satisfies a growth condition similar to the one in 13.8 and that the constant C can be chosen to depend only on the dimension n and the maximal dilatation K . The proof of such a growth inequality can be based on the Harnack inequality. 13.9. R e m a r k .

Let u: H 2 --~ R + be a harmonic function. By a well-known

property of positive harmonic functions u satisfies the Harnack inequality (4.11) with C8 _< (ll-~_,s)2 for each s E (0,1) (see e.g. [GT, p. 281). If f : G E = {z E G : If(z)[ < 1}, and G \ E

--+

R n

is K - q r ,

-# 0, then log If] satisfies (4.11) in each

component of G \ E with a constant Cs depending only on n , K , and s (see [SE], [MOS], JR12, pp. 232-239], and [GR]).

167

By virtue of T h e o r e m 13.6 a qr m a p p i n g satisfying the hypothesis of the following t h e o r e m is normal. 13.10. T h e o r e m .

Let f : B n -~ R '~ be a qr mapping with 0 E E = { z E B ~ :

If(z)] _< 1 } and suppose that there exists a positive number T such t h a t p(x, E) <_ T

implies [f(x)l <_ e. Then there are positive numbers ~ and ~ , of which "~ depends only on n and K ( f )

and ~ also on T , such that

l+lxl for all x E B • . Proof.

We m a y assume t h a t B n \ E ~ 0 , since there is nothing to prove if

E -- B n . For x C B '~ \ E let u(x) = log I f ( x ) l . We wish to find an u p p e r bound for

If(z)l when z is a fixed prescribed point. Case 1. z C B ' ~ \ E t h a t p(z, zl) = M .

and u ( z ) > 1. Now p ( z , E ) - - - M

Clearly u(x) > 0 for x E D ( z , M )

u satisfies the Harnack inequality (4.11) in D ( z , M ) only on n , g ( f ) ,

>T.

Fix Zl e E such

and it follows f r o m 13.9 t h a t

with a constant C8 depending

and s E ( 0 , 1 ) . Denote

F = { y C B ~ : u(y) < 1 } ,

FI -- F (~ J[z, Zl].

Select z2 E F1 with p(z, z2) = p(z, F1) • It follows f r o m the hypothesis of the theorem t h a t p(z2,zl) >_ T , while z2 E J[z, zl], 0 C E , implies t h a t (see (2.17)) (13.11)

1 + tzi

p(z, z2) < p(z, zl) - T < p(z,O) : log 1 - Izl

Denote by ~" the hyperbolic metric of D ( z , M )

(see 4.25). We are next going to

apply 4.12 to u t D ( z , M ) and to the points z and z2 • For this purpose we need an u p p e r b o u n d for ~'(z, z2). To find such a bound, m a p z to 0 by Tz (see 1.34). In order to avoid notational ambiguity denote h = Tz. From the conformal invariance of p it follows t h a t h m a p s D ( z , t ) onto D(0, t ) , for each t > 0 . Hence by (2.24) and (2.25) we see t h a t

hCz2) e S '~-1 (th ½p(z, z2)) , hDCz, M ) = B '~ (th ½pCz, zl)) •

168

Diagram 13.1. T h e proof of T h e o r e m 13.10. In view of the conformal invariance of ~" it follows t h a t (see (2.17) a n d 4.25)

ZD(z,M)(Z,Z:) =ZD(O,M)(O,h(Z:)) (13.12)

1÷ r : log - ;

1-r

D e n o t e !p :

r -

½p(z, z2) th½P(Z, Zi) th

l p ( z , z , ) , T = lp(z, z2). Because ! o - r _> ½T w e o b t a i n by (13.11),

(13.12), a n d 2.29(1) the inequalities l+r 1- r (13.13)

th~o + t h z thto- thT 1 < th(~o - ~)

th(!p + ~) th(tp-T)

1 + th!othT 1-th!othT

1 + th io th r 1 - th 2 ~o

1

< ~th- - "~T

By 2.29(3), (13.12), (13.13), and (13.11) we get (13.14)

p(z, z2) < p(z, zl) + 2 a r t h e - T .

Because 0 E E it follows f r o m the choice of Zl t h a t

(13.15)

p(z, zl) <_ p(z,O)

1 + Izl = log -[z------1 T"

1 e 2~ 1th------~ < ~th " ~T

169 Since u satisfies the Harnack inequality, (13.14) and (13.15) together with 4.12 imply that

l+lzt

u(z) <_ Csu(z2) exp["/'~(z,z,)l < I3(~). (13.16) ~=Cs

(1 +e-T'~"t l_e_T] ,

/ "~=(logC~)

.

;

l+s lOgl_ s

Note that u(z2) = 1 by the choice of z2. In Case 1 the desired inequality follows from (13.16). Case 2. z E B n \ E

and u(z)_< 1 or z C E .

In this case If(z)]_<e and hence

the assertion is trivial. We shall next give an alternative proof for 13.10 in the particular case of bounded mappings, since in this case the proof is very short. 13.17.

C o r o l l a r y . Let f: B " -~ R '~ be a non-constant qr mapping with

f B ~ c B ~ , yo E Bn \ f B ~ . Then

(1)

I:(z) - u01 > 2exp(-A C1 + tzlh -

\ _1- - Z - ~ ]

~)

holds for z e B " where A > 0 and q = ( l o g C , ) / t o g ~_~. Moreover, (2)

1

where 6 = Kx(/)

-II(z)l

>_

~ ~--S-~)

a n d a > O. F . r t h e r m o r e ,

i : ,~ = 2 a n d / ( 0 )

(3)

1 - I f ( z ) l > 4,_~ ( ~ ) 1 + If(z)l for all x E B " where 8 = K I ( f ) . P r o o f . (1)

Denne v: B" -+ It+

= 0 then

8

by v(x) = - l o g ( ½ 1 f ( x ) - ~0[) for z ~ I3 =. It

follows from 13.9 that the function v satisfies the Harnack inequality (4.11) in B ~ with a constant Cs • Now 4.12 yields v(x) < v(O)Cs exp('yp(0, x)) where ~ = (log C s ) / l o g ~_8s . The desired bound with A = v(O)C8 follows. (2) This was proved in Exercise 11.9 with a = 2-2Kz(Y)(1 - I f ( 0 ) [ ) . (3) By 11.3, 5.61(3), and 7.47(2)

~,2(Iz{)

1 -4-lY(z)l < 1 + _ 1/991/8,2(1 - { x l " ~ 1 - I f ( z ) { - 1 - ~,2({z{) \ 1 + {z{] ~1 -Izl: which yields the desired inequality.

170

The above results 13.10 and 13.17 depend on a p a r a m e t e r s which can be chosen arbitrarily in (0, 1) . 13.18. R e m a r k . The exponential function g: B 2 ~ B 2 \ { 0 } , g(z) -- e x p ( ~ ) , z E B 2 , shows t h a t an exponential rate of decrease in 13.17(1) can be attained, even by analytic functions.

g is not uniformly

Recall t h a t it was shown in 11.4 t h a t

continuous as a m a p of (B ~,p) into (G, k c ) , G -- B 2 \ {0}. 18.19. E x e r c i s e .

Let G be a proper s u b d o m a i n of R n and let f : B n --+ R n be

a qr m a p p i n g such t h a t f : (B~,p) --+ ( G , k c )

d(f(x),OG). and K / ( f ) ,

is uniformly continuous. Let d l ( x ) =

Show t h a t dy(x) has a lower b o u n d in terms of dy(O), p(O,x), n , similar to t h a t in 13.17(2). [Hint: Observe t h a t dy satisfies the Harnack

condition (see 12.5 and 12.16). Next apply 4.12.] We shall next give some corollaries to 13.10. T h e first one is a Picard t h e o r e m for qr mappings. For the s t a t e m e n t of this result we call a q m m a p p i n g f : R n --+ R n of the entire space R n normal if there exists a function wI : (0, co) -+ (0, co) such that w f ( t ) - + O

as t--+O and q ( f C z ) , f ( y ) ) <_~ : ( p R ( x , y ) )

for all R > 0 and x , y C B'~(R) where PR is the hyperbolic metric of B'~(R) (cf. 4.25). Equivalently, f : R '~ ~ R n is termed normal if

q ( : k ( x ) , f k ( y ) ) <_ w y ( p ( x , y ) ) for all z , y E B '~ and all k > 1 where fk(z) -- f ( k z ) , 13.20.

f:

R n -+ R n

Theorem

z EBn.

(Picard's theorem for qr mappings). A n o r m a l qr mapping

is a constant. In particular,

if

c a r d ( R n \ f R n) is at least the number

p of 13.8, then f is a constant. Proof.

Without loss of generality we may assume t h a t

R " \ B n . Applying 13.10 to f]Bn(2tzl)

If(0)] _< 1.

Fix z E

we see t h a t

tf(z)l < exp(3"Z + 1) where q depends only on n and K ( f ) for all z E R n \ B

and fl also on w I .

This inequality holds

n and hence, by the m a x i m u m principle, for all z E R n. Thus

f is bounded, in contradiction to T h e o r e m 11.15. T h e second p a r t follows from the Schottky t h e o r e m for qr maps, T h e o r e m 13.8.

O

171

By a deep recent result of Rickman [Rill] the n u m b e r q in 13.20 tends to oo as K - - * c~ if n = 3 . 13.21.

Lemma.

Let f : B'* -* R n be a n o r m a l q m m a p p i n g , tet (bk) be a

s e q u e n c e in B '~ w i t h b k ~ b E O B " , let f ( b k ) ~ y E R n \ f B ' * M > O. T h e n

Proof.

as k --* oo , a n d let

f ( x ) --* y as x --+ b a n d x E E = U D ( b k , M ) .

By performing a preliminary sense-preserving MSbius t r a n s f o r m a t i o n if

necessary, we m a y assume t h a t y = oo. Assume now t h a t the result is false. T h e n there exists a s e q u e n c e (ak) in E with ak--~ b as k--* oo and I f ( a k ) l < A < oo for all k = 1 , 2 , . . .

and some A .

Let gk E JM(B") with gk(O) = ak (see 1.34).

T h e n f o gk is normal in the sense of 13.1, Wfog~ = w f

in view of 13.2(1), and

I ( f o g k ) ( O ) ] / A _< 1. By passing to a subsequence and relabeling if necessary we may

assume t h a t p ( a k , b k )

< M

for all K . It follows from 13.10 t h a t

I f ( b k ) l / A = I f ( g k ( g - ~ l ( b k ) ) ) l / A <_ C < oo

for all k where C does not depend on k .

This inequality yields a contradiction,

since f ( b k ) --* oo as k --~ c ¢ . Hence the antithesis is false and the result is proved. O In the case of normal meromorphic functions of the unit disk in C , L e m m a 13.21 can be deduced also from Hurwitz' theorem.

An alternative proof of the n -

dimensional result 13.21 can be based on the notion of the local topological index and on n o r m a l families (see [VU3, 6.3]). The hypotheses of L e m m a 13.21 can be much weakened. This appears from T. Kuusalo's recent result [K2], which shows t h a t Hurwitz' t h e o r e m (and hence also 13.21) holds for discrete open normal maps. Note t h a t quasiregularity is not needed here. For a similar result see G. T. W h y b u r n [WIll]. 13.22. R e m a r k s .

T h e hypothesis V E R n \ f B

n in 13.21 can be replaced by the

slightly weaker requirement t h a t N ( 9 , f , B n) < oo. T h e proof of such an extended version of 13.21 is left as an exercise for the reader. We now give an example to show t h a t the hypothesis 9 E ~ n \ f B n cannot be entirely dropped. For this purpose we consider the b o u n d e d analytic function g: B 2

--+ B 2

\ {0}, g ( z ) ~- e x p ( ~ ) .

Fix

a E B 2 \ {0} and choose a sequence (bk) in B 2 with bk --* el and g(bk) = a for all k = 1,2,....

By studying the properties of g we see t h a t g ( z ) 5/~ a as z --~ el and

172

z E [.J D ( b k , 1), i.e. the conclusion of 13.21 fails for this function g if a E B 2 \ {0}.

Therefore the assumption y C O r B '~ cannot be dropped from 13.21. 13.23. E x e r c i s e . In the particular case when O f B '~ is a non-degenerate continuum, one can deduce 13.21 by applying the Kl-inequality. Give the details. [Hint: Apply 11.5(1).] 13.24.

N o t e s . This section is taken from IVU10]. For 13.21 see [BS]. An account

of Schottky's theorem can be found in [BU, Ch. XII]. In the particular case of analytic functions, 13.17 can be found in [HM].

Chapter IV BOUNDARY BEHAVIOR

In the present chapter we shall study the behavior of qc and qr mappings near the b o u n d a r y of the domain of definition. There is a fundamental difference in the study of these two classes of mappings: in the case of qr mappings the maximal multiplicity of the mapping may be infinite even in every neighborhood of a given b o u n d a r y point. Consequently, one cannot apply the important K o - i n e q u a l i t y to the qr case in the same way as to the qc case because of the presence of a multiplicity factor which may be infinite. Many differences in the theories of qc and qr mappings are more or less directly connected with this fact. The fundamental problem which we are going to study in this chapter is the following. Problem.

Let f : B '~ --* R '~, n_> 2, be a qc or qr mapping, b E 0 B '~, and

let E c B '~ be a s e t with b E E and f ( x ) --* a as x--~ b, x E E .

Under which

conditions on f and E is a in fact an angular limit of f at b? By Lindel6f's well-known result this is the case if f is a bounded analytic function and E is a curve terminating at b. We shall show by exploiting the K o - i n e q u a l i t y that if E is thick enough at b in the sense of n - c a p a c i t y and if f is qc, then f has an angular limit a at b. We give an example to show that the thickness condition is in a sense best possible. We shall also discuss the case where f is qr. Under the additional assumption t h a t f be Dirichlet-finite we shall extend the above result a b o u t qc maps to the case of qr mappings. We shall investigate also some other properties of Dirichlet-finite qr mappings.

174

14.

Some

properties

of quasiconformal

mappings

We shall introduce some notation and terminology, useful in the discussion of b o u n d a r y behavior, and then prove some results a b o u t qc mappings. T h e presentation is aimed to be self-contained also in this chapter. Those readers who wish to find some background, motivation, or further results on b o u n d a r y behavior of analytic functions are referred to [CL], [NO], [LOH], or [PO]. Of these, [LOH] contains an extensive bibliography. For the b o u n d a r y behavior of qc mappings see IN1] and IN3]. 14.1. D e f i n i t i o n .

Let f: H ~ --+ R n be continuous. The m a p p i n g f is said to

have

(1)

a sequential limit a E R." at 0 if there exists a sequence (bk) in H " with bk --~ 0 and f(bk) --* a ;

(2) an asymptotic value a E R "

at 0 if there exists a curve ~/ : [0,1) --* I-In ,

t e r m e d an asymptotic curve, such t h a t -~(t) --* 0 and f(.~(t)) --~ a as t --~ 1 ; (3) an angular limit a E R n at 0 if, for each ~ E (0, ½~r), f ( x ) a as x tends to 0 in C ( ~ )

approaches

(for the definition of the cone C ( ~ ) see the

definition preceding T h e o r e m t 1.17); (4) a l i m i t a E R as x - + 0

~ at 0 t h r o u g h a s e t

E,if

0EECI-I

~ U { 0 } and f ( x ) - - * a

and x E E .

T h e set C ( f , b) of all sequential limits of f at a b o u n d a r y point b is t e r m e d the

cluster set of f at b. If A c 0 H n is n o n - e m p t y , we denote C ( f , A ) = ObEA C ( f , b ) . In the literature an angular limit is sometimes called a n o n - t a n g e n t i a l or (if n _> 3 ) a conical limit. It is clear t h a t C ( f , b) is always a c o m p a c t n o n - e m p t y subset of f H n . From the well-known fact t h a t o n e - t o - o n e mappings preserve open sets, it follows t h a t C ( f , b ) C Of H " for o n e - t o - o n e m a p p i n g s f (see also 9.12). 14.2.

Remarks.

(1) The cluster set of f : H '~ --* R'~ at b E a H '~ can

alternatively be defined as

c(f, b) = N f(u n H-) u

175 where U runs through all neighborhoods of b. From this definition it follows that

C(f,b) is connected whenever f is a continuous mapping of H '~ . (More generally, C(f,b) is connected if f is defined on a domain G which is locally connected at b • OG [CL, p. 3].) (2) For x • R

'~, e > 0 , 1 e t

E~ : { z • G :

q(f(z),x) < e } .

AsecondequivaIent

definition of C(f,b) is

c(/,

b) : { z • Kin: b • ~

for all ( > 0

}.

(3) It is clear that q(f(x),C(.f,b)) ---+0 as q(x,b) ~ 0 , x • I I a . In particular,

C(f,b) = {b'} iff f h a s a l i m i t b' at b. 14.3. E x a m p l e s .

(1) By the Riemann mapping theorem there exists a confor-

real mapping f : H 2 -~ D , where D is as pictured.

D A

Diagram 14.1. It follows from the theory of prime ends (cf. [CL], [D], [PO], [NO]) that the segment A corresponds to a single point b E 0 H 2 under f , i.e. C(f,b) = A . In this case f has no asymptotic value, hence no angular limit at b. (2) A conformal mapping f: H 2 -~ G with C(f,O) = B , where G is the domain and B the segment pictured, has an angular limit but not an ordinary limit at 0 (cf. [CL], [O]).

G

B

Diagram 14.2.

176

(3) It seems difficult to give an explicit expression for the conformal m a p p i n g f in (2). We now exhibit an example of a real-valued function with properties s o m e w h a t analogous to those of (2), i.e. this function will have an angular limit but not a limit at a b o u n d a r y point. Let u: H 2 -* (0, oo) be defined by

u(x,y) = (x 2 + y2)/y for

(x,y) E H 2 . T h e n u has no limit at 0 , in fact C(f,O) .... [0, c c ] , b u t it does have an a n g u l a r limit 0 at 0 and an a s y m p t o t i c value c > 0 t h r o u g h the circular arc 1 2 , x > O} . {(x,y) ell2: x2+(y½c) ~ : ~c (4) T h e h a r m o n i c function u: H ~ --* (0, 7r),

u(x, y) : arctan(y/x), (x, y) e H 2 ,

has a c o n s t a n t value on each ray in H 2 e m a n a t i n g f r o m the origin. (5) T h e function v: H 2 ~ [0, 1],

v(x, y) = s i n 2 ( 1 / V / ~ + y2 ) has no a s y m p t o t i c

value at 0 , hence no angular limit at 0 . T h e function vl: H 2 --+ [0,11, v l ( x , y ) =

v(x,y)y/v/-~-ff4_ y2 has an a s y m p t o t i c value 0 at 0 b u t no angular limit at 0 . A conformal m a p p i n g of H 2 m a y fail to have an angular limit at a b o u n d a r y point b E 0 H 2 (cf. 14.3(1)). However, the set of all such points of 0 H 2 is very small; it is of c a p a c i t y zero by Beurling's t h e o r e m (see 14.7 below). A b o u n d e d analytic function of the unit disc B 2 has an angular limit at each point of 0 B 2 except possibly for a subset of 0 B 2 of linear measure zero (Fatou's t h e o r e m [CL, p. 17]). A set E C I-I n is said to be E C C(~) E¢C(p) 14.4. a at 0. f

non-tangential at 0 , if 0 E E C I-In L 3 { 0 } and

for some ~ E (0, 7~r), 1 and

tangential at 0 if 0 C E C

H ' ~ U { O } and

for each ~ C (0,½rr). Remark.

Suppose t h a t a m a p p i n g

f : I-I n --÷ ~ n

has an angular limit

It follows almost immediately from the definition of an angular limit t h a t

approaches

a

not only t h r o u g h each n o n - t a n g e n t i a l set b u t also t h r o u g h a set

E which is tangential at 0 . In fact, by the definition 14.1(3), for each k = 1 , 2 , . . . there exist r k , rk+l E (0, ½rk), such t h a t x E E k , Ek

=

C(2(__(~_TU) A~k

B'~(rk) , implies

q(f(x), a) < 1/k. Clearly f approaches a t h r o u g h the tangential set E = U~°_I E k . However, the "degree of c o n t a c t " between E and O H n d e p e n d s on f . In the preceding discussion we have considered r e a l - or v e c t o r - v a l u e d continuous m a p p i n g s of H n . Some of the above definitions have n a t u r a l c o u n t e r p a r t s for m a p p i n g s of an a r b i t r a r y d o m a i n G in R ~ , which we shall use if necessary.

Now

we are going to consider n o r m a l m a p p i n g s (cf. 13.1). T h e following l e m m a should be c o m p a r e d to 13.21.

177 14.5.

Lemma.

Let f : l i ' ~ -* R n be normal, let bk E t t "~, bk -4 O, a n d l e t

f(bk) -4 fl. For every e > 0

there exist M e

q(f(~),Z) <E

for

(O,c~) and p >_ l such that

• e E u = U D(bk,M). k>_v

Moreover, i[ there exists an angle ~ E (0, ½~r) such that bk E C ( ~ ) for all k , then

m,([0, bk] A EM) >_ d ( ~ , M ) > 0. P r o o f . The first part follows from the definition of a normal function. For the second part note that B'~(bk,bkn(1 -- e - M ) ) C D ( b k , M )

by (2.11), where bkn is the nth coordinate of bk. Since bk E C ( ~ ) , we see that bkn :> Ibkl cos ~ , and the assertion follows with d(~, M ) = (1 - e T M ) cos ~ . 14.6.

Lemma.

0

If f: I-I n --~ G' = f H n is a qc mapping then the following

assertions hold.

(1) The set OG' = C ( f , OII n) is a non-degenerate continuum. (2)

I f ~ j C I-I n , 0 ~ E-j, and f E 1 N r E 2 = ¢ , f;hen [ ~ ( A ( E 1 , E 2 ; I I n ) )

< oo.

P r o o f . (1) Because one-to-one continuous maps (and their inverses) preserve open sets the set-theoretic equality OG ~ -- C ( f , a H n) is clear. It follows easily from 14.2 that

OO

c ( / , 0 l i n) = ~ / u j

;

uj = Hn \ D(e~,j)

]=2

Because f is continuous, the sequence { f U j : j = 2, 3 , . . . } is a decreasing sequence of connected compact sets of R'~ and thus C ( f , 0 H n) is connected by a well-known topological result. By 8.6(1), (7.31), (2.6), and 10.19 0 < ~n--1 en log 2 ~_~~ t t - ( e n , 2en) ~ g o (f) #fH~ (f(en), f ( 2 e , ) ) ,

and hence C ( f , Oli '~) contains more than two points; that is, C ( f , OH '~) is nondegenerate.

(2) Because f is one-to-one fA(E~,E~;H") = A(fE~, fE2; f H ~) and thus by 5.23 or by 6.20 and (10.11) M(A(E,,E2;IIn))

< K o ( f ) M ( A ( f E I , f E 2 ; f l i n ) ) < c~.

0

The following result is a generalization of Beurling's theorem on conformal mappings (see [CL, p. 56, Theorem 3.5], [N3]).

178

14.7. L e m m a .

Let f: B a --~ G' be qc and

E = { b E 0 B a : f has no a s y m p t o t i c value at b} . If F C E is compact, then cap F = 0. Proof.

Assume t h a t F C

F = A(K,F;Ba).

E is c o m p a c t and c a p F > 0 .

Let K =--aB (~)1 and

Denote by Fr the family of all rectifiable p a t h s in F and by F"

the family of all rectifiable paths in f F ~ . T h e n by 5.8, 5.20, 6.1(5)

M(r',) = M(/r~) _> M ( r , ) / K ( / ) = M ( r ) / K ( f ) > 0 because cap F > 0. Hence F'r ¢ ~). Thus there exists a rectifiable p a t h '7 E F r

such

t h a t f o "7 is rectifiable, i.e. f has a limit through ]'7]. This contradicts the choice of F .

El

14.8. A n o p e n p r o b l e m .

This problem, due to F. W. Gehring, has been studied

by P. C a r a m a n [C2]. Let f : B n -+ G' be a qc m a p p i n g and Ear = { b E OB n " f[½b, b) is non-rectifiable ) . For a Borel set A c E a r let F A = { [½b, b) : b e A } . T h e n every p a t h in f F A is non-rectifiable and hence M ( f F A ) = 0 by 5.8. It follows from (5.13) t h a t also M(FA) = m . _ i ( A ) ( l o g 2 ) 1-a = 0 and hence r n a _ l ( A ) = 0 whenever A C E a r is a Borel set. Problem: Is it true t h a t cap F = 0 for every c o m p a c t subset F of E a r ? For the following chapters we shall need a convenient criterion for the thickness ofaset

E C R n at a p o i n t

x E R n . The lower and upper capacity densities of E

at x are defined by (cf. [VU2], [VU3]) (14.9)

cap dens(E, x) = lira inf M ( E , r, x ) , r--+o cap dens(E, x) = l i m s u p M ( E , r, x ) , r--*0

where M ( E , r , x )

is as in (6.2).

Set Az = { r > 0 : S " - l ( x , r )

•E

# 0} for

x E R a . If Az is measurable we define the lower and upper radial densities of E at x , respectively, by

(14.10)

raddens(E,x) = l i m i n f m l ( ( O , r ) M A~) , r--o r rad dens(E, x) = lim sup m l ((0, r) N Az) , r---*O

r

where m l is Lebesgue measure on R . It is not difficult to see t h a t Ax is measurable for every x C R n if E is open or closed.

179

If E is a compact subset of R '~ with r a d d e n s ( E , 0 ) > ~ > 0 ,

14.11. L e m m a .

then c a p d e n s ( E , 0 ) > c(n,~) > 0 , where c(n,~) depends only on n and 6. The proof of this lemma is a straightforward application of spherical symmetrization. The details can be found in [VU3]. It is clear that a similar result holds also for upper densities. 14.12. E x a m p l e s . let E = {0} U ( U S k ) .

(1) Let Sk----S"-'(2-k) N { x : x r ~ _ > O } ,

k= 1,2,...,and

It follows from 5.34 that cap dens(E, 0) > 0 , while clearly

rad dens(E, 0) = 0 = rad dens(E, 0). (2)

There exists a compact set

cap dens(E, 0) > 0.

E

of zero Hausdorff dimension such that

By a well-known result, see 7.15(1), there exists a compact

C a n t o r - t y p e set E1 C Bn(2) \ B ~ of positive capacity and zero Hausdorff dimension.

Exploiting this fact we construct a set E with the desired property.

Let

h: R n --+ R '~ be the mapping h(x) = ~1x , x C R,~ , and denote Ek+1 = h E k . The set E = {0} U ( [.J Ek) is compact and of zero Hausdorff dimension. Since cap E1 > 0, also M ( E 1 , 4 , 0 ) = 5 > 0 (see 6.1(5)). Hence also cap dens(E, 0) > 5. 14.13. R e m a r k s .

(1) It is possible to construct a compact Cantor set E on the

positive x l - a x i s such that rnl(E ) = 0 , cap dens(E, 0) > 0 , and r a d d e n s ( E , 0) = 0 . Therefore, in some cases there are no positive lower bounds for the capacity density in terms of the radial density. Sometimes one can exploit other lower bounds for the capacity densities, see [M4]. (2) The condition cap dens(E, 0) > 5 > 0 is sometimes used in the following way.

First fix r0 > 0 such that M(E,r,O)

> ¼5 for r C (0, r0).

Next choose

A = A(n,5) > 2 such that w,~_l(log2A) 1-~ = ¼5. Then

M(-Bn(r/A), r, O) ~ Wn_l(lOg2A) 1-n -- 1~ f o r a l l r E (0, r o ) . Let E l = E n ( - B n ( r ) \ B n ( r / A ) )

and E 2 = E N - B n ( r / A ) .

Further,

by 5.9,

M(E,r,O) < M(EI,r,O) + M(E2,r,O) and hence M(EI,r,O) > ½~. The next lemma gives a condition for a curve family to have infinite modulus generalizing 5.33 (cf. [VU2]).

180

14.14.

Lemma.

If capdens(E~,0) = 51 > 0 and c a p d e n s ( E 2 , 0 ) = 52 > 0,

then M ( A ( E i , E 2 ) ) = c o . P r o o f . Fix r0 E (0,1) such that M(EI,r,O) > 36 for all r E (0, ro) and let A1 = Al(n,61) be the number in 14.13(2). Fix a sequence rl > r2 > ... such that rl e (O, ro) and M(E2,rj,O) > 362 for j =

1 , 2 , . . . and let A2 = A 2 ( n , 6 2 ) be as in

14.13(2). Denote A = max{,\l,A2}. Then

wn- l (l°g 2A) 1-n = -i1 min{ 61, 62 } •

(14.15) Fix j

and denote Fi = Ei A (B'~(rj) \ Bn(rj/A)),

Applying 5.41 to the triple F1,F2,F3 we obtain as in

i = 1,2,

F3 = S n - l ( 2 r j ) .

14.13(2)

M(A(F1, F2)) _> 2 d ( n ) m i n { 61, 52 } where d(n) = 2-23 -'~ min{1, c,~ (log 2)'~/w,~_ 1}. Next we are going to select a positive number # = #(n, 51,5~) such that (14.16)

M(A(F1,F2;R;) ) > d(n) min{61, 62 }

where R~ = Br~(2#rj) \ Bn(rJ(2A#)). Since F1,F2 C B n ( r j ) \ Bn(rj/A) it follows from 5.9 and (5.14) that it suffices to choose /~ so that

(14.17)

2 w , _ l ( l o g 2 # ) 1-n < d(n) min{ 51, 52 } .

We shall next find an upper bound for tt in terms of A and n . It follows from (14.15) that wn_l(log(2A)v) 1-n _< ~1 p_ l - n min{51, 52 } Hence (14.17) is fulfilled as soon as 2 , _> (2A)v and ½pl-,~ <_ d(n). Let p0

>_ 1 be

the least integer satisfying this last inequality and set ~ = (2A) p° . With this choice of ~ (14.17) holds. By passing to a subsequence of ( r j ) , if necessary, we may assume that the rings R ;

are separate and that (14.16) holds for all j .

and (14.16) that OO

M ( A ( E 1 , E 2 ) ) _> E M ( A ( E 1 , E 2 ; R ; ) ) - - - ( x ) . j=l

It follows from 5.4

181 14.18.

Example.

There exist sets E

cap dens(F, 0) > 0 and M ( A ( E , F ) )

and

F

with cap d e n s ( E , 0 )

> 0,

< 1: Let ro = 1 and choose r j + l e (0,½rj)

such that c~

(

rj ) l _ n

2Ewn_l log j=l c~)

< 1. 7'3-t-1

oo

Set E = U j = I s n - - l ( r 2 j - 1 )

and F = Uj=I S n - l ( r 2 j )

• By 5.9 and (5.14)

O(3

M ( A ( E , F ) ) _< E

M(A(E'Fi))

j=l o0

1--n

~_ E W n _ l [ ( l o g j=l

14.19.

Exercise.

rj )

+ ( l o g r j _ , ~ 1-,~] < 1 .

rj+l

rj

/

Applying 14.6(2) and 5.33 show that a qc mapping of H n

cannot have two distinct asymptotic values at a point b E 0 H n . Applying 14.6(2) and 14.14 one can generalize this observation as follows. If a qc mapping of I I '~ has a l i m i t a i through a set E j at 0 , j =

1,2,andif

al ¢ a2 , then it is not possible

that both cap dens(E1, 0) > 0 and capdens(E2,0) > 0 hold. 14.20. E x e r c i s e .

Let E C H n be non-tangential at 0 and let f: R n --~ R ~

be a K - q c mapping with f H '~ = H '~ and f(0) = 0. Show that f E

is non-tangential

at 0. [Hint: Apply 12.12 to f I R ~ \ {0} and make use of the fact that f ( 0 H ~) = OH ~ .] See also [MOR2].

15.

Lindel6f-type theorems

From a result of E. Lindelgf it follows that a conformal mapping of B 2 having an asymptotic value a at a boundary point b also has an angular limit a at b. A similar result was proved by Gehring [G3, p. 21] in the case of qc mappings in R 3 , and the same proof applies to the n-dimensional case. The following result weakens the hypothesis about the existence of an asymptotic value. 15.1. Theorem. Let f: H n-+ G' be a qc mapping, and let E C H n be such that O E E angular limit

a n d capdens(E,O) > 0 . If f ( x ) --~ a as x ~ O , x E E , t h e n a at O.

f

h a s an

182 P r o o f . Suppose, on the contrary, t h a t there exist ~ 6 (0, ~1 r ) (bk) in C(9~) with f(bk) --~ i

and a ~ i -

and a sequence

By performing an auxiliary Mhbius

t r a n s f o r m a t i o n we m a y assume t h a t a, i ~ oo. Let 3r = la - ill. As a qc m a p p i n g of I-In , f is normal (cf. 13.7(1)) and it follows from 14.5 t h a t there exist numbers M > 0 and r0 > 0 such t h a t

(15.2)

rE1 c B n ( a , r ) ,

E1 = B n ( r 0 ) N E ;

f e z C B'~(i,r) ,

E2 = B~(r0) N ( U P ( b k , M ) ) .

We denote F = A(E1, E 2 ; H ~ ) .

By (5.14) and (5.2) M ( f F ) < oo. Since bk • C(~)

it follows f r o m 14.5 t h a t r a d d e n s ( E 2 , 0 ) > 0.

On the other hand we get by 5.22,

14.11, and 14.14 t h a t

M(r) >

½M(A(E1,E2;R'~)):c~.

This inequality contradicts (15.2) and M(£) _< K o ( f ) M ( / r ) . 15.3.

Remarks.

(1) The condition cap dens(E, 0) > 0 in 15.1 cannot be

replaced by c a p d e n s ( E , 0 ) > 0.

To prove this s t a t e m e n t we consider a conformal

m a p p i n g f: t t 2 --~ G t having no limits along the y-axis at 0.

For the existence

of such a m a p p i n g the reader is referred to the theory of prime ends (cf. references given in 14.3).

Let C ± ( f , 0) be the cluster set of f

fix a • C±(f,O).

at 0 along the y-axis and

By the definition of C±(f,O) there are numbers tk "N 0 with

f(tke~) ~ a . By 13.23 f(x) --* a as x --~ O, x •

UD(tke2,1),andweseeby

14.5

(or more directly, by (2.11)) t h a t

raddens(UD(tke~,l),O ) > O, and hence also the u p p e r capacity density is positive by the proof of 14.11.

The

function f has the desired properties, since it fails to have an angular limit at 0. (2) T h e main interest in 15.1 lies in the case of a tangential set E .

If E

is

n o n - t a n g e n t i a l at 0 and if cap dens(E, 0) > 0 then, as we shall show in 15.7, E contains a sequence (bk) with bk --* 0 and limsupp(bk,bk+l) < oo. From this fact and from 13.21 and from Gehring's result [G3] one gets a simple proof of 15.1 in case E is tangential at 0 . To ensure the measurability required for the definition (14.10) of a radial density we assume in the following theorem t h a t

E

is either open or closed.

This is no

183 restriction of generality, since from the fact that f ( x ) ~ a , x --* 0, x ~ E it follows by elementary properties of continuous mappings that f has a limit a at 0 through an open set F with E C F whether E is open or not. A result analogous to 15.4(1) for bounded analytic functions is due to T. Hall [HI. 15.4. C o r o l l a r y . Let f: H a --~ G r be a qc m a p p i n g , let E c H ~ be an open or closed set with 0 • E and f ( x ) --* a ,

x -~ 0 , x •

E.

T h e n f has an angular

limit a at 0 if one o f the following conditions is satisfied.

(1)

E is a curve t e r m i n a t i n g at 0 or, m o r e generally, E is a set with

raddens(E,0) > 0. (2)

E={bk:

k = 1 , 2 , . . . } where bk • I t '~ and b k - + O , a n d

limsupp(bk,bk+l)

(3)

< oo.

c a p d e n s ( E M , 0 ) > 0 , where E M = [ . J ~ E D ( x , M )

and 2vl • (0, o0).

P r o o f . Part (1) follows from 14.11 and 15.1. For the proof of (2) suppose that p(bk,bk+i) < M

for k > ko. T h e n the set E M = Uk>_ko D ( b k , M )

is connected and

f has the same limit a through E M by 13.21 (or by Exercise 13.23). After this observation, part (2) follows from (1). Part (3) follows from 13.23 and 15.1.

O

When we compare the above condition 15.4(1) with (3), the following question arises. Suppose (1) holds. Does there exist M • (0, c<~) and a sequence (bk) in E with bk --* 0 and p(bk, bk+l) < M ? The answer is negative, as the following example shows. 15.5.

Example.

There exists a set E c H n with r a d d e n s ( E , 0 ) = 1 such

that (15.6)

limsupp(bk, bk+l) = co

for every sequence (bk) in E with bk--~ 0. Set Ek = [ 2 - k - l e l , 2 - k e l ] + t k e n

,where

t k / t k + l ---- k , and E = [.J Ek • Then it follows from (2.11) that p ( E k , U E j ) ----* c~ j#k

as

k-----~c~.

Hence (15.6) is clearly satisfied, and E has the desired properties. An essential feature of the above example is that the set E is tangential at 0. Indeed, we shall show that such an example is impossible if E is non-tangential.

184

15.7. Non-tangential

sets.

Let the set E C H n be n o n - t a n g e n t i a l at 0 with

cap d e n s ( E , 0) = 25 > 0 and ~ E (0, ~1r ) such t h a t

such t h a t E C C ( p ) .

Choose ro E (0,1)

M(E,r,O) >_ ~ for r E (0, r0) and ,~ > 1 such t h a t

_<

._l(lOg2a)

T h e n it follows f r o m R e m a r k 14.13(2) t h a t for each r E (0, r0) there exists a point

br E En-R(r,r/3,,O) where R(r,r/,~,O) = B'~(r) \-B"(r/3,). Let rk = ro/(23, k) a n d bk = br~ • By 4.23 we get

p(bk,bk+l) <_p(C(~) N R(r,r/)~2,0)) = c(p,n,(~) < oe . This inequality shows t h a t (bk) is the desired sequence. We shall next c o m p a r e the hypotheses of 15.1 with those of 15.4(1). 15.8. Example.

If the dimension n > 3, then there exists a set E C H '~ such

t h a t cap d e n s ( E , 0) > 0 but r a d d e n s ( E M , 0 ) = 0 for all M > 0 . For simplicity let n = 3 and define E by

s = 0 {

Y, z) •

+

= 2

z = 2-

/k

}.

k=l

Fix M > 0 .

Let

E M = U~EED(x,M) and m = { r > 0 : Sn-I(r) NEM 7£0}.

Clearly cap d e n s ( E , 0) > 0 (the dimension n > 3 ) . By (2.11) the lengths of the corn-

eM2-k/k and it follows t h a t rad d e n s ( E M , 0) = 0 (for more details, see [VU2, 6.9(3)]). It seems to be an o p e n question w h e t h e r a set

p o n e n t s of A have an u p p e r b o u n d

with similar properties can be c o n s t r u c t e d in H 2 tOO. We shall next prove a generalization of the above L i n d e l S f - t y p e t h e o r e m 15.1, which is m o t i v a t e d by a t h e o r e m of J. L. D o o b [D]. Consider a qc m a p p i n g f : H '~ --+ G ' with 0 E

C(f, 0) (this condition is just a normalization). We w a n t to find a condition,

as general as possible, which implies t h a t f has an angular limit 0 at 0 . Denote (15.9)

Ee=f-lBn(e),

be=capdens(Ee,0),

for e > 0 . We are going t o prove a t h e o r e m , which shows t h a t f has an a n g u l a r limit 0 p r o v i d e d t h a t the n u m b e r s ~f~ satisfy either (1) liminf~e > 0 or (2) l i m i n f ( ~ = 0 with ~

t e n d i n g to 0 sufficiently slowly as e -~ 0 .

A result of this c h a r a c t e r was

proved by J. L. D o o b [D] in the case of b o u n d e d analytic functions.

185 15.10. Theorem.

Let f: H '~ ~ G' be a qc mapping, e > O, Ee = f - l B ' ~ ( e ) ,

and /i~ -- cap dens(E~, 0) . If limsup 5~ log

= ee ,

~.--"*0

then f has an angular limit 0 at the origin. Proof.

Suppose, on the contrary, t h a t there exist ~ • (0, ½~r) and a sequence

(bk) in C ( ~ ) with bk ~ 0

and f ( b k ) - ~ ¢ 0

as k - - ~ c ~ .

Let O < 2r0 < I~1. By

relabeting if necessary we m a y assume, in view of 14.5, t h a t fD(bk, M) (2 R'~\B'~(ro), k = 1, 2 , . . . for some M > 0 .

G ' = f i t '~

D(bk,M) h C(~)

f qc

0

Diagram 15.1.

T h e proof of 15.10.

For every e E (0, ro) there exists t~ such t h a t (15.11)

M(Zc, r,O) ~ ½~ for

r•

(0, t e ) .

Fix e • (0, r o ) . For Ibkl < te denote

F~ = -B~(Ibkl) ;~ Be, F k =-B~(tbkt) N ( UD(bk,M)) ,

By (15.11) we have for [bkt < te

186 From 14.5 and 5.34 it follows that

M(r~3) _> c ( n ,

~, M)

for all k . Let £~ = A(E~, UD(bk,M);H'~).

= c > 0

By virtue of the s y m m e t r y principle

5.22 and the comparison principle 5.41 one obtains

(15.12)

M(F,) >_ ½ M(r~z) _> ~t . 3 - n m i n {

t ~5~, c, cn log 2 } >

AS~

for Ibkl < t¢ where A is a positive number depending only on n , p , and M . From (5.14) we get the upper bound

This inequality, together with (15.12) and M ( r , ) _< K o ( f ) M ( f r , ) , yields

<

oU)

Letting e --~ 0 we get a contradiction. 15.13. R e m a r k s .

(log r°) '-° E

O

(1) Theorem 15.1 is a special case of the above result 15.10

when liminf 5~ > 0 . (2) Theorems 15.1 and 15.10 seem to be among the best results implying the existence of an angular limit, even in the particular case when f is conformal and n = 2 (cf. [VV2], [VV3]). 15.14. A n o p e n p r o b l e m .

For E c I-I ~, 0 E E ,

and a E R 2 let C ( E , a )

be the class of all conformal mappings of H 2 into B 2 having limit a at 0 through the set E . Assume now that E has the following property:

(15.15)

If f E C ( E , ~ ) then f has angular limit (~ at 0.

In particular, if cap dens(E, 0) > 0 , then E has this property by 15.1.

Denote

EM -= UxeE D(x, M ) , M > 0. Does it follow from (15.15) that cap dens(EM, 0) > 0 for some M > 0 ? 15.16. A n o p e n p r o b l e m . Tq={xEH

For q > 1 let n : x~> (x~+...+z~_l)

q/2}

187

and

T q ( z ) - - T q + { z } for z E R n with z n = 0 .

Let f : H n - + t t a b e a q c m a p p i n g

and let q > 1 be given. Does there exist z E OH n such t h a t

Tq(z) at z ? In the limiting case q = l

f has a limit along

this is true by 14.7. See also 14.4. If n = 2 ,

q = 2 , one can c o n s t r u c t a conformal m a p p i n g of H 2 having an a n g u l a r limit a at a single b o u n d a r y point 0 b u t failing t o have limit a at 0 t h r o u g h T~ [GAP]. (It is well k n o w n t h a t the answer is negative in the case of b o u n d e d analytic functions, n

=

2

431).)

(see [CL, p.

16.

Dirichlet-finite mappings

T h e goal of this section is to e x t e n d T h e o r e m 15.1 so t h a t it applies to c o o r d i n a t e functions f j ,

1 <j


mapping f:H '~--*G',

f=

(fl,..-,fa).

Such an

extension is m o t i v a t e d by a result of F. W. G e h r i n g and A. J. L o h w a t e r [GL], which reads as follows. Let f : H 2 --~ R 2 , f = ( f l , f ~ ) ,

be a b o u n d e d analytic function,

let ~/j be a curve in I-I 2 t e r m i n a t i n g at 0 , and let f j have a limit a j j = 1 , 2 . T h e n f has an angular limit a =

along -yj,

(al,a2) at 0 .

It follows f r o m an example due to S. R i c k m a n [RI5] t h a t a similar result is not true for b o u n d e d qr m a p p i n g s in H a , if n > 3.

In the present section we shall

show t h a t the result in [GL] has a c o u n t e r p a r t for quasiregular m a p p i n g s with a finite Dirichlet integral. Let u: H n - +

P~ be a continuous A C L n function. T h e n u is said to be

Dirichlet

finite, or to have a finite Dirichlet integral, if (16.1)

=[

Dir(u)

JH We say t h a t u has a

]Vu[adm < oc. n

locally bounded Diriehlet integral if there exist n u m b e r s B > 0 ,

M > 0 such t h a t (16.2)

f~

for all x E H a where

(x,M)

tVul'" drn ~ B

D(x,U) is as in (2.11).

Let G c R '~ be an o p e n set.

A continuous f u n c t i o n u: G --* R

monotone (in the sense of Lebesgue) if the equalities (16.3)

max D

u(x) ----m a x u(x) OD

and

mi___nu(x) = rain D

OD

hold w h e n e v e r D is a d o m a i n w i t h c o m p a c t closure D C G .

u(x)

is said to be

188 16.4. R e m a r k .

It follows from the above definition t h a t if t E R , then each

c o m p o n e n t A ~ ~ of the set { z C G : u ( z ) > t } fails to be relatively compact, i.e. A A O G

=fi q}. A similar s t a t e m e n t holds if >

is replaced by > ,

< , or < .

Hence m o n o t o n e functions satisfy a weak m a x i m u m principle. The class of monotone functions is wide: it contains harmonic functions as well as solutions of certain elliptic partial differential equations associated with qr mappings. 16.5. E x e r c i s e .

(1) The function u: H 2 --+ (0,~r), u ( z ) = arg z , is a monotone

ACL 2 function. Show by c o m p u t a t i o n t h a t u fails to satisfy (16.1) but t h a t it does satisfy (16.2). (2) Construct a monotone function u: H 2 --+ R + which has no a s y m p t o t i c value at any point z E O H 2. The next result is a f u n d a m e n t a l property of functions with locally bounded Dirichlet integral. Some results of this kind were proved already by D. Hilbert and H. Lebesgue in the beginning of this century (see [LF1] and the references given there). These ideas have also found frequent application in geometric function theory in connection with the so-called l e n g t h - a r e a method. For further references see 5.72. 16.6. T h e o r e m .

L e t u: B n ~ R

be a m o n o t o n e function with locally b o u n d e d

Dirichlet integral. T h e n lu(x) -- u(y)t n <_ C

where r = t h } p ( x , y )

log

(l-r)

1-n

and C is a positive constant d e p e n d i n g only on the n u m b e r s

n , M , and B in (16.2). In particular, u : ( n " , p ) -* (R, I l) is uniformly continuous.

Proof.

Clearly we m a y assume t h a t u(x) < u ( y ) . Since the right side depends

on x and y only through the MSbius invariant quantity p(x, y) , we m a y assume t h a t x = re1 = - y ,

r = th¼P(x,y)

E={zEB"

(see (2.25)). Let

: u(z) <_u(x)} ,

and denote Fr = A ( E , F ; B n ( v / r ) ) .

F={zEB

'~ : u ( z ) > u ( y ) } ,

It follows from 16.4 and 5.32 t h a t

M(rr) > e. log

1

L e m m a 7.4 yields f M(r~) _< lu(x) - u(y)l - " / IW?d~ JB -(v~)

•

189 In view of (16.2) the integral can be estimated from above in terms of B and the number

k inf{ k : Bn(x/7 ) c U D ( x j , M ) , j=l

Ixjl <_ v / r } .

It follows from (4.19) and 4.22 that we now obtain I V u l n d m <_ B d ( n ) ( 1

/-E

~v/~) 1 - n ~ 2 ' ~ - l B d ( n ) ( 1 - r) 1-'~ .

-

"(47)

In conclusion, the above inequalities yield ]u(x)-u(y)]n
(log})-l(l-r)

1-'~

C =

16.7. C o r o l l a r y . Let u: B'~--+ R

for all x, y E B n

be a m o n o t o n e ACL n function. Then

where cn is as in 5.34 and r = t h ¼ P ( x , y )

P r o o f . Clearly we may assume that Dir(u) < c~ and u(x) < u ( y ) . Define the sets E and F as in the proof of 16.6 and let F = A ( E , F ; B '~) . By 8.6 and 8.7 M(F) > A , , , ( x , y ) >

½r(sh 2 } p ( x , y ) )

- C n logth l p ( x , y ) .

Lemma 7.6 yields

M(r) _< lu(~) - u(y) l - " and hence the result follows.

Dir(u)

CI

We remark that the upper bound in 16.6 or 16.7 is not accurate for large values of p(x, y ) . A better estimate for large values of p(x, y) can be derived from 16.6 and the fact that u is uniformly continuous, see 4.13. 16.8. T h e o r e m . let E C H as x - * O ,

'~ b e a s e t

Let u: H ~ -+ R with 0 E E c H

be a m o n o t o n e Dirichlet finite function and

' ~ u { 0 } and cap dens(E, 0) > 0 .

x c E , then u has an angular limit a at 0.

If u(x) - ~ a

190

P r o o f . The proof is similar to that of 15.1. Fix ~ E (0, ½7r) . Suppose, on the contrary, t h a t there exists a sequence (bk) in C ( ~ ) with bk ~ 0 and u(bk) --+ j3 ~ a . We shall assume t h a t - o o < a < fl < c~; in other cases the proof is similar. Let Bk be the b k - c o m p o n e n t of the set B -- { z E I-In : u(z) > ( a ÷ 2 / 3 ) / 3 } A = { z E H ~ : u(z) < ( 2 a ÷ ~ ) / 3 } .

and p E N

and let

By 16.7 and the proof of 14.5 there are M :> 0

such t h a t D ( b k , M ) c B k

for all k ~ p

and

raddens(B,0) > d(~,M) > 0 ; hence cap dens(B, 0) > 0 by 14.11. Since c a p d e n s ( A , 0 ) > cap dens(E, 0) > 0 it follows f r o m 14.14 and 5.22 t h a t M(A(A,B;H~))

> ½M(A(A,B;Rn))--oo.

M(A(A,B;H"))

< 3"(/3- a)-nDir(u) < oo,

From 7.6 we have

which is a contradiction. 16.9.

Corollary.

O Let f: H n ---* R n be a qr mapping and assume that there

are sets E j C H n such that f j ( x ) --* a j

as x --~ 0 ,

x E Ej,

c a p d e n s ( E j , 0 ) > 0 and D i r ( f j ) < o0 for each j = 1 , . . . , n , limit a = ( a l , . . . , a , )

aj asx---~O, x E E y ,

If

then f has an angular

at O.

P r o o f . The proof follows from 16.5(2) and 16.8. 16.10. C o r o l l a r y .

j = 1,...,n.

O

Let f: H " -+ R n be a qc mapping and assume that f j ( x ) -+ EiCH

'~ j = l , .

n

Ifcapdens(Ej,O) >O,j=l,

n

then f has an angular limit a - - - ( a x , . . . ,an) at 0.

P r o o f . Let h C ~ ( R n) be such that h(en) = oo and h D ( e ~ , l ) -- R n \ B

n.

By considering the m a p h o f , if necessary, we m a y ~ s u m e t h a t f is bounded by 1 in H n n Bn(¼) = D (note: here we use the fact t h a t f is injective). Moreover,

Since IOSs(x)/axkl __

1 < i,k <

we see that D i r ( L )

1 , . . . , n , and hence the proof follows from 16.8.

[]

<

j =

191 It follows from a well-known formula for change of variables that all K - q r mappings f: I-I" ---* B n have a finite Dirichlet integral, that is Dir(f) < K m ( B '~) = KR,~. More generally this holds for K - q r mappings f: H '~ ~ B '~ with finite maximal multiplicity Y ( f , I-I '~) < c~, that is Dir(f) < g N ( f ,

H n) i2n. It is easy to give examples

of bounded analytic functions with an infinite Dirichlet integral (for instance, the exponential function in the left plane). However, bounded qr mappings have a locally bounded Dirichlet integral according to the following theorem of Reshetnyak (proof omitted) [R13, p. 127]. 16.11.

T h e o r e m . For n >_ 2 , K >_ 1, and r • (0,1) there exists a n u m b e r

c(n, K , r) such that each K - q r m a p p i n g f: B ~ -+ B n satisfies

f. -(r) If'(x)l'~dm<_c(n,K,r) 16.12. T h e o r e m .

D

L e t f: B n -~ R '~ be qr. T h e n the following conditions are

equivalent:

(1)

f: (B n, p) --+ (R n, I I) is uniformly continuous.

(2)

There are n u m b e r s M > O and B > 0 such that /o

(z,M)

Iff(x) I~dm -< B

holds for every z E B n .

(3)

There are n u m b e r s T > 0 and C such that I f ( x ) - f ( y ) l <_ C w h e n e v e r x, y E B n and p ( x , y ) < T .

Proof.

(1) ~ (2): Fix t > 0 such that p ( x , y ) <_ t implies I f ( x ) - f ( Y ) l <- 1.

It follows from (2.23) that B'~(x, (1 - I x l ) t h ½t) c D ( x , t ) .

Let h • Jvi(R '~) with hB '~ = B ' ~ ( x , ( 1 - 1 x l ) t h ½ t ) . Then f o h : B

'~ ~ B '~ is K - q r

and

/

JB .(1/2)

I(f ° h)'l ' dm

=

/

Je~(x,~)

If'l d

½)

by 16.11, where s = $1( 1 --I~1) t h ls t . Now choose a number M such that D ( x , M ) C Bn(x,s)

for all x. Because for all x • B ~ D(x,M)

C B n ( x , (e M - 1)(1 - I x l ) )

192 l t h l 7t implies that D ( z , M ) by (2.23), the choice e M - 1 = -~

. With this

C B"(z,s)

choice of M and with B = c ( n , K , ½) the condition (2) holds. (2) =~ (1) : It suffices to show that each coordinate function of f is uniformly continuous.

On the other hand the coordinate functions are monotone.

Now the

uniform continuity of coordinate functions follows from (2) and 16.6. It is clear that (1) =~ (3). So it remains to prove (3) ==~ (1). It follows from the Schwarz lemma 11.2 (see also the proof of 13.4) that

If(x)-f(y)I


\

th½T

/

where a = K z ( f ) 1 ~ ( l - n ) . The desired conclusion follows.

' UI

16.13. C o r o l l a r y . S u p p o s e t h a t f : B '~ ~+ R n is a q u a s i r e g u l a r m a p p i n g w i t h f B - ] f ' ( x ) [ '~dm < ~ "

Then 1

]f(x)l < If(0)] + 1 + F l o g /'oral1 x E n

n where M = s u p { T

: p(x,y)
1+

1 -]x'-"----~ If(x)-f(y)]_< 1}.

P r o o f . The proof follows from the proof of 4.13 and from 16.12. 16.14. R e m a r k .

0

Theorem 16.11 yields an upper bound for the growth of the

Dirichlet integral of a bounded K - q r mapping f: B n --~ B n . Indeed, 16.11 combined with 4.22 shows that B

.(r)

] f ' ( x ) [ ' ~ d m ~ A(1 - r) 1 - n ,

where A depends only on n and K . 16.15. N o t e s . is [VU10, 4.29].

The results of this section are from [vug] except for 16.12 which

Some open

problems

(1) Find an explicit expression for qn(s) when n > 3 (see Sections 5 and 7). (2) Let E , F c H n be c o m p a c t and disjoint, let F* = {(xl,...,x,~-l,-X,~) :

(xl,...,x,~) E F } , F = A ( E , F ) , F* = A ( E , F * ) . Is it true t h a t M(F) > M(F*)

(cf. 7.59)? (3) Find all domains D such t h a t AD(x,y) 1/(l-n) is a metric on D . Is this true for D = R

' ~ \ { 0 } and n = 2

(cf. S e c t i o n 8 ) ?

(4) Let f : B n --+ f B n c B n be discrete, open, and proper. Assume t h a t n > 3 and B I is compact. Is f o n e - t o - o n e (Section 9)? T h e answer is yes if f B '~ = B '~ . (5) Find an u p p e r bound for the linear dilatation H(x, f) of a K - q c m a p p i n g

f: G ---* f G , G C R '~ , such t h a t the b o u n d tends to 1 as K --- 1 (cf. Section 10). (6) Does there exist an absolute constant C , independent of n and K , such t h a t T h e o r e m 11.40 holds with C in place of MI(n,K) ? (7) For given n > 2 , K > 1, and 6 C (0, 1), does there exist a n u m b e r A(n, K, 6) with the following property: if f: B '~ --~ f B n C B n is K - q r and ]f(0)l > 6 then card{zCB (8) Let f : B n ~ B

n

1 (~) : f ( z ) = 0 } <_A ?

n, n>3,beqr.

Show t h a t f has at least one radial limit.

(The case of Dirichlet-finite f is well known [MIK2], [MR1].) (9) Prove or disprove the following assertion. For each n :> 2, r E (0, 1), and K > 1 there exists a n u m b e r d(n,g,r)

with d(n,K,r)--* d(n,g)

as r - + 0 and

d(n,K) --+ 1 as K --~ 1 such t h a t whenever f: B " --~ R n is K - q c , then fBn(r) is a d(n,K, r ) - q u a s i b a l l . More precisely, the representation fBn(r) = gB n holds where g: R ~ --- R '~ is a d(n,K,r)-qc m a p p i n g with g(cc) = c~. (Note: It was kindly pointed out by J. Pecker t h a t we can choose d(2, 1, r) = (1 + r ) / ( 1 - r) either by [PC, pp. 39-40] or by a more general result of S. L. Krushkal' [KRI. ) Additional open problems can be found in [BAM], [G9], and IV10].

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Index A list of special symbols is given on pages XVII-XIX. absolute ratio 9 absolutely continuous 83 ACL, ACL n 81, 82 admissible family 41 admissible function 49, 82 angular limit 174 antipodal point 5 asymptotic curve 174 asymptotic value 174 Bernoulli's inequality 34 bilipschitz mapping 11 branch set 122 capacity, p-capacity 82 capacity density 178 capacity zero 85 chordal metric 4 closed mapping 126 closed path 49 cluster set 174 condenser 82 comparison principle 6I conformal mapping 2 conformal metric 103 conical limit 174 cross ratio 9 curve 49 dilatation, inner 128 dilatation, linear 134 dilatation, maximal 128 dilatation, outer 128 Dirichlet integral XIII, 187 discrete mapping 122

elliptic integral 66 euclidean isometry 6 exceptional curve family 50 extremal length 49 formal derivative X V H functional identity 67 functional inequality 68 geodesic segment 21, 23, 33 GrStzsch condenser 88 Gr6tzsch constant 89 Gr6tzsch ring 65, 88 ttarnack inequality 44 Hausdorff dimension 86 Hausdorff measure 86 HSlder continuity XI, 137, 141 hyperbolic ball 22 hyperbolic distance 20, 23 hyperbolic length 20 hyperbolic metric 19, 20, 23 hyperbolic voiume 20, 23 inversion 2 isometric decomposition 15 isometry 6 LV-integrability X H light mapping 122 Liouville's theorem 19 Lipschitz constant 11 Lipschitz mapping XI, 11 local index 123 locally rectifiable 49 locus 49

209 maximal multiplicity 125 maximum principle 127 Mgbius transformation 3 modulus of a curve family 49 modulus of a ring 65 modulus of continuity 134 modulus metric 103 monotone function (in the sense of Lebesgue) 187 Mori's ring 117 multiplicity 125 NED set 112 non-tangential limit 174 non-tangential set 176 normal condenser 131 normal domain 123 normal mapping 163, 170 normal neighborhood 123 normal representation 49 open mapping 122 open path 49 orientation-preserving mapping 3, 122 orientation-reversing mapping 3, 122 orthogonal mapping 2 path 49 path lifting 126 Picard's theorem for qr mappings 170 Poincar6 extension 4 Poincar6 half-space I, 19 Poincar6 metric (distance) 19, 27 proper mapping 126 Ptolemy's theorem 70 Pythagorean theorem 7, 14 QED set 112 quasiconformal (qc) mapping 128 quasihyperbolic ball 35 quasihyperbolic metric (distance, length) XV1, 33 quasiisometry 11

quasimeromorphic (qm) mapping 128 quasiregular (qr) mapping 127 radial density 178 rectifiable path 49 reflection 2 Riemann sphere 4, 5 ring 65 Schottky theorem 166 Schwarz lemma 120, 137, 144 sense-preserving mapping 3, 122 sense-reversing mapping 3, 122 separate curve families 50 sequential limit 174 similarity transformation 3 spherical bail 7 spherical cap inequality 59 spherical isometry 6 spherical metric 4 spherical ring 53 spherical symmetrization 87 stability theory X I I stereographic projection 4, 6 stretching 2 symmetric ratio 38 tangential set 176 Teichm~ller condenser 88 Teichmfiller ring 65, 88 topological degree 121 translation 2 uniform domain X I V , 35