Lecture Notes in Mathematics Edited by A. Dold and B. Eckmann
1319 Matti Vuorinen
Conformal Geometry and Quasiregular ...
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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 }
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 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 -
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 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) 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 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.
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)) 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 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)) [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) 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
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_ 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 ) 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))) 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)
_ 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)
_ 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 ) _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) 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)) 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)
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~ 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 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 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
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) 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 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 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)
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 ) ) ~
•
Since E1 c B n we get by 6.5
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 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 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.
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- 1 ,
(2)
t + 1 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-" T(la, b,c, dt) Here equality holds if b = slel , a = s2el ,
c
~
83ei
,
d
~
s4e
1 ,
and
s l < s2
_ 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)
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)}.
1.
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 ) 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
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 _ 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- 1 and t E (0, tK]
_
1 and t C (0, tK]
O~K(t) 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)) 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)
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 , (2) p(x) _ 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 ) ) 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 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) _ 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) 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(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) 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 ) 2.
T h e n for distinct x, y E D
2 ~ - ~ ( m ) _< ~ ( m : + 2m) 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 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/ 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 _ 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_ 1 such t h a t
(lO.2)
If(x)[ ~
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 ) _ ~ 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) _ 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)) 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 ) ) 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 r . Since A,~ _> 1
(in fact, An > 4 , see 7.22) we have
]f(x) - f ( Y ) l 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 ") _
[f(Y)l- By 5.27 and 8.5 we obtain
ASB, 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 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)) _ 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) _ 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 ) 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 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))
~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 ) } 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[
' by) ' = log 3 :> 1 , (av,
kv (av, bp) _ 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)
_
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)
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 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) 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-,~ _ 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 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
_
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
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
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 } 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