MOBIUS T~SFO~~TIONS IN SEVERAL DIMENSIONS ~-====~=~===~=====================~~~==~====;=

. Lars V. ~hlfors

Copyright 1981 by Lars V.. Ahlfors

INTRODUCTION ======~=====

These are day to day notes from a course that I gave as Visiting

Ordway Professor at the University of Minnesota in the fall 1980. I

had used some of the material earlier for a similar course at the

university of Michigan and for a month each at the University of

Vienna and the University of Graz. The present notes are a little more

detailed, but still not in a definitive ferm.

My aim was to give a rather elementary course which would

acquaint the hearers with the geometric and analytic properties of

Mobius transformations in n real dimensions which could serve as

an introduction to discrete groups of non-euclidean motions in hyper-

bolic space, especially from the viewpoint of solutions of the

hyperbolically invariant Laplace equation as well as quasiconfor:mal

deformations.

I wish to thank the University of Minnesota and the Chairman of

the Mathematics Department for having invited me, and above all my

friend and former student Albert Marden for having initiated my visit.

I also thank the many faculty rrembers and students who had the patience

to sit in at roy lectures to the very end.

I am also indebted to Dr. Helmut Maier who helped me edit the

notes and supervised the typing. The typist, unknown to me, has also

my sympathy.

Lars V. Ahlfors

'TABLE OF CONTENTS ===~=~===========

I. The classical case

II. The general case

III. Hyperbolic geometry

IV. Elements of differential geometry

v. Hyperharrnonic functions

VI. The geodesic flow

VII. Discrete subgroups

VIII. Quasiconformal deformations

p.l

p.13

p.31

p.41

p.57

p.72

p.79

p.101

I. The classical case.

1.1. Everybod¥ is familiar with the fra.ctional linear transformations

(1) az +b

y(z):= cz + d

where a, b, c, dEC and ad - be f O. They act on the extended complex plane

C = C U (CO)} which we identify with the 1 - dimensional proj eeti ve space P 1 ( C) •

In terms of homogeneous coordinates w = 'V(z) can be expressed through

the matrix equation

(2 )

This has the advantage that the Mldbius group of all y can be represented

as a matrix group either by means of the ~eneral linear group G~(C) or

by the unimodular or sRecial linear group S~ (C). More precisely the

Mobius group is isomorphic to

* where C is the multiplicative group of nonzero complex numbers and to

where 1 0 r=(Ol) • In spite of this identification we shall denote the

Mobius group by ~ ( C) rather than PS~ (C) for the simple reason that

the identification does not carryover to higher dimensions.

We shall of'ten sittplify y( z ) to yz and we think 0'£ Y as a matrix

normalized by ad - bc = 1. We should be aware, of course, that y and -y

2

represent the same M"obius t=:-a.n.sformation. It is useful to memorize the

formula for the inverse in S~ :

(3)

1.2. There is a natural topology on G~ (C) and hence also on S~ (C) and

}~ (C). G ~ ( C) is connected for the space of all complex 2 X 2 matrices has

eight real dimensions and the space of singular matrices has only six. The

mapping

ad~ be )

of G~(C) on s~(C) shows that S~(C) is likewise connected. We can

consider S~{C) as a double covering of ~~(C) •

The subgroups with real coefficients, are denoted. by G~(JR) , S~(:R)

and ~(:R) • There is also a subgroup Gr;C1R) with positive determinant.

+ G ~ (:Ia) and S ~ (JR ) are connected, but G~ (JR ) is not for one cannot

pass continuously from a ma~rix with positive determinant to one with negative

determinant.

Any YEM2(JR) maps the real axis, the upper half-plane, and the lower

half-plane on themselves. This is obvious from

(4) yz - yz =

1.3. From (1) one computes

(5) y( z) - y( z ') :=

... z-z

2 jcz+d·1

and in the limit for zt-+z

(6} y'(z) = ad. - be

2 (Cz + d)

In particular, z ~ yz is copf'Orr.lal. We rewrite (5) as

(7) 1/2 1/2

y(z) - Y(ZI) = Y'(z) l' Y'(ZI) I' (z- z')

where it is understood that

:= Jad- be cz+d

with a f:xed choice of the square ~oot.

3

1 .. 4. The cross-ratio of' four points z, z I , s, S' in that order is

d,e:fined by

(8) zt - 6 z' - (; I

when tbis makes sense, i.e. When at most two points coincide. B.lf use of (7)

it follow's that

(z , z' , " ,I)

:for the ·derivatives cancel against each other.. In other words, the cross-

ratio is invariant under all Mobius tr&nsi'Qrmations ..

The cross-ratio is real if and. only if the four points lie on a circle

or strai~ht lines. Note that our definition makes

(z , 1 , 0 ~ (0) = z

1.5. From (6) one obtains

(10)

and hence

(11)

y"(z) yl (z)

~:it: ~ =

2c cz+d

if c i O. It follows that

(12)

and for l!'" Z

(13 ) D ~: t: ~ = - ~ •

More explici t,ly, this is equivalent to the vanishing of the Schwarzian

derivative:

(14) III 3 It 2 II 1." 2 :L _ _ (.Y.:.) = (Y...)' __ (.Y:.) ::: 0 y' 2 "VI y' 2 yl

4

There is obviously a close link between the cross-ratio and the Schwarzian

(15 )

for arbitrary meramorphic f. For those who like computing, I recommend

proving the formula

(16) (f(z + ta) ,f(z + tb) , f{z + tc) , fez + td)):;:

t2

(a",b,c,d)(l+ 6' (a-b) (c-d)S:r(Z» + o(t3) .

5

1.6. The half-plane. We return to the action of S~(E) on the, upper

half-plane H2:;:: {z :;:: X + iy ; y > OJ. We use the notation G;;: S~ (JR)

(or l\~(1\) ).

Let K denote the isotropY' group of i. From

ai+b = i ci +d

2 2 we get a ;;: d, b:;:: - c , a. +b ;;: 1 • Hence g E K has the form

(17) (COS G

g;;: - sin e sin e) cos e

In other words, K ~ 80(2) • More precisely, e and 8+ n give

different elements of K but the same element of M2 (1&) •

Let A be the group of homothetic transformations (: t ~ 1) , t > 0 ,

and N the group of translations 1 u

( 0 l) • The group NA is the group of

simila.ri ties and also the isotropy group of co. It is simply transitive on

H2 as seen from

( lu),t 0 )(') 2 o 1 t -1 ~ :;:: u+t i o t

Hence G:;:: NAK ; this is tile lwasawa decomposition. Each g E G has a unique

decomposition g:;:: nak. Rxplicit1y, if g(i):;:: x + iy , y > a , then

(18) g:= (l X ) (y1/2 0) ( cos e sin e ) o 1 0 y-l/2 -sin e cos e

Every function f on If can be lifted to a fUnction ! on G

defined by !(g) :;:: f(gi) Conversely, a f\L~ction on G becomes a function

on ~ if it is constant on the left cosets g K. ~ is identified with

G / K. (many authors prefer K \ G ).

6

1.7. 111e circles orthogonal to the real axis, including vertical lines, are

mapped on each other by all. g E G. They are the straight lines in Poincare t s

model of pyPerb~lic or non-euclidean geametry. Any two distinct pOints

z=-, Z2Erf lie on a unique n.e. line (Fig. 1)

Fig .. 1

A distance can be d.efined by the point-pair invariant

(19)

the invariance is seen from the relation

(20)

For z2 ... zl we obtain the infinitesimal invariant

(21) ds ;:;: ~ y

2 ('ire have dropped a factor 2 ). This defines the Poincare m.etric on H

It leads to a new distance function d.(zl ' z2) defined as the minimum

(22 )

7

over all paths from zl to z2. The shortest curves are the ~eodesics.

To find the geodesic we use agE G which maps zl' z2 on iy 1 ' iy 2 •

It is readily seen that the vertical line segment from iy 1 to iy2 is

minimal, and thus

(23 )

It follows tb.at the geodeSics are the orthogonal circles.

The relation between 8 and d is deterrrl.ned by

from which we conclude that

d 5 := tanh-

2

The distance function d is ad.ditive on fiwe .. lines, i.e. d(Zl' Z3} = d(zl' z2)

+ d( z2 ' Z3) if' z2 lies between zl and z3 .

We can use (24) to verify that 8 is a distance :function, :ror if

5 = tan hS!' < 2

1.8. There is a dif:'erent way of computing d(zl' z2) which shows the

A.ciditivity at once. We refer to Fie· l and compute (z2' zl' ~l' S2) -

If the circle is mapped on the imaginary axis the cross-ratio becomes

8

This proves the relAtion

In this form "We may even regard d(z~, z2) as a signed. distance which becomes

negative when the order of' zl' z2 is reversed~ The additivity on a geodesic

::ollows at once from

1.9. The length of an arc is

J c

M y

and the area of' a measurable set E is

J dx~Y E Y

It is a useful exercise to compute the area of a n.e. polygon P

(Fig. 2)

"-I

..... / .....

" I I " ,

I ,', f \' \

( \ \ , I , I \ I \

1 r , t I I 1

Fig. 2

By Stokes I formula

A = r u p

dx Ady 2

Y

9

Each side is an arc o~ i8 z - a = re and, ~ ;::: - ae

y J de measures the

change 0= the a.ngle of the tangent. For a simply connected polygon the total

change~ including the jumps at the angles, is 2n. If the outer angles are

and the inner angles a v one obtains

A =-1:: f3 - 2:1( = (n - 2)1C - E a v v

' .. 10. Tt is easy to :;>ass from the half-plane tf t,o the uni.t disk B2;;;

('E C ; I 'I < I}. We want to choose a canonical Meoius mapping .; ~B2 •

A good choice is to let z = 0 ,. i ,CJ) correspond to ,= -i , 0 "i. This

gives

(26) z-i ,=i-;+i'

r+ . • .;, 1. Z = -1. C-=i'

We introduce the notation ,*;::: 1 = ...L­, 1,1 2 for the symmetric point of

, with respect to the unit circle. B.1 the reflection principle

over into "C* and by the invariance of the cross-ratio

(27)

This shows that the point-pair invariant in B2 is

1'1- '21

(1- tl '2 I

-z ,z go

10

a..l1d the correaponding Poincare metric is

(28)

(r'emember that we had suppressed a factor 2).

It is again trivial that diameters are geodesics and hence the same is

true of all circles orthogonal to the unit circle. The di stance from 0 to

!: > 0 is

r d ;;; d(O,r) = J 2dt2 o I-t

l+r ;;; log l-r' '

Note that 8(o,r); r .

Another observation is that

1+r (r , 0 , -1 , 1) ;;; -1 -r

r = tanh ~

If the geodesic through '1" 2 meets the unit circle in (1)1' ~ we have

thus

From now on we prei'er to use the variable z in the unit disk, and

with the notation of Fig. 3 we have thus

Fig. 3

11

(29)

l.li. The self-rnap'pings of B2 which take a into 0 are of the form

(30) ie

l! Z = e z-a

1- az

We want t:l derive this formula in a geometric way.

Fig. 4 For this pur~ose we construct a* and the orthogonal circle with

center a* as shown in the picture. The reflection in this orthogonal circle

carries a into 0 and an arbitrar.y point z into

I 12 * a cr z = a* + (a* -1) (z - a*) = - '::" a a

z-a 1- aZ

To obtain a sense-preserving mapping we let as. be followed by reflection

in the line through the origin perpendicular to a.. If' a.rg a = a this

reflection is expressed by

(31) zl-?Tz = a z - a

1 -' 8.z

and we end up with the mapping

The most general mapping is T followed by a rotation about the origin, a

and it is hence of the farm (30). We make a note of the formulas

2 l-IT z l a

T' (z) a

2 , 1- laJ (1- az)2

1 2

1 .. fz{ ,

all well known to conformal mappers. The last formula expr'esses the irtvariance

of the Poincare metric. The norwAlization o~ T is such that T'(a) > 0 and a a

T' (0) > 0 • a.

Note that TO:; -a and hence T = T- 1 a -~ a

13

II. TPe general cas,e.

2.l. Before passing to the general case we shall pay special attention to

the group M(If) of' !'-1'obius transformations acting on the upper half- space

H3 : (x ::; (Xl' x2 ,X3

) ,x3

> 0) ~ Already ?oincare was well aware that the

action of M(C) can be lifted to H3 or if' one pref'ers to all of 3 *) JR •

In fa.ct, any y E M( C) is a product of reflections in circles (or straig..ht

lines), the circle determines an orthogonal hemisphere (or half-plane) and

the reflection extend.s to a ref'lection in the hemisphere. One has to show,

of' course, that the end result does not depend on the particular factoriza-

tion of' y.

The three-dimensional upper half-space is very special because of' the

.fact that one can make use of quaternions in a very elegant manner.

The quaternions can be identified with matrices

(1) (~ ~) -v u

where u, vEe In fact, if' we write i=(10) O-i '

k= (~~) and u=,\ +i~ , v=v1 +1v2 we obtain

(2 ) ( u y) +' . k -v u = ul ~~ + v IJ + v 2

= u + vj

- -The conjugate of z::; u + vj is z = u - vj and the absolute va,lue

Izl is given by Izl2 = zz = luJ2 + Iv/ 2 • When computing tr_€ product we

have used the rule aj = ja for arbitrary complex a.

We have replaced the notation M2(C) by M(C) •

14

Points i!l. 1R3 will ~ow be denoted by z:;: x + y j with x E C , y E lR

(y > 0 if z E~). Suppose yES~(C) is given by

a b Y = (c d)' ad - be = 1 •

We define the action by

-1 -1 yz == (az+b)(ez+d) = (zc+d) (za+b)

We need to show that these expressions axe equaJ., and that yz. is a special

quaternion or the same form as z.

In the first place, the two expressions (3) are equal if and only if

(zc+d)(az+b) - (za+b)(cz+d) = 0 •

This works out to

zcb + daz - zad - bcz = 0

which is true because ad - be is real.

Here

The next step is to actually compute yz.. To begin with

(az + b)(cZ+d)

yz = ~az+b)(cz+d)

I cz +d12

cz + a. = ex + d + cyj

cZ"+d. = ex + a - cyj

az +b ;::; ax+b + ayj

- 2 = (ax+b)(cx+d) + acy + [-(ax+b) cy+ay(cx+d)] j

: (ax + b)(Ci: + d) + a'i:.y2 + yj

15

and thus

(4 ) -- - - 2 yz.::; ( ax + b )( ex + d) + acy + -yj

Icz + dJ 2

which is of the desired form. Also, fez +d12 == ICX+dI2 + Jcl

2 y2 0

2.2. We shall also compute a. formula for the difference "fZ - yz t. We

write it in the form.

y.z - yz. T = (az + b)( cz + d) -1 - ( z 'c + d.) -1 ( Z f a + b)

(5) = (z'c+df1[(ztc+d)(az+b)- (z'a+b)(cz+d)](cz+d(l

::; ( Z I C + d ) -1 (z _ Z f ) (cz + d) -1 •

This is at least similar to the corresponding formula (1.4) in the ccmplex

co,se.

On passing to the absolute values we obtain

(6)

and infinitesimally

(7)

I j - Iz-z'l yz - yz t - I cz + d I I ez' + d [

fdy(z) I :;; laz\ 2

Icz +dl

Comparison with (4) shows that

(8)

is invariant.

ds::; ~ y

16

The cross-ratio should be defined by

By use of (5) one obtains

Thus the matrices correspond.ing to the quaternions (YZI' YZ2 ' "(Z3 ' YZ4) and

(Zl ' z2 ' Z3 ,z4) are similar. Because the absolute value is the square root

of the determinant and the real part is half of the trace of the matrix (1)

it follows that the absolute value and ~he real part of the cross-ratio are

invariant.

The stabilizer o:r j is the group of unitary matrices

The full group M(H3 ) is ~ransi ti ve, for yj::; u + vj, v > 0 , by taking

uv -1/2) -l/2 v (,

2.3. The quater nion technique works only for :M(~ ) and even if it looks

elegant it is not particularly useful. We shall now again deal with actions

on the entire space JRn • We shall use the notation x:;::o (Xl' •.• ,Xn

) E lRn

and when operating with ma-;rices we treat x a.s a column vector. The g.coup

of similarities consists of all mappings

x~mx+b

17

where bEEn and ill is a conformal matrix, i. e. m:;;;).,k with A > 0

and .k E O(n)

Reflection in the unit sphere is defined by

(10) * x 17 x :::: Jx :;: (JO = m , J~ :;;; 0)

where of Course I 12 2 :2

x :;: ~ + ••• +xn • We use mostly the notation x * , but

sometimes one needs a letter symbol for the mapping, and then we use J.

Obviously, ;l = I , the identity mapping. I will also stand for the unit

matrix I n

We adopt the following definition:

Definition. The full MObius group M(~n) is the group genera.ted by all

similarities together with J. The M"obius group M(T) is the subgroup

whose elements contain an. even number of' factors J and sense-preserving similarities.

In other 'Words, M(lRn ) is the sense-preserving Mooius group. Note that

I do not use M(En ) as being too J)edantic.

The derivative of a. dif'ferentiable map:ping f from one open set in JRn

to another is the Jacobian matrix

f' (x) or Df(x)

with the elements

of. (11) f' ' (x). . = ':!.v-~ ;::; D. f. (x)

lJ O-CO-j J 1

The derivative of a similarity yx. = mx + b is the constant conformal

matrix m. The matriX Jf (x) bas the components

(12)

for x 'I 0 .

2x. x. ~ J

JXJ2

18

It is almost indispensable to adopt a special notation far the ubiquitous

lLatrix Q(x) with the entries

(l3) x .. x.

Q() = ...1:......J. x .. 2 lJ Ixl

This enables us to write (12) in the for.m

(14 ) 1 Jt(x) = ~ (I - 2Q(x)} IxJ

This is perha:gs the most importa.nt formu.la in the whole theory of Mooius

transformations.

2 From Q = Q we obtain

(I - 2Q)2 = I

which means that I - 2Q E O(n). Thus J' (x) is a conformal matrix for each

x"l 0 •

It now fellows by the chain rule that y. (x) is a con£orm.a1 matrix for

any Y E MA(JRD) • 1 ~~'b t ant In other words, al lYlO ius transform.a ions are C ormal

With a suitable interpretation at co ar..d. -1 Y co.

Definition. For any y E M(mn ) we denote by I yt (x) I the positive number

~ such that1Y'(x)1 E O(n) In other words, iy'(x;] is the linear change

cf scale at x, the same in all directions.

19

Another application of the chain rule proves that

(l5) I/:2 1;:2 [yx-yyj = Iv'(x)1 IVf(Y)1 !x-yl

In fact, this is triv-ial when 'V is a simila.rity, and for J 'tore obtain

2 *) IJt(x)IIJI(y)llx-yl by (14) •

If 'V1 and 'Y2 satisfy (15) then it is also true for VI '\(2 :

I 'VI Y2X - 'Vl 'V2yl = I Vi ('V2X ) 11./2 I 'Vi (Y2Y) [1./2 I Y2X - Y2yl :=

1/2 1/2 11/2 11(:2 1/2 Ij:2 lyi(Y2X )I {' 1V2(x)1 1Yi(Y2Y) I' (Y2(Y) Ix-yl:= I (YlY2)'(X)( f' (Y1 'V2)'(X)(

Ix-YI

As in the complex case we wish to use (15) "jO prove the invariance of

the cross-ratio, but this time there is only an absolute cross-ratio

(16) I I -~ ~J a,b,c,d - la::df : Tb-dT

-which is obviously invariant in the sense that

(17) I ya , 'Vb , yc , yd I = I a , b , c , d I

We can use the invariance to prove that circles are mapped on circles.

Indeed, it is a classical theorem in JR2 and JR3 , extendab1.e to lRn ,

20

that a, b , c ) d lie on a. circle in cyclic order if and only if

la-blle-d1 + Ib -clld-a! = fa -cllb ... dl • This condition can be -written

in the invariant form. J a , d , b , c I + Ie, d , b ,al = 1 •

As a second application of (17) we prove the :following important lemma.

Lemma 1. If Y leaves <D fixed, then y is a similarity.

Proof'. If ym ;:: 00 the.:1. yx ... yO leaves 0 and. eo :riXed and it is

sufficient to prove that 'V0 = 0, y co = QO implies yx. = mx wi th a

constant conformal matrix m. In other 'Words, we need. to show that yl (x)

is constant.

First,

or

Next

which implies

gives

222 r yx - yy I ;:: it I x-y J

2 (yx,yy)::: ~ (x,y) •

It follows that

so that

21

V(x + y) = yx + VY

'Vi (x+ y) :::; y' (X) :;; canst.

~ A n Corollary. If a T b are both fini te ~ then the most general y E M( JR )

with ya:;::> 0, Vb;::: CD is of the form

(18) . * * yx ;::: m[ (x - 0) - (a - b) ]

where m is a consta..r;t conformal matrix.

In fact, it is clear tha.t (x - b)* ... (a- b)* takes a into 0 and b

lnLa CD.

2.4. A. n We denote by M( 13 ) the subgroups Which keep

Bn = (x; Ixl < l} invariant. Because Mobius transformations are bijective

the unit sphere Sn-l;::: {x; lxl ;::: I} and the ext~ior of "En are also in-

variant.

Lemma 2. If' Y EM( aD) and yO:; 0 ,then y is a rotation (i. e. yx;::: kx ,

k E O(n) }.

Proof. If y~ = CD we know by Lemma 1 that V =!OX and beca,use Imxl = 1

for Ixl = 1 it follows that In;:;; k EO (n)

Assume now that '1. 100 ;::: b i CD. Then (18) implies

* * I (x - b) + b I ;;;; const.

for Ixl;::: 1 But

22

Ixl Ix-bllbl

Hence Ix - b I := canst. for Ixl = 1 , and that is impossible since b f 0 •

2 .. 5. We shall now determine the most general V E M( Bn) • This is done almost

exactly as in l.l1 for n == 2 eKcept that we do not have the complex notation.

We begin by proving a non-trivial identity:

Lemma 3.

?~oof. As it stands (19) does not make sense for y = 0 , but it is obvious

that both sides tend to 0 for y ~ O. We assume now that y f 0 and

write

** Ax == (x - y )

(20)

In other words, we trea.t y a.s a. constant and have to show tha.t Ax = Bx •

It is immediate that

* * Ay ==By = 00

AfYJ ::;: Boo = 0

Therefore, 0 and co are fixed points of' AB- 1 and by Lemma. 1 (AB- l ) 1 is

~onstanttl But then At (x) Bt (x)-l is also constan~, for

(21)

-l -l -1 = AI(E X)· B'(B x} =

-1 (AT B' )oBK:: canst.

From (20) and (14)

B' (x)

AT(X);: 1-2Q(X-~*) Ix- Y*I

:: lY12{1 - 2Q(y)) (I - 2Q,(x* -~)) (I - 2Q(x) ~ l:x:* _y1 2Ix\2

and. for x = y we find

23

Hence AI (x) = Bt (x) for all x, and since AO = BO = -y it follows that

Ax,=Bx.

Direct comparison of' AI (x) and Bt (x) a.s given by (21) yields

(22) (I - 2Q(y» (I - 2Q(x - y*) = (I - 2Q(x* - Y»)(I - 2Q(x» •

This is an important identity.

2.6. We repeat the construction that was given in 1.11 and. refer to the

same :figure. Given a E ~ (a f 0) we construct a * and the sphere

n-1 * (1 * 2 l/2. * . j 1}2 / I I . S (a, a I -1) ) 'WJ.th center a and radl.us 1- la al wh~ch

intersects n-l S orthogonally. The reflection in this sph~e is given by

(23 )

24

* I *12 * * cr x = a· + (a. - l)(x - a ) a

We let cr be followed by reflection in the plane through the origin a

perpendicular to a. It is easy to see that this second ref'lection amounts'

to multiplication with the matrix 1- 2Q(a) " In fact

y' = {I-2Q(a»y=y- ~ lal

and Fig. 5 shows the location of y' .

We now define the canonicaJ. mapping

T x = (I - 2Q( a») (j x a a

Fig. 5

and conclude that the most general y E M{lBn

) with ya = 0 is of the form

kTa with k E O(n) •

The explicit expression for

identi ty ( 19 ) • It is clear that

T x be~omes much simpler if we use the a

(1-2Q(a»)a* = -a* and thus we obtain

25

* I -*1 2 * * TaX = -a + (a - 1) (1 - 2Q(a»)(x - a) •

Replace Y' by a in (19) and multiply by I - 2Q( a). We attain

* * I 12( * * (1 -2Q(a»(x- a) := a + a x - a)

and

. 2 2 * * T x := -a * + (I a *1 .. 1)a + (1- 1 a I )(x - a) • a

Fina.l.l.y,

(25 )

* This is as simple as one can wish, but if cae wants to avoid the x

notation it can easi~ be brought to the form

(26) T x = a

12 2 (1- la )ex- a) - Ix-al a [x,a]

2

where [x,a] = Ixllx* .. al := I allx - a *1 and

We collect the various expressions for T x that we heve derived. y

a)

b)

c)

d)

T x = (r-2Q(y»(y*+(jy*1 2 -1)(x_y*)*) y

T x = -y + (1- lyJ 2 ){x* - y)* y

T x = (1 - I Y 12)x .. ( 1 .. 2eY + I y [ 2 U"

y [x,y]2

T x = (1_\y\2)(x_y) - lx- yJ 2y

y [x,y]2

26

Recall that [JL,y} = Ixlly - x*J ;:;: IYlix - y*1 . a) and b) are easy to differentiate and yield

(27)

1 _ 1 v 12 « ( *» tr'(X) = '''"2 (I-2QY» I -2Q x-y y [x,y]

2 l .. IYI * =, 2 (I - 2Q( x - y) )( I - 2 Q.( x» .. [x,y}

We have already remarked on the id.entity of the two matrix products. We

choose to introduce the notation

* ll(X,y) ;; (I - 2Q.(y) )(I - 2Q(x - y »

which leads to

T ~(x,y) = ~(y,x) •

Note that this can also be written

(28) ~(x,y) !:l(Y,x) ;::; I

It is helpful to View the formula

(29) Tf (x) = 1- )X~ ll(X,y) Y [x,y]

as a representation in polar coordinates:

( 30) IT'(X)I = l ... lyl: y [x,y]

is the "absolute" value and a(x,y) is the "argument" of Tt (x) • y

We make a note of the special values

(31 )

Also,

2 T' (0) :;; 1 - [;.rl

y

T 0 = -y .. and hence T = T-1

. Y ~ -y y

27

1

Direct computation of IT xl from any of the formulas a) - d) is verY' y

laborious. Fortunately, we can use the difference formula to obtain

(32) IT xl :;;Ix-yl y -rx,yr

NOW a short com~taticn leads to

(33) 2 2

I (2 (1- Ixl )(1- Iyl ) 1- T x = - .' -- 2 . .,.... -y [x,y}

Together with (29) we have thus

(35)

IT'(X}! - y

We have thereby proved the invariance of the Poincare llletric

ds = 2}wd 2

1- Ixl

lt is again seen-that every orthogonal circle is a geodesic. The

hyperbolic distance from the origi~ is

~ d( O,x) = log "l=liT

and more generally

28

(37) d (x, y ) = log I y , x , g , 11 I

where S, II are the end points of the directed. geodesic from x to y.

2.8. There is a close relation between TyX and rxr. To derive it we shall

first prove:

Proof: Let Ix and Rx be the left and right hand side of (38). Clearly,

Iq = By = 0 and hence U- l ( C) == o. But

where the left hand equation is obvious, whereas the right hand equation

.follows by application of (34). Thus

-1 I r r_1 {La )(0); L (Y)R (y) = 1

which proves that

side is

and one obtains

IR-1 = I . Apply the Lemma with y == T •

x

T' (y)

T~ =- {~(y) , TyX.

The le:rt hand

29

or

T y:;:; -A(y,x)T x x y

(39)

Remark. T is not defined for I y{ :;:; l but by formula b) we should write y

T x = -y ~ by continuity. According to l39 ) it follows that y

(40)

T~ :;:; 8{y,X)Y for Iyl = 1

T x = 6(x,y)x for [xl:;:; 1 y

One more formula. Differentia.tion of (38) with respect to x yields

on equating the arguments in (29) one obtains

(41) ~~ 6( 'VX , 'Vy) TY'1iTT ::, TY'T:YTf 8( x, y )

2.9. Although the formulas for the selfmappings of tF are simple enough it

is sometimes more advantageous to concentrate on the geometric picture ..

Recall that every 'Y E M( Bn) , Y f I , has a canonical representation

(42)

where

(43)

y = kT a

-1 So - Y 0 o..nd kE 00(.0) • Since Jkl = l we have

30

The set tx; l v' (x) j = l} is nothing e.lse than the orthogonal splu:!l'e with

* center a Nothing prevents us from considering the full sphere rather

than only the p~t inside Bn. It is called the isometric sphere of V,

and we denote it by K(Y). Moreover, we denote the tnside of K(Y) by

I(Y) and the exterior by E('Y). Observe that the identity has no isometric

sphere.

The chain rule applied to -1

y[y xl = x yields

and of course just as well

From this we draYT the following conclusions:

-1 1(y .. l) 'V maps on E( V) and E( 'V-l

) on I( y) •

Of course this does not determine 'Y completely, but only up to rotations

about the conmon axis of the unit sphere and the isometric sphere.

Furthermore, the restriction -1

Y : K( V) -. K( Y ) is an isometry and hence

a congruence mapping both with respect to the eucl:.dean and non- euclid.ean

geometry.

31

III. HYPerbolic geamet;r

3.1. This is .a. short chapter devoted to two separate tasks:

1 0

Derive some formulas of hyperbolic trigonometry ..

2° Show how to represent Mooins groups as matrix groups.

3.2. Let ABC be a non- euclidean triangle in Bn

• We denote the angles by

A , B ,C and the n.e. lengths of the opposite sides by a, b , c. we shall

prove:

I. The hyperbolic law of cosines:

(1) cosh c = COM a cosh b - sinh a sinh b cos C

(2 )

II. The byperbolic la.w of siBes

sinA sinh a

= sinB sinh b

= sine sinh c

3.3. PrDOf of I. We may assume that C = 0 , that a falls along the positive

Xl - axis and that b lies in the x1x2 - plane!' It is thus sufficient, to

consider the case n = 2 and we may use the complex notation.

Fig. 6

. a b The points B and A are at euclidean. distance tanh '2 and tanh 2' from o.

a b iC Hence B ~~d A are ~epresented by tanh 2' and tanh 2 e The point-

pair invariant is

(3) B(A,B) = [tanh ~ ... tanh ~ e

iC I I a; I b -iC I :;. 11 - tanh '2 tanh '2 e

tanh .£ 2

Just as in ordinary trigonometry all the hyperbolic fUnctions can be expressed

x rationalJ.y through tanh 2' . The formuJ..as are

From (3)

cosh X :;.

sinh X :;

2 x J..+tanh '2

1- tanh2 ! 2

2 tanh ! 2

( - 2 a 2 b 4 a b

cosh c ;; .L + th 2') (1 + th '2)... th '2 th 2' cos C

{l- th2 ~ ) (1 ... th2 ~ )

::: cosh a cosh b ... sinh a sinh b cos C

3.4. Proof of II. From (1)

cos C

. 2 C S1n :;

8in2

C 2

sh c =

::;::; cosh a cosh b - cosh c sinh a sinh b

222 (ch a. ... 1) (ch b ... 1) - (ch a ch b ... ch c)

2 2 ' sh a sh b

222 1 ... ch a ... ch b ... cit c + 2ch a ch b ch c

222 sh a sh b sh c

The synmetry prove s (II),

33

3.5. In this section we want to show that MO~.n-l) is isomorphic with

M(Bn). For n = 2 this was already done in Ch.I by observing

(tacitly) that M(R1 ) and M(H2) are both equal to PSL2 (R), while

on the other hand M(H2 ) and M(B2 ) are isomorphic because there is

a conformal mapping from H2 to B2

In the general case we begin by extending an arbitrary

• • (-'rTIl) h· "d . f Rn- 1 to a mapp1ng 1n M n • For t 1S purpose we ~ ent1 y

subspace {x =o} of n

n R • Recall (Corollary 2.5) that any

with the

n-1 ye:M(R ~

with ya = a , yb = ro , b F ~ ~as a unique representation of the form

* *' (4) yx = rorex-b) - (a-b) ]

,.... where m = Ak, A > 0, kE O(n-l). If we replace m by m we obtain a

mapping of Rn whose restrictio'n to Hn is an extension of y •

(In case b = 00 (4) should be replaced by yx = m(x-a». This

construction shows that M(Rn- 1) is isomorphic to M(Hn) •

It remains to prove that M(Hn ) is isomorphic to M(Bn). For this

it suffices to find a standard Mobius transformation 0: Hn+Bn which

we choose so that x = ( 0 , en ' (0) carre spo·nd to y = ox = (-e ,O,e) n n

where e n

is the last coordinate vector. The restriction of to

Rn- 1 is the usual stereographic projection.

The following pictures illustrate the choice better than words:

x

x-e n

LII,I

- e", '.

. I I / /' I

* e + 2(x ... e ) n n

* * y = aX = [e + 2(x - e ) ] n n

I

e n

.... e n

The correspondence is thus given by

(5 ) ** y = (e + 2(x - e ) ) n n

* * X :=. e + 2 (y - e ) n n

In terms of coordinates one finds

(6)

and

* y = 1

if­Y :;;:

n

2x. ].

2 Ixl -1

Ix - e l2 n

(i = ~, ••• , n - l)

34

Fig. 7

(71

When x = 0 one n

(8) Yi

and for Iyl ;::: 1

(9) X. J

35

2yi 2 \y- e I

(i ~ 1 , ••• , n ... 1)

x = n

n

2 1- Iy\

2 ly- e I n

verif':'es that lyl2 = 1

2x. ~

J. Y' =

Ixl 2 + 1

n

y. J. ::; 0

l-y X

n n

* , and (6) reduces to Y =y

2 Ixl - 1

Ixl2 + I

one recognizes these f'o~mulas.

3.6. The stereographic projection (8) mapa n-l { 1 . B = xl'···' ~-l ,0 w~ th

2 2 xl + ••• +xn_1 < I on the lower hemisphere of' n-1 {n I I S ,i.e. yElR; y =1,yn<O}

Let it be followed by the projection (Y1 ,···, Yn ) ~ (Y1 -' .... ,Yn- 1 , 0) ~ The

result is a ma];>ping of Bn

_1 on itself' defined by

(IO) 2x y :;;;

"'-ERn- 1 t \ 1 4\e , IX < . This is of course not a Mobius transformation, for the

mapping (y 1 ' ••• ,y n) ~ (y I ' •• Ii ,Y n-l ,0) is not conforlnal.

What is the nature of the mapping (10)? The stereographic projection ~

n-2 confonnal and leaves the spbere S = (xI! = 0 , 1xl = I} fixed. Therefore

it maps orthO@onal circles of Bh

-1

, i. e. circles orthogonal to

circles on the' hemisphere, which are MSO orthogonal to Sn-2.

n-2 S on

Fig. 8

But an orthogonal. circle on the hemisphere lies in a plane parallel to en and

its !)rojection is a straight line segment" In other 'Words, (lO) maps every

n-l geodesic in B on the segment between its end points. This is also obvious

by a simple computation. The equation of' an orthogonal circle is of the form

<Ia\ >1)

or

Ixj2 + l = 2x a

and thus (lO) is equivalent to ay = 1 , the eqp.ation of' a straight line.

This can be us.ed to construct the Klein model of hyperbolic space. In this

n-1 model the noneuclidean lines are the line segments in B The distance

of' two points y' ,yll is defined as d(x', XU) between the corresponding

points in the Poincar~ model.

b

c Fig. 9

37

"'2 . A glance at Fig.9 shows that la',b',c,dj=la,b,c,dl • Thus the n.e.

distance between x,y in the ~lein model is ~ loglx,y,t.ni.

3.7. We pass now to the matrix representation~ Recall that the group

O(n,l) acts on Rn+1 and leaves the quadratic and bilinear forms

O:::X,x> 2 2 2 = xo - xl - - x n

<x,y> = xoYo - x1Y1 - .... - xnYn

invariant. In ma.trix language A e: O(n,l) if

(11) AT (1 O)A =(1 O) . o -In 0 -In

It follows from (11) that (det ~)2 = 1 . The subgroup with det ~ ~ 1

is denoted by SO(n,I).

Geometricg.lly, the lightcone {<x,x>=O} and its interior {<'x,x> >O}

are invariant under a11 A E O(n,l) _ The same is true of the hyper-

boloid

x <0 • o

{<x,x:;, = I} with two mantles, one with x >0 o and one with

Theorem 1.- The groups M(Rn - l ) and SO'(n, I} are isomorphic ..

Proof: The upper hyperboloid

{ n+11 . 1 0) u = x g R <x,x>= I X > o

and the plane s~ction

{ n+l1 V = X'ER <XI,X I > >0, Xl = II o

are in bijective correspondence through the central projection

Xl = x/x. Fig.40 o shows x,y E U and the intersections

38

of the

line through x and y with the lightcone together with the projec-

tions x',y' and ~I,n' •

By elementary geometry the cross-ratios Ix,y,~/nl and Ix',yt,~I,ntl

are equal. To determine their common value we observe that ~ and n

are both of the form x+ty 1+ t

This leads to the equation

with real t satisfying <x+ty,x+ty> =0.

(12) 2 <x,x>+2t<x,y>+t <y,y> = 0

which consequently has two real roots t 1 ,t2 corresponding to ~,n.

The cross-ratio turns out to be tl/t2 which is positive.

The section V can be identified with the Klein model of Bn.The

straight line segments are the geodesics and d(x',y'} = ~Ilog t1/t21.

By conjugation with the central projection each A E SO(n,1) deter.rnines

a bijective mapping AI of V on itself. Because the coefficients

of (12) are invariant the same is true of the cross-ratios and hence

also of the n.e. distances d(xl,y'). This means that each At is a

n.c. motion, and it is obviously sense-preserving.

Conversely, if A' is a n.e. motion it can be lifted to a

mapping A of U which carries collinear points (x,y,s,n) into

collinear points (AX/Ay,A~,An) with the same cross-ratio. Because

of this the coefficients of (12) will remain proportional to each

39

other, and since <Ax,Ax> ~ <x,x > = 1 it follows that <Ax,Ay>=<x,y>

a~ well.

Fig.I0

40

The mapping A can be extended to all x inside the double

COne by defining Ax = <XtX>% A«X,X>-~ x) in the upper cone and

subsequently Ax = -A (-x) in the lower cone. One verifies that this

extension still satisfies <Ax,Ay> = <x,y> as well as

<A(x+y)-Ax-Ay,A(x+y)-Ax-Ay> = 0 •

It follows that A{ax) = nAx and A (x+y) = Ax + Ay • In other words,

A is linear, at least inside the double cone. Because every X E Rn+1

is a linear combination of vectors in the cone it is clear that A

extends to a linear mapping of Rn+1 on itself, and becauSe <Ax,Ay>=

<x,y> it is represented by a matrix A E DCntl) ; since the upper cone

maps on it,self it is even in SO (n, 1) •

It is evident from the construction that the correspondence

between A and AI is bijective and product preserving. We have thus

proved, so far, that

showed that M (B) n

proof of Theorem 1.

M eE) n

is iso:norphic to SO(n,l). In (3.:5)

is isomorphic to ( n-1,) d h· M.R ran' t 1S completes the

The passage from M(Rn- l ) to O(n,l) means a l,QSS of two

dimensions. For instance, the classical Mobius group M(C) is repre-

Rented by the Lorent~ group of 4x4 matrices. For practi~al purpose~

this is much too complicated, but for theoretical reasons it is impor­

tant to know that every M(Rn ) can be written as a matrix group.

41

IV. Elements of differential geometg.

4.1. I want to take a very 1lllsophisticated look at d.ifferential geometry and

regard it simply as a set of rules for changing coordinates.

A differentiable n-manifold is a set covered by coordinat~ patches such

that over lapping parts are connected by coordinate changes of the form

(1) - - ( x . ;: X. Xl ' ••• ,x ) , l.. l.. _ n i;;;l, .•. ,n

The change of coordinates shall be reversible, and this means that the Jacobian

matrix

ax. II ox~ "

J ex)

is non- singular. The coordinate changes will alwB¥s be C

The typical contravariant vector is a differential

(2) . oX..

d? = L: J. dxJ j oXj

Observe that we are using upper indices for the differentials, but not for

the variables.

An arbitrary contravariant vector is a s.ystem of vector-valued functions,

one· for each coordinate system, connected by the same relation as in (2),

namely

(3 ) -i aXi a

j a = ax:-

J

where we are using the summation convention, as always :from now on.

Similarly, a covariant vector is typified by a derivative (or gradient)

(4) of' '::."'1r" :;::: AX.

~

We use lower indices for covariant vectors, and the general rule is

b. ~

OX. b. ~

J uX. ~

42

The idea carries over to tensors of higher order. Here are tm rules for

contravariant, covariant and mixed tensors of second order:

;J-j = aX. --2:. OX

h

OX. -i a. ;;; l.

J Ox h

hk a

It makes sense to speak of ~etric and. skew-symmetric tensors, for if

ahk = akh , then 'aij = a:ji •

The Kronecker 8~ is a mixed tensor with the same components in all J

coordinate systems. In fact,

h o~ ax. o~ 8~ l.

8k OX. = - 3i":" =

J o~ J J

Ox. OX. l- and ~ are inverses.

dX, OX. J 1.

because the matrices

4.2. An important way to form new tensors is by CO:ltraction. Example: if

k aij is a twice covaria.nt and once contravaria.nt te.:lsor, then

;:: j a. a. •• ~ l.J

(summation! )

is a cO'JaX'lant vector, for

-j 'Ox. h o~ Ox

h h Ox,k -.-J. == ~ ---a

ij = ~ Nf" oX. OX. o~ l. J ).

4.3. A differentiable manifold becomes a Riemannian space by the choice of a

metric tensor 'Whose matrix is positive definite. It serves several purposes,

1. It defines arc length acd volume.

2. It serves to push indices up and. down.

3. It defines covariant differentiation.

4. It defines paraliel displacement •

.Arc length is defined by set..tiIlg

1 i . '2 as = (g .. ax rucJ)

l.J

and. dcf'ining the .length of an ~c 'V by

B~ca.use

t(y) = r ds ~

y

is covariant of" ord,er two and the differentials axi

variant, ds has a meaning independent of the local coordinates.

are cont.rR.-

The determinant of g.. is denoted by g. fi transforms like a l.J

density in the sense that

R = (det

This follows by the determinant multiplication rule. This defines an invariant

volnme by

-rr:: 1 n V- tJ...;g dx ••• dx

(we consider only sense preserving changes of coordinates.)

4.4. The inverse of the matrix g.. is denoted by gij l.J

44

It is clearly a

contravariant tensor of order two. Multiplication with gij or gij followed

by contraction pushes indices up and down. For instance

a j = g. . a. l.J l.

The mixed components of gij are

= 8~ J

af 4.5. The gradient a of a scalar function is a covariant vector. For oc

i other tensors differentiation of the components does not by itself lead to a

new tensor. It has to be replaced by covariant differentiation, which we pro-

c eed. to d.efine.

The Christoffel symbols are

(6)

a.."1.d

(7) rl:j

=: ghk r. . k • l. l.J ,

In order to find out how they transform we start froll

and d.ifferentiate using the chain rule

ag •. ogab OX o~ oX .....:2:J. a c :: ~ - OX. o~ OXi ~ c J

o~ ~ o2;t oXb + (

a + a ) gab Oxidxk ax. Oxj~ ax.- ,

J :1

where we have interchanged a, b in the last term.

Permutation of the subscripts leads to

oX ~ oX (8) r .. k =T a c +

l.J, ab,c ax. ax:- ax; gab J. J

The extra term on the right shows that r .. k l.J,

Multiply both sides of' (8) by

hk o~ ~ -cd g = oxd

-. g axc

This gives

ox o~ a~ 2-

(9) r:'-. d a + '0 xd

= rab ax.- rx:- ~ OXiOxj l.J l. J

oXd

Again, there is an extra term.

We use (9) to find the mixed derivatives

multiply (9) by ~ to obtain ~

(10) oa­~

ax.Ox. :1 J

a~ a

Oxi

Oxj

is not a

Oxh ~ d

~ a~

tensor.

• For that purpose

4.6. If yj is a contravariant vector we shall show that

(11)

is a mixed tensor. From

we obtain

Use (10) to eliminate the cross-derivative:

T-a Tte very last term sim:plifies to r . v • We :pull it over to the left hand a:L

side Which became~

On the rig'h t we are left with

oJ1 aXk

ax. ~k

oX. ~ h - - -l. + -..J. --- V Ox.,lt Ox~ ox OXa ax.

:L h J.

Ir!terchange h and. a in tl1e second. term. It becoltes

and vre have shown tha.t

which is the rule for a mixed tensor.

The formula for the covariant d.erivative of a covariant vector is s:.milar:

We skip the invariance proof.

The ~eneral rule is to add one term for each contravariant index and

subtract one term for each covariant index.

Example:

k V

h V."

l.J

k avo "~ a k h -Y + r=-h ·.va., _ - r v ... ra r . ~ a. 1J ih aj jh ai

One verifies t...hat the covariant differentiation of a product f'ollows the usual

rules. Another practical rule is that the metric tensors

behave like constants. For instance

_ Ogij V~ gij - ~

a f k " g .

1. aJ

ag .. ~ - f1.-- j ... rkj . = 0 v~ Q..~, }l.

It is customary to write

4.7. If ~ is a scalar, then

i:Jf V V f == V k j k Oxj

is symmetric in j and ~, so that

48

For other tensors this is no longer true. For instance, operating on

a covariant vector one obtains

h where R" k is a tensor, the curvature tensor of the metric ~J

given by the somewhat complicated formula

(J2 ) h

R. 'k ~J

h ar ;; ~+ ~

It is

We make no attempt to carry out the computation. Contraction with respect to

h and k leads to the Ricci curvature

(l3 )

and double contraction to the scalar curvature

(14) R - Ra = gab R a ab

4.8. Vectors and tensors are sitting in the tangent s~ce attached to each

point of a differentiable manifold, but there is no automatic way of comparing

vectors and tensors at different points. The notion of parallel displa.cem.ent,

or connection, serves this purpose. We shall use only the stmplest connection,

known as the Riemannian connection.

Let x = x ( t ) repres ent an arc y in a Riema..nnian space with the metric

tensor g .. ; we assume that x(t) ECOl. Let set) be a contravariant vector ~J

at x(t) •

49

Definition. We say that ~(t) remains parallel along y it

(15) ~si(t)~(t) = 0

for all t .

More explicit~, (15) reads

(16) or

= 0 •

This is a system of first order linear differential equations. Because

the r!j and the ~ are C- it has a unique so1ution with giVen initial i dt values ~ (t~) . In other words, ars vector ~(to) determines a vector

~(t) obtained by parallel displacement along V •

4.9. The length of a vector ~ , measured in the Riemannian metric, is .. 1/2

(;,~)1/2 = (i~j gij '~CJ)

Similarly, the angle between two vectors .5 ~ on is given by

cos e

Theorem. The inner product (~'11)' and hence length and angle, remain

constant under parallel displacement.

The proof is a straight forward cow;putation making use of (16) and tte

identity

50

4.10.

,Definiti~n. An arc x(t) , 0':: t .:5 to ' is a geodesic if the tangent vector . dx.

x' (t) remains parallel to itself. If we put t;l. = dt in (1.6) we obtain

the condition

(18)

for x(t) to be a geodesic.

i ~ dx. fkj - ..-J. = 0

d.t dt

It is not only the shape of the arc but also the parametrization that

I!1akes an arc geodesic. By the theorem the length of x' (t) remains constant

along a geodesic. If we replace the parameter t by or, the length would be

dt mUlt:!.plled by d5 and would no longer be constant w:less the change were

linear: t = a7 + b •

We can regard (18) as a system of differential equations for the functions

x. (t). According to the general theory we can prescribe the initial values 1.

x.(O) and the initial derivatives x!(O) . In other words, there is a geodesic 1. ~

tram every point in every airection. We ma¥ even nor.malize so that the tangent

vector has length one, in which case t becomes arc length, usually denoted

by s.

The existence theorem produces x(t) only in some small interval [0, t ) , o

but we can start anew from the point x(t). Thus a geodesic can be continued o

forever unless it "tends to the ideal boundary".

A geodesic arc is, at least locally, the shortest arc between its end

points. This is proved in calculus of variations.

4.11. We shall now applY these notions of differential geometry to the unit

ball. Bn

with the Poincare metric

51

(19)

We are in the unusual situation where we can use the same coordinate system

in the whole space Bn , namely by identifying each point JC E BO with its

coordinates (xl' ••• ' xn). Any diffeomorphism of Bn would lead to

another coordinate system, but we choose to consider only coordinate changes

of the form

(20) x = yx

where y E M(Bn). This means that we are interested in the co-O.rormal structure

o£ Bn.

The metric (19) corresponds to the metric tensor

(21) 48 ..

~J 8 .. ~J

A little more general~ we consider an arbitrary conformal metric

and have then

(in this connection 5 .. ~J

ij -2 g = P 8 ..

~J

is an element of the unit matrix and not a tensor.)

We wish to compute the curvature tensor. It is convenient to use the

notation

u = log p ,

On.e obtains

+ 5ij(~ - ~~) - ~(uik - ui '\.)

+ (8ij \k - 3ik\j)l vu l2

R .. = 8 •. bou + (n - 2) (u •. - u. U. + B. j I vul 2) •

~J ~J 1J 1 J ~

For the Poinca;r9 metric

u = log P == log 2 - log (1 _ I x 12 )

u = i

u .. = 1J

2B .. 1J

2 1- (xl

28 .• 1.]

2 1- (xl

52

Rh ;; . 'k ~J

~ijk

4 2 2 (B.. 5. . - 8. kB .)

(1 _ I x I) nk l.J ~ nJ

16 = (l- Ix12)4 (\k5ij - 5ik\j) = ~gij - gikgjh

4(n - 1}8 .. l.J

R = gi jR .. = n(n- 1) ~J

53

The scalar curvature is constant) but ~ = -1. However, the formula for

~ijk shows that the sectional curvature is constantly equal to -1.

4.12. The Beltrami parameter

There are three important invariant differential operators which generalize

the grad.ient, the divergence and tile Laplacian.

a) The gradient maps functions on vectors. If f is a scalar

V.f 1.

:::; of Ox.

1.

is a covariant vector. The corresponding contravariant vector is

(22)

i- i' of gJ Vf =gJ_ j Ox

j

The square le~th can be written either as a d.ot product Vif· V.f or as :L

i j gij V f V :f = gij Vir v j:f :::;

JVfl2 = I: ij ...2!. of

g ox. Oxj i,j J.

ThiS is the first Beltrami parameter and it used to be called V:f. It 1

obviously generalizes the square of the gradient.

* See L.P. Eisenhart: Riemannian Geometry, Princeton Uhiversity Press t

1960, p.Bl, (25·9).

b) The d.ivergence leads from. vectors to scalars. If v is a vector-

valued function we define

(23) divv i = V.v

~

i = V v. ~

These expressions are identical for ij g behaves like a constant and we

obtain

ij == 9.g v.

1. J

From

avj ;;: - + ox.

J..

we have

(24 )

iJ' ;;: g V.v, ;;: ~ J

j k fik v

There is a sim?le formula for By the well known rule for dif'f'erentia-

tion, of a determinant.

Btlt

and hence

and thus

(25 )

~ log g = tr (g-l ~)

~ij = '::I.x.. f k ,· + rkj . u It ~,J ,,~

iJ' ag .. =- g -3J.

~

ij Ogij _ ij + ij r = 2 ri g o~ - g f k1 ,j g kj,i ki

55

From (24) and (25) we get

(26) 1 0 .

divy :::: ~ (jg yl. )

Ji ~

c) When both operators are combined we obtain the second. Beltrami

P8:!ameter

~f = d.iv gradf :::: 1

.fi

This is a direct generalization of the Laplacian and it is frequently

referred to as the Laplace-Beltram:.. operator.

I-ts importance lies in the i'act that it is independent of the local

coordinates. For instance, if f is d.efined on Bn

a.ad if we use the

Poincare metric, then for n

yEM(B ) ~

(28) ~(foy) = (~f)OY •

For short, a. solution of ~f = 0 on Bn will be called hyperbolically

harmonic (h .. h).. TNe see from (24) that if f is h .. h , so is fO y

This is not true for ordinary harmonic i"unctions, except, as we shall see,

when n;;:; 2 •

On a Riemannian space a solution of' ~f will be called harmonic si.'lce

this does not make sense in aQy other meaning.

4.13. We compute ~f for a conformal metric ds:::: pldxl . One gets simply

(29) f -n 0 (1'I11- 2 _C>f' ) I'i... =p .. ""'2 oXi oX i

2 P ;:: D

1 ... Jxlc one finds In the special case

( 30) ~f ~ (1- ~j222 [ Ar + 2{n - 2~ xi :: 1 , l-Ixj 1.

WIth the cusLolll.ary I,lOtation IxJ::; r

(31) A.2f'

For the half-space

( 32)

x. of :::; of

~ r '-"-

J- ar J..

== ~l- r2}2

4 [llf

1 P :;:-­x n

and

+ ~{n-2) 2 r l ... r

M] or

57

v. ;Hyperharmonic :functions

5.1. We shall now study hyperharmonic functions on Bn. our first task

will be to find all solutions u(x) of ~u = 0 that depend. only on r:= Ix] •

Fo!' a f\mction uCr) one obtains

and thus

oU Oxi

x. = u ' (r) ~ .....

x.x. 8;; x.x. ull(r) -¥ + u l (r) (~- .2:.....sl)

r r r3

00. ;;:> ull(r) + (n -1) ut (r) r

According to equation (32 ) in 4.l3 u(r) will satisf'y

(1) u"(r) + (n-I) ur~r) + 2(n-22) rul(l") = 0 .

l ... r

If ut(r} f 0 this can be written as

unCr) en -1) + 2(n - 2)r = 0 u t (r) + r 1 2 -r

or

d 2 dX [ logu'(r) + (n-l) logr - (n- 2) log (1- r ) ] = 0

from which we conclude that

u l (r)

This lea:ls to the general solution

.0-1 r

= const.

r (2 ) u(r) = aJ ( 2 n-2

1- t) dt + b n-l

t

We see at once that no solution can stay finite for r:: o. As a

normalized solution we introduce

n>2 g(r)...... 12 r2- n for r ~ 0 and

n-For n =2 g(r};:: log % . For

g(r) = O«l_r)n-l) for r _1 • For n = 3 T:::' 1 2 1

g(r)=C"r - -) ::-+r-2. F. r

5.2. Toget.her wiLh g(r) any g( 11'x\) n with l' EM(B) is again h - h

because by (29) in 4.13 y comnmtes with the Laplace-Beltrami operator. In

particular

(4)

has a. singularity at y and Will be regard.ed as the Green' s function with

pole at y.

5.3. The fact that a h - h function that depends only on r is either a

constant or has a rather stro~~ singularity suggests that any isolated

singula.ri ty of a h - b function. should be equally strong. In :f'unction theory

we are used to study such questions by use of integral formulas, especially

Cauchy t S integra.l formula and for ordinary harmonic functions Green t s formula.

For h - h functions there are two ways of approaching the question.

Ei ther one can derive Green t s formula in the general setting of Riemannian

spaces or, in the hyperbolic case, one can apply the classical Green's formula

to sui tabJ..y chosen functions that a.r e naturally connected with the hyperbolic

metric.

59

We shall use both methods, but we begin with the second which is by

far the simplest.

5.4. As a preparation it is useful to collect a few facts about multiple

integrals especially these connected with spheres.

(5 )

(6)

Recall Euler's r - function and. B - function

to

J a-l -t r(a):::; t e dt,

o Re a> 0

1 B(a,b) = J ta-l(2._t)b-l dt ,

o Re a > 0 ,

They are closely related. 2 The substitution t:;: x gives

(7) OCt 2

rea) = 2 I x2a- l e -x dx o

and t = sin2~ in B(a,b) gives

'JC/2 ( 8 ) B( a, b) = 2 S sin 2a-1. cp cos 2b-l cp d cp

o

~"\rom (7)

and in polar coordinates

Re b > 0

to 2 ~j:2

S 2a+2b - 1 -r ~ 2a-l 2b-l rea) reb) :::; 4 r e dr J cos cp sin r.pdcp

and. thus

(9)

o 0

= r(a+b) B(a,b)

B(a,b) :;: rCa) reb) r(a+b)

60

In particular, since and r(l):= 1 ,

(10)

We shall denote the "surface area·' or (n - 1) - dimensiona.l euclidean

m.easure of 8n

-1

by 0,) (some authors prefer c.o 1.). The area of Sn-l(r) n n-

n-l n 1s then (J)n.itl. r and the volume of B (r) is

r V (r) := J co r n- 1

dr := nOn

To compute the area of a spherical cap of radius cp (mea.sured alopg the

unit sphere) we project on the equatorial plane x. = 0 and note tha.t the area. n

element on the sphere is

dO' =

The area of the cap is thus

dx1

... dxn

_1

x n

sin cp A(cp) - en S - n-1 o

n-2 r dr

Fig. 11.

61

With r ;; sin e , ax = cos e de , x = cos 8 n we find

(11)

In particular,

and thus

This makes

follows that

(J2)

ij) n-2 A(cp) = {Dn_l S sin e de

o

1(/2 ill = 2A ( !.) :;; 2m S

n 2 n-1 0 n-2

sin a de

w n

(D n-1.

= B( n;l , ~) = r (E. )

~

(l) n

=

independent o£ n and since ill2 = 2n it

For comparison we shall also compute the area of' a sphere and volume of

a ball in hyperbolic space. Recall that a hyperbolic radius s corresponds

s to the euclidean radius r = tanh 2 of a ball with center o. Tbe n.e.

area of' the sphere is tben

and the voiume is

(13)

ill n

n-1 n-1 2 r . n-l = ill S1M s n

s V. (8) = ill J sinhn- 1

t dt h n 0

(Vh stands for n. e. vOlume). The formula is thus the same as be£cre with

sinh instead of sin and of course n + 1 in the place of n.

62

5.5 We return to h - h functions. Rec~ll the classical Gree.o, I s formula

(14) s u ~vdx: = r U &! dO' - S (Vu .. V'v)dx D ~D~ bn D

Here D is a region in JRn, dx = dxl ...... dxn , ~ is the derivative in

the direction of the outer :!l.ormal of the smooth boundary ~D and (VU" vv)

is the inner product of the gradients ..

Let us now assume that D lies in Bn. We shall repJace everything by

its hY.Perbolic counterpart, using the f'ol~ow1ng notations: for the hyperbolic

volume element:

the ar ea element: 2n- l d g

= (1- txI2)ll-I

the normal derivative: = 1- Ixl2 Q!

2 bn

the gradient:

the 1aplac1an:

= 1- tx I2 Vu 2

~v ..

With these notations we clatm that

Lemma 1:

It is ea.sy to chec.k that this is exactly the same as (14) with u replaced by

-u = 20-

2 (1_ 1 .... 12 )2-n u .. .A. As customary, one can eliminate the ItDirichlet, integraltl

on the right by interchanging u and v and subtracting. The resulting formula

(16) S eu l\ v - v~ u) ~ D

is the one in most common use.

The special case v = 1 leads to

(17) r - J bu .J t\ udx - - d~ D bD ~

- v

and it s:3.ows above ali that an h - h function has vanishing flux

(18)

In euclidean terms (18) reads

(19) r ~ d g~ 2 = 0 • "bD v

n (1- Ixl )n-If D = B{r) we obtain *)

S bu dO' = 0 S (r) br

In other words, the mean value

m{r) == 1:.. JUdo-(On s(r)

is constant. In particular, if u is h - h in a full neighborhood of the

origin, then

(20) uCo) = I

*) From now on n will be fixed and we w.ri te B for Bn and S for 8n- 1 .

for all sufficiently small r. This is the mean-value property. The mean-

value can also be taken with respect to volume, and because of the :cotational

symmetry it does not matter whether we use euclidean or non-euclidean measure.

However, the n.e. average is invariant and we obtain;

I.ennna. The val·ole of a h - h function at the center of a n. e. sphere or ball

is equal to the n.e. average.

Remark. The mean-value formula can also be proved without 8.J:lY computatiou

whatsoever. In fact, it is possible to write

m( Ix!) = r u(kx)d.k "o(n)

where dk is the Haar' measure of O(n) Because u(kx) is h - h for every

k it follOWS that m(lxl) is a h-b. f'tulctionwhict depends only on Ixl and hence must be a constant,. As a result of the mean-value property h .. h

functions satisfy the maximum-minimum principle:

A non-constant h - h fwlction cannot have a relative maximum or minimum

on an open connected set.

The proof is the same as for n = 2 •

5.6. The question of removable singularities has the :following answer,

Theorem.. Suppose that DcB is open a.nd a ED. If u(x) is h-h in

D \ ra} and. if'

limu{x) J In-2 x-a :::: 0 if n>2

x-ta,

lim. u(x) 1 0 if n=2 ::::

1 , x-+a. log lx-at

then u has an h - h extension to D.

Proof ~ Tliithout loss of generality we may assume that a = 0 and D = B(p) ,

p > o. We choose a fixed y E B(p) , y I 0 a.r:.d apply Lemma 1 to u:= u(x) ,

V = g(x,y-) in the region n(p) \ (n(r) U D(y,r» with r < min (iyl , p - lYl> • Since ~u = ~v := 0 we obtain

(21) r (u(x) bg~Y) - g(x.y) b~) ) ~ (x) ~ 0 •

S(p) +S(r) + S(y,r)

We look for the limit as r .... 0 and start by considering the integral

s (y ,r). It is evident that the integral of the second term will tend to 0,

for g(x,y) = 0 (---1..-2) (or O(log 1: ) ) while >.'Ou is bounded and dah n- r un r n-1 n

contains the factor r • The integral of the first term can be witten as

(22) -s (1- Ixl 2 )2-n u(x) bg(x,y) rn-1dro ~r

Ix- Yl =r where di.D is the element of "solid angle" (or the measure an Sn-1).

We recall that g(x,y):;: g( ~'X xl) and I T xl = Jx -yJ so that y y ~

( , blTxl - bgb~~Y) = -g < I TyXJ ) J =

Here [x,y] -+ \1- ly(2 as x-+y and

r ~ log l'l x I = r -2.. log r ... I . Or y ~r (x,y)

It follows a.t once that (22) tends to ill u(y) • n

66

We have now to investigate the part of (21) that re~ers to S(r) • We

make first the observation that by Green's formula the integral

(23 ) I (u(x) ?$(X>Y) - g(x,y) ~) d'b (x) S(r) ~ b~,

is in fact independent of r and hence defined for all y E B(p J \ (o} .

Moreover, because g(x,y) = g(y,x} it is;evidently a h - h function of

y. We wish to show that the boundedness condition of Theorem 1 makes it

identically zero.

Let us rewrite (23) as

(24)

Observe that g and ~ remain bounded when r - O. Hence 'l:ihe condition

I In-2 u(x) x - 0 makes the limit of the first term zero.

For the second term we write

~(r) = r g(r~,y) bu(r') ameS) .. 8(1) ", br

and obtain 2r 2r S jJJ(t) dt == r den f g(ts,y) bn(t~) dt r JS(I) 'r ~t

= f .. S(l) ~ 2r 2r J g(ts,y) n(tg) I - S u(tt:) ag(t~2Y) dt dm(g)

r r bt

67

This is o~ the order

with ~(t):: o( ~l) t

1 o( -.n:2) • Hence there is a t between r and 2r

r and since lim r n-l. L (r ) . t . t t b .- ex~s s l. mus e zero.

We can now conclude from (2l) that

u(y) = - 1. S (u{x) eg(x,y) - g(x,y) :o.~ } dO"h • a;- S(p ) ~~ U n

n

The right hand side is a h - h function in S(p) and. defines the desired

extension of u .

5.7. Bo".mdary values. If u is h .. h in B and has a. continuous extension

to the closure B, then the mean value property implies

(25) u( 0) = t S u(x)dro. n n-l.

S

We obtain. a more general formula if we appl¥ (25) to uaY where y EM(Bn ) •

Indeed, uoy is again b - h and extends continuously to the boundary. Hence

(26) 1 u(VO) = -

ron s u( y x) CIm(x)

n-l S

We specialize to 'V == T-1 for a fixed y E B . Y

Because

u(y)

and replace x by T x in the integral. y

This leads to

u (y) :: 1:.. r u (x) I T 1 (x) I n-l dill ( x) ron .~ S Y

or explicitly

(27) u(y)

68

where 'We have used the simplification [x,y} = Ix -YI when Ix] = 1 •

This is the Poisson formula far the ball. Note that the kernel is just

the (n - l)st power of the Poisson kernel for n = 2 .

5.8. We shall use the notation

k(X,y) = 1- IYI~ Ix-yl

with the understanding that Ixl = 1 and Iyl < 1. Formula (27) lets us

suspect that k(x,y) is always a h - h function of y.

The direct verification of this is not difficult, but it is much quicker

to pass to the half'- space Jf-. Reca 1.1 that the laplacian fo,.. the half' space

is

It follows without computation that

A xa :: 0:(0: - n + 1) xo. h n n

ThLlJ::l e ver"y °. ~l x Is all E::dgerl ... f'unctlon and oX. is h ... h • n n

We recall (see 3.5) that the canonical mapping Bn

-+ tf is such that

x = n

k(e , y) n

where x E Hn , y E Bn. We conclude that k (e ,y) ex is an eigen:f"unction of n

n n-l n-l ~ for B and that k( en ' y) is h - h. Given any xES (apologies

for using x twiee) there exists a rotation ~

-1 is obvious that k(x,y) = k(f3e ,y) = k(e ,t3y) n n

we conclude that

such that px = e , But it n

Because of' the invariance

(28) ~ k(X,y)O: = a(o:- n + 1) k(x,y)CX

and n .. l

~ k(x,y) = 0 •

5.9. Suppose now that f(x) is of class L1(S) , i.e.

Then we can form

J I f(x)r da>(x) < 00

s

(29) 1 S ( n-l u(y) :;; k x,y) f(x) dro(x) , (1)n Sn-l

which is obviously h - h •

Theorem. u(y) has rad.ial limits rex) a.e.

Proof.. It is .known that

J f(x)dco(x)

(30) lim B(s,8) = f(~) 8-0 S dm(x)

B(~, 8)

for a .. e. ~ E S (see e.g. Rudin, ReaJ. and. Complex AnalysiS, Tbm. 8.8). We

shall prove that u(r~)'" f(~) for r ... l whenever (30) is ful.filled. We

may assume that ~ = e so that n

r n-l J_ k(x, re) f(x) dm(x) S n

We introduce the colatitude ~

of the kernel to

defined by x = cos cp and change the notation n

K(r, tp) 2 n-l

:;; 1 ( 1- r 2 )

(l)n 1- 2r cos cp + r

Moreover, we S~1.all write

F(cp) ;;:; r t:J

x > cos cp n

70

f(x)d£n(x)

for the integral of' .£ over the polar -cap of' radius tp • Clearly, the

formula for u(re ) n

can be w-ri tten in the form

(31) -:rr

u(ren

) ;;:; r K(r,cp) Ft (cp) dqJ a

where Ft (cp) exists a.e. and the measure F t (cp)dtp is a.bsolutely continuous.

Integration by parts yield.s

We wish to show that u(re) -+ fee ). Without loss of' generality we may n n

assume that f(e):;;::: 0 , for otherwise we need only replace f by f - fCe ) .. n n

It is :tmmediate that K and ~ tend uniform1¥ to zero for r - 1 in

any interval 8.:s tp .:s 1C , 5 > 0.. Hence u(re) has the same limit as n

8 u{5(r) :; - J F( r.p) fro K(:r, cp)d-C9 •

o I

Because of (30) we can choose 8 so small that

= e c.o 1 Jq> sinh

-2 e de

n- 0

I~ follows that

:for M > 0 • bcp

Another integration by parts leads to

a lua(r)1 ~ - eA(8) K(5)+' e S KA'(cp}dcp

o :n:

< € r KA' (tp)dcp = e r K(r,cp)dro = e Jo S

by virtue of the fact that

71

as a special case of the Poisson formula. We conclude that lim u(re ) = r-l n

lim U o (r) = 0 . 8-0

5.10. A little more generally u(y) -+ f(en ) in any "Stolz cone It characterized

by

(32 )

Write yl = lyle for short. ~ (32) n

Ix ... yl.s Ix-Y'j + Iy-enl + Iyt-enl <

!x-y" + (M+l)(l-Iyl)

Lr Ixl = 1 this gives

!x-yl.:s (M+2)Jx-yl

Similarly,

Hence the ratio 1k~fX'Y~) lies between fixed bounds and we conclude that the x~y

limit of u(y) - f(e) is also zero when y - e inside the cone. n n

72

VI. The geodesic flow.

6.1. We shall now pass from B = En to its unit tangent space T1(B) •

The points of Tl (B) consist of a point x E B and a direction at that pOint ..

The diTection will be given by a unit vector ~ E S = Sn-l. Thus Tl (B)

can be id.entif'ied with B X S , but W'e prefer to think of it as the spa.ce of

directed line elements (x,S). Because of connections with dynamics Tl(B)

is sometimes referred to as the phase space.

The Mobius group M(Bn

) acts in an obvious way on Tl (B).. If y E M(Bn

)

i-:. is clear that x shoulci be mapped on y x • At the same time g should

be transformed by -:.he matrix y'(x) to give the new d.irection yl(X)~ for

the line element at 'V x , but in order to obtain a W.ll t vector we have to

divide by ,y' (x) I ..

(1)

(2)

Hence we define the action of y by

y : (x,~) -+ ('IX, X' (x} e; ) Iv' (x) I

There is an obvious invariant volwne element J namely,

dm = ~ am (~) ,

~

where m(g) denotes the solid angle. In fact, the Poincare metri~ is invariant

under y and ~ und.ergoes a rotation which keeps the spherical measure in-

variant.

6.2. It is of interest to introduce a point-pair invgriant for the action (1).

Let (x,S) and (y,l1) be two line elements. We claim that

73

(3 )

is a suitable invariant.

We observe first tha.t this expression is symmetric. In f"act, multiplica-

tion by A(y,x) does not change length and

~(y ,x){ll- A(x,y )~) ;:; 6(y ,x) '\1- ~

by virtue of the identity A(x,y) A(y,x) ;::: I. Thus

Now let us check the invariance. Simultaneous application of ~ on

, , (4) Y (y) il- A(yx,yy) Y (x) ~ I

ly' (y) I Iy"(x) I

But we have proved (see (41) in (2.8)) that

Thus (4) becomes

I y'ey) (ll-A(x,Y)~)I;::: 11l-A(x,Y)~1 lv' (y) I

and. the invariance is established.

6.3. We shall now determine the infinitesimal form of (3). In otber words,

we want to compute

l~+d~ - A(x,x+dx)1g1

Here is the computation:

(6)

X~j Q(x) .. ::: 2

lJ Ixl x.dx. + x.dx.

dQ( ) ~ J J ~ x .. = 2 ~J Ixl

x.dx, 1. J

Ixl2 -x.x.(xdx) ..2;J

2x,x.(xdx) ~ J

2x.x.. (xdx) J K

where we made use of the summation .convention.

(7)

Here

and thus

[(I - 2Q(x) ) dQ,(x~ij = x ,<ix, - x.dx,

J 1. 1. J

Ixl 2

* 6(x , x + dx) = (I - 2Q(x - x - dx)}(I - 2Q(x)) •

2 ( * ) 1 .. Ixl(;.

Q, X -x - dx = Q ( ~ x .. dx) J:x;l

= Q(x- Ix12'2 dx) l-lxJ

Ixl2

= Q(x) - ., 2 d Q(X) 1- Ixl

~(x,x+dx) = (I-2Q(x) + 2!x12

2 dQ(x)(I-2Q(x» 1- Ixl

2 = I - 21xl 2 (I - 2Q(x» d Q(x)

1- JxJ

*) Note that (I - 2Q)2 = I implies dQ, (I - 2Q) = -(I - 2Q)dQ .

74

a.nd by (7)

(8)

Now we substitute (8) in (5) to obtain

[(~ + d~ - 6(x, x + dx)~). :::: ~

;;; d.~. - 2 :L

(dx~)x. - (x~)dx. ). ).

2 1- Ixl

The in:fini tesimal invariant is hence

(9) Id~ _ 2 (~dx)x - (~X)dx

1- Ixl

2(x.dx. - x.d.x.) ~ J J ~

2 l-Ixl

An invariant Riemannian metric on T1 (B) can now be introduced by

(10) 2 ds ;:::;

75

6.4. We shall now define the geodesic flow which is a one parameter group of

dif'feomorphisms Vt of Tl (B) which satisfy VsVt = Vs+t •

Every line element (x, S ) determines a geodesic ray wi cb starts from

x in the d.irection ~. Fix a real number t. Let x move along the i

geodesic from x to a. point x at distance t from x; distances are

counted. positive in the direction of the ray, negative in the opposite

direction. At the same time we let the vector ~ slide to the positive I I

tangent vector ~ a.t x ; (see Fig. 1.5)

76

Fig. 12

We define

(1J.)

Mar eover ~ the

construction is clearly invariant with respect to Mobius transfornations in

the sense that

(J2)

for each y E M(B) .

6.5· The most important property of the V is that they define a flow in t

t~e sense that each Vt leaves the volume element dm, defined by (2) ,

invariant. We shall prove tbe invariance by introducing an other volume element

din on Tl (B) which is at the same time invariant under Mobius transformations

and. under the transformaticns Vt

.

If dm. and din are both invariant u.nd.er M(B) , then their quotient is

an automorphic :function. But M(B) is transitive in the sense that any line

element (x,~) can be mapped on any other. But this means that the ratio

cii : dm is constant. Since din is invariant lmder every 'Vt

the same is

then true of dIn.

77

To construct dm we introduce a new set of parameters on T1(B) •

As we have already pointed put every line element (x,~) determines an

o~iented. geodesic through it. This geodesic has an im tial point u and a

terminal point v. Clear ly , u and v can be calculated explicitly from

x and ~ (by Sr complicated fol"lrll)..la).

Conversely, u and v determine the geodesic, but we need still another

parameter to find the position of x on the geodesic. Let the midpoint of

the geodesic from u to v be denoted by a = a(u,v). To locate x we

use the signed non-euclidean distance s frbm a to x (see Fig. 16).

Fig. 13

It is now clear that there is a bijective correspondence between the pairs

(x,~) and the triples (u,v,s). This correspondence is in fact a diffeo­

morphi sm between B X SandS X S X m. .

The action of' Vt

on (u,v,s) is Cluite oQvious: (u,v,s) is replaced

by (u, v, s + t). As a result the volume element

&D( u) den ( v) ds

and more generally any element of the form

feu, v) am(u) aro(v) ds

is invariant..

78

We want to find a dID that is !nvariant also against all YEM(B).

y moves x to a point yx at distance s from ya and thus to the

directed distance s + d(y~,a(yuiYv~ from the midpoint a(yu,yv). This

means that s is invariant ll;P to an added term that depends only on

u and v. Because [YU_yv[2n-2= Iy' (u) In-lly'(v) In-llu_vI2n-2 it

-follows that

din = dw{u)dto(v}ds

I : 2n-2 u-vi

is invariant with respect to y • The conclusion is that

i

dm idOl (u) dw (v) ds = C !!--~~~~--

lu-v f2n-2

with constant c • Actually, one shows by suitable specialization that

c = 1 '.

79

VII.Discrete Subgroups.

1.1. As seen in 2.6 the most general . 'V EM (If) is of the form kTa I

(kE SO(n) .aE Bn) Therefore. MeBD) can be topologized by 80(n) XBn

A subgroup pc M (BD) is discrete it the identity I has a. neighbourhood

whose intersection With r reduces to ! From now on r will always

denote a discrete subgroup.

We know that any '" E M(BD) which fixes the origin is in 80(0) Because

80(0) is compact the subgroup of r which stabilizes 0 is a finite subgroup

of SO(n)

The points yO , 'V Er , are isolated. If not there would exist an

infinite sequence of distinct Yn,E r sucb that V-I

0 = a ~a E B We know n n

that 'Yn =J3 uTa Wi th P. n E O(n) n

that 13 .... f3 and hence 'V, _ 131' n n a

B.Y passing to a subsequence we may assume

But then . V. 'Y -1 ~I when m, l' .... 0:: which mo

implies Y. = 'Y except for finitely many pairs m,n m n This is a contradiction.

. n-l We canc1udc that yJ tends to S when 'V runs through an infinite group r .

n-l It is equally true that Va tends to S for any a E B In tact,

since d( -va,~) = d( a.,b) is a finite constant the n. eA distance from 0 to

Ya . will also tend to infinity.

More generalJ.:y, if KC: B is compact there are only finitely many

y E r such that K n V K r .¢'

1.2. Two points a and Ya "e r ,are called equ1va1entJ> and one can

pass to the quotient B Ir by identification of equivalent :points. we

shall denote it by _r) It is definitely a Hausdorff' space, and for

n = 2 and n = 3 it is known to be a manifold; for n > 3 it is still referred

to as the quotient-manifold. but it probably need not be a mapifold.

The di:fficulty comes from the fixed points. Suppose a is a fixed

point of Y Then r 'Y I (a)f = 1 as seen from

lv' (a) I l-I'Va/ 2

1

•

80

In other words, a lies on the isometric sphere K(V) (see 2.9), provided .

that ala If a = 0 then y E soC n) and the fixed points of V fill some

k -dimensional. plane with 0:5k.:5n-2 Hence an arbitrary V fixes some

k-dimensional geodesic subSJ;lace. As is seen by the above equation we have

I y' (a) I -0 uniformly for all a from any compact set inside Bn Therefore

these subspaces tend to the boundary. We conclude that every point in B

has a neighbourhood which meets onlY a finite number of these fixed subspaces.

7.3. We adopt the followipg definition:

Derini tion: A point b E:B is called a limit point of r if there exists an

infini te sequence of Vv e r and a point a E B such the. t Y, a"'b 'V

The set of all limit points of r is the limit Bet A = A(ro) The

set of accumulation points of ra 1s denoted by A(a) Clearly, A = UA( a)

The following is true:

Theo1"em. A=A(a) for all aEB (with a few trivia~ exceptions);

Proof. If a,b E B it is trivial that A(a) =A(b) because the distance

d( 'Va, 'Yb) = d(a, b) is fixed. In particular A(a) = A( 0)

Consider now the case a E S We shall assume that ra ~ (a} , i."e.

a is not a fixed point for the whole group. There is then a b =y a~ a o

'I E r o

1) We pr'ove first that A(O) C A(a) FaT this purpose let c be an

interim point of the geodesic (a,b) . We know that A( c) = A{ 0) For

every e E 1\( 0) there is thus

_-___ b 81

Fig. 14

a seq:uence 0 f Yv E r with Y c-e 'J We can pass to a subsequence for which

Y'J~ and Y'Jb converge to limits at and b ' If at =b' it is clear that

e=a' On the other hand, if a' fbi the 'tv!! must tend to the geodesic

(a I , b') . but a~so to S , and benet! ei the.!· to a t Or b' Because b = Y a a

both a' and b l belong to A(a) Thus e E A (a) and we have :proved that

2) For the opposite conclusion choose an arbitrary dE A(a) and a sequence

} -1 ( ) [OY'\) suchthat 'YV,a.-d while YvO-e and. ~v O-e l Recall that K'Y,

E( 'V), I( 'V) are characterized by I,,' (x) I = 1 I 'VI (x) I < 1 and r y' (x)! > 1

(see 2.9). Because OEE(y.) its image 'V'\) 0 lies in I (''J-1) and 'V -10 'J

in I(y ) It follows lies that Ih'" v -1) tends to e and I (y\» to e'

If a~ e' then aEE(y ) for large v and hence yaEI(y-l) .... e \) \I 'V

i.e. d=eEA(o) But if a=e' EA(o) then Y'Ja E A(O) b·~ca.use 'Vvt\.(O) =: A(O)

But A(O) is closed and we find again that dE A(O) Since d was an

arbitrary point in A(a) we have proved that A(a)cA(O)

The proof fails when a is a fixed point for all of r There are at

most two such points and all such groups can be classified (the elementar,y

groups) .

7.4. We list the following ~roperties of A

L) A is closed because A(O) is closed. The complement 0 =:8 \ It is

open and is called the set of discontinuity.

82

2.) A is invariant unde~ r , and every invariant closed set contains

A

3. ) Either A = S or A is nowhere dense on S r is said to. be or

the' first kind if A =8 , otherwise of the second kind.

4~) A has no isolated points; it is a perfect set~

5. ) Let A and B be compact sets in 0 Then A meet s only a finite

number of yB,V£r

The proOfs of these statements are all very easy and are left to the

reader.

1.5. Assume tha.t the identity is the only element of r n SO(rt); in other

words, 0 is not a fixed point. Then every Y Er\I has an isometric

sphere K(~) With correspottding E(Y) and I(V) In what follows these

notations will refer onlY to the part contained in B Thus lC('V) is

a non-euclidean hyperplane, and E{Y) I (y ) are hyperbolic half-spaces"

We shall consider the set

p =n E{'V) 'Y~r\I

and prove that it has the following properties:

(i) P is open

( i1) bPc u te( 'V ) (here bP is the boundary in B ).

(iii) every x t B is equivalent to either a single point in P or to

at least one and at most finitely many poin~s on ~p

(i) and (ii) follow from the fact that every intersection pnB(r) is

automatically contained in all but a finite member of the E(Y) and is thus a

tini te intersec"tion.

Recall that

I y , he) I < 1 and

= 1/(1-lxI 2}.

K(y),E(y) and I(y) are characterized by Iyl (x) 1= 1,

IY'(~) I > 1 respectively, and that [yJ(x) 1/(1-[yxI2

)

8~

It follows that x E E(Y} is equivalent to 'vxl > r xl while jvx , = I xl

if and only if x E K(V) In other words, a point is in P if and on~ if

it is strictly closer to 0 than all its images~ and it lies an rY if there

are several closest points (including the case of a closest fixed point). Since

the ~oints in an orbit are isolated the existence of at least one and at most

finitelY many c10sest points is obvious, and property (iii) follOWS. Observe

the rather remarkable fact that equivalent points on ~p are all equidistant

f"rom 0

Property (iii) characterizes P as a fundamental set. As an intersection

of half-spaces it is convex in the n.e. sense. P is a n.e. po~hedron and

it is referred to as the Poincare'(or Dirichlet) fundamental polyhedron of r

with respect to the origin. The faces of P are the (n-I)-dimensional

( ) ( ) ( -1 intersections ?)p n K 'V The faces on K y and K Y ) are equivalent

and congruent in the euclidean and non-euclidean sense.

7.6. Convergence and divergence. We are interested to know how fast the

points in an orbit tend to S or, which is the same thing, how fast

the orbits tend to infinity in the hyperbolic sense.

The first observation is that any two orbits fa and rb are comparable

in the sense that the ratios

lie between finite limdts.

In fact, from d(Ya,Vb) ==d(a,b} it follows that

or

log ,1 + iybl < log .!..t IvaI + d(a"b) l_jybj = l~-Iyal

1 + Iyb(

l-/Yb/

from which we deduce that

for all 'V

8L

A good w~ to study the density of an orbit is to investigate the divergence

or convergence of series of the ~orm

~ (1- ·';lal fi yEr

for different po~ers a

We :prove first:

Lemma 1. Every discrete group r satisfies

(1) - a

~ (1- IvaI ) <co yEr

for all CX>n-l ..

Proof. It is sufficient to consider the case a = 0 . As before, P

denotes the Poincars polyhedron of r

Because a>n-l

J (1_ f xJ2) a--ndx = c < ex)

B

Since B = U V P up to a null-set

e.=L r (1_lxI2)a:-nclx=;~ r (1_lyxI2)Ct-nr~1(x)lndx . 'VEryp y; .

. -1 As usual we write 'f 0 = a ..

and

This leads to

C:;;I: r .j

OVEr P

Then

"!- _ [a[2

[x,a]2

Here [x,a]~2 as seen from [x,a]2= 1+ /xl 2 laf 2 -2xa::; 4

Therefore we obtain

(2) 2 2a

L (1- I a/ )0<: 2 c < 00

oyer = J( 1- [xl 2 )a-n dx

p

and the Lemma is proved.

Observe that (2) even gives a computable upper bound for the series.

Moree'ver, because

We conclude that

,

7.7. For ex = n-1 the series mayor may not converge, and accordi~ly the group

r· is said to be of convergence type or divergence type. There is a nice

geometric characterization of the two types.

Consider the orbit of a ball B ,for instance one that is so small o

that the images 'VB 0 are disjoint.

Let a be the n.e. center and p the n.e. radius of B o

For present and.

future use we shall determine the euclidean radius and center of B o

The diametrically opposite points of Bo on the line from 0 through

a are at the (signed) euclidean distances

tan h 1 (log 1 + , a' + 2" 1- 'al - P

= I al :tt~ ~

1 :!." I a/ th ~

from 0 By use of the$e expressions we see that the euclidean radius is

(4)

r=l 2

=

( I af '+ th ~

1+ la./ t h ~

(1 - r aJ2)th ~

22,. 1- 'al 2

th 2

and that the center c is given by

87

As I al -1 we see from (h.) that r behaves asymptotically like a

constant times (1- I a[)

times (1- I al )n-l

and the surface area of B behaves like a constant o

We conclude:

r is of convergence type if and only if the sum of the surface areas

of the y Bo is finite.

Instead of looking at the areas of the y B one can also look at the :)

areas of their central projections or IIshadows" on S(l)

7.8. Recall that r is said to be of the secon:i kind if S \ A is not

empty. We prove now:

Lemma 2. Every group of the second kind is of convergence type.

Proof. There exists a spherical cap C with a en

C meets only a finite number of 'VG It follows that

(6) .L ..r f y' (x) I n-1dru(x) < co

\{ C

-1 Wi th the usual notation 'V ° = a we know that

I 'V I (x)[ = 1 - I al 2

2 [jr,a]

>

Because C is compact

} I I n-l From (6) and (7 we conclude at once that L: (1- a) < Q:)

a = 'YO

7.9. The type of a group r is closely related to the existence of a

Green's function on ~(r) Any function on 'fII.r)· -can be viewed as the

projection of a function on B which is automorphic with respect to r . ~

The Green's function on m(~) with pole at the projection of x is defined o

to be the projection of a function g r (x) with the following prcrperties:

is h-h for xEB\rxo

2° ~ is automorphic: ~(yx) =~(x)

3° lim (gr(x) - g(x,xo}) = exists X"" X o

4° ~ is the smallest positive function with these properties.

For simplicity we specialize to the case x = 0 The function on o

m (r) which corresponds to ~ will be denoted by ~

88

Theorem 1. r is of convergence type if and on~ if m{r) has a Green's

function.

Proof:

1) We assume that r: ( 1- I al ) n-l < ()O

_lr where, as before, a = 'V 0 We show that

converges and possesses the properties 10 - 4°. The convergence follows from

and

000 It is obvious that ~ has the ~roperties 1 - 3. To prove 4 let

h be any function with the properties 10 - 20 and let g ~ be a partial sum

cf the series g r We have g~ = 0 at the boundary. By the maximum

:principle for h - h functions g~ ~ h ,and hence also gr (x) ~h(x)

2) Assume now that gr exists. We denote by n(r) the number of points

a = 'Y 0 in the ball B(r); we choose r so that no I a I = r . v v V

Choose p so ama11 that the balls B(n ,~) do not intersect and are \)

contained in B(r} . We applY y17) in (5.5) to conclude that

(8) S ~ ~ =0

s(:d U Sea ,p )h~

We let p -0

v

and observe that the integral over Sea ,p) v

S bg(x , e..) Sea ,p) b~

v

and this in turn has the swne limit as

n-2 2

(\

j Sea ,p)

'" Here g(x,a) = g(' T xl) and

v a

bg(x,a ) \)

v

1 n-l p

n-2

is the same as

It follows that (9) tends to ro and we conclude from (8) that n

l. b8r 1. n-1 ber n(l') = - - S s- dab = - ID (~)n-2 S b~

(l)n S(r) ~ - n ~l-r S(r)

90

We can now integrate to obtain

Here gt" (rg) >0 while gr- (1' oF;J is independent of r . We have proved that

the integral

1

J (1_t2 )n-2 n(t)dt

r 0 t n - 1

converges. This means that

1

(10) S(1-t)n-2 n(t)dt<c» .

o

It is a very familiar fact, proved by integration by parts, that the integral

(10) converges together with

1 f (1- t)n-ldn( t) a

which is nothing elBe than

The theorem is proved.

7./0. In the theory of Riemann surfaces it is customary to say that a surface is

ot class 0G it it has no Greents function. There is no reason wqy this

terminology should not be carried over to arbitrary Riemannian manifolds and

even to those quotient manifolds m(r) which are perhaps no manifolds. Our

theorem states that r is of divergence type if and only if m(r)coG

Similarly, a Riemann surface is saic. to be of claBs 0HB if it carries

no bounded harmonic functions other than the constants. An important theorem

of P. J. ~rberg states that

In othel.' words, il.' °l;here is a non-tri via~ bounded harmonic function, then there

is also a Green's function with an arbitrary pole. The opposite inclusion

is not true.

The termino1ogy and the theorem carry over to arbitrary Riemannian spaces.

The proof remains essentiallY the same as for Riemann surfaces. We refer to

the proof in Ah1fors - Sario, Riemann surfaces, p. 204 - 206.

We can therefore state:

91

Theorem 2. If r is of divergence type there are no non-constant bounded harmonic

functions on ~(r)

7.11. Let G be a transformation group acting on a measure space m . One says that G acts ergodically on m if every invariant subset of m is either a null-set or the complement of a null-set.

Theorem 3. If' r is of divergence type, then r acts ergodicallY on S

Proof. If not there would exist an invariant measurable set EC S with

o <m(E) <m(S) Use the Poisson formula (27) in (5.7) to construct the harmonic

fUnction whose radial boundary values are given by the characteristic function

X of E The function is

u(y} = 1 ill

n

Because 'X,('Vx) =X{x) for y E r one finds

(

1- ru.f.) n-l n 1 J C. ,2 X(yx)ryl(x)1 - dal(x) s yx - yY

= (J)l S -1 2 X{x) aro{x) = u(y) (

1. I 12) n-l

n S Ix - yl where we have used the iq.entities l-/'VY12 = IV'Cy)1 (1-ly/2) and

2 2 lyx -yyl = lv' (x)'''v'(y)flx - yr

The function u is automorphic ~ h - h , and non-constant. This contradicts

Theorem 2. :aence the set E cannot exist.

7.12. Cons1der a ball :Bo =B{ro ) =Bh(P o ) about the, orig1n. We denote by

L(E) the set of all ~ E S such tb,at the radius to ~ meets infinitely manY" o

'VB , y E r . Another way of saying the saIne thing is that S € L(B) if there o ~

are j;nfin1tel¥ m.a.n.y points VO at non-euclidean distance <p from the o

radius (O~~) . Geometric,aily" this means ,that there are infinitely many

'10 =8. inside the lens-shaped region shown in Fig.15 arbitrarily close to t:: • In 'V

Fig. 15

particular, there are then intini te1¥ many a in -the Stolz cone with the 'J

o:pening C'.po = 2 ~c tan r 0 • If this is true for some B 0 then !; is

said to be a conical 1imit point and the set

L =U L(B } o

is the conical limit set of r Obviousl¥, LeA .

92

93

If ~ is a conical limit ~oint for the orbit ro ~ then it is also a

conical limit for any orbit ra Indeed, if yO is at n.e. distance

~ Po from the radius (0, 1S) , then Va i Bat distance -;; Po + d( 0, a) from

the same -radius.

This leads to a.nother characteriZaLion of conical limits. The radius

(o,s) is a geodesie and it projects to a geodesic on ~r) If ';, is a

conical limit this geodesic will came again and again within some fixed distance

from any given point. On the .contrary, if' S is not a conical limit, then the

geodesic will eVentually leave any aompact set (the geodesic tends to the

"idea.l boundary"). In :particular, if m ( r) is compact, then all g E S are

conical lints.

Lemma 3. If r is of convergence type, then m(L) = 0

Proof. As usual the orbit ro will be written as a sequence {a 1 v.J Assum.ing

convergence there exists,. for every 'S >0 J a v such that o

(11) t: (1- I a I }n-l < e 'V>v 0 v

There is then an a with \I> \)0 inside the '\)

Stolz cone with angle ~o arbitrari~ close to Let ~ be the angle v

between g and a \I

trigonometry to

0:'V 2 sin 2 1-1 ~ ,.

'V

The fact that a is in the Stolz cone translates by '\)

Fig. 16

In the limit

lim ctv _ 1- 'a , 'V

< tan CPo

T.he point ~ is contained in the gpherical cap ~ith cente~ avl1av' and

radius a This 'V

area of each cap is

by

S is covered by all these caps ~th

n-l asymptotically ~roportional to a 'V

Hence the measure of L(B) is < o

'V >V The o

which is majorized

constant times e

by (11), and consequently equal to zero. F1nal~, L is a union of countably

many L(B ) o

, and we have proved that m(t) =0

7.13. The conical limit set is in any case invariant under r For if

s ELand "E r then y map's a Stolz cone at S on a set contained in a

slightly laxger Stolz cone at y;

If r is of divergence type we conclude that either m ( L) = 0 or

In ( L) = OJ . n Thus, for !BZ grQup r onl.:y the extreme cases can occur.

We shall eventually prove that m ( L) -= ill for all r of divergence type n

so that the distinction between convergence and divergence is the same as

betw~en m ( L) = 0 and m ( L) = (Jjn However~ it is appropriate at this time

to say something about the history qf the problem.

OriginallY the problem dealt with the geodesics on a closed Riemann surface.

In the 1930' S important groundwork was done by Morse, Hedlun~ Myrberg and

95

many others. A very decisive step was taken by Eberhard Hopf (1936) who proved

that r acL1:l e.rgodlcally on S x S i~ tbe case of' a compact Riemann surface

with or without ptmctures • The action is the one that takes (;tt}) to (y ~ ~y 11 )

This is of course very much stronger than ergodicity on S

It did no'L take long for Hop£' to realize that his theorem can be extended

to several d~mensions in a much more general form, and that finite volume is

not the right condition. In 1939 he proved the following:

Theorem. (E. Hop£') r acts ergodic ally on S r S if ann only if m(l,) = m n

The proof makes essential use of Birkhoff's individual ergodic theorems

and cannot be considered elementary.

The most recent development of the theo:t"Y is due to Dennis Sullivan (1978).

Theorem. (D. Sullivan) r acts ergodically on S Y. S if and only if f'

is of divergence type.

His proof uses Markov chains, and I must confess.,that I do not understand

it. However, there are now in existence relatively elementary proofs of the

fact that every r of divergence type satisfies meL) = (1) n Such proofs

have been given both by Sullivan and Thurston. In what follows we shall give

a rather detai1ed version of Thurston's proof.

7·14" The action of r on S x S is defined by y( s, 11) = ('VS' "(1) This action

is said to be dissipative if there exists a measurable set ~~S x S such that

An 'Vll == fJ for all 'V E r \ I

m(lJ .. 1 A) = m(S l' S) = of-, , n y6"

Briefly, A is a measurable fundamental set.

Lemma 4, If meL) = 0 then the action of r -on S x S is dissipative.

Proof.. Suppose tha.t (~'T!) ~ (8: L) x (9 - L) \ diagonal.. Then

11 . 1- I a I m V

V-tct\ I r - a f .> 'V

.... 0 l ' 1- I a I , l.m , \J

V .... ea I" - a: J v

by the definition of L

-1 As before y 0 = a and hence '\) v

.... 0

Either II! - Eo v' ~ .1 I:: - il' or I 'T\ - a v I ;;- 1. 'g .. 111 2 2

This implies

Theref"ore, by (13), r Y" I (~i' . r'V" '(11) r -0 and it ,follows that there is VoE r

for Which 1"10 I (~) J .. ('Vo

' (nH is a maximum.

Equivalently t ~.ince

this also means that r'V g -' ry nJ is: a maximum. In other words, if o 0

~ :: *f; , 11 = Y 11 then o 0 0 0

(14) I ~ -11 I ~ r'V~ -''Y1l I o '0 - 0 0

for all 'V E r ,an4 this is equivalent to

97

For simplicity, let us assume that the origin is not a fixed point. We claim

that the set

has properties 1) and 2). In the first place if (~,~) and (~o'~o) are equivalent

they cannot both be in A for that would imply r So -110

1 >, g-1l1 and

Secondly, our construction has shown that every

a) S = 11

c) I Vi (;) J . I 'V' ('Tl)' = 1 for some y E r \ I

It is clear that a) and c) are true only on a null-set, and if m(L):=: 0 the

same is the case of b). The proof is complete.

(The idea of this proof is due to Sullivan)

7.15. We embark now on Thurston's proof of the following fact:

Theorem 4. If' m(L) = 0 , then r is of conyergence type.

Remark. This is a consequence of Sullivan's ~heorem. On the other band,

Theorem L makes Sullivan's theorem a consequence of Hopfts theorem.

Proof. (Thurston). We consider again a ball Bo = B(p,o) centered at the

origin and we denote its images y-~ with B and the shadow of B with ~ 0 v V

B' The measure of a set Ec S will now be denoted by I E/ rather than \)

m(E)

The condition m(L) = 0 can be written in the form

lim I II B f I = 0 'Vo-ttIJ' ~. \)

\)""'-vo

Indeed, let ~o denote the characteristic function of U B'

V>Va

v "'neans that

lim f \, (~ ) dro( ~) .:: 0 Va-IXI S 0

whi le' I L I = 0 means that

I lim X'J (~)~(~),; 0

S 'VO-M 0

Then (16)

These conditions are simultaneous~ fulfilled by virtuE of Fatou's lemma.

On the other hand. the asse~tion that r is of convergence type is. as

we have seen, equivalent to

1: <00 ,,=0

It is evident that (16) implies (17) provided that the B' do not overlap too v

much, The idea of the proof is to show that this is the case.

Step 1. As a first simplification we show that one can discard many of the

B Fqr this purpose we ~hoose a number p > 0 J which will ultimately b~ \)

large, and use it to define a subsequence (vk}~ as follows:

10 Choose \l = 0 o

5,0 Suppose that ~o, .. "~ have been chosen so that the mutual distances

d(a ~a ) are all > p Then .choose 'V k + 1 as the smallest 'V such that Vi Vj

d ( a , a ) > P for i = 0, ... " k. '\)i vk + 1

other a IS v

Fig. 17

This choice is always possible and we see that the following is true:

a) d(a, a ) > P for all h f: k \/h '-'k

b) to every \/ there exists a vk such that d(a ,a ,< r. 'J 'it = f'

99

There are only a finite number N of a ~

wi th d(a\l' 0) ~ p and hence also

only N points a with d(a,a ) ~ p When this is the case then v v 'it

1- is. I " <M

=

where M depends only on p

For simplicity we can return to the notation a for (16) remains true v

and it is sufficient to prove (17) for the subsequence.

Step 2. We choose p so large that the B dQ not overlap~ We place an 'V

observer at the origin and speak of total or partial eclipses when two shadows

overlap. The B will be subdivided into classes depending on the number 'V

of times they are eClipsed.

The cla.ss I o

consist,s only of B o We remove B 0 and define lIas

the class of all B that are completely visible from 0 '\)

Next we remove

all B v Ell a.nd define ~ as the class of those B v which are now completely

visible, and so on. Clearly, every B will belong to a class I and the v m

shadows of t}:le B" E 1m are disjoint.

It will be shown that

t IB~I. ~ K J. IB~' r

1m + 1 1m.

where n < 1 This will obviously imply (1.7) ~

Step ~. Every Bj

E 1m + ~ is partially or totally eclipsed by a

B·EI 1. m Let

We shall need an

100

upper bound for the ratio rj/ri

Fig. 18

We refer to the picture in which ai

and aj are at n.e· distance

< Po from the s~e radius. Let bi,bj

be the o.e, orthogonal projections of

on the radius. We have then

p < d(al.' ,a ... ) < d(bi , bJo) +2po ;: d(O,b ) - d(O,b.) +2p < d(O,a.) - d(O,a.) +lJp '" j 1. 0 J 1. 0

or

d(O,aj

) > d(O,a.) +p - 4p 1. 0

and

1+ /80.1 J

which implies

> 1 + I a.1

1.

p - 4p e 0

1-1 a.1 1.'

By use of' formula (4) we final4r find

(18) r~ _ +4 2Po ....Ii. < 4e P Po cos h --r

i 2

101

The main thing is that this quotient can be made arbitrarily small by choosing

p large enough.

Consider now all the Bj Elm + 1. Wh:i:ch are partially ecl.ipsed by Bi Elm P

Their shadows Bj are disjoint and the,y 2ie asymptotically within distance

most

from the rim of B! 1.

Their total area is therefore asymptotical~ at

(We have again used the notation A(r) for the area of a s~herical ca~ with

radius r ). In view of the estimate we can thus choose p so that, for

su:fficiently large m ,the total area o:f the shadows Bj of aJ,.l Bj

E 1m + I

that are partially' but not totalJ..y eclipsed by some B1 Elm will be less than

102

Step. h. We pa.ss now to the Bj E Im + 1 which are totally ec lipsed by a

B. € I 1. m We need an auxiliary lemma.

Letnma. 5. If ~ IE B" n B I - ':. i j with B. EI , 1. m

(ai'S) intersects the ball ~j with center a. J

Y is the radius of" B o 0

B j E Im + l. ' then the geodesic

and n. e. radius 20 ,where '0

Proof. We map the unit ball eonformally on the half-space If1 so that 0

goes to . en and ~ to ~ We keep the names 07* th~ points a1

, aj

The

geodesic (O~S) becomes a vertical. line through en and (ai'S) a vertical

through at whose intersection With Jan-l we shall denQ~e by c

o a

Fig. 19

Let bi,bj

be the closest points to ai,aj

on the vertical through 0

and let cj

be the closest point to b j on the vertical through c (the

picture is misleading in a~ much as aj

need not be in the S~e plane ~s the

two verticals). The non-euclidean distances are computed by

d(a.. ,b.) ~ ,J

But

cos

"Jt

2 ~i -J~ -logcot-2 sin (0

ct)i n

= J ~ '" log cot .f CPj sin q>

so that to. >cPi

and 'J

It follows that

d(aj,CJ.) < d(aj,bJ +d(b ,c.) < 2p

= J j J = 0

and the lemma is proved.

The next picture shows on the left a B. and so~e totally eclipsed B. ~ J

together with their shadows. On the right the whole configuration has been

transformed by Vi

The image 'V.B~, is not the same as ~ J - ,

is contained in (ViBj ) .

y.B~ 1. J

104

Fig. 20

However, the lemma tells

It is clear that condition (16) remains true when 'A is Tf?placed by R o 0

Therefore, as soon as m is big enough,

t 1 'V. B ~ J :5 ~ I ('Yo; B.) I r < e j 1. J - j ... J

On. the other hand, if p r ()I) then y.O approaches the boundar,y while the 1.

area of the shadow of Bo viewed trom 'ViO decreases· to a positiv~ limit

a In other worJa, o

But

where the minimum and may.imum m~ be taken with respect to B! 1.

From (19), (20) and (21) it follows that

Here the maximum of

~05

is taken at the center and the minimum on the rim. The ratio tends to the same

limit as ri/(l- I air) and we know this limit to be finite There is thus

a constant K such that

~ 1 Bj' I < K £ I B~ r j ~n 1

We choose E so small that the factor on the right is <,1 . 3

This lets us conclude that "the shadows of'the totally eclipsed BjE 1m + 1

make up at most one third of the shadows of the B. E I Together wi th 1 m

our previous result for the partially eclipsed Bj

we have proved that

I: lBj, < 2!: 1Bir BjElm+ 1 3 Bi Elm

~or a11 surrlclent~ large m

/B'I is finite, and hence that i

This in turn proves that the sum of all the

~o6

The theorem is ~roved.

10'7

8. Quasicomformal Deformations.

8.1 By w~ of motivation we shall first recall the basic properties of

quasicamformal (q.c.) mappings in n-dimensions. A q.c. mapping is first of all

a homeomorphism.

F: 0 - 0'

from one open set in JRn to another. One of several ways to impose the right

regularity conditions is to require that F is absolutely continuous on lines

(a.c.l.) This means that the restriction of F to a.e. coordinate parallel

line is absolutely continuous.

Under these conditions the partial derivatives D.F. exist a.e. and we can form J. J

the Jacobian matrix

DF =: " D i F j II

a. e. The Jacobian determinant will be denoted by JF and

1 --(1) XF = (JF) n DF

is the normalized Jacobian, while

T MF = (XF) .XF

is the symmetrized and normalized Jacobian. (XT is the transpose of X)

F is ca,lled K- q.c. if

( 3) , I XF 112 =: tr MF ~ nK2 .

It is possible to ~ite ).,

(4) T MF = U (

where UE SO( n) and "1> A2 > ... >).n > 0

Clearl¥,

XF = v( . ) u An

108

where V Is another orthogonal matrix, although U and V are not llDique1.y

determined in the case of equal eigen-values. Observe that ~lA2' "An = 1

and that

(6) ~ > II XFI12 = li + .. +).~ ~ n

-1 / This implies upper bounds on "1' "n and '\ An , and sometimes K is

~sed to denote one of these bounds~

8.2. Because q.c. mappings in n dimensions are difficult to handle it is

reasonable to linearize by passing to the infinitesimal case. Let F(x,t)

be a one-parameter family of a.e. differentiable mappings Which for t ~O

bas the development

(7) F(x, t) = x + tr(x) + o( t)

We denote differentiation with respect to t by a dot and write

(8) F(x) = F(x,O) = rex)

We assume that (7) can be differentiated to yield

DF(x, t) = 1+ t D.f(x) + o( t)

or

. (9) (DF) (x) =Df(x)

Furthermore,

(10)

. J(:X) = trDf

. 1 (XF) =Df -- trDf. I . n

(MF) • .:: Df + Dfl] - g, trDf . I n"

All thi~ motivates introducing th.e matrix

or, in terms of indices;

1.09

Observe that trSf = 0 Thus Sf is obta.ined by first syuunetrizing Dr and

then ~ubtra~ting the multi~le of I which makes the trace 0 The space of'

syumetric nx n matrices with zero trace will,in these lectures, 'be denoted

by SMn . We shall use the square n~rm defined by

Definition 1. f is called a k- q.c. deformation if f ISf'll ~ kJO a .. e.,

in ()

For n;:::? we can use the contp'Le'lr notation f'=u+iv , z=x+iy It

turns out that

Sf ~I ;Ref' Imf'" ..

(14) z £

Imf .. -Ref Z -Z

and I lsrlf2 = 2rfJ 2 so that f is k-q.c. if and only if If -' < k a.e . • Z z

110

In view of (14) it is reasonable to regard Sf as a natural generalization

of the comple}~ derivative :r_ z

8.3. Our definition is still deficient because it does not spell out the

regularity conditions. We replace it by the following:

Definition 1. A homeomorphism f: n _ JR n is called a q.c. deformation

if f has local~ integrable distributional derivatives Djfi and if the matrix­

valued function Sf formed with these derivatives has norm II sfll E Lt». It is

k-q. c. if the L03 norm of 11Sfli is ~ k.rn

For n = 2 a 0 - q. c. ma;pping is conformal (this is Weyl' s lemma)

and for n>2 all conformal mappings are Mabius transformations. It can be

expected that for n > 2 there are only a few deformations with Sf = 0

Lemma lw If n>2 then Sf = 0 if and only if f is of the form

where a and b are constant vectors and S is a constant matrix which is

skew outside and constant on the diagonal. ?

Proof. We assume first that f E C-· For this proof only we denote

components by superscripts and derivatives by subscripts. The hypothesis means

i' . i ~-; that fj = -q for i J J and 1'i = fj If i , j , k are all d.ifferent

fik" = -~k = -~i = ~i = -f~.. and hence fJ~k = 0 j i k j J~ i h

If' j f k we can find h + j ,k and obta.in f'i;jlr = f"""hjk. = 0

~kjj = -~kk = -~kk = f~hh = ~hh = -i!jj = -~ii = 0 Similar~, both for j = i and j! i

Also,

foj - 0 iii -

Consequently a~~ third and higher derivatives are zero, and it is easy to

Ul

check that the lower terms have to be as in (15).

If f bas merely distributional derivatives one uses the ,trick of convolving

with a radial function 8 with sUPI>ort in B(e;) Since 8(1'* 6 ) = Sf* 8 = a e 'e s

we conclude that f* 6 has the form (16) and in the limit for € .... o f e

has the sa.me form.

8.lt. Let ~ be a differentiable function with values in SMn We

* define S ~ to be a vector-valued function with the components

where we are again using the summation convention.

It is readilY seen by the Green-Stokes' formula that

provided that either f or ~ has compact support. The dot products are

abbreviations of Sf. '~i. ~J J

* Because of (17) we regard S a.s

the adjoint of S Note that the formula remains valid as soon as ~ is

continuous with locally integrable distributional derivatives, and this will

henceforth be the required degree of regularity.

It is again instructive to look at the case n::;:: 2 If

cp = (CPll

CfJ2

we write r.p::;:: u + i'V wi tb u = '1'1.1' v = Cf12 and we let z = x + iy = Jl't + iX2

* s Lawl :for the- lnde pendeut va.!" iab Ie. If we identify the vectoL' S cp -wi'Lh

* * (s CP)l + i(3 cp) 2 we find that

(18)

Together with Sf = f_ z

it follows that S*Sf = 1: ar P-

In the general case

Operators of this type are familiar from the theory of elasticity-

8.5. * It is natural to look for fundamental solutions of S Sf = 0 '<-

112

We start with solutions of s* y:::: 0 where 'V shall have values in mf! It

k turns out that ~here are n linearly independent solutions y k = 1 , , n ,

with the weakest possible singularity at 0 They are given explicitly by

(20)

A short computation rev€als that each term is separately annihilated by

* S The reason for the linear combination is to make the trace equal to zero.

The singularity is weak, enough to make integrable, but the derivatives

are not.

For n = 2 one finds that yL and y2 are represented by liz and ilz

reS];)ecti vely. The connection with the Cauchy kernel is obvious and W'e shall

k find that Vij does indeed play very much the same role as the Cauchy kernel.

8.6.

(21)

We shall now solve the equation

k k Sg = 'V

113

* k which would. imply S Sg = 0 .

k It is a good guess that the components of g

should. be of the form

k g. =a(r)5·

k+b(r)X.x.. ,

1 11K

where t = brl as usual.

Elementary computation gives

(Sllij = ~ (a~(r) +b(r») ( 6ikXj + 6jkX t ) + b' ~r) "iXj"k.

- 1:: ~\j~(al(r) +b(r) +rb'(r)) n r

Comparisor. with (20) gives the conditions

1: (_ a' (r) 2 r

b l (r) = r

+ ber )=-1.... n r

n-2 n+2 r

-1 (al(r) +b(r) +rb'(r» =-1 n r n

r

The last is a consequence of the first two and one finds

~(x) = -[n-2 g~ n n-2) n-2 r

n-2 n

if n>2 and g~ == (log r) BUt for n=2

8.7. The f'ollowing problem arises naturally:

ProblePJ,. If one knows Sf ,how can one find :r ~ and what condi tiona

must the matrix-valued function Sf satisfY?

In other words, when does the inhomogeneous equation

(24) Sf' == "

have a S oluti on:J and how can one find i t1 :ror n = 2 it is known that the

equation is always solvable and the solution is given by the generalized

Cauchy integral formula, also known as Pompeiu's fo~ula.

For simplicity we shall treat only the case where v is of class

114

Lm(~) and bas compact support. From 8.3 we know that the solution of (24), if:

it exists, is unique up to ~unctions of the form (15).

The first theorem is a generalization of Pompeiu's formula which connects

f and Sf

Theorem 1. Every quasiconformal deformation :f satisfies

cnrk(Y) = - r Sf(x) yyk(x:..y)dx Bty,r)

+ S [(I + (n - 2)Q,(x - y) )f(x) ]kdw(x)

S(y,r)

with c = 2(n - I)m n n If f has compact support the formula reduces to n

cnfk(Y) = - S Sf(x) .yk(x - y)dm(x)

Rn

Proof. It is sufficier:.t to consider y=O .. Integration by parts*)givea

*) If f has only distributional derivatives one has to consider integrals of

the form

where A has a graph like

o o

S Sfij yij dx = j Di fjY~j dx = B(r)-B(ro ) B(r)-B(r~)

r I r f xi

r str) j r o

.U5

In the limit for r ~ 0 the integral over o

tends to

ill fk

( 0) + (n - ~)fi( 0) r x,xk

dill n Stl)l.

o~viously,

ro n n

so that the limit adds up to 2(n-l) ron fk(O) (fx)x n

Because 2 = Q(x)f the lemma follows. r

8.8. A little more generally we shall define

(25) rk-v(g)=IVij(X) VllX-y)dx

R

to be the k th component of the potential I \) of \.l

to be of class L 1

with compact support.

Lemma 2. rkv has the distributional derivatives

her.p.- \J is supposed

and b :::: 4CJ.)n

n n+2

Before proving this lemma we write down the explicit expression for

% y~. (x) = eik. 8jh + 8jk 5ih ... _a1j 5hk ~J Ixln-

(27) - n "ikXjxh + 8jkXi~ _ 5ij~Xn Ixln+2

+(n-2) 6ihXj~ + 8jhXi~ + \\txixj

Ixln+2

_ (n2 .4) XiXj~~

Ixln+4

We claim that this is a Cald,eron-Zygmund kernel. Since it is obviously

bomogeneous of degree -n we need only check. that

(28) S DhY~j(x) dw(x) = 0 8(1)

This can be checked by computation with only a little bit of trouble

being caused by the last term in (27).

lli5.

Wi thout a:lY computation at all we can a.lso reason as follows: By Stokes'

formula

Here the integrals over S(r].) and s(r2} are equal simply because the integrand

is homogeneous of order 0 It follows that the volume integral on the left

vanishes, and this implies (28).

The Cald~ron-Zygmund theory informs us that the principal value in (26) exists

a. e. and represents a function in LP for any p > 1 ,

Proof of the Lemma. We assume first that

r: \I(Y) ::: f Vij(x) ~(x-y)dx Ix-y! >p

= S Vij(X+Y) Y~j(X)dx Ixl >P

eo 'V E C and write

for the translated ~otential. It follows at once that

Dhik(y) = S DhV (x+y) l'~.(x)d.y. = p '1 ij l.J

X >p

For p - 0 the surface integral tends to

The evaluation of this integral requires a formula for

J xi Xj~Xk dm(x)

S(l)

118

The easiest way to get this formula is to substitute (27) in (28). One obtains

ill

S XiXjXh~ dill = n(~+2) (oij~ +5ih5jk +oUBjh)

8(1)

and finally

It now follows =rom (29) that

It is known from the Calderon-Zygmund theory of singu1ar integral operators

that the truncated integral converges to its pt'. v. in rJ for every p> 1

From this it is trivial to conclude that the limit of DhI: is the distributional

derivative DbIk ,a.nd the lemma. is proved for all ~ e: CCIO •

To prove it for general \} we pass a.gain to a convolution 'V '*5e where

8eEC

oo is supported in B(e) and J 5e: dx = 1 • From what we have proved

The pr. v. is a bounded 1inear opera tor on LP " ~ > 1 . Therefore the right

band side of(30)tends in LP to the right band side of (26). Hence the lett hand

side has a limit in LP and that limit is the distributional derivative.

The lemma is p:roved.

8.9. We regard Iv as a vector-valued function with the components

k I 'V • It makes sense to form S (I \2)hlt and by Lemma 2 we find a.t once!

where

k in the sense of 'V~j sta.nd.ing for a vector with the components V

ij

The explicit expre,ssion is

n+2 S'ij,hk(x) = (oik8jb +8JkBib -2 8ijohk} I~

+ n(5ijxh%1t + \.It Xi!jJ r !r n+2 ... (6ikXj~,+ 8ihXj~ + °jhXiXk + BjkXi~) I!I n+2

Observe that S'Vij , hk = Syhk, ij It is also a C. - Z • kernel.

Suppos e that we apply (31) to 'V ::: Sf . Becaus e ISf = -c f we have n

SIv = -c Sf = -c" and thus n n

with

a = c -b = 2(n+l)(n-2) n n -po n(n+2)

CI) n

Conversely, if ~ (with cQmpaet sup.p~rt) satisfies (32) then (31) implies

so that Sf = \i 1

is satisfied by -r = - - Iv . ·c n We have proved:

Theorem. 2. If \) € Let:> has compact support a necessary and sufficient

120

condition that the equation Sf:= 'J have a solution f with compact support

is that ~ satisfies (32).

As for the sufficiency we are not claiming that f = --1- I~ bas compact c

support, but since Sf" = 0 in a neighbourhood of infinity f1 is trivial of the

form (15) and by subtractir.g that trivial solution of Sf = 0 we obtain a

solution of' Sf =" with compact support.

8.10 We sha11 now applY Theorem 1 to show that every q.c. deformation

satisfies a near-Lipschitz condition.

Lemma 3. Every q.c. deformation f satisfies a condition of the ro~

Proof. We may choose y' = 0 From Theorem 1 we obtain an obvious

estimate of the form

+ C{R) Iyl

Repla.ce x by 171 x in the integral. Because 'Vk(x) is hcmogeneous

of degree I-n we obtain

If r yl' >R the integral on the right is smaller than the same integral over

B(2) which is a finite constant by rotational symmetry. If Iyl < R we must

also estimate the integral over 2 ~ I xl ~ ~I ~

It is readily seen that f Ivk(x -rtF) _...,.k(x) II = o(fxf-n)

the integral is bounded by a constant times l~

(33) is true for a suitable A(R)

as x'" QO Therefore

and we conclude that

8.11. We shall now prove that every q.c. deformation r(x) with compact

support gives rise to a one'·para;nleter group of conformal mappings Ft (x) = F(x~ t)

such that F sOFt = F s+t - and F{~, 0) = f(x.) . .,More precisely:

Theorem 3. If f' is a k - q.,c. deformation then Ft is a kit! e -g.c.

mapping.

(~his t..heorem was firsi. proved by M. Reimann).

co Proof. Assume first that f' is C For fixed x we consider the

differential equation

. (3~) F(X, t) = f(F(x~ t) }

with initial condition F(x~O) =X The eJl"ist'ence of a solution is classica~

and the uniqueness fo~lows because r satisfies (33) wnich is an "Osgood

Condi tionll• Actually, the solution wi~l e""ist for all f because f is bOl,lllded.

Moreover,

1.22

F(X,s +t) =F(F(x,s) "t)

for both sides are solutions of (3~) with initial value F(x,s) . This

means that FtoFs = Fs+t . In particular, FtoF:-t = x and since F(x,t) is

continuous iD x it is a homeomorphism.

Differentiation of (3'+) with respect to x yields

. (DF) (x,t)=(DfoF)DF(x,t)

Also

· 1· (log JF) = tr(DF)- (DF) = tr(Df'oF)

-1 We o.ppq this to XF = (JF) n: DF to obtain

and

. 1 (XF) = ( - - tr(DfoF) +DfoF)XF

n

T • ([' T 2 [eXF) (D)] = (XFT[DfoF +Df of-- tr(DfoF) 1XF n

from which it follows that

By the Cauchy-Schwarz inequality

Together with II sf'.1 < k we now obtain

and by integration

where we have used the initia1 condition

11 XF(x,O) I r =.jn

If f is not differentiable we form the convolution and

use it to generate Fe Because lls~el I ~ k it follows as before that

Fe (x, t) is K - q.c. with K = ekl tl We write the diffe~ential equation for F in integrated form

E!

t :r (x,t)=x+S f (F (x,t))dt

e e e o

1?3

Because the Fe are K - q. c. with a fixed It they are equicontinuous on every

compact set. Hence one can choose a sequence e(N) - 0 such that Fe(N)

converges to a limit F (x,t) which satisfies o

t F (x, t) = x + S r(F ex, t) )dt o 0

o

This means that F (x,t) is the unique solution of (34) and hence equal to o

Since it is a homeomorphism. and limit of K - q.c. mappings with

it is itself K - q.c. and the theorem is proved.

8.12. The quantity MF undergoes a very simple and predictable change

When F is composed with a Mobius transformation. The corresponding rules

124

* for the operators Sand S are not quite as simple, but of vital importance.

Because we have used the letter ~ in an important role we shall now use

capital letters A, B, etc. for Mobius transformations. Recall that a Mobius

transformation A defines a change of coordinates that we prefer to write as

x = Ai (rather than x = Ax ) A contravariant vector r(x) expressed in

the i-coordinates acquires the co~onents

which we can also write as

We shall denote the function f by fA Explicitly,

The formula (35) defines a groUJ) repres'entation t for

as seen by the computation

The basic transformation formula for Sf reads:

Lemma. 4.

Proof. We remark first that differentiation of the identi ty X-~ = I

yie'l.d:s

and hence

-1 -l( )-1 D .X = - X D.X X #

J J

Therefore, differentiation of'

leads to

(Dr A) ij = - ((DA) -\ DA] ik(foA)k

(38) +( (DA) -l(DfoA)DA)ij

In the first term on the right

where we have used the identity

2

(39) (DA)TDA = (JA)ii I .

-2

125

]26

Differentiation of (39) gives

g g -1

(DkDA)TDA + (DA) TDkDA =~ (JA)n (nit log JA)r =~ (JA)n (tr(DA)-~kDA) I

2

= ~ (JA) n (tr(DA) TDkDA) I

This shows that the symmetrization and normalization of the first term in

(38) contributes nothing. On account of (39)

while

tr[ (DA) -~foA DA~ = tr DfoA

Therefore the second term of (38) contributes exactly (DA)-l(SfoA)DA •

Observe that this computation has made essential use of the fact that DA

is a conformal matrix. More generally, we shall say that a function vex)

with values in s~f transformB like a mixed tensor if

Note that VA has again values in mf1 .

8.13. In order to continue this discussion of the invariance properties

we want to take a good look at the relation

(41) * SSf'. cp dx = - Jf. S cp dx

We would like these inner products to be invariant when f is replaced

127

by fA and ~ by $A - aut this is not so if Sf and ~ are both

transformed by (40) _ Indeed, we would have

and the integral in (41) would not ~tay invariant. TO remedy the

situation we regard the integral as an inner product between the

function Sf and the measure ~dx, and We transform the measure

according to the rule

This amounts to transforming $ as a mixed teqsor density according

to the rule

(42)

Similarly, the right hand side of (41) makes sense only if we

* regard S ~ as a contravariant vector density. In order to derive

* the transformation rule for S 4> let f be a test-function and con-

sider that on one hand

and on the other hand

-f Sf· <I> dx ;;; - J (SfoA) (<poA) 1 A I (x) t ndx =

128

= - J At (x) -l(SfoA)A' (x) • tfIA (x)dx

The comparison shows that

Here A1 (x)T == /A'(x)/2A,(x)-1 and if we were to regard S~ as a. contravariant

vector, then the transformation rule would read

* , In+2 * (43) S (~A) = At (x) (s -..p) A •

* However, it is more natural to regard v=S (CPA) as a covariant vector density

which transforms according to

To sum up: The pairings

are defined and Mobius invariant when f is a contravariant vector, v is a

129

covariant vector density,"\J is a mixed tensor. cp is a mixed. tens'or density

which transform according to the following rules ~

v A (x)= A' (x) -1 ,,(Ax)A r (x)

a>A (x) ;:: 'A I (xl InA' (x) -lq>(Ax)A I (x} •

The invariance is in the sense that

The operators S s* satisfy

S(fA) ;:: (Sf) A

S {CPA' = ,A' (x) I n+2 (8' 'cp) A

* Sf' is a rUxed tensor, "S ~ is a covariant vector density. We do not define

Sv or S*"

8.14. The fact that Sf is not a density while S* 'V is defined only for

* densities makes the operator 8-)(8 meaningless except in the euc/lidean case.

To rescue the situation we need an invariant density. In hyperbolic space

such a density is given by the PoincarE! metric. We use the notation ds = p' dxl with

2

n the density itself being given by p

Now we can pass from tensors and vectors to densities simply through

multiplication by pn For instance pnSf is a mixed tensor density~

130

S*"nSf i t d -n-2 * n t ~ is a coyar an vector density, an p S P Sf is a contravarian

vector. To account for these changes it is convenient to define two new operators)

n * -n-2 namelY p = p Sand P :;: p P They satisfy the relations

and thus also

The basic invariant second order differential operator is P*P and in

this connection we shall say that a vector-valued function f is harmonic if

P*Pf=O

(44)

*

It is convenient to split this into two pieces, namely

* c:p=Ff, P ~=O

or S tp = 0 For n = 2 it turns out that cp is a holomorphic quadratic

differential.

* How does F P compare with the Laplace-.Beltrami ~. In the first place

~ applies to scalars and P*P to vectors. But in the Hodge-DeRham theory

131

there are two invariant operators dB and 8d which apply to differential

forms of any order and in particular to first order differentials

which may be identified with f A differential is harmonic if (do + od)a= a

whereas in our termdnology f is harmonic if

Here

1 ) 1 (- - 1 d 60: - - 8~ - ID = 0 n 2

i ID=R. f.dx

04 J 3. c..

where R~ is the Ricci curvature tensor.

8.15. We sball now study the theory of functions

in some detail.

LeD1tJ19. 5. If P '*P.r = 0 then

a)

c)

S SfijX j dm(x) = 0

B(r)

r Sf .• x. Xjdlc(x) = 0 .. l.J 3.

S(r)

Jr Sf .. X.X.x. d:'Jl{x) =0

l.J 3. J K S(r)

Proof. By Green's formula

and this im,plies (45a). Similarly~

* f that satisfy P 'Pf = 0

and

(46)

:x r nSf' x ~ S (n , J n JP i'J" '1 r = Dj P Sf".x. I dx = p Sf .. 0 .. dY. = 0

S ( r J' ~J 1. l.J l.J B(r) B(r)

xxx. r. nSf' i;i a: A_ r D ( I1-.p ) dx JP 1j r '-q = ~ J. P ~.LijXi '~ .

B(r)

= J pnSfij(8ijXk +\:jXi)dx

B(r)

= S p~fikxi dx = C by (45a)

B(r)

Cor. :[=*Pf = 0 implies

S Sf •• Y~j (x) dO' (x) = 0 1.;] 1-

S(r)

In fa::t,

k 2Sf.k" Sf . (x) = . J

iJ :Vij r n . x + (n-2) j n+2

r

Now Theorem 1 (8.2) leads at once to the

Center formula.

c f(O) = r (I + (n-2)Q(x»f(rx)dro(x) , 0 $ r < 1 . n Btl')

Recall that c 2(n-1) ill n n n

If f has a continuous extension to S(l) ,or even if it has radial

limi ts a. e", the formula ramains true for r = 1:

(46' ) c :reO) = r (I + (n-2)Q(x) ):r(x)dm(x) n J

S

-133

There is a simplification if f(x) is ta.ngential for J xl = 1 for then

(Q(X) f(x)) i ~ X.X .f .. (x) = a ~ J J

and the formula reduces to

(~6ft ) :r(o) =2 S f(x)dm .

cn S

8.16. Just as in the case of Poisson's formula for harmonic functions we

can use (1.6') to express f'(Y) in terms of the boundary values. Clearly,

we need or..ly apply (46') to the function

fT-l(x) = (T -1), (x)-lr(T -\r) y y y

Because Ty -I( 0) = y we obtain

On the left

and on the right we replace x by T x to obtain y

cnr(y) = (1- ry( 2) r (I + (n ... 2)Q,(T x)T • (x)f(x) IT f (x)J n-ldm(x) t,I y Y Y S

Recall that T I ex) = I. T '(x)j A(x,y) and y y

'T '(x) I = 1- I yf 2 Y [x,y]2

or 1. - ly(2 because ix' = 1 . We have also shown (see 2.8~ (40») that Ix-yJ2

T x = A(X,y)x when Ixl = 1 . Thus y

or

Q(T x) ~(x,y) = ~(x,y)Q(x) y

Taking these simplir1cat1ons into account we end up with

Theorem 4 ..

I [2 n+l c fey) = j (1-;,) A(x,y) (I + (n-2)Q(x) )f(x)dc.o(x) n 1 I-n

S x-y

lah

There is again a simplification if f is tangential, for then Q(x)f(x) = 0 •

Moreover, .6(x,y) = (I - 2Q(x,y» (I - 2Q,(x») so that, if we -prefer, (47) can be

written as

(47)' 'bf(Y) =.r (1- 111:!n+l(I -2Q(x-y: )f(x)dm{x). s Ix -y I

8.17. Evidently we can get a corresponding formula for Sf(y) by diff-

erentiating(47) or (l7'). For arbitrary y the campu~ation becomes almost

prohibitive, but we can use the device of ccmputing the d~rivRtives only at y=O .

Ignoring quadratic and higher terms in y we have

( J 12 n+1 l-y) f'>J ,1 0'" 1+2n(xy)

J 12n ( 1 - 2xy ) x-y

and

so that

Q(X-Y\j = (Xt -'1i )(:fY} '" (xi "'j - .xil'j - XjY i){l +2xyl

\](" -yl rv xixj +2xi Xj(XY):'XiYj-XjYi

(1._ly!2,n+l (I - 2Q(x .. ,y)\j '"

Ix _ yf2n

(I - 2Q.(x)) .• +2n(I - 2Q(x)). j (x.y) - 4x. x. (xy) +2{x' Yj + x.y.) ~J ~ 1. J ~ J J.

It follows that

For simplicity we shall do only the tan,gentia~ ca.&;le. By virtue of

Xj f j = 0 on the boundary the result reduces ~o

cnDkfi (0) = S (2nfi~ + 2fkx i )din(x)

S

Symmetriz~tion leads to

Cn (Dk f i (O)"+Di f'k(O»)=2(n+l) S (fj~+f1tXj)dm .

S

The trace on the right is already zero a.nd we find

(48) CnSf( 0) ik = (n+1) J (fi~ + fkX i ' dro ,

S

135

This is already quite neat, but we get an even nicer formula if we replace

the integral by a volume integral. By the Green-Stokes formula

I (fi%k +f'kxi)Cko= j (Dkf'i +D1

:f'k)dx ~ S B

and B imi laxly

0== ! fhXhda> = I J)hfhdx

S B

Accordingly, we cO::lclude from. (liS) that

(49) cnSf(O) ::; 2(n+l) J Sf(X)d,... .

B

It is now very easy to pass to the general formula. ~ before, we applY

(49) to iT -1 if

Sf(X} is then replaced by

(T -1) '(x)-lSf(T-1 x)(T -1) '(x) Y' y Y

For x = 0 the left and right factors cancel against each other and the middle

factor is Srey) In the integral on the right we replace the integration

variab Ie by T x y

If we observe that

becomes

S 'fy' (x}S:f(X)l'y' (x) -1 .

B

The scalar factors in Ty'(x) ) -1 and T' (x y

(50) c grey) ==2{n+l) r b(Ji:,y)Sf(x)A(y,x) n u •

B

the integral

cancel, and we obtain

It is again preferable to f'OCUB the attention on

We obtain the reproducing formula for ~ in the form:

Tkeorem '5. If' cp =p nSf wiiih tang-cntinl f and S*cp= 0 , then

It is instructive to compare (51) with the Bergma'I,l kernel formula t'~

analytic functions in the unit disk. It ~(z) is complek analYtic ~or

I zl <1 ,then

tp{ ,) =; S S spe z ) (:- i~ I z 12 ) 2

dxdy

rzl<l (l-Cz)

for a.l1 1,1 < 1 ,provided that

S S , cp(~) I (1 - J z r 2) 2 dxdy < ~

Izi <1 One checks that

does indeed reduce to 1. when n = 2 1C

Clearly, the matrices A(x~y) and

6(Y,X) account for the argument of the kernel.

Remark. We have proved Theorem 5 under much weaker eonditirns" but it does

remain true as soon as

(52) J ! 1~(x)11 Cl_lxI2)n d.x<,(O

B

8.18. * Because of the invariance properties of P and P

considerations are easily adaptable to discrete groups.

all our

Let r be a. discrete subgroup of' M(aD)

definitions:

We introduce the following

Def.l. If the vector-valued fUnction f(x) satisfies

for all A E r l then f is said to be au.tomorphic under r

Def.2. A mixed tensor density (D{x) is automorphic under r if

for a.ll A Er

Det'. 3'. A mixed tensor v (x) is a. B ~ltrami differential if

for all AEr .

If f is automorphic, then Sf is aBeltra.mi differential a.nd p nSf is an

au;omorphic mixed density. Conversely, if (J)~ Pf is automorphic it does not

follow that f is automorphic, but only that S(fA - t) = a +, In other words,

fA .., f is trivial, i. e. the components are quadratic polynomial1J. We write

£;. - f =PAf and call PAt' t21e period. of f under the mapping A

The periods satlsf,y the cocycle condition

for

139

A vector function with Zero periods. defines a vector function on the quotient

manifold M • . r 8.19. As far as boundedness is concerned there are two important conditions.

Condition L A mixed tensor 'Y (x) is said to be of class Leo(r) if it

is measurable, if 11" ~x) r , is (essential1y) boun:led and vA =\)

We can use the same terminology for tensor densities ~(x) when

for a.ll "E r 1 I 12 n

" = cp(x)( - x ) 2

is in L<lO(r)

Condition 2. The tensor density cp(x) is said to be of class Ll(r) if

Here the integrand is invariant ,so that the integral ean be taken over arry fundamental set , for ;instance the Poincare r fUndamental polyhedron p(r)

Remark. Recall that "q:>(x)" reters to the square norm of the matrix.

In particUlar, ~ E Ll(I) ~f

S "CPJJ dx < ex)

B

without ary automorphy condition. In this situation the Poincare' e-series

applies ar..d leads to e cp E L l(r)

Theorem 6. If" q>E LI(I) then

1 converges absolute1y a.e. and belongs to L (r)

Proof. Let P be a fundamental set :for r

J II cpr I elY = E S II cpll dx < 00

B AEr AP

By assumption

13ut

and

p

CPA {x} = r A· (X), nA t (x) -lcp(Ax) A' (x)

f f q>A (x) , , = 'A' (x) ,n, , tp(~) r ,

From this we conclude that

(53) r L; "qJAfI dx < 00

p

and the convergence of the series follows. The same estimate shows that

Fina.lly, e cp is automorphic because

((i)rp)B=L: (CPA)B=~CPAB::::®tp A A

8.20. It is convenient to use the inner product

( cp, ,) = J cp. 1jr (1 - f x, 2 ) n dx

r B/r

~40

when cpEL1Cr) , ,ELooCr) (i .. e. \) =.(1- Ixr2)D E Loo

) • The invariance of

the inner Pl":>duct is obvious. Also, if co ELI (I) , • E L oo( r) then

Proof~ Let X. be the characteristic function of P ~ Then

Remark. r:r tp and ~ ELI n L Cb then (® qhi.~ ) r ~ (cp, ® ,) r .

8.21. We return to Th~orem 4 and 'Poisson's formula (47). Let us fix k

and x, r xl == 1 -' and consider the vector-valued kernel K(x.y) with the

components

There is good reason to believe that K(y)

direct computation is almost prohibitive.

* satisfies P F.K=O •

Let us fIrst drop tte condition I xl = 1 and !'eplace (54) by

Then

Again,

-1 (l-jx[ )nK.{y)= " - ._. -"" 2 (1- [xI2)n+1 (1- L~d2)n+1 .( (1-l x

212)

1. [x,y]2n+2 [x,y] A(x,y) ok

1. •

B.¥ our usual formulas

and

1_/xl2 Tx' (y) ;:; 2 A(Y,x)

[x,y]

x y - 2 u\x,y T '( )_~_(l- !x,2) -1 ~,f )

[xtY]

so that

142

The right hand side is nothing else than [(1-lyI2)n+lekJT so that x

Ki(y) = (1-lxI 2)-n [(1-ly}2)n+1ekJT x

where e k is the unit ~ector with components 0ik.

Almost without computation one obtains

p«1..-ly1 2)e

k), , canst. (o'ky.+6'kY.-

2 = -1J 1. J J l. 1)

#: 2 ( s p ( (1- 1 y 1 ) ek ) i = canst. t>ik #: 2 (1-l yI 2)n+2 (P P ( (1-1 y I ) e

k) i = canst. 5 ik

0, 'Yk ) 1.J

where the values of the constants are irrelevant. By the fact that

* P P is an invariant operator it now follows that

P*PK(Y)i;:; (1-lxI 2 )-n {P*PL (1-1 12)n+lekJT}i = x

canst. !J,(X'Y)ik •

This is an algebraic identity which must stay in force when Ixl=l.

* It follows that P PK(~,y) = 0 for Ixl= 1 , and hence any integral

of the form

f(y) = f (1_lyI2)n+l A(x,y)h(x)dx

S tx,y]2n

is a harmonic function of y . 8.22. There is a similar fact for Theorem 5 • More expljcitly,

(51) reads

One can show that

A(X'Y)ihA(X'Y)jk

[x.y]2?l

·l4-3

is of the form 'I"hk, with 'Tl1k = - ""kh for hfk and 'rb.b. = 'Tkk (skew-symu:.etric

outside the diagonal, constant on the diagonal). This has the effect that for

.!!!l 'J{x) the function

w(y) = r A(x,Y)\J(x)A(Y,x)dJc . 2 B [x,y] n

* satisfies S L\II(Y) = 0 .

The operator L has other interesting properties. 1) If vex) is

automorphic (as a mixed tensor) then L \J is automorphic as a mixed tensor

density).2)If vELC»(r) then the same is true of Lv .

is in LICr) so is Lv

Proof of 3. It is clear that

and hence

lIILv(y)/ldy ~ r dY S Ilv(x~ll dx .

p P B [X,lJ

On partitioning B into copies AP one sees that

I' •• dy r .. dx= r .. dx r .. dy . ., tJ ~ oJ

p B p B

144

In this way

But

and we obtain

8.22. A weak finiteness theorem.

We shall now introduce a rather special class Q(r) defined as follows:

Definition qJ E Q(r) if the following 113 true:

* *) I . S cp::: 0 and cp == Pf for some f wi th xf ::: 0 on S .

II. CPA = cp for all A E r .

III. CD E L;J,.(r) n LQ)(r)

(In other words, J f f tpl , dx < 0:1 and r f C!>(x) If (1-1 x12)n is bounded. )

B/r IV. q:; has a smooth extension to S no and this extension satisties

More explicitly,

cp •• x.x. = XiX .CO-4k lJ J It J'o.J

on S\A.

*) It is understood that f has a continuous extension to S .

2 Condition IV needs a motivation. For n:=; 2 q> dz is a holomorphic

quadratic differential. The condition means that cp dz2 is real on I z! = 1

outside the limit set. When this is so qJ can be extended to 1 zl > 1 by

symmetry.

For arbitrary n the symmetric extension of ~ would be defined by

* 'PJ

= C!> where Jx = x This condition reads

cp(x *) = I xl2n(I - 2Q(x) hp(x~ (I - 2Q(x) )

and for Ixl = 1 it reduces to Q.(x;cp(x) =q>(x)Q(x}

The following is an analYtic finiteness· theorem whose topological and

geometric consequences for the quotient manifold M(r) are not clear.

Theorem 7- If r is finitely generated, then the dimension of the linear

space Q(r) is finite.

Sketch of proof. As already mentioned (see 8.18) if CPA =cp then PAf=

fA-f is a trivial vector function. As such each component of PAr is a

quadratic polYnomial in the Xi determined by a finite number of coefficients.

Suppose that r is generated by Al , ... ,.AN . There is a linear map

o.f Q(r) on a l'inite dimensional vector space which takes each cpt:: Q(r) into

the coefficients of PA f"."PA f The kernel. of this homomorphism consists 1 n

of all ~ such that the corresponding f is automorphic with respect to r

The thp.orem wi1.1 be provpd if we show tha.t t=1U~h s. (,fI must be idfmtjcaJly l7.ero.

For this purpose we study the ~ntegral

which can also be written as

.r Djfi CPijdX= -.r fiDjqlijdy.+! fi(Oijnjdcr

P P bP

* is the outer normal of oP .)

~46

The boundary oP consists of pairwise congruent faces of the polyhedron,

Plrt of n and points on A . Let A and A~ be a pair of corresponding

faces. Then

r foi tn~J.nJ.d a = f f. (Ax)cp •• (Ax)n. (Ax)' A I (x) In - Ida .1 ..... -1. •. 1 lJ J AA A

On the right we substitute

f.(Ax) = (A'(x)f(x»). ::;A'(x). f (x) 1 1 1a a

The minus sign is because A maps the outer normal at x on the inner normal

at Ax .

The integrand ~ecom~s

*) lie are temporarily assumming that Green's formula is applicable.

147

- f ti) 4> (x) n (x) a ac c

and thus

S' f. 4l • • n . dO' = - f f. ~ . . n . dO' • A6 ~ 1J J A 1 1) )

The integrals cancel against each other.

Suppose now that P is a finite polyhedron and that no points

of A belong to ap. The remaining part of the boundary integral

in (56) is then

S nnP

f . cp •• x. do • 1 1J )

Now we make use of Condition IV to write

f.~ .. x.;dw 1. .l..J J

2 = f. <f>'. • x . x~ dw = .l.. 1.J J K

which is zero because f.x. = 0 by assumption.This proves ($,~) = 0 1. ].

and bence ~ = 0 under strong regularity assumptions.

In order to get rid of the extra assumptions we use the same

mollifier technique as in the standard proof of the finiteness theorem

for Kleinian groups. Suppose that 00 -C (P) I A ~ 0 I has compact

support on P \ A and satisfies A (Ax) = A(X) at equivalent points

of ap. ~he earlier reasoning is sufficient to prove that

(57) (CP , ). <P ) = - Sf. ~ A <p.. dx • P l. aX j l.)

The theorem will follow if we can find a sequence

A +1

" and f. caA lax. )+0

~ v J boundedly on P .

Let <5 (x) be the euclidean distance from. x

148

of ). V

such that

to A . At corres-

ponding points x and AX e: ap the distances <5 (x) and 0 (Ax) are

equal. To see this observe that Jxl=IAX~ and hence IA'{x)1 =

(1-IAxI 2 )/(1-rxf 2) = 1 • Furthermore, for any y

!AX-Ayl = /x-ylIA' (x) I~IA' (y) I~= Ix-y/IA' (y) r~

JAX-YI= IAX-AA-lyJ= /x-yl IA'(y) I-~ .

and hence min (I Ax-Ay I , (Ax-y r) ~ r x-y I. As y varies over A it

follows that o(Ax) ~ o(~) and by symmetry o (Ax) = o(x} as asserted.

d is not smooth, but 'o(x)-o(y) J ~ Ix-y/ and Jgrad ol~ 1 a.e.

Let hv(t) be 0 in [O,l/vJ~ linear in [1/v,2/v] ana 1 for

t > 2/v . We shall choose

h [(log log v

Although this function is not smooth it is not hard to see that (57)

(57) remains in force when A is replaced by ~v

It is evident that A v

the derivative one finds that

tends boundedly to 1 as v-+-co • As for

149

lax/ax.1 < 2'Je . v J = 2e ( 2e 2

~(x)log o(x) log log o(x»

a.e. Actually, the derivative is zero except when

2e v/2 < log log o(x) < v •

This leads to the estimate

(58) I I 2e aA v/8x < 8e/vo(x)log o(x) •

Let us now extend f to all of Rn by the symmetry relation

f J = f , that is to say according to the rule IxI 2 f(x*)= (I-2Q(x»f(x).

Because of Condition I Q(x)f(x) = 0 on S and the extension is

* consequently continuous. Moreover, Sf(x) = (I-2Q(x»Sf(x) (I-2Q(x»

so that I ISfl I renains bounded. By Lemma 3 (8.10) we may therefore

conclude that f satisfies a near-Lipschitz condition If(x)-f(y) I =

OClx-yllOglx:yl) . We observe next that f is identically zero on A • In fact,

for any A £ r , f(AO) = AI (O)f(O) and hence It{AO) «IAI (0) I If(O) 1= (1-IAOI 2) If(O) I. Choose a sequence of A such that AO tends to

x € A • It follows by continuity that f(x) = 0 •

Together with the near-Lipschitz condition we conclude that

If(x) 1= 0(0 (x) log l/o(x». In view of (58) this implies that

f.OA lax. tends boundedly to 0 as \.1700. This is precisely what ~ \.I J

was stil+ needed in order to conclude from (57) that (<I> ,4» == 0 I

and Theorem 7 is proved.

Remar~. The theorem is of course rather me'aningl'ess unless one

can show, in the oppositja direction, that groups with a finite

ditIl~nsiQnal Q (1') have some rather speci,al p:r;operties, i,or instance

with respect to the ~opology of BIT •