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
p.l
II. The general case
p.13
III. Hyperbolic geometry
p.31
IV. Elements of differential geometry
p.41
v.
p.57
Hyperharrnonic functions
VI. The geodesic flow
p.72
VII. Discrete subgroups
p.79
VIII. Quasiconformal deformations
p.101
I.
The classical case.
Everybod¥ is familiar with the fra.ctional linear transformations
1.1. (1)
y(z):=
where
a, b, c, dEC
C = C U (CO)}
and
ad - be
f
az +b cz + d O.
They act on the extended complex plane
which we identify with the 1 - dimensional proj eetive space
In terms of homogeneous coordinates
w = 'V(z)
P1 ( C) •
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 by the unimodular or sRecial linear group
S~ (C).
G~(C)
or
More precisely the
Mobius group is isomorphic to
where
C*
where
r=(Ol) •
is the multiplicative group of nonzero complex numbers and to
1 0
Mobius group by
In spite of this identification we shall denote the
~ ( C)
rather than
PS~ (C)
for the simple reason that
the identification does not carryover to higher dimensions. We shall of'ten sittplify normalized by
ad - bc
= 1.
y( z )
to
yz
and we think 0'£
Y as a matrix
We should be aware, of course, that
y
and
-y
2
represent the same M"obius t=:-a.n.sformation. formula for the inverse in
S~
It is useful to memorize the
:
(3)
1.2.
There is a natural topology on
}~ (C).
G~ ( C)
G~ (C)
S~ (C)
and hence also on
is connected for the space of all complex 2
X
2
and
matrices has
eight real dimensions and the space of singular matrices has only six.
The
mapping
ad~ be ) of
G~(C)
consider
on s~(C) S~{C)
shows that
S~(C)
is likewise connected.
We can
~~(C) •
as a double covering of
The subgroups with real coefficients, are denoted. by G~(JR) , S~(:R) and
~(:R) • There is also a subgroup Gr;C1R)
+
G~ (:Ia)
and S ~ (JR )
G~ (JR )
are connected, but
pass continuously from a
ma~rix
with positive determinant. is not for one cannot
with positive determinant to one with negative
determinant. Any
YEM2(JR)
maps the real axis, the upper half-plane, and the lower This is obvious from
half-plane on themselves.
...
(4)
yz - yz =
z-z
jcz+d·1 1.3. (5)
From (1) one computes y( z) - y( z ')
:=
2
3
and in the limit for
zt-+z
y'(z) =
(6}
ad. - be
(Cz + d) z ~ yz
In particular,
2
We rewrite (5) as
is copf'Orr.lal.
1/2 1/2 y(z) - Y(ZI) = Y'(z) l' Y'(ZI) I' (z- z')
(7)
where it is understood that :=
with a f:xed choice of the square 1 .. 4.
Jad- be cz+d
~oot.
The cross-ratio of' four points
z, z I
,
s,
S'
in that order is
d,e:fined by
zt - 6 z' - (; I
(8)
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
4
From (6) one obtains
1.5.
(10)
y"(z)
2c
yl (z)
cz+d
and hence
~:it: ~
(11)
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 It 2 II 1 (.Y:.) . " 2 ::: 0 :L _ _3 (.Y.:.) = (Y...)' __ y' 2 "VI y' 2 yl
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)):;: (a",b,c,d)(l+
t
2
6' (a-b) (c-d)S:r(Z»
+ o(t 3 ) .
5
1.6.
The half-plane.
half-plane
S~(E)
We return to the action of
H2:;:: {z :;::
X
+ iy ; y > OJ.
on the, upper
We use the notation
G;;:
S~ (JR)
(or l\~(1\) ). Let
K denote the isotropY' group of
From
i.
ai+b = i ci +d
we get
a ;;: d ,
2
a. +b
b:;:: - c ,
2
;;: 1 •
Hence
G
sin cos
COS
(17)
g;;:
- sin
K ~ 80(2)
In other words,
different elements of
Let
(
•
e
e
More precisely,
and
8+ n
M2 (1&)
•
1 u ( 0 l) •
The group co.
give
~
A be the group of homothetic transformations
simila.rities and also the isotropy group of
( : t 1) , NA
t
>
0 ,
is the group of
It is simply transitive on
as seen from ( lu),t o1 t
0 )(')
ot
Hence
2
~
-1
:;:: u+t i
G:;:: NAK ; this is tile lwasawa decomposition.
decomposition
g:;:: nak.
(18)
g:=
Every function
(lo
Rxplicit1y, if
X )
(y1/2
f
on
~
G / K.
Each
0) (
cos e sin -sin e cos
If
can be lifted to a fUnction
Conversely, a f\L~ction on
if it is constant on the left cosets K \ G ).
has a unique
e) e
y-l/2
(many authors prefer
g EG
g(i):;:: x + iy , y > a , then
0
1
defined by !(g) :;:: f(gi) on
e) e
but the same element of
K
and N the group of translations
H2
g E K has the form
g K.
~
!
on
G
G becomes a function is identified with
6
1.7.
111e circles orthogonal to the real axis, including vertical lines, are
mapped on each other by all. g E G.
model of
pyPerb~lic
z=-, Z2Erf
They are the straight lines in Poincare t s
or non-euclidean geametry.
lie on a unique
n.e. line
Any two distinct pOints
(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 ds
(21) ('ire have
dropped a factor
2 ).
;:;:
~ y
This defines the Poincare m.etric on
It leads to a new distance function d.(zl ' z2) (22 )
2
H
defined as the minimum
7
over all paths from
zl
to
z2.
The shortest curves are the
To find the geodesic we use agE G which maps
zl' z2
It is readily seen that the vertical line segment from
~eodesics.
on iy1 ' iy2 • iy1
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 5
The distance function
+ d( z2 ' Z3)
if'
z2
d
:=
d 2
tanh-
is ad.ditive on
lies between
We can use (24) to verify that
fiwe .. lines, i.e.
zl and 8
d(Zl' Z3} = d(zl' z2)
z3 .
is a distance :function, :ror if
5 = tan hS!' < 2
1.8.
There is a dif:'erent way of computing
A.ciditivity at once.
d(zl' z2)
We refer to Fie· l and compute
which shows the
(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
negative when the order of'
d(z~,
zl' z2
is
z2)
as a signed. distance which becomes
reversed~
The additivity on a geodesic
::ollows at once from
1.9.
The length of an arc is
Jc My and the area of' a measurable set E is
J E
dx~Y Y
It is a useful exercise to compute the area of a n.e. polygon
P
(Fig. 2)
"I
/
I
,',
I
, 1
I
(
I
r I
f
.....
,
.....
,
"
" \'
\ \
\
\
, 1
I I
t
\
I Fig. 2
9
By Stokes I formula
u
Each side is an arc change change~
0=
dx Ady 2
r
A =
p
Y
z - a = re
o~
i8
~
and,
the a.ngle of the tangent.
y
;::: -
A =-1:: f3
a
v
2n.
If the outer angles are
one obtains
v - 2:1( = (n - 2)1C - E a v
' .. 10.
Tt is easy to :;>ass from the half-plane
('E C ;
I 'I
< I}.
measures the
For a simply connected polygon the total
including the jumps at the angles, is
and the inner angles
J de
ae
tf
t,o the uni.t disk
We want to choose a canonical Meoius mapping
A good choice is to let
z = 0 ,. i ,CJ)
correspond to
,=
B2;;;
.; ~B2 •
-i , 0 " i .
This
gives
z-i
(26)
,=i-;+i' We introduce the notation
Z
,*;:::
, with respect to the unit circle. over into
=
r+ 1.. • .;, -1.
C-=i'
1 =
,
...L-
1,1 2
for the symmetric point of
B.1 the reflection principle
"C* and by the invariance of the cross-ratio
(27)
This shows that the point-pair invariant in
B2
'2 (1- tl '2 I 1'1-
1
is
-
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
> 0 is r
d ;;; d(O,r) =
J
o
Note that
l+r 2dt ;;; log l-r' 2
I-t
r = tanh ~
'
8(o,r); r .
Another observation is that
1+r (r , 0 , -1 , 1) ;;; - 1 -r 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
to
11
(29)
l.li.
The self-rnap'pings of
(30)
l! Z
B2 which take = e
ie
a
into
0
are of the form
z-a 1-
az
We want t:l derive this formula in a geometric way.
Fig. 4 For this center
a*
carries
a
pur~ose
we construct
a*
as shown in the picture. into
0
and the orthogonal circle with
The reflection in this orthogonal circle
and an arbitrar.y point
z
into
2 * a cr z = a* + (a* I 1 -1) (z - a*) = - '::" a a
To obtain a sense-preserving mapping we let in the line through the origin perpendicular to
reflection is expressed by
(31)
zl-?Tz =
a
as. a..
z-a
1- aZ
be followed by reflection If'
a.rg a =
a
this
and we end up with the mapping
z- a 1 -' 8.z
The most general mapping is and it is hence of the farm (30).
T
a
followed by a rotation about the origin,
We make a note of the formulas
2
T' (z)
a
l-ITa z
, 1- laJ (1- az)2
l2
1 1 ..
all well known to conformal mappers.
of the Poincare metric. T' (0) > 0 a.
fz{
,
2
The last formula expr'esses the irtvariance
The norwAlization
o~
Ta
•
Note that
TO:; -a
a
and hence
T
-~
= T-
a
1
is such that
T'(a) > 0 and a
13 II. 2.l.
TPe general cas,e.
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
,X ) ,x > 0) ~ Already ?oincare was well aware that the 3 3 action of M(C) can be lifted to H3 or if' one pref'ers to all of JR3 • *)
H3 : (x ::; (Xl' x
In fa.ct, any
2
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 factorization 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 (~ ~) -v u
(1) where
k= (~~)
and
i=(10)
In fact, if' we write
u, vEe
u=,\ +i~ , v=v1 +1v2
O-i
'
we obtain
u uy) = u + ' . k ( -v ~~ + v IJ + v 2 l
(2 )
= u + vj
The conjugate of
Izl
is given by
have used the rule
Izl2 aj
z::; u + vj = zz = =
ja
luJ2
is +
-z = u- -
vj
and the absolute va,lue
Iv/ 2 • When computing tr_€ product we
for arbitrary complex a.
We have replaced the notation
M2(C)
by
M(C) •
14
Points i!l. (y
>0
if
1R3
~ow be denoted by z:;: x + y j
will
z E~).
Suppose
yES~(C)
a b Y = (c d)'
x E C , y E lR
with
is given by
ad - be = 1
•
We define the action by yz == (az+b)(ez+d)
We need to show that these quaternion
-1
= (zc+d) -1 (za+b)
expressions axe equaJ., and that
yz.
is a special
or the same form as z.
In the first place, the two expressions (3) are equal if and only i f
(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.
The next step is to actually compute yz =
yz..
To begin with
~az+b)(cz+d)
Icz +d1 2 Here cz
+ a. = ex + d + cyj
cZ"+d.
= ex + a - cyj
az +b ;::; ax+b + ayj
- 2 + [-(ax+b) cy+ay(cx+d)] (az + b)(cZ+d) = (ax+b)(cx+d) + acy : (ax + b)(Ci: + d) + a'i:.y2 + yj
j
15
and thus
yz.::;
(4 )
- 2 ( ax + b )( -ex + d) + acy + -yj Icz + dJ 2
which is of the desired form.
2.2.
Also,
fez
+d1 2
==
ICX+dI 2 + Jcl 2 y2
We shall also compute a. formula for the difference
"fZ - yz t .
0
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
Iyz
(6)
- yz t
j --
Iz-z'l
Icz + d I Iez' + d [
and infinitesimally fdy(z) I
(7)
Comparison with
(8) is invariant.
(4)
:;;
laz\ Icz +dl
shows that ds::;
~ y
2
16
The cross-ratio should be defined by
By
use of (5) one obtains
Thus the matrices correspond.ing to the quaternions (Zl ' z2 ' Z3 ,z4)
are similar.
(YZI' YZ2 ' "(Z3 ' YZ4)
and
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
The full group M(H3 )
j
is the group of unitary matrices
is ~ransi tive, for
yj::; u
+ vj,
v > 0 , by taking
-1/2)
uv
v -l/2 (,
2.3.
The quater nion technique works only for
elegant it is not particularly useful. on the entire space
JRn •
:M(~ )
and even if it looks
We shall now again deal with actions
We shall use the notation x:;::o (Xl' •.•
and when operating with ma-;rices we treat
of similarities consists of all mappings x~mx+b
x
a.s a column vector.
,X
n
) E lRn
The g.coup
17
where
bEEn
and
ill
is a conformal matrix, i. e.
m:;;;).,k
with
>
0
x*
,
A
and .k E O(n) Reflection in the unit sphere is defined by
(10) 2
where of Course
Ix 1
:;:
x 17 x * :::: Jx :;:
(JO
:2 + ••• +xn •
We use mostly the notation
2
~
=
m ,
J~
:;;; 0)
sometimes one needs a letter symbol for the mapping, and then we use Obviously, matrix
I
;l
= I , the identity mapping.
I
but
J.
will also stand for the unit
n
We adopt the following definition: Definition.
The full MObius group
similarities together with
J.
M(~n)
is the group genera.ted by all
M(T)
The M"obius group
whose elements contain an. even number of' factors
similarities. In other 'Words, I
do not use
n M(lR )
n M(E )
J
is the subgroup
and sense-preserving
is the sense-preserving Mooius group.
Note that
as being too J)edantic.
The derivative of a. dif'ferentiable map:ping
f
from one open set in
to another is the Jacobian matrix f' (x)
or
Df(x)
with the elements
(11)
f' ' (x). . lJ
=
of.
~
':!.v-
O-CO-j
The derivative of a similarity matrix
m.
The matriX
Jf (x)
;::;
D. f. (x) J
1
yx. = mx + b
is the constant conformal
bas the components
JRn
18 2x. x.
for
J
~
(12)
JXJ2 x
'I
0 .
It is almost indispensable to adopt a special notation far the ubiquitous lLatrix
with the entries
Q(x)
x .. x. = ...1:......J. Q() x .. 2
(l3)
Ixl
lJ
This enables us to write (12) in the for.m 1
~
Jt(x) =
(14 )
IxJ
(I - 2Q(x)}
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
which means that
x"l
I - 2Q E O(n).
Thus
=I
0 •
It now fellows by the chain rule that any
is a conformal matrix for each
J' (x)
A(JRD) YEM
•
For
is a con£orm.a1 matrix for
~~'b ius transform.a t ions are In other words, al1 lYlO
With a suitable interpretation at Definition.
y. (x)
any y E M(mn )
~ E O(n) such that1Y'(x)1 cf scale at x, the same in
co
ar..d.
-1
Y
we denote by
In other words, all
directions.
C ant ormal
co.
I yt (x) I the positive number
iy'(x;]
is the linear change
19
Another application of the chain rule proves that
[yx-yyj
(l5)
=
In fact, this is triv-ial when
Iv'(x)1 I/:2 'V
IVf(Y)1
1;:2
!x-yl
is a simila.rity, and for
IJt(x)IIJI(y)llx-yl
If
'V 1
and
'Y 2
satisfy (15) then it is also true for
I'VI Y2X lyi(Y2X )I
2
'Vl 'V2yl =
J
'tore obtain
by (14) •
VI '\(2 :
IVi ('V2 X ) 11./2 I'Vi (Y2Y) [1./2 IY2 X -
{' 1V2(x)1 1/2 1Yi(Y2Y) 11/2 I'
1/2
(Y2(Y)
11(:2
*)
Y2yl
Ix-yl:= I (YlY2)'(X)(
1/2 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)
Ia,b,c,d I
--~ la::df
~J : Tb-dT
-which is obviously invariant in the sense that
(17)
Iya , 'Vb ,
yc , yd
I = Ia , 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 ,
Ij:2
20
that
a, b , c ) d
la-blle-d1
+
lie on a. circle in cyclic order i f and only if
Ib -clld-a!
fa -cllb . . dl • This condition can J a , d , b , c I + Ie, d , b ,al = 1 •
in the invariant form.
=
be -written
As a second application of (17) we prove the :following important lemma. Lemma 1.
Proof'.
Y leaves
If
If
ym ;::
00
fixed, then
yx ... yO
the.:1.
sufficient to prove that
'V0 = 0,
constant conformal matrix m.
y
is a similarity.
leaves y
co
=
QO
0
and.
implies
First,
or
Next
gives
222 r yx - yy I ;:: it Ix- y J
(yx,yy):::
It follows that
so that
~
2
:riXed and it is
yx. = mx
wi th a
In other 'Words, we need. to show that
is constant.
which implies
eo
(x,y) •
yl (x)
21
V(x + y) = yx + VY
'Vi (x+ y) :::; y' (X) :;; canst.
Corollary.
If
ya:;::> 0,
with
~
a
T
are both fini te ~ then the most general
b
Vb;:::
CD
is of the form
yx ;::: m[ (x - .0) * - (a - b) *]
(18)
m is a consta..r;t conformal matrix.
where
In fact, it is clear tha.t lnLa
CD.
2.4.
We denote by
n B =
n
A
y E M( JR )
(x; Ixl
A.
n
Sn-l;::: {x;
takes
a
into
0
and
b
the subgroups Which keep
M( 13 )
< l} invariant.
the unit sphere
(x - b)* ... (a- b)*
Because Mobius transformations are bijective
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.
for
Ixl
If = 1
y~
=
CD
we know
by Lemma 1 that
it follows that
Assume now that
'1. 100 ;::: b
In;:;;
i
Ixl;::: 1
But
and beca,use
k EO (n)
CD.
Then (18) implies
I (x - b) * + b * I ; ; for
V =!OX
const.
Imxl
= 1
22
Ixl
Ix-bllbl Ix - b I
Hence
2 .. 5.
:=
for Ixl
canst.
= 1 , and that is impossible since
We shall now determine the most general
exactly as in l.l1 for
n == 2
b
f
0 •
V E M( Bn) • This is done almost
eKcept that we do not have the complex notation.
We begin by proving a non-trivial identity: Lemma
?~oof.
3.
As it stands (19) does not make sense for
that both sides tend to
0 for
y ~ O.
y
= 0 , but it is obvious
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
* ==By* = AfYJ ::;: Boo = 0
Ay
Therefore,
~onstanttl
0
and
But then
co
00
are fixed points of'
At (x) Bt (x)-l
AB-
1
and by Lemma. 1
is also constan~, for
(AB- l ) 1
is
23
-l
= AI(E (AT B'
From (20) and
-l
x}
X)· B'(B
-1
-1
=
)oBK:: canst.
(14) AT(X);:
1-2Q(X-~*)
Ix- Y*I (21) B' (x)
and.
for
Hence
x = y
lY12{1 - 2Q(y)) (I - 2Q,(x* -~)) (I - 2Q(x) ~
::
l:x:* _y1 2Ix\2
we find
AI (x) = Bt (x)
x, and since
for all
AO = BO
= -y
it follows that
Ax,=Bx.
Direct comparison of'
AI (x)
and
(I - 2Q(y» (I - 2Q(x - y*)
(22)
Bt (x)
a.s given by (21) yields
= (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.
* Sn-1 (a,
Given
a E~
(a
(1 a * I2 -1) l / 2 . ) 'WJ.th
intersects
n-l
S
f
0)
center
orthogonally.
we construct a*
a * and the sphere
. and radl.us
j 1- 1}2 la /
The reflection in this
sph~e
IalI
. wh~ch
is given by
24 cr x = a·* + (a. I *12 - l)(x - a * ) * a
(23 )
We let
cr
a
perpendicular to
be followed by reflection in the plane through the origin a.
It is easy to see that this second ref'lection amounts'
to multiplication with the matrix
y' and Fig. 5
=
1- 2Q(a) "
{I-2Q(a»y=y-
In fact
~
lal
shows the location of y' .
Fig. 5
We now define the canonicaJ. mapping T x
a
= (I - 2Q( a»)
x a
n y E M{lB )
and conclude that the most general kTa with
(j
with
ya
=0
is of the form
k E O(n) •
The explicit expression for identi ty ( 19 ) •
It is clear that
Tx a
be~omes
much simpler if we use the
(1-2Q(a»)a* = -a* and thus we obtain
25
I -*1 2 - 1) (1 - 2Q(a»)(x - a) **
TaX = -a* + (a
Replace
Y'
a
by
in (19) and multiply by
** (1 -2Q(a»(x- a)
:=
I - 2Q( a).
•
We attain
a + Ia 12( x * - a) *
and T x a
:=
-a* +
(I a .*1 2 .. 1)a
+ (1- 1a 2 )(x* - a) *
I
•
Fina.l.l.y,
(25 ) This is as simple as one can wish, but if cae wants to avoid the
notation it can (26)
easi~
T
[x,a] =
where
a
be brought to the form (1-
x =
la 12 )ex- a) - Ix-al 2 a [x,a]
Ixllx* .. al
Iallx - a *1
:=
2
and
We collect the various expressions for a)
T x
b)
T x
= -y
c)
T x
=
(1 -
=
(1_\y\2)(x_y) [x,y]2
y
y
T x y
Tyx
= (r-2Q(y»(y*+(jy*1
y
d)
x*
2
that we heve derived. -1)(x_y*)*)
2 + (1- lyJ ){x* - y)*
IY12)x
.. ( 1 .. 2eY + [x,y]2
Iy [ 2 U"
lx- yJ 2y
26 Recall that
[JL,y} =
Ixlly - x*J
IYlix - y*1 .
;:;:
a) and b) are easy to differentiate and yield v 12 (I-2QY» « tr'(X) = 1 _ 1 '''"2 y [x,y]
I -2Q (x-y
*»
(27) l .. =,
IYI2
2
[x,y}
(I - 2Q( x* - y) )( I - 2 Q.( x»
..
We have already remarked on the id.entity of the two matrix products. choose to introduce the notation
ll(X,y) ;; (I - 2Q.(y) )(I - 2Q(x - y *» which leads to
= ~(y,x) T
~(x,y)
•
Note that this can also be written
(28)
~(x,y)
!:l(Y,x) ;::; I
It is helpful to View the formula
(29)
)X~
Tf (x) = Y
1[x,y]
ll(X,y)
as a representation in polar coordinates: ( 30)
IT'(X)I
=
is the "absolute" value and
l ...
lyl:
[x,y]
y
a(x,y)
is the "argument" of
We make a note of the special values
Tt (x) • y
We
27
[;.rl 2
T' (0) :;; 1 -
(31 )
y
Also,
T 0 = -y .. and hence Y
~
laborious.
T
= T-y y
ITy xl
Direct computation of
1
1
.
from any of the formulas
a) - d)
is verY'
Fortunately, we can use the difference formula to obtain
ITy xl :;;Ix-yl -rx,yr
(32)
NOW a short
com~taticn
1- IT
(33)
y
leads to
x (2
=
(1Ixl 2 )(1- Iyl 2 ) . ' -- 2 . .,.... -
[x,y}
Together with (29) we have thus
-
IT'(X}! y
We have thereby proved the invariance of the Poincare llletric
(35)
ds =
2}wd 1-
Ixl 2
lt is again seen-that every orthogonal circle is a geodesic. hyperbolic distance from the
origi~
is
d( O,x) = log ~ "l=liT and more generally
The
28
I
(37)
d (x, y ) = log y , x ,
S, II
where
2.8.
g , 11 I
are the end points of the directed. geodesic from
There is a close relation between
TyX
and
rxr.
x
to
y.
To derive it we shall
first prove:
Proof: Iq
Let
= By = 0
Ix
and
be the left and right hand side of (38).
Rx
l
o.
U- ( C) ==
and hence
Clearly,
But
where the left hand equation is obvious, whereas the right hand equation
.follows by application of (34). -1 I {La )(0);
which proves that
IR-
1
= I
Thus r_1
r
L (Y)R (y) .
Apply the Lemma with
side is
and one obtains
T~
T' (y)
=-
= 1
{~(y) ,
TyX.
y == T • x
The le:rt hand
29
or T y:;:; -A(y,x)T x
x
y
(39)
is not defined for
Remark.
T y
T x = -y
~ by
y
continuity.
I y{
:;:; l
According to
T~ :;:; 8{y,X)Y for
but by formula b) we should write
l39 ) it follows that
Iyl
=
1
(40)
Tx y
One more formula.
on
=
6(x,y)x for
[xl:;:; 1
Differentia.tion of (38) with respect to
x yields
equating the arguments in (29) one obtains
~~ 6( 'VX , 'Vy) TY'1iTT ::, TY'T:YTf 8( x, y )
(41)
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
(42) where
(43)
'Y E M( Bn)
, Y f I , has a canonical representation
y = kT So -
-1
Y
0
o..nd
kE 00(.0) •
a
Since
Jkl = l
we have
30
tx;
The set
l v' (x) j
a*
center
= l}
Nothing prevents us from considering the full sphere rather
than only the p~t inside and we denote it by
I(Y)
is nothing e.lse than the orthogonal splu:!l'e with
K(Y).
It is called the isometric sphere of
Bn.
Moreover, we denote the tnside of
and the exterior by E('Y).
K(Y)
V, by
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:
'V-1
maps
1(y.. l)
on
E( V)
Of course this does not determine
and
'Y
l E( 'V- )
on
I( 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
This is .a. short chapter devoted to two separate tasks:
3.1. 1
0
2° 3.2.
Let
A , B ,C
Derive some formulas of hyperbolic trigonometry ..
Show how to represent Mooins groups as matrix groups. ABC
n be a non- euclidean triangle in B •
We denote the angles by
and the n.e. lengths of the opposite sides by
a, b , c.
we
shall
prove: I.
(1)
cosh c II.
(2 )
3.3.
The hyperbolic law of cosines:
= COM a
cosh b - sinh a sinh b cos C
The byperbolic la.w of siBes sinB sinh b
sinA = sinh a
PrDOf of I.
=
sine sinh c
We may assume that lies in the
Xl - axis and that
b
consider the case
n = 2
C
=0
, that
x x 2 - plane!' 1
a
falls along the positive
It is thus sufficient, to
and we may use the complex notation.
Fig. 6
The points
Hence
B
B ~~d
and
. A are at euclidean. distance
A are
~epresented
2'a
tanh
by
tanh
and
'2a
and
b tanh 2
e
iC
tanh
2'b
from
o.
The point-
pair invariant is
[tanh
B(A,B) =
(3)
~ ... tanh ~ eiC
I
11 - tanh
a;
'2
I
tanh
b
I
-iC
'2 e
tanh .£
I :;.
2
Just as in ordinary trigonometry all the hyperbolic fUnctions can be expressed rationalJ.y through
tanh
2'x .
The formuJ..as are
2 x
cosh
X
:;.
J..+tanh
'2
2 1- tanh
!
2 tanh
sinh X :;
2
!2
From (3) 2 a 2 b ( .L + th 2') (1 + th '2)... cosh c
;;
2 {l- th
~)
4
th
'2a
(1 ... th2
th
~
b 2'
cos C
)
::: cosh a cosh b ... sinh a sinh b cos C
3.4.
Proof of II. cos C
. 2 S1n
C
:;
2
8in C 2
=
From (1) : ;: ;
cosh a cosh b - cosh c sinh a sinh b
2 2 2 (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
sh c
The synmetry prove s (II),
222
sh a sh b sh c
33
3.5. In this section we want to show that MO~.n-l) is isomorphic with M(B n ). For n = 2 this was already done in Ch.I by observing (tacitly) that
M(R1 )
2 M(H )
on the other hand
M(H 2 )
and
and
a conformal mapping from
H2
are both equal to PSL 2 (R), while 2 M(B ) are isomorphic because there is
to
B2
In the general case we begin by extending an arbitrary • • to a mapp1ng 1n
ya
n
=a
, yb
=
•
For t h·1S purpose we ~"dent1. f y
Rn • Recall (Corollary 2.5)
{x =o} of
subspace
with
M(-'rTIl) n
ro
,
b F
~
~as
Rn- 1
that any
with the ye:M(R
n-1
~
a unique representation of the form
yx = rorex-b) * - (a-b) *' ]
(4)
,....
=
Ak, A > 0, kE O(n-l). If we replace m by m we obtain a n mapping of R whose restrictio'n to Hn is an extension of y • where
m
should be replaced by yx = m(x-a». This construction shows that M(Rn - 1 ) is isomorphic to M(H n ) • It remains to prove that M(Hn ) is isomorphic to M(B n ). For this it suffices to find a standard Mobius transformation 0: Hn+Bn which (In case
b =
00
we choose so that
(4)
x
= (0 , en ' (0) carre spo·nd to y = ox = (-e n ,O,e) n
where e n is the last coordinate vector. The restriction of Rn - 1 is the usual stereographic projection.
to
The following pictures illustrate the choice better than words:
x
LII,I
x-en
- e", '.
.
I
I / /' I
34
e + 2(x ... e )* n
n
I
e
n
y
= aX = [e
n
+ 2(x - e ) *] *
n
.... e
Fig. 7
n
The correspondence is thus given by
(5 )
**
y = (e + 2(x - e ) )
n
X :=.
n
e + 2 (y* - e )*
n
n
In terms of coordinates one finds y
*= 1
2x.].
(6) if-
Y :;;: n
and
Ixl 2 -1
Ix - en l2
(i =
~, ••• , n -
l)
35 2yi
(i
2
\y- e n I
~ 1 , ••• , n ... 1)
(71 1-
xn =
When x n = 0
Iy\ 2
ly- e n I2
lyl2
one verif':'es that
= 1
Yi
Iyl ;: :
and for
J.
~
Ixl 2
Y' = n
+ 1
*
=y , and (6) reduces to
Ixl 2 - 1 Ixl 2 + I
2x. (8)
Y
1
y.
(9)
J.
X.
l-y n
J
one recognizes these
3.6.
X
::; 0
n
f'o~mulas.
n-l { 1 w~.th B = x l ' · · · ' ~-l ,0
The stereographic projection (8) mapa
2
2
xl + ••• +x _ < I n 1
on the lower hemisphere of'
(IO)
B _ n 1
n 1 "'-ER -
mapping
,
t \ IX
< 1 .
The
on itself' defined by 2x
y :;;;
4\e
II
{ n yElR; y =1,yn
(Y1 , · · · , Yn ) ~ (Y1 -' .... ,Yn - 1 , 0) ~
Let it be followed by the projection result is a ma];>ping of
Sn-1 ,i.e.
This is of course not a Mobius transformation, for the
(y1 ' ••• ,yn) ~ (yI ' ••
,Yn-l ,0)
Ii
What is the nature of the mapping (10)? n-2 S
confonnal and leaves the spbere it maps orthO@onal circles of
h
B -
1
circles on the' hemisphere, which are
=
(xI!
is not conforlnal. The stereographic projection ~
=0
,
1xl = I}
fixed.
, i. e. circles orthogonal to MSO
orthogonal to
Sn-2.
Therefore S
n-2
on
Fig. 8 But an orthogonal. circle on the hemisphere lies in a plane parallel to
its !)rojection is a straight line segment" geodesic in
n-l B
and
In other 'Words, (lO) maps every
on the segment between its end points.
by a simple computation.
en
This is also obvious
The equation of' an orthogonal circle is of the form
>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.
model the noneuclidean lines are the line segments in of' two points
y' ,yll
is defined as
d(x', XU)
n-1
B
In this
The distance
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 distance between
x,y
in the
~lein model is
3.7. We pass now to the matrix O(n,l)
acts on
representation~
O:::X,x>
=
2 xo
AT (1
-
It follows from (11)
2 xl
-
-
-
x1Y 1
- ....
A
that
x
2
n - xnYn if
e: O(n,l)
O)A =(1
o -In
O) .
0 -In (det ~)2
=
1 . The subgroup with
are invariant under a11 {<x,x:;,
det ~ ~ 1
SO(n,I). and its interior {<'x,x> >O}
Geometricg.lly, the lightcone {<x,x>=O}
boloid
Recall that the group
and leaves the quadratic and bilinear forms
invariant. In ma.trix language
is denoted by
~ loglx,y,t.ni.
n 1 R +
<x,y> = xoYo
(11)
• Thus the n.e.
=
I}
A E O(n,l)
_ The same is true of the hyper-
with two mantles, one with
x o >0
and one with
x <0 •
o
Theorem 1.- The groups
n M(R - l )
and
SO'(n, I}
Proof: The upper hyperboloid
u = {x and the plane
s~ction
g
. 1I Rn+11 <x,x>=
X
o
> 0)
are isomorphic ..
38
V = { X'ER n+l1
<XI,X
I
> >0,
Xl
o
=
II
are in bijective correspondence through the central projection Xl
= x/x. o
line through
x
and
and
x',y'
tions
x,y
shows
Fig.40
y
~I,n'
E U
and the intersections
of the
with the lightcone together with the projec-
•
By elementary geometry the cross-ratios Ix,y,~/nl and Ix',yt,~I,ntl
are equal. To determine their common value we observe that are both of the form
x+ty 1+ t This leads to the equation (12)
with real
t
<x,x>+2t<x,y>+t 2
=
which consequently has two real roots
V
0
can be identified with the Klein model of
Bn.The
= ~Ilog
t1/t21.
d(x',y'}
By conjugation with the central projection each
of
(12)
~,n.
t
straight line segments are the geodesics and
a bijective mapping
AI
of
V
A
E
SO(n,1)
deter.rnines
on itself. Because the coefficients
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
is a
At
n.c. motion, and it is obviously sense-preserving. Conversely, if mapping
n
<x+ty,x+ty> =0.
1 ,t 2 corresponding to tl/t2 which is positive.
The cross-ratio turns out to be The section
satisfying
and
~
A
of
collinear points
U
A'
is a n.e. motion it can be lifted to a
which carries collinear points (AX/Ay,A~,An)
of this the coefficients of
(12)
(x,y,s,n)
into
with the same cross-ratio. Because will remain proportional to each
39
other, and since a~
~
<x,x >
=1
it follows that =<x,y>
well.
Fig.I0
40
The mapping COne by defining subsequently
Ax
A
can be extended to all
Ax = <XtX>%
=
-A (-x)
extension still satisfies
A«X,X>-~ x)
A
= <x,y>
= nAx
A{ax)
inside the double
in the upper cone and
in the lower cone. One verifies that this
It follows that
x
and
=0
as well as
•
A (x+y) = Ax + Ay • In other words,
is linear, at least inside the double cone. Because every
1 X E Rn +
is a linear combination of vectors in the cone it is clear that A n extends to a linear mapping of R +1 on itself, and becauSe = <x,y>
A E DCntl) ; since the upper cone
it is represented by a matrix
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
M eE) n
is iso:norphic to
is isomorphic to
SO(n,l). In (3.:5)
( n-1,) ran'd t h·1S completes the M.R
proof of Theorem 1. The passage from
M(Rn - l )
to
O(n,l)
means a l,QSS of two
dimensions. For instance, the classical Mobius group Rented by the
Lorent~
group of
4x4
matrices. For
M(C) is repre-
practi~al purpose~
this is much too complicated, but for theoretical reasons it is imporn tant to know that every M(R ) can be written as a matrix group.
41
IV.
4.1.
Elements of differential geometg.
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
- x l... ;:
(1)
X. ( Xl ' ••• l.. _
,xn ) ,
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
. d? =
(2)
L: j
oX..J
oXJ.j
dx
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 a =
aXi
ax:-J
a
j
where we are using the summation convention, as always :from now on. Similarly, a covariant vector is typified by a derivative (or gradient)
42
of'
(4)
'::."'1r"
AX.
:;:::
~
We use lower indices for covariant vectors, and the general rule is OX. b. ~
b.
J
~
uX.
~
The idea carries over to tensors of higher order.
Here are tm rules for
contravariant, covariant and mixed tensors of second order:
;J-j
-i a. J
=
;;;
aX.
hk
--2:.
OX
a
h
OX.l. Ox
h
It makes sense to speak of a
hk
= a
kh
, then
'aij
The Kronecker coordinate systems.
=
a:ji
ax.
o~ o~ l. = 8~ = OX. 3i":" J o~ J J
-
Ox.l-
OX.
and
dX,
J
ij
all
In fact,
because the matrices
k
•
J
8k
a
and. skew-symmetric tensors, for if
8~ is a mixed tensor with the same components in
h
4.2.
~etric
~
OX.1.
are inverses.
An important way to form new tensors is by CO:ltraction.
is a twice covaria.nt and once contravaria.nt te.:lsor, then
a. ;:: a. j•• ~
l.J
(summation! )
Example:
if
is a cO'JaX'lant vector, for -j a ij
4.3.
Ox,k 'Ox. h o~ Oxh h -.-J. == = ~ OX.). o~ ~ Nf" oX.
---
J
l.
A differentiable manifold becomes a Riemannian space by the choice of a
metric tensor 'Whose matrix is positive definite.
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 •
It serves several purposes,
.Arc length is defined by set..tiIlg 1
as
= (g .. l.J
and. dcf'ining the .length of an
t(y)
=
r
~
i
.
~c
'V
ax rucJ)
'2 by
ds
y
is covariant of" ord,er two and the differentials
B~ca.use
variant,
ds
axi
are cont.rR.-
has a meaning independent of the local coordinates.
The determinant of g.. l.J
is denoted by
g.
fi
transforms like a
density in the sense that
R
= (det
This follows by the determinant multiplication rule. volnme by
-rr::
V- tJ...;g
dx1 ••• dxn
This defines an invariant
44
(we consider only sense preserving changes of coordinates.)
4.4.
The inverse of the matrix g..
is denoted by gij
l.J
contravariant tensor of order two.
Multiplication with
by contraction pushes indices up and down.
g. . a
j
gij
followed
For instance
are =
aaf oc
or
l.
The mixed components of gij
The gradient
gij
= a.
l.J
4.5.
It is clearly a
8~ J
of a scalar function is a covariant vector.
For
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 l.
=:
ghk
r.l.J. , k
•
In order to find out how they transform we start froll
and d.ifferentiate using the chain rule
ogab OXa
ag •.
.....:2:J.
::
o~
-OXi
~ c
o~
oXc
OX.
~
J
o~
+
a gab ( Oxidx k
~ ax.
o2;t
+
J
a Oxj~
oX
b ax.- ) , :1
where we have interchanged a, b in the last term. Permutation of the subscripts leads to (8)
r l.J, .. k
=T
oXa ~ oXc ab,c ax. ax:- ax; + J.
a~a gab
J
The extra term on the right shows that
r l.J, .. k
~
Ox Ox i
a~
j
is not a tensor.
Multiply both sides of' (8) by g
hk
o~ ~ = ox axc d
-. -cd g
This gives
d
r:'-. l.J
(9)
= rab
oxa
o~
ax.- rx:-J l.
a~
oXd
~
2-
+
'0 xd
Ox
h ~ OXiOxj d
Again, there is an extra term. We use (9) to find the mixed derivatives
~
multiply (9) by
~
oa-
to obtain
ax.Ox. :1
(11)
For that purpose
~
(10) 4.6.
•
If
yj
J
is a contravariant vector we shall show that
is a mixed tensor.
From
we obtain
Use (10) to eliminate the cross-derivative:
Tte very last term sim:plifies to
side Which
T-a r . v a:L
•
We :pull it over to the left hand
became~
On the rig'ht we are left with
-oJ1
Ox.,lt
Ir!terchange
h
and.
aXk
-
Ox~
a
:L
ax.
-l. +
oxh
~k
oX.
-..J.
OXa
~ --ax.
in tl1e second. term.
and vre have shown tha.t
which is the rule for a mixed tensor.
V
h
J.
It becoltes
The formula for the covariant d.erivative of a covariant vector is s:.milar:
We skip the invariance proof. ~eneral
The
rule is to add one term for each contravariant index and
subtract one term for each covariant index. Example: k
avo "~ -Y + r=-h
k V V." h l.J
~
a ·.v ., _ a. 1J
raih
k ... aj
v
rajh
r
h
ai
.
One verifies t...hat the covariant differentiation of a product f'ollows the usual
rules.
Another practical rule is that the metric tensors For instance
behave like constants.
_ Ogij
V~ gij -
a
~
fk" g . 1.
aJ
ag ..
~ - f1.-- j v~
Q..~,
... rkj .
It is customary to write
4.7. If
~
is a scalar, then V V f k j
is symmetric in
j
== V
and
k
~,
i:Jf
Oxj so that
}l.
= 0
48
For other tensors this is no longer true.
For instance, operating on
a covariant vector one obtains
where R"h
is a tensor, the curvature tensor of the metric k the somewhat complicated formula
~J
given
by
arh;;
h R.~J'k
(J2 )
~+
~
We make no attempt to carry out the computation. h
and
It is
Contraction with respect to
leads to the Ricci curvature
k
(l3 ) and double contraction to the scalar curvature a
(14)
R - R
a
= gab R 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.
or connection, serves this purpose.
The notion of parallel displa.cem.ent,
We shall use only the stmplest connection,
known as the Riemannian connection. Let
tensor
x = x (t )
repres ent an arc
g .. ; we assume that ~J
at x(t) •
y
x(t) ECOl.
in a Riema..nnian space with the metric Let
set)
be a contravariant vector
49 Definition.
We say that
~(t)
remains parallel along y it
~si(t)~(t)
(15)
= 0
for all t .
More
explicit~,
(15) reads
(16) or =
0
•
This is a system of first order linear differential equations. Because the 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 •
r!j
4.9.
The length of a vector
~
(;,~)1/2
, measured in the Riemannian metric, is . . 1/2 = (i~j gij '~CJ)
Similarly, the angle between two vectors
cos
.5
~
on is given by
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.
x' (t)
An arc x(t) ,
.:5 to ' is a geodesic
0':: t
remains parallel to itself.
if the tangent vector
. dx. t;l. =
dt
If we put
in (1.6) we obtain
the condition i
(18)
fkj
x(t)
for
~ d.t
dx.
..-J. = dt
0
to be a geodesic.
It is not only the shape of the arc but also the parametrization that I!1akes an arc geodesic. along a geodesic. mUlt:!.plled by linear:
t =
dt d5 a7
By the theorem the length of
t
If we replace the parameter and
by
x' (t)
remains constant
or, the length would be
would no longer be constant w:less the change were
+ b •
We can regard (18) as a system of differential equations for the functions According to the general theory we can prescribe the initial values
x. (t). 1.
x.(O) 1.
and the initial derivatives x!(O) . ~
In other words, there is a geodesic
tram every point in every airection.
We ma¥ even nor.malize so that the tangent
vector has length one, in which case
t
by
becomes arc length, usually denoted
s.
The existence theorem produces but we can start anew from the point
x(t)
only in some small interval
x(t). o
[0, t ) , o
Thus a geodesic can be continued
forever unless it "tends to the ideal boundary". A geodesic arc is, at least locally, the shortest arc between its end points. 4.11. ball.
This is proved in calculus of variations. We shall now applY these notions of differential geometry to the unit
n B
with the Poincare metric
51
(19) We are in the unusual situation where we can use the same coordinate system n B , namely by identifying each point
in the whole space coordinates
(xl' ••• ' x ) . n
Any diffeomorphism of
n B
JC
E BO with its
would lead to
another coordinate system, but we choose to consider only coordinate changes
of the form (20)
x = yx
where
o£
n y E M(B ) .
This means that we are interested in the co-O.rormal structure
Bn. The metric
(19) corresponds to the metric tensor 48 .. ~J
(21)
8 .. ~J
A little more
general~
we consider an arbitrary conformal metric
and have then g
(in this connection
5 .. ~J
ij
-2 = P 8 .. ~J
is an element of the unit matrix and not a tensor.)
We wish to compute the curvature tensor. notation u
=
log p
,
It is convenient to use the
52
On.e
obtains
+ 5ij(~ - ~~) - ~(uik - u i '\.) 2
+ (8ij \k - 3ik \j)l vu l R .. = 8 •. bou ~J
~J
+ (n - 2) (u •. - u. U. + B. j 1J
1
J
~
For the Poinca;r9 metric
I 12 )
u = log P == log 2 - log (1 _ x u
i
u ..
1J
=
=
2B .. 1J
1-
(xl
28 .• 1.]
1-
(xl
2
2
Ivul 2 )
•
53
Rh. 'k ;; ~J
~ijk
4
2 2 (B.. 5. . - 8. kB .) nk l.J ~ nJ
(1 _ Ix I)
16 = (l-
Ix12)4 (\k5ij - 5 ik \j)
=
~gij - gikgjh
4(n - 1}8.. l.J
j
R = gi R .. = n(n- 1) ~J
The scalar curvature is constant) but
~
= -1.
However, the formula for
~ijk
shows that the sectional curvature is constantly equal to
4.12.
The Beltrami parameter
-1.
There are three important invariant differential operators which generalize the grad.ient, the divergence and tile Laplacian.
a)
The gradient maps functions on vectors. V.f
is a scalar
: :; of
Ox.1.
1.
is a covariant vector.
If f
The corresponding contravariant vector is ii' of gJ Vf = g J _
Oxj
j
The square
le~th can be written either as a i j gij V f V :f
JVfl2 =
(22)
= gij Vir
I: g i,j
ij
v j:f
d.ot product
Vif· V.f :L
:::;
...2!. of ox.J. Oxj
ThiS is the first Beltrami parameter and it used to be called
V:f. 1
It
obviously generalizes the square of the gradient.
*See L.P.
Eisenhart:
1960, p.Bl, (25·9).
Riemannian Geometry, Princeton Uhiversity Press t
or as
b)
The d.ivergence leads from. vectors to scalars.
If
v
is a vector-
valued function we define i
divv = V.v
(23)
~
i
= V v.
~
These expressions are identical for
g
ij
behaves like a constant and we
obtain == 9.g
ij
;;: g
v.
J
1.
iJ'
V.v, ;;: J
~
From j
;;: -av
ox.J..
j
+ fik v
k
we have
(24 ) There is a sim?le formula for
By
the well known rule for dif'f'erentia-
tion, of a determinant.
~
log g
~ij
=
= tr (g-l ~)
=-
g
iJ'
ag ..
-3J.
~
Btlt
'::I.x.. u
f k ,· ~,J
It
+
rkj .
,,~
and hence ij
g
and thus
(25 )
Ogij _
o~
-
g
ij f k1 ,j
+ ij r g
= 2 ri kj,i
ki
55
From (24) and (25) we get divy ::::
(26)
c)
0
1
~
Ji
(jg
~
.
yl. )
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
Poincare metric, then for
f
is d.efined on
For short, a. solution of
harmonic
(h .. h)..
a.ad if we use the
n yEM(B ) ~
~(foy) = (~f)OY
(28)
n B
~f = 0
• n B
on
TNe see from (24) that if
will be called hyperbolically 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. (29)
We compute
~f I'i... f
""'2
for a conformal metric =p -n
0
oXi
(1'..I 11- 2 _C>f' )
oXi
ds::::
pldxl .
One gets simply
2
P ;::
In the special case
D
1 ...
( 30)
~f ~
(1-
Jxlc
~j222
one finds
[ Ar + 2{n -
2~
xi : : 1 ,
l-Ixj WIth the cusLolll.ary I,lOtation
of
x.J- ~ J.. (31)
A.2f' ==
For the half-space
( 32 )
:::;
IxJ::;
r
of
r '-"-
ar
~l- r2}2 [llf + 4
1 P :;:-x n
1.
and
~{n-2)
l ... r
2
r
M] or
57
v. 5.1.
;Hyperharmonic :functions
We shall now study hyperharmonic functions on Bn.
will be to find all solutions
Fo!' a f\mction
u(x)
of
~u = 0
our
first task
that depend. only on
uCr) one obtains x.
oU
= u ' (r) ~ .....
Ox
i
x.x. -¥ r
ull(r)
8;;
x.x.
r
r3
+ u l (r) (~- .2:.....sl)
and thus 00.
t
;;:>
ull(r) + (n -1) u (r) r
According to equation (3 2 ) in 4.l3 (1)
u"(r) + (n-I)
If ut(r}
f
ur~r) +
u(r)
will satisf'y
2(n- 2) rul(l") 2 l ... r
=
0
.
0 this can be written as unCr) u t (r) +
en -1) + 2(n - 2)r = 0 r
1
-r
2
or d dX [ logu'(r) + (n-l) logr - (n- 2) log (1- r 2 )
from which we conclude that r
u l (r)
.0-1 =
const.
This lea:ls to the general solution r
(2 )
u(r) = aJ
2 n-2 ( 1- t ) dt + b t
n-l
] = 0
r:=
Ix] •
We see at once that no solution can stay finite for
o.
r::
As a
normalized solution we introduce
For
n =2
g(r) = O«l_r)n-l)
5.2.
%.
g(r};:: log
Toget.her wiLh
r _1 •
for
g(r)
because by (29) in 4.13
any y
g(r)......
n>2
For
g(
For
11'x\)
n
=3 with
2 n 12 r n-
for T:::'
g(r)=C"r n
l' EM(B)
-
r ~0
and
1 2 1 - ) ::-+r-2.
F.
is again
r
h - h
comnmtes with the Laplace-Beltrami operator.
In
particular
(4) has a. singularity at
pole at
5.3.
and Will be regard.ed as the Green' s function with
y
y.
The fact that a
h -h
constant or has a rather singula.ri ty of a
h - b
function that depends only on
stro~~
r
is either a
singularity suggests that any isolated
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 suitabJ..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.
r - function
Recall Euler's
to
(5 )
r(a):::;
(6)
B(a,b)
a-l
Jo t 1 = J o
They are closely related.
and
t
= sin2~
(8 )
in
-t
dt,
Re a> 0
ta-l(2._t)b-l dt , The substitution
= 2
t:;: x
Re a 2
> 0 ,
Re
b
>
0
gives
2
OCt
rea)
(7)
e
B - function
and.
I
x
2a l - e-x dx
o
B(a,b)
gives
B( a, b) = 2
'JC/2
S
sin2a-1. cp cos2b-l cp d cp
o
~"\rom
(7)
and in polar coordinates
rea) reb) : :;
2
to
4
So
r
= r(a+b) B(a,b) and.
(9)
~j:2
2a+2b - 1 e -r dr J~ 0
thus B(a,b) :;: rCa) reb) r(a+b)
cos
2a-l cp sin2b-l r.pdcp
60
and
In particular, since
r(l):= 1 ,
(10) We shall denote the "surface area·' or (n - 1) - dimensiona.l euclidean n 1 m.easure of 8 by 0,) (some authors prefer c.o 1.). The area of Sn-l(r) n nn-l n 1s then (J)n.itl. r and the volume of B (r) is r
J
n 1
V (r) := co r nOn
dr :=
To compute the area of a spherical cap of radius unit sphere) we project on the equatorial plane
x.
n
=0
cp
(mea.sured alopg the and note tha.t the area.
element on the sphere is dx dO' =
1
... dx _
n 1
x
n
Fig. 11.
The area of the cap is thus
A(cp) -- enn-1
sin cp r n-2
S o
dr
61
With
r ;; sin
e ,
ax
(11)
= cos e de
, x
= cos 8
n
we find
S sinn-2 e de ij)
A(cp) = {Dn_l
o
In particular,
ill
n
1(/2
S n-1 0
= 2A ( !.) 2
n-2 sin
:;; 2m
a de
and thus
w n (D
= B( n;l , ~) =
r
n-1.
(E. ) ~
independent o£
This makes
n and since ill2 = 2n it
follows that (J2)
(l)
n
=
For comparison we shall also compute the area of' a sphere and volume of
Recall that a hyperbolic radius
a ball in hyperbolic space. to the euclidean radius
r = tanh
2s
of a ball with center
s
o.
corresponds Tbe n.e.
area of' the sphere is tben n-1 n-1
ill
2
r
=
n
and the voiume
(Vh
n
.
n-l
S1M
s
is
s
(13)
ill
V. (8) = h
ill
stands for n. e. vOlume).
n
J0
n 1 sinh - t dt
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.
su ~vdx:
(14) Here
D
=r
&!
U
~D~
D is a region in
Rec~ll
S (Vu .. V'v)dx
dO' -
bn
the classical Gree.o, I s formula
D
JRn,
dx = dx
l
...... dx , n
~ 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.
its hY.Perbolic counterpart, using the
We shall repJace everything by
f'ol~ow1ng
notations:
for the hyperbolic
volume element: 2 n- l d g
the ar ea element:
=
(1-
txI2)ll-I = 1-
the normal derivative:
Ixl 2 Q! 2
bn
= 1- t x I2 Vu
the gradient:
2
the 1aplac1an:
~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 2 -u = 2 0 - 2 (1_ 1.A. .... 1 )2-n u
..
u
replaced by
As customary, one can eliminate the ItDirichlet, integral tl
on the right by interchanging
S eu l\v -
(16)
u
and
v
and subtracting.
v~ u) ~
The resulting formula
- v
D
is the one in most common use.
The special case v = 1
r
(17)
.J D
leads to
t\ udx -- J
- bu
~
bD
and it s:3.ows above ali that an
h- h
d~
function has vanishing flux
(18) In euclidean terms (18) reads
r
(19)
"bD If D = B{r)
we obtain
~ n v
d
g~
(1-
Ixl )n-
bu dO' br
=0
2
= 0 •
*)
SS (r) In other words, the mean value m{r) ==
1:..
(On
is constant.
In particular, if
u
JUdo-
s(r) is
h - h
in a full neighborhood of the
origin, then
uCo)
(20)
*)
From now on
n
=
I
will be fixed and we w.rite
B for
n B
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
r
m( Ix!) =
u(kx)d.k
"o(n)
where k
dk
is the Haar' measure of
it follOWS that
m(lxl)
is a
and hence must be a constant,.
O(n) h-b.
Because
u(kx)
is
h- h
for every
f'tulctionwhict depends only on
As a result of the mean-value property
Ixl 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
5.6.
n = 2 •
The question of removable singularities has the :following answer,
Theorem.. D \ ra}
Suppose that
DcB is open
a.nd
a ED.
If u(x)
and. if'
limu{x)
Jx-a
In - 2
::::
0
if
n>2
if
n=2
x-ta,
lim. u(x) x-+a. log
1 1
lx-at
::::
0
,
is
h-h
in
then u
has an h - h
Proof ~
extension to
D.
Tliithout loss of generality we may assume that
p
> o. We choose a fixed y E B(p) , y
V
= g(x,y-) ~u
Since
~v :=
=0
and
a.r:.d apply Lemma 1 to with r
D
= B(p)
,
u:= u(x) ,
< min (iyl
, p - lYl> •
0 we obtain
r
(21)
0
n(p) \ (n(r) U D(y,r»
in the region =
I
a
bg~Y)
(u(x)
- g(x.y)
b~) ) ~ (x) ~
0
•
S(p) +S(r) + S(y,r) We look for the limit as s (y ,r).
for
r .... 0
and start by considering the integral
It is evident that the integral of the second term will tend to
g(x,y) = 0 (---1..-2)
r
n-
contains the factor
-s
(22)
r
(or
n-1
(1-
•
1:r ) )
>.'Ou is bounded and dah un n The integral of the first term can be witten as O(log
Ixl 2 )2-n u(x)
Ix- Yl =r where
di.D
is the element of
We recall that -
"solid
g(x,y):;: g( ~'X
( bgb~~Y)
0,
y
,
I
xl)
= -g < TyXJ )
while
bg(x,y) rn-1dro ~r
angle" (or the measure an and
ITy xl
blTxl
J
=
= Jx -yJ ~
Sn-1).
so that
66
Here
[x,y] -+ \1-
ly(2 as x-+y and
r ~ log
l'ly x I
Or
= r
-2.. ~r
log
It follows a.t once that (22) tends to
ill
n
r ... I . (x,y)
u(y) •
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
I
(23 )
S(r)
is in fact independent of
y.
b~,
and hence defined for all
r
y E B(p J \ (o} .
it is;evidently a h - h
g(x,y) = g(y,x}
Moreover, because
~) d'b (x)
(u(x) ?$(X>Y) - g(x,y) ~
function of
We wish to show that the boundedness condition of Theorem 1 makes it
identically zero.
Let us rewrite (23) as (24)
Observe that n-2
u(x) Ix I
- 0
~
and
g
remain bounded when
r - O.
Hence 'l:ihe condition
makes the limit of the first term zero.
For the second term we write
~(r)
=
r
g(r~,y) bu(r') ameS) br
.. 8(1) ",
and obtain
2r
S
jJJ(t) dt ==
f
.. S(l)
2r
den
JS(I)
r
=
r
~
g(ts,y) n(tg)
f
'r
g(ts,y) bn(t~) dt ~t
I2r - S2ru(tt:) r
r
ag(t~2Y) bt
J
dt dm(g)
67
This is with
o~
1
o( -.n:2)
the order
•
Hence there is a
r and since
~(t):: o( ~l)
lim r n-l..-L (r )
between
t
. t s l.. t mus t
ex~s
r
and
2r
b e zero.
t
We can now conclude from (2l) that u(y) = -
1.
a;-
S
n
The right hand side is a
extension of u
5.7.
h- h
function in
U
S(p)
B,
(25)
If
u
is
t S
u( 0) =
is again
h .. h
in
B and has a. continuous extension
u(x)dro.
n-l. S
b- h
u(VO) = ron
We specialize to
'V == T-
1
Y
uaY
S
for a fixed
y EB .
Because
u(y) and replace
x
by
T x y
u (y) ::
or explicitly (27)
u(y)
1:.. ron
in the integral.
r
.~
S
where
y EM(Bn )
and extends continuously to the boundary.
sn-l u( y x) CIm(x)
1
(26)
•
and. defines the desired
We obtain. a more general formula if we appl¥ (25) to uoy
dO"h
then the mean value property implies
n
Indeed,
}
n
.
Bo".mdary values.
to the closure
:o.~
(u{x) eg(x,y) - g(x,y) ~~
S(p )
This leads to
u (x) IT 1 (x) In-l dill ( x) Y
•
Hence
68
where 'We have used the simplification
the
[x,y}
= Ix - YI
Ix]
when
= 1 •
This is the Poisson formula far the ball.
Note that the kernel is just
(n - l)st
n
power of the Poisson kernel for
2 .
=
5.8. We shall use the notation k(X,y) =
1-
IYI~
Ix-yl with the understanding that
suspect that
k(x,y)
Ixl
= 1
is always a
Iyl
and
< 1.
Formula (27) lets us
function of y.
h - h
The direct verification of this is not difficult, but it is much quicker to pass to the half'- space
Jf-.
Reca1.1 that the laplacian fo,.. the half' space
is
It follows without computation that a A x :: 0:(0: - n + 1) xo. h n n x
ThLlJ::l e ver"y
°.
n
~l
Is all E::dgerl...f'unctlon and
oX.
n
is
We recall (see 3.5) that the canonical mapping
x
n
where ~
n x E H , y E Bn.
for
for using
n
B
and that
=
n B
-+
tf
is such that
k(e , y) n
We conclude that k (e ,y) ex is an eigen:f"unction of n
k( en ' y)
n-l
is
h - h.
x
twiee) there exists a rotation ~ -1 is obvious that k(x,y) = k(f3e ,y) = k(e ,t3y)
n
we conclude that
h ... h •
n
Given any xES such that
px
n-l
= en
(apologies ,
But it
Because of' the invariance
~ k(X,y)O:
(28) and
~ k(x,y)
5.9.
n .. l
= a(o:- n + 1) k(x,y)CX
= 0 •
Suppose now that
f(x)
is of class
J If(x)r da>(x) s
<
L1(S) , i.e.
00
Then we can form
S
u(y) :;; 1
(29)
(1)n
(
k x,y) n-l f(x) dro(x) ,
Sn-l
which is obviously h - h • Theorem. Proof..
u(y)
has rad.ial limits
a.e.
It is .known that
J lim
(30)
S
B(~, ~
ES
= f(~) dm(x)
8)
(see e.g. Rudin, ReaJ. and. Complex AnalysiS, Tbm. 8.8). u(r~)'" f(~)
shall prove that may assume that
f(x)dco(x)
B(s,8)
8-0
for a .. e.
rex)
~
= e
n
for
r ... l
whenever (30) is ful.filled.
We We
so that
r
J_ k(x, re) S n
We introduce the colatitude
~
n-l
defined by
f(x) dm(x) x
n
= cos cp
of the kernel to
2
1- r 2 ) K(r, tp) :;; 1 ( (l)n 1- 2r cos cp + r
n-l
and change the notation
70
Moreover, we
S~1.all
write
r
F(cp) ;;:; x for the integral of'
formula for
.£
f(x)d£n(x)
t:J
n
> cos cp
over the polar -cap of' radius
tp
•
Clearly, the
u(re ) can be w-ri tten in the form n
ra K(r,cp) Ft (cp) dqJ -:rr
u(re ) ;;:; n
(31) Ft (cp)
where
exists a.e. and the measure F t (cp)dtp
is a.bsolutely continuous.
Integration by parts yield.s
We wish to show that
assume that
u(re)
Without loss of' generality we may
fee ) .
n
f(e):;;::: 0 , for otherwise we need only replace
n
It is :tmmediate that any interval
-+
n
8.:s tp
.:s 1C
5
> 0.. Hence u(re) has the same n
8
u{5(r) :; -
Jo F( r.p) fro K(:r, cp)d-C9
•
I
Because of (30) we can choose
8 so small that
= I~
:for
e c.o
n-
1
Jq> sinh - 2 e de 0
follows that
M
bcp
by
f - fCe ) ..
~ tend uniform1¥ to zero for r - 1
K and
,
f
> 0 • Another integration by parts leads to
limit as
n
in
71
lua(r)1 ~ - eA(8) K(5)+' e
:n:
<
€
r
o
a
S KA'(cp}dcp
KA' (tp)dcp = e
J
o
r K(r,cp)dro S
= e
by virtue of the fact that
as a special case of the Poisson formula.
We conclude that
lim u(re ) =
r-l
n
lim U (r) = 0 . o 8-0
5.10.
A little more generally
u(y)
-+
f(e ) n
in any
"Stolz cone It characterized
by
(32 ) Write
yl =
lylen
for short.
Ix . . yl.s
Ix-Y'j +
!x-y" Lr
Ixl = 1
~ (32)
Iy-enl
+
Iyt-enl
<
+ (M+l)(l-Iyl)
this gives
!x-yl.:s
(M+2)Jx-yl
Similarly,
Hence the ratio limit of
1k~fX'Y~) x~y
u(y) - f(e) n
lies between fixed bounds and we conclude that the is also zero when y - e
n
inside the cone.
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
~ ES
= 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. n M(B )
The Mobius group i-:. is clear that
shoulci be mapped on
x
be transformed by -:.he matrix the line element at
,y' (x) I
divide by
acts in an obvious way on
'V x
y'(x)
If
At the same time
to give the new d.irection
n y E M(B )
g should
yl(X)~
for
, but in order to obtain a W.ll t vector we have to
.
Hence we define the action of y y :
(1)
yx •
Tl (B)..
(x,~)
-+
('IX,
by
X' (x} e; )
Iv' (x) I
There is an obvious invariant volwne element J namely, dm = ~
(2)
where m(g) under
y
am (~) ,
denotes the solid angle.
and
~
~
In fact, the Poincare
metri~
is invariant
und.ergoes a rotation which keeps the spherical measure in-
variant.
6.2. Let
It is of interest to introduce a point-pair invgriant for the action (1).
(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.
tion by A(y,x)
In f"act, multiplica-
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 .
Now let us check the invariance.
,
~
Thus
Simultaneous application of
on
,
Y (y) il- A(yx,yy) Y (x) ~ ly' (y) I Iy"(x) I
(4)
~
I
But we have proved (see (41) in (2.8)) that
Thus (4) becomes
I
y'ey)
lv' (y) I
(ll-A(x,Y)~)I;::: 11l-A(x,Y)~1
and. the invariance is established.
6.3.
We shall now determine the infinitesimal form of (3).
we want to compute l~+d~ - A(x,x+dx)1g1
In otber words,
74
Here is the computation: X~j
Q(x) .. ::: lJ
2
x.dx. + x.dx. J
~
=
dQ( x ) .. ~J
Ixl
Ixl
2x,x.(xdx)
~
J
2
J
~
2x.x.. (xdx) J K
(6) x.dx, 1.
x.x.(xdx)
J
Ixl2 -
..2;J
where we made use of the summation .convention.
(7)
[(I - 2Q(x) )
x ,
dQ,(x~ij
J
=
1.
Ixl
1.
2
J
6(x , x + dx) = (I - 2Q(x* - x - dx)}(I - 2Q(x)) •
Here Q,
=
(X * -x - dx ) = Q (
Q(x-
Ixl(;.2 J:x;l
1 ..
~ x .. dx)
Ix12'2 dx)
l-lxJ 2
=
Ixl . , 2 d Q(X)
Q(x) -
Ixl
1-
and thus
~(x,x+dx) = (I-2Q(x)
12
+
2!x 2 dQ(x)(I-2Q(x»
1-
= I -
21xl 1-
*)
Note that
(I - 2Q)2 = I
Ixl
2
2 (I - 2Q(x» d Q(x)
JxJ
implies
dQ, (I - 2Q) = -(I - 2Q)dQ
.
75
a.nd by
(7) 2(x.dx. - x.d.x.) ~
(8)
~
J
J
l-Ixl
2
Now we substitute (8) in (5) to obtain [(~
;;;
+ d~ - 6(x, x + dx)~).~ ::::
d.~.
:L
-
(dx~)x.).
- 2
(x~)dx.).
Ixl 2
1-
The in:finitesimal invariant is hence
_2
Id~
(9)
(~dx)x - (~X)dx
1-
Ixl
An invariant Riemannian metric on
(10) 6.4.
ds
2
can now be introduced by
;:::;
We shall now define the geodesic flow which is a one parameter group of
dif'feomorphisms
V t
in the d.irection
geodesic from
(x, S) determines a geodesic ray Fix a real number
~.
to a. point
x
VsVt = Vs +t
which satisfy
Tl (B)
of
Every line element x
T1 (B)
x
i
t.
at distance
Let
from
t
•
wi cb starts
from
x move along the x; distances are
counted. positive in the direction of the ray, negative in the opposite direction.
At the same time we let the vector
tangent vector
~
I
I
a.t
x
; (see Fig. 1.5)
~
slide to the positive
76
Fig. 12
We define
(1J.) Mar eover ~ the
construction is clearly invariant with respect to Mobius transfornations in the sense that (J2)
y E M(B) .
for each
6.5· t~e
The most important property of the
sense that each
V t
V
t
is that they define a flow in
leaves the volume element
dm, defined by (2) ,
We shall prove tbe invariance by introducing an other volume element
invariant.
din on Tl (B) which is at the same time invariant under Mobius transformations and. under
If
the transformaticns dm.
din are both invariant u.nd.er M(B) , then their quotient is
and
an automorphic :function.
element
cii : dm
(x,~)
Vt .
But
M(B)
is transitive in the sense that any line
can be mapped on any other.
is constant.
then true of dIn.
Since
But this means that the ratio
din is invariant lmder every 'V
t
the same is
77
dm
To construct
we introduce a new set of parameters on T1(B) • (x,~)
As we have already pointed put every line element o~iented.
geodesic through it.
terminal point x
and
~
v.
Clear ly ,
This geodesic has an u
and
v
determines an
im tial
point
u
and a
can be calculated explicitly from
(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 the geodesic from
u
to
v
on the geodesic.
be denoted by
use the signed non-euclidean distance
s
a
=
Let the midpoint of
a(u,v).
frbm a
to
x
To locate
x
(see Fig.
16).
we
Fig. 13
It is now clear that there is a bijective correspondence between the pairs (x,~)
and the triples
The action of' (u, v, s + t ) .
This correspondence is in fact a diffeo-
B X SandS X S X m. .
morphi sm between
by
(u,v,s).
V
t
on
(u,v,s)
is Cluite oQvious:
As a result the volume element
&D( u) den ( v) ds and more generally any element of the form
feu, v) is invariant..
am(u) aro(v) ds
(u,v,s)
is replaced
78
We want to find a y
x
moves
to a point
directed distance means that
u and
s
s +
that is !nvariant also against all
dID
at distance
yx
d(y~,a(yuiYv~
s
from
ya
YEM(B).
and thus to the
from the midpoint
a(yu,yv). This
is invariant ll;P to an added term that depends only on
v. Because [YU_yv[2n-2=
Iy'
(u) In-lly'(v) In-llu_vI2n-2 it
-follows that
din = dw{u)dto(v}ds
: 2n-2 I u-vi
is invariant with respect to
y
i
dm
=
C
idOl
• The conclusion is that
(u) dw (v) ds
!!--~~~~--
lu- v f2n-2 with constant c = 1 '.
c • Actually, one shows by suitable specialization that
79 VII.Discrete Subgroups. is of the form
1.1. As seen in 2.6 the most general . 'V EM (If)
(kE SO(n) .aE Bn)
Therefore. MeBD)
can be topologized by 80(n) XB
is discrete it the identity I
A subgroup pc M (BD)
whose intersection With r
kTa I
has a. neighbourhood
r will always
From now on
reduces to !
n
denote a discrete subgroup. We know that any '" E M(B
Because
which fixes the origin is in 80(0)
)
r
is compact the subgroup of
80(0)
of
D
which stabilizes
is a finite subgroup
0
SO(n) The points
yO
Yn,E r
infinite sequence of distinct that
'Yn =J3 uTa
that
13 .... f3 n
implies
Wi th
n and hence
Y. = 'Y
m
n
We canc1udc that
n
n
n
yJ
But then . V. 'Y-1 ~I mo
131'
a
tends to
~a E B
We know
B.Y passing to a subsequence we may assume
except for finitely many pairs m,n . n-l
It is equally true that since
I V- 0 = a
sucb that
P. n E O(n) 'V, _
If not there would exist an
, are isolated.
'V Er
,
S
Va tends to
when
n-l S
'V
when m, l'
....
which
0::
This is a contradiction.
runs through an infinite group
for any a E B
In tact,
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
yE r
such that
1.2.
Two points
nVK r .¢'
K
a
pass to the quotient shall denote it by n =2
and n
=
and
Ya
B Ir _r)
"e r
,are called equ1va1entJ> and one can
by identification of equivalent :points.
we
It is definitely a Hausdorff' space, and for
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
r
.
80 Then r'Y I (a)f = 1
point of Y
lv' (a) I
l-I'Va/
as seen from
1
2
•
In other words, a lies on the isometric sphere K(V)
ala
that
If
a =0
y E soC n)
then
k -dimensional. plane with
(see 2.9), provided
.
and the fixed points of V fill some Hence an arbitrary
0:5k.:5n-2
V fixes some
As is seen by the above equation we have
k-dimensional geodesic subSJ;lace.
I y' (a) I -0 uniformly for all a from
n B
any compact set inside
these subspaces tend to the boundary.
Therefore
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
infini te sequence of
b E:B Vv
er
is called a limit point of a E B such
and a point
r
The set of all limit points of set of accumulation points of ra
r
the. t
Y,'V a"'b
A = A(ro)
is the limit Bet
1s denoted by
if there exists an
The
Clearly,
A(a)
A =UA( a)
The following is true:
Theo1"em. Proof.
A=A(a)
for all
(with a few trivia~ exceptions);
If a,b E B it is trivial that
d( 'Va, 'Yb) = d(a, b)
is fixed.
Consider now the case
a
aEB
In particular
A(a) =A(b)
because the distance
A(a) = A( 0)
We shall assume that
a ES
is not a fixed point for the whole group.
ra ~ (a}
There is then a
, i."e.
b =y a~ a o
'I E r o
1)
We pr'ove first that
interim point of the geodesic every
e E 1\( 0)
FaT this purpose let
A(O) C A(a)
there is thus
(a,b)
.
We know that
c
A( c) = A{ 0)
be an For
_ -___b
81
Fig. 14
a seq:uence Y'J~
0
(a I , b') . but
2) [OY'\) }
a'
a~so
to
and b l
S
belong to A(a)
the
'tv!! at
suchthat
'YV,a.-d
while
Because
lies in If
I(y ) v a~ e' then
d=eEA(o) A(O)
I,,' (x) I = 1
its image
OEE(y.)
It follows that
-1)
a=e' EA(o)
v
A(a)
The proof fails when
we have
a
and hence
proved that
a
( ) K'Y,
r y' (x)! > 1
and
1 I (''J- )
and 'V'J -10 and I (y\» to e'
e
yaEI(y-l) .... e 'V
\I
then Y'Ja E A(O) dE A(O)
b =Y a
and a sequence
Recall that
lies in tends to
is closed and we find again that
arbitrary point in
dE A(a)
I'VI (x) I < 1
'V'\) 0
for large
aEE(y\) ) But if
Ih'"
Because
and we have :proved that
and. ~v -1O-e l
YvO-e
(see 2.9).
b'
Or
e E A(a)
Thus
it is clear that
at =b'
must tend to the geodesic
For the opposite conclusion choose an arbitrary
are characterized by
But
fbi
If
, and benet! ei the.!· to
E( 'V), I( 'V)
i.e.
and b '
at
a'
On the other hand, if
e=a'
We can pass to a subsequence for which
with Y'J c-e
converge to limits
Y'Jb
and
both
Yv E r
f
b·~ca.use
Since
'Vvt\.(O)
=:
A(O)
was an
d
A(a)cA(O)
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 L)
A
is closed because
A(O)
A
is closed.
open and is called the set of discontinuity.
The complement
0
=:8 \ It is
82 2.)
A is invariant
unde~
r
, and every invariant closed set contains
A
r
3. ) Either A = S or A is nowhere dense on S the' first kind if
A =8
or
, otherwise of the second kind.
4~)
A has no isolated points; it is a perfect
5. )
Let A and
number of
is said to. be
set~
Then A meet s only a finite
B be compact sets in 0
yB,V£r
The proOfs of these statements are all very easy and are left to the reader.
Assume tha.t the identity is the only element of
1.5. words,
0
is not a fixed point.
sphere
K(~)
With correspottding
Then every Y Er\I E(Y)
and
I(V)
r nSO(rt);
has an isometric
In what follows these
notations will refer onlY to the part contained in B a non-euclidean hyperplane, and E{Y)
I (y )
in other
Thus
lC('V)
is
are hyperbolic half-spaces"
We shall consider the set
=n
p
E{'V)
'Y~r\I
and prove that it has the following properties:
(i) ( i1) (iii)
P
is open
bPc u te( 'V ) every
x
(here
tB
bP
and
(ii)
B
).
is equivalent to either a single point in
at least one and at most finitely many (i)
is the boundary in
poin~s
on
P
or to
~p
follow from the fact that every intersection pnB(r)
automatically contained in all but a finite member of the E(Y)
is
and is thus a
tini te intersec"tion.
Recall that
Iy , he) I < 1 and = 1/(1-lxI 2 }.
K(y),E(y)
IY'(~) I
> 1
and
I(y) are characterized by
Iyl (x) 1=
1, 2 respectively, and that [yJ(x) 1/(1-[yxI )
8~
It follows that if and only if
x E E(Y}
is equivalent to
> r xl
jv x , =
while
In other words, a point is in P
x E K(V)
it is strictly closer to
'vxl
0
than all its
images~
if and
and it lies an
Ixl on~
rY if there
are several closest points (including the case of a closest fixed point). the
~oints
if
Since
in an orbit are isolated the existence of at least one and at most
finitelY many c10sest points is obvious, and property (iii) follOWS. the rather remarkable fact that equivalent points on
~p
Observe
are all equidistant
f"rom 0 Property (iii) characterizes
P as a fundamental set.
of half-spaces it is convex in the n.e. sense.
P is a n.e.
As an intersection po~hedron
and
it is referred to as the Poincare'(or Dirichlet) fundamental polyhedron of r with respect to the origin. intersections
?)p nK('V )
The faces of P are the The faces on K(y )
(n-I)-dimensional
and K(Y
-1
)
are equivalent
and congruent in the euclidean and non-euclidean sense.
7.6.
Convergence and divergence.
points in an orbit tend to
We are interested to know how fast the
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
in the sense that the ratios
lie between finite limdts. In fact, from
d(Ya,Vb) ==d(a,b}
it follows that
and
rb
are comparable
8L
or log ,1 +
iybl
l_jybj
< log .!..t IvaI = l~-Iyal
+ d(a"b)
1 + Iyb(
l-/Yb/ from which we deduce that
for all 'V
A good
w~
to study the density of an orbit is to investigate the divergence
or convergence of series of the
~ (1- ·';lal
yEr
for different
po~ers
~orm
fi a
We :prove first:
Every discrete group
Lemma 1. (1)
~
(1-
yEr
r
satisfies
-
IvaI )a
for all CX>n-l .. Proof. It is sufficient to consider the case denotes the Poincars polyhedron of
Because a>n-l
J (1_ fxJ2) a--n dx = c <
ex)
B
Since
B = U VP
up to a
null-set
r
a =0
.
As before,
P
r (1_lxI2)a:-nclx=;~ r (1_lyxI2)Ct-nr~1(x)lndx
e.=L
y;
'VEryp As usual we write
. -1 'f 0
=a
..
.
.
Then "!- _ [a[2
[x,a]2
and
This leads to C:;;I:
r
.j
OVEr P
Here
[x,a]~2 as seen from [x,a]2= 1+ /xl 2 laf 2 -2xa::; 4
Therefore we obtain 2
(2)
L (1-
oyer
Ia/ )0<: =
2a
c J( 1- [xl 2 )a-n
<
2
00
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,
r·
is said
and
to be of convergence type or divergence type.
accordi~ly
the group
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 Let
a
'VB 0
are disjoint.
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
from 0
1
(log 1 + , a' +
2"
1-
'al -
=
P
1 :!."
I a/ th ~
By use of the$e expressions we see that the euclidean radius is
I af '+ th ~ r=l 2
(4)
Ial :tt~ ~
(
1+
la./ t h ~
(1 - raJ2)th ~ =
1-
and that the center
'al 2 c
22,. th 2
is given by
87
Ial
As
-1 we see from (h.) that
(1- I a[)
constant times
times
r
behaves asymptotically like a
and the surface area of B
o
behaves like a constant
Ial )n-l
(1-
We conclude:
r is
of convergence type if and only if the sum of the surface areas
y Bo
of the
is finite.
y B:) one can also look at the
Instead of looking at the areas of the
areas of their central projections or IIshadows" on
7.8.
Recall that
empty.
is said to be of the secon:i kind if
Every group of the second kind is of convergence type. There exists a spherical cap C
.L
..r f
\{
C
y' (x) In-1dru(x)
Wi th the usual notation
I'V
I
(x)[ = 1 -
'V
Ial
[jr,a]
Because C
is compact
It follows that
< co
°a
-1
aen
with
C meets only a finite number of 'VG (6)
S \ A is not
We prove now:
Lemma 2. Proof.
r
S(l)
=
we know that
2
>
2
From (6) and (7 } we conclude at once that
L: (1- Ia) I
n-l
<
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
projection of a function on
B
The Green's function on
m(~)
'fII.r)· -can
be viewed as the
which is automorphic with respect to
with pole at the projection of x
to be the projection of a function
g
r
(x)
o
r .~
is defined
with the following prcrperties:
88
is
~
2°
xEB\rxo
h-h for
~(yx) =~(x)
is automorphic:
3° lim (gr(x) - g(x,xo })
=
exists
X"" X
o ~ is the smallest positive function with these properties.
4°
For simplicity we specialize
m(r)
which corresponds to
r
Theorem 1.
~
to the case
x =0
The function on
o
will be denoted by ~
m{r)
is of convergence type if and on~ if
has a Green's
function. Proof: 1)
We assume that r: ( 1- I al )n-l <
where, as before,
a
= 'V
_lr 0
()O
We show that
0
converges and possesses the properties 1
-
4°.
The convergence follows from
and
It is obvious that h
~
0 0 0
has the ~roperties 1
be any function with the properties 1 0
cf the series g r :principle for 2)
h-h
We have
gr
2
0
To prove 4 let
g~
and let
g~ = 0 at the boundary.
functions g~ ~ h
Assume now that
-
- 3.
exists.
,and hence also We denote by
n(r)
be a partial sum
By the maximum
gr (x) ~h(x) the number of points
a = 'Y 0 v
v
in the ball B(r);
Choose
p
.
\)
.
do not intersect and are
We applY y17) in (5.5) to conclude that
~ ~
S
IaV I =r
so that no B(n ,~)
so ama11 that the balls
contained in B(r} (8)
we choose r
=0
s(:d U Seav ,p )h~ We let
and observe that the integral over
p -0
S
Sea ,p)
v
is the same as
bg(x , e..)
b~
Sea ,p) v
and this in turn has the swne limit as 2
n-2
(\
j Sea ,p)
'"
Here
g(x,a) v
=g(' T a xl)
and
v
bg(x,a\) )
n-2
1
n-l p It follows that (9) tends to ro
n
l.
b8r
S s-
n(l') = - dab = (l)n S(r) ~ -
and we conclude from (8) that 1.
n-1
ID (~)n-2 n ~l-r
S S(r)
ber b~
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 r
0
converges.
n(t)dt
t n- 1
This means that 1
(10)
S(1-t)n-2 n(t)dt
.
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) theorem states that r
which are perhaps no manifolds.
Our
is of divergence type if and only if m(r)co G
Similarly, a Riemann surface is saic. to be of claBs
0HB
if it carries
91
no bounded harmonic functions other than the constants.
of P. J.
~rberg
An important theorem
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:
Theorem 2. If r
is of divergence type there are no non-constant bounded harmonic
functions on ~(r) 7.11.
m. subset of m
Let G be a transformation group acting on a measure space
One says that G acts ergodically on
m
if every invariant
is either a null-set or the complement of a null-set. Theorem 3.
Proof.
If'
r
is of divergence type, then
r
acts ergodicallY on S
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 Er
one finds
ru.f.)
1J ( C. ,2
s
yx - yY
n-l
n 1
X(yx)ryl(x)1 -
dal(x)
92
I 12) = (J)l S 1. - 1 2 n S ( Ix - yl
n-l
where we have used the iq.entities 2 2 = lv' (x)'''v'(y)flx -
l-/'VY12 = IV'Cy)1 (1-ly/2)
The function Theorem 2.
u
is automorphic
:aence the set
~
h-h
Cons1der a ball :Bo =B{ro ) =Bh(P ) o
L(E)
the set of all ,
o
yEr .
are j;nfin1tel¥
, and non-constant.
This contradicts
E cannot exist.
7.12.
'VB
and
yr
lyx -yyl
o
= u(y)
X{x) aro{x)
~ ES
about the, orig1n. ~
such tb,at the radius to
We denote by
meets infinitely manY"
S € L(B)
Another way of saying the saIne thing is that
VO at non-euclidean distance
m.a.n.y points
~
o
if there
from the
radius
(O~~)
'10 =8.
inside the lens-shaped region shown in Fig.15 arbitrarily close to
'V
.
Geometric,aily" this means ,that there are infinitely many
Fig. 15
particular, there are then intini te1¥ many a o:pening
C'.po = 2
~c
tan r 0
•
'J
in -the Stolz cone with the
If this is true for some
said to be a conical 1imit point and the set L =U L(B }
o
is the conical limit set of r
Obviousl¥,
LeA
.
B0
then
!;
is
t::
• In
93
If
~
~oint
is a conical limit
from the radius
(0, 1S)
~
then it is also a
Indeed, if yO is at n.e. distance
conical limit for any orbit ra ~ Po
ro
for the orbit
, then
i Bat distance
Va
-;; Po + d( 0, a)
from
the same -radius. This leads to a.nother characteriZaLion of conical limits.
(o,s)
is a geodesie and it projects to a geodesic on
~r)
The radius
If ';, is a
conical limit this geodesic will came again and again within some fixed distance from any given point.
S
On the .contrary, if'
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.
Proof.
r
If
is of convergence type, then
As usual the orbit
ro
will be written as a sequence {av.J1
convergence there exists,. for every (11)
m(L) = 0
'S
>0 J
a
v
o
Assum.ing
such that
t: (1- Ia I }n-l < e 'V>v0
v
There is then an
a
with
\I> \)0 inside the
'\)
Stolz cone with angle between
g and a
\I
trigonometry to 2 sin
0:'V
2
1-1 ~'V ,.
~o
arbitrari~
The fact that
close to a
'\)
Let
be the angle v is in the Stolz cone translates by ~
Fig. 16
In the limit _
lim ctv
1-
T.he point
a
radius
~
tan CPo
'V
is contained in the gpherical cap ~ith cente~
S is covered by all these caps
This
'V
<
'a ,
area of each cap is asymptotically
~roportional
many L(B o )
7.13.
, and we have proved that
The conical limit set is in
s ELand
"E r
then
y
In (.
is
L) = OJn
F1nal~,
The
< constant times
e
L is a union of countably
m(t) =0 case invariant under
map's a Stolz cone at
slightly laxger Stolz cone at If
any
>V o
'V
o
by (11), and consequently equal to zero.
'V
to a n-l which is majorized
Hence the measure of L(B)
by
~th
and
avl1av'
r
For if
S on a set contained in a
y;
r is of divergence type we conclude that either m ( L) =0 or Thus, for !BZ grQup
We shall eventually prove that
r
onl.:y the extreme cases can occur. m ( L) -= ill
n
for all
r
of divergence type
so that the distinction between convergence and divergence is the same as betw~en
to say
m( L) = 0 and m( L) = (Jjn
However~
it is appropriate at this time
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. that r
A very decisive step was taken by Eberhard Hopf (1936) who proved
acL1:l e.rgodlcally on
i~
Sx S
tbe case of' a compact Riemann surface
with or without ptmctures • The action is the one that takes This is of course very much stronger than ergodicity on
(;tt})
to
(y ~ ~y 11
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 In 1939 he proved the following:
not the right condition.
r
(E. Hop£')
Theorem.
acts ergodic ally on
Sr S
if ann only if
m(l,)
=mn
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
have been given both by Sullivan and Thurston.
meL) = (1)
n
Such proofs
In what follows we shall give
a rather detai1ed version of Thurston's proof. 7·14"
The action of r
on
Sx S
is defined by y( s, 11) = ('VS' "(1)
is said to be dissipative if there exists a measurable An 'Vll == fJ
for all
m(lJ , ..,1 A) = m(S y6"
Briefly,
l'
S)
'V E r \ I
=of-n
A is a measurable fundamental set.
set
~~S
This action xS
such that
)
Lemma 4, Proof..
If meL) = 0
I I
Ir
V-tct\
.>
by
-on
(~'T!) ~ (8: L) x (9 - L)
Suppose tha.t
. 1- a 11m V
r
then the action of
.... 0
- a'V f
I I I" - a:v J
' 1-, a \J l l.m
,
V .... ea
\
Sx S
is dissipative.
diagonal..
Then
.... 0
the definition of L As before
Either
y
-1 0=a
'\)
~ .1
II! - Eov'
2
and hence
v
I:: - il'
I'T\ -
or
av I ;;-
1. 2
'g . 111
This implies
Theref"ore, by (13),
1"10
for Which
I
:: *f; 0
(14)
for all
,
(~i' . r'V" '(11) r -0 and it ,follows that there is VoE r
(~) J .. ('Vo ' (nH
this also means that
o
I
is a maximum.
~.ince
Equivalently t
~
rY"
11 = Y 0
0
r'V
g -' ry
o
0
nJ
is: a maximum.
11 then
I~ o -11'0I ~- r'V~ 0 -''Y1l 0 I 'V E
r
,an4 this is equivalent to
In other words, if
97 For simplicity, let us assume that the origin is not a fixed point.
We claim
that the set
has properties 1) and 2). they cannot both be in
(~,~)
In the first place if
rSo-11
A for that would imply
0
(~o'~o)
and 1
>, g-1l1
are equivalent
and
Secondly, our construction has shown that every
a)
S =11
c)
IVi (;)
J
.
I'V' ('Tl)'
=1
for some
y Er \ I
It is clear that a) and c) are true only on a null-set, and if m(L):=: 0 same is the case of b).
the
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.
Remark.
If'
m(L) = 0
r
, then
is of conyergence type.
This is a consequence of Sullivan's
~heorem.
On the other band,
Theorem L makes Sullivan's theorem a consequence of Hopfts theorem. Proof.
We consider again a ball Bo = B(p,o)
(Thurston).
origin and we denote its images B'
\)
The measure of a set
y-~
Ec S
~
0
with B
v
and the shadow of
will now be denoted by
m(E)
The condition m(L) lim
I
'Vo-ttIJ'
=0
II B f I = 0 ~. \)
\)""'-v
o
centered at the
can be written in the form
IE/
B
V
with
rather than
Indeed, let
denote the characteristic function of
~o
B' V>V v
Then (16)
a
"'neans that
lim Va-IXI
f \, (~ )dro( ~) .:: 0 S
IL I = 0
whi le'
U
0
means that
I
lim
S
'VO-M
X'J (~)~(~),; 0 0
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
<00
1:
,,=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
Fqr this purpose we
B
~hoose
a number
\)
(vk}~
large, and use it to define a subsequence 0
p > 0
J
which will ultimately
b~
as follows:
o =0 5,0 Suppose that ~o, .. "~ have been chosen so that the mutual distances 1
Choose
d(a
~a
d(a
,a
Vi
'\)i
Vj
)
vk + 1
\l
are all > p ) >P
for
Then .choose i
'V
k +1
as the smallest
'V
= 0, ... "k.
other a
v
IS
Fig. 17
such that
99
This choice is always possible and we see that the following is true: a)
d(a, a \/h
>
)
P
for all
h
'-'k
f:
k
,<
r. there exists a vk such that d(a ,a 'J 'it = f' There are only a finite number N of a wi th d(a\l' 0) ~ p and hence also
b)
to every \/
~
a
only N points 1-
with d(a,a
v
v
is. I
'it
) ~
p
When this is the case then
<M
"
=
where M depends only on
p
For simplicity we can return to the notation
a
Step 2.
dQ not
v and it is sufficient to prove (17) for the subsequence.
We choose
p
so large that the B
'V
for
(16) remains true
overlap~
We place an
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
the class of all
consist,s only of B
o
o
B '\)
all
Bv Ell
that are completely visible from
a.nd define
visible, and so on.
We remove
~
as the class of those
Bv
B0 0
and define
Next we remove
which are now completely
Clearly, every B will belong to a class I
v
lIas
m
and the
B" E 1m are disjoint.
shadows of t}:le
It will be shown that
IB~I. ~
t
1m + 1
where n < 1 Step
B·EI 1. m
~.
K
J.
IB~' r
1m.
This will obviously imply (1.7) ~ Every
Let
Bj E 1m + ~ is partially or totally eclipsed by a We shall need an
100
upper bound for the ratio rj/ri
Fig. 18
We refer to the picture in which a i and a j are at n.e· distance < Po from the s~e radius. Let bi,bj be the o.e, orthogonal projections of on the radius. p
We have then
< d(al.' ,a... ) < '"
d(b , bJo) +2po ;: d(O,b ) - d(O,b.) +2p < d(O,a.) - d(O,a.) i j 1. 0 J 1.
or d(O,a ) > d(O,a.) +p - 4p j
1.
0
+lJp
0
101
and 1+
/80.1 J
1+
>
Ia.1 1.
e
p -
4p
0
1-1 a.1
1.'
which implies
By
use of' formula (4) we final4r find
r~
(18)
....Ii.
r
< 4e
_ +4 P
i
Po cos h
2Po
--
2
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 Their shadows Bj from the rim of
by
Bi Elm
P
are disjoint and the,y 2ie asymptotically within distance B!
1.
Their total area is therefore
asymptotical~
at
most
(We have again used the notation A(r) radius
r
).
for the area of a s~herical ca~ with
In view of the estimate we can thus choose p
su:fficiently large m ,the total area o:f the shadows
Bj
so that, for 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
Letnma. 5. -
Y
o
which are totally ec lipsed by a
We need an auxiliary lemma.
B.1. € I m
(ai'S)
Bj E Im + 1
~
If
':.
IE B"i
n Bj
with
I
intersects the ball ~j
is the radius of" B
Proof.
goes to . en geodesic
m , Bj E Im + l. ' then the geodesic with center a. and n. e. radius 20 ,where 1.
J
'0
0
We map the unit ball eonformally on the half-space
and
(O~S)
through at
B. EI
~
to
We keep the names 07*
~
becomes a vertical. line through
th~
en
If1
so that
points
and
a ,a The 1 j (ai'S) a vertical
whose intersection With Jan-l we shall denQ~e by c
o
a Fig. 19
Let bi,b j
be the closest points to
ai,a
j
on the vertical through
0
0
and let
c
j
be the closest point to b j
picture is misleading in two verticals).
a~
much as
a
j
on the vertical through need not be in the
S~e
c
(the
plane
~s
the
The non-euclidean distances are computed by "Jt
2 ~i d(a.. ,b.) -J~ -logcot2 ~ ,J sin (0 ct)i
n =
J ~ '"
CPj
log cot
sin q>
.f
But
cos
so that
to. >cP i 'J
and
It follows that d(aj,C .) < d(aj,bJ +d(b ,c.) < J j J = J =
2p 0
and the lemma is proved. The next picture shows on the left a together with their shadows.
transformed by Vi
B.
~
and
so~e
totally eclipsed B.
On the right the whole configuration has been
J
104
y.B~ 1. J
The image
'V.B~, ~
J
is not the same as
-,
is contained in
(ViBj )
However, the lemma tells .
It is clear that condition (16) remains true when
Therefore, as soon as
t j
1'V. B ~ J 1.
J
:5 ~ I ('Yo; B.) - j
... J
On. the other hand, if
area of the shadow
a
o
In other worJa,
But
p
I
r
r
Fig. 20
'A
o
is Tf?placed by
R0
m is big enough,
< e
()I)
then y.O approaches the boundar,y while the 1.
of Bo viewed trom 'ViO decreases· to a
positiv~
limit
~05
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
The ratio tends to the same
is taken at the center and the minimum on the rim. ri/(l- I air)
limit as
and we know this limit to be finite
There is thus
a constant K such that
~ 1Bj' I < K £
j
We choose
I B~ r
~n
E
1
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 1
our previous result for the partially eclipsed B
j
lBj,
I: BjElm+1
~or
/B'I i
a11
< 2!:
surrlclent~
3
m
Together wi th
we have proved that
1Bir
BiElm
large m
is finite, and hence that
This in turn proves that the sum of all the
~o6
The theorem is
~roved.
10'7 8. 8.1 By
w~
Quasicomformal Deformations.
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. regularity conditions is to require that
One of several ways to impose the right
F is absolutely continuous on lines
This means that the restriction of F to a.e. coordinate parallel
(a.c.l.)
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
a. e.
=:
Di F j
"
II
The Jacobian determinant will be denoted by JF
and
-XF = (JF) n DF 1
(1)
is the normalized Jacobian, while T MF = (XF) .XF
is the symmetrized and normalized Jacobian. F
( 3)
is ca,lled
K- q.c. if
I
, XF 112
It is possible to
=:
~ite ).,
(4)
T
MF = U (
tr MF ~ nK2 .
(XT is the transpose of
X)
108
where
UE SO( n)
and
"1> A2 > ... >).n > 0
Clearl¥, XF =
where
V
v(
.
An
) u
Is another orthogonal matrix, although
determined in the case of equal eigen-values. and that (6)
~ > II XFI12
=
li
This implies upper bounds on ~sed
8.2.
to denote one of these
+ ..
+).~ ~
-1 "1' "n
U and
V are not llDique1.y
Observe that
~lA2'
"A n = 1
n
and
' \ / An
, and sometimes
K
bounds~
Because q.c. mappings in n
dimensions are difficult to handle it is
reasonable to linearize by passing to the infinitesimal case.
Let
be a one-parameter family of a.e. differentiable mappings Which for
bas the development (7)
F(x, t) = x + tr(x) + o( t)
We denote differentiation with respect to
(8)
F(x) = F(x,O)
t
by
a dot and write
=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)
is
F(x,t) t
~O
1.09
Furthermore,
. J(:X) = trDf
.
(10)
1
(XF). =Df -n trDf. I (MF) •.:: Df + Dfl] -
All
thi~
g, trDf . I n"
motivates introducing th.e matrix
or, in terms of indices;
Observe that
then
~ubtra~ting
syumetric
by SMn
nx n
.
Thus
trSf = 0
the
multi~le
Sf
of I
is obta.ined by first syuunetrizing Dr and which makes the trace
0
The space of'
matrices with zero trace will,in these lectures, 'be denoted
We shall use the square n~rm defined by
Definition 1.
f
is called a
k- q.c.
deformation if
f
ISf'll
~
kJO
in () For
we can use the contp'Le'lr notation
n;:::?
f'=u+iv
,
z=x+iy
It
turns out that
(14)
Sf
~I
;Ref'
z
Imf.. Z
and
I lsrlf2
=
2rfJ 2 Z
Imf'".. £
-Ref
-Z
so that
f
is k-q.c.
if and only if
If z-' <• k
a.e .
a .. e.,
110
In view of (14) it is reasonable to regard of the
comple}~
8.3.
We replace it by the following:
valued function
L03
n>2
is called a q.c. deformation Djfi
formed with these derivatives has norm
11Sfli
norm of
n =2
For
f: n _ JR n
integrable distributional derivatives
Sf
if the
and for
A homeomorphism
local~
has
k-q. c.
z
Our definition is still deficient because it does not spell out the
Definition 1. f
as a natural generalization
:r_
derivative
regularity conditions.
if
Sf
a
0 - q. c.
is
~
II sfll E Lt».
ma;pping is conformal (this is Weyl' s lemma)
all conformal mappings are Mabius transformations. n >2
there are only a few deformations with
Lemma lw
n>2
then
a
If
and b
It is
k.rn
expected that for
where
and if the matrix-
Sf = 0
if and only if
are constant vectors and
S
f
It can be
Sf = 0
is of the form
is a constant matrix which is
skew outside and constant on the diagonal. ?
f E C-·
We assume first that
Proof.
For this proof only we denote
components by superscripts and derivatives by subscripts. that fij k"
-q
i' fj =
for
i
J J.
= - ~k = ~i = -f~.. i = -~i k j J~ If'
j
f
k
we can find
i 1'i
~-;
= fj
If
and hence
fJ~k = 0
and
h
+j ,k
i
and obta.in
~kjj = -~kk = -~kk = f~hh =~hh = -i!jj = -~ii = 0 both for
j =i
and
Consequently
, j
, k
The hypothesis means are all d.ifferent
i h f'i;jlr = f"""hjk. = 0
Similar~,
Also, foj
- 0
iii -
j! i
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
bas merely distributional derivatives one uses the ,trick of convolving
f
with a radial function we conclude that
8(1'* 6 ) = Sf* 8 = a 'e s has the form (16) and in the limit for € .... o f
8
with sUPI>ort in
e
f* 6
e
Since
B(e;)
has the sa.me form.
8.lt. define
Let
S*
~
n ~ be a differentiable function with values in SM
We
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 abbreviations of Sf.
~
has compact support.
The dot products are
Because of (17) we regard S* a.s
'~i.
J
~J
the adjoint of S
or
f
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'
s Lawl :for the- lnde pendeut va.!" iab Ie.
v = Cf12 If we
z = x + iy =Jl't + iX 2 identify the vectoL' S *cp -wi'Lh and we let
112
(s *CP)l + i(3*cp) 2
we
find that
(18) Together with
Sf = f_
it follows that
z In the general case
S*Sf =
1:P- ar
Operators of this type are familiar from the theory of elasticity-
8.5.
We start with solutions of s*y:::: 0 turns out that with the
*
It is natural to look for fundamental solutions of
~here
are
n
S Sf = 0
'V shall have values in
where
linearly independent solutions
weakest possible singularity at 0
y
k
'<-
mf!
It
k=1 ,
, n ,
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. integrable, but the derivatives
The singularity is weak, enough to make
are not.
For
n=2
reS];)ecti vely. find that
k
Vij
one finds that
yL
and
y2
k
liz
and
ilz
The connection with the Cauchy kernel is obvious and W'e shall does indeed play very much the same role as the Cauchy kernel.
8.6. We shall now solve the equation (21)
are represented by
k
Sg = 'V
113
k
* k
S Sg = 0
which would. imply
It is a good guess that the components of g
.
should. be of the form k
g. =a(r)5·
where
t =
+b(r)X.x..
1 k 1K
1
brl
,
as usual.
Elementary computation gives (Sllij = ~
(a~(r)
- 1::
+b(r») ( 6ikXj + 6 jkX t ) + b' ~r)
~\j~(al(r) +b(r)
n
"iXj"k.
+rb'(r))
r
Comparisor. with (20) gives the conditions
1: (_ 2
a' (r) r
l
b (r) r
-1 n
+ ber )=-1.... n r
n-2
=
r
n+2
(al(r) +b(r) +rb'(r»
r
=-1 r
n
The last is a consequence of the first two and one finds
~(x) = -[n-2
g~ if
n>2
8.7.
n n-2)
and g~ ==
n-2
r
n-2
(log r) BUt
n
for
n=2
The f'ollowing problem arises naturally:
ProblePJ,.
If one knows
Sf
must the matrix-valued function
,how can one find
:r
~
Sf satisfY?
In other words, when does the inhomogeneous equation (24)
Sf' == "
and what condi tiona
114
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 For simplicity we shall treat only
Lm(~)
and bas compact support.
fo~ula.
the case where
v is of class
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.
cnrk(Y)
+
Every quasiconformal deformation = -
:f
satisfies
r
Sf(x) yyk(x:..y)dx Bty,r)
S [(I + (n - 2)Q,(x - y) )f(x) ]kdw(x)
S(y,r) with
If f
c = 2(n - I)m n n n
cnfk(Y) = -
has compact support the formula reduces to
S Sf(x) .yk(x - y)dm(x)
n
R
Proof.
*) If f
It is sufficier:.t to consider
y=O
..
Integration by parts*)givea
has only distributional derivatives one has to consider integrals of
the form
where
A has a graph like
o
o
.U5
S
yi
Di fjY~j dx j dx = j B(r)-B(r~)
Sfij
B(r)-B(ro )
r
I
r
o
r
xi
f
str)
=
r
j
r
In the limit for
o
~
0
tends to
the integral over
r
f ( 0) + (n - ~)fi( 0) x,x dill n k Stl)l. k
ill
o~viously,
ron n
so that the limit adds up to 2(n-l) ron fk(O) Because
(fx)x
2
r
= Q(x)f
n
the lemma follows.
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
to be of class
Lemma 2.
L
1
rkv
I \) of
\.l
with compact support.
has the distributional derivatives
her.p.-
\J
is supposed
lli5.
:::: 4CJ.)n
b
and
n
n+2
Before proving this lemma we write down the explicit expression for
%y~. (x)
=
~J
(27)
- n
eik. 8jh +
"ikXjxh +
8jk 5ih ...
Ixl n-
_a1j 5hk
8jkXi~ _ 5ij~Xn
Ixln+2 +(n-2)
6ihXj~ + 8jhXi~ +
\\txixj
Ixln+2 _ (n2 .4)
XiXj~~
Ixl n+4
We claim that this is a Cald,eron-Zygmund kernel. bomogeneous of degree
SDhY~j(x)
(28)
-n
Since it is obviously
we need only check. that
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). Wi thout a:lY computation at all we can a.lso reason as follows:
formula
By Stokes'
Here the integrals over S(r].) is homogeneous of order 0
and s(r } are equal simply because the integrand 2 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 a. e. and represents a function in LP for any p > 1
Proof of the Lemma.
r:
\I(Y) :::
f
Ix-y!
We assume first that
Vij(x)
, eo
'V
EC
and
~(x-y)dx
>p
S Vij(X+Y) Y~j(X)dx
=
Ixl >P for the translated Dhik(y) p
For
p - 0
~otential.
=S
DhV
'1 X >p
ij
It follows at once that
(x+y)
l'~.(x)d.y. = l.J
the surface integral tends to
The evaluation of this integral requires a formula for
Jxi Xj~Xk S(l)
dm(x)
write
(26) exists
118 The easiest way to get this formula is to substitute (27) in
S XiXjXh~ dill =
(28).
One obtains
ill
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: derivative
DbIk
,a.nd the lemma. is proved for all
To prove it for general \}
~ e:
CCIO
is the distributional •
we pass a.gain to a convolution
'V
'*5e
where
J 5e: dx = 1
•
From what we have proved
The pr. v. is a bounded 1inear opera tor on LP
"
~>1
oo
8 EC e
is supported in
B(e)
band side of(30)tends in side has a limit in LP
LP
and
.
Therefore the right
to the right band side of (26).
Hence the lett hand
and that limit is the distributional derivative.
The lemma is p:roved.
8.9. I
k
'V
•
We regard
Iv as a vector-valued function with the components
It makes sense to form
S (I \2)hlt and by Lemma 2 we find a.t once!
where
'V~j
in the sense of
sta.nd.ing for a vector with the components
k ij
V
The explicit expre,ssion is n+2
S'ij,hk(x)
+ n(5ij x h%1t + \.It Xi!jJ
Observe that
r!r
n+2 ...
(6ikXj~,+ 8ihXj~ + °jhXiXk + BjkXi~) I!I n+2
It is also a C. - Z
S'Vij , hk = Syhk, ij
Suppos e that we apply (31) to SIv = -c Sf = -c" n n
8ij ohk} I~
= (oik8 jb +8 JkBib - 2
'V :::
Sf
.
Becaus e
• kernel. ISf = -c f n
we have
and thus
with
a
n
=c
Conversely, if
n ~
-b = 2(n+l)(n-2) -po
CI)
n(n+2)
(with cQmpaet
sup.p~rt)
n
satisfies (32) then (31) implies
120
so that
is satisfied by
Sf = \i
Theorem. 2.
If
~
1
Iv .
We have proved:
n
\) € Let:> has compact support a necessary and sufficient
condition that the equation is that
-r = - ·c-
Sf:= 'J
have a solution
with compact support
f
satisfies (32).
As for the sufficiency we are not claiming that Sf" =0
support, but since
="
= --1-
I~
bas compact
c in a neighbourhood of infinity
form (15) and by subtractir.g that trivial solution of solution of' Sf
f
f1
Sf = 0
is trivial of the
we obtain a
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
Proof.
From Theorem 1 we obtain an obvious
We may choose
y' = 0
satisfies a condition of the
ro~
estimate of the form
+ C{R) Repla.ce
of degree
x
Iyl by
171 x
I-n we obtain
in the integral.
Because
'Vk(x)
is hcmogeneous
If
B(2)
r
yl' >R
the integral on the right is smaller than the same integral over
which is a finite constant by rotational symmetry.
Ixl ~ ~I ~ f Ivk(x -rtF) _...,.k(x) II = o(fxf- n )
also estimate the integral over
It is readily seen that
2
Iyl < R
If
we must
~
the integral is bounded by a constant times
l~
as
Therefore
x'" QO
and we conclude that
(33) is true for a suitable A(R)
We shall now prove that every q.c. deformation
8.11.
r(x)
support gives rise to a one'·para;nleter group of conformal mappings such that FsOFt = Fs+t - and Theorem 3. If
f'
is a
F{~, 0) = f(x.) k - q.,c.
with compact F (x) = F(x~ t) t
. .,More precisely:
deformation then
F t
is a
ekit! -g.c.
mapping. (~his t..heorem
Proof.
was firsi. proved
Assume first that
f'
by M. Reimann).
is
co
C
For fixed x we consider the
differential equation (3~)
.
F(X, t) = f(F(x~ t) }
with initial condition
F(x~O)
and the uniqueness fo~lows because Condi tionll •
Moreover,
The eJl"ist'ence of a solution is classica~
=X
r
satisfies (33) wnich is an "Osgood
Actually, the solution wi~l e""ist for all
f
because
f
is bOl,lllded.
1.22
F(X,s +t) =F(F(x,s) "t)
for both sides are solutions of means that
FtoF s = Fs+t
continuous iD
x
.
(3~)
with initial value
In particular,
FtoF:-t
=x
it is a homeomorphism.
Differentiation of (3'+) with respect to
x yields
.
(DF) (x,t)=(DfoF)DF(x,t)
Also
· =tr(Df'oF) (log JF)· = tr(DF)-1 (DF) -1 We o.ppq this to
XF = (JF) n: DF
to obtain
. 1 (XF) = ( - - tr(DfoF) +DfoF)XF n
and T
•
(['
T
2 n
[eXF) (D)] = (XFT[DfoF +Df of-- tr(DfoF) 1XF from which it follows that
By the Cauchy-Schwarz inequality
Together with
II sf'.1
< k we now obtain
F(x,s) and since
.
This F(x,t)
is
1?3
and by integration
where we have used the initia1 condition 11 XF(x,O) I r
If
and
is not differentiable we form the convolution
f
use it to generate Fe (x, t)
=.jn
is
Fe
K - q.c.
We write the
lls~el
Because
I
~ k
it follows as before that
with K = ekl tl
diffe~ential
equation for
F
E!
in integrated form
t
:re (x,t)=x+S o
Because the
Fe
compact set.
f
e
(Fe (x,t))dt
are K - q. c.
with a fixed It they are equicontinuous on every
Hence one can choose a sequence
converges to a limit
F (x,t) o
e(N) - 0
such that
Fe(N)
which satisfies
t
F (x, t) o
=x + S r(F 0 ex, t) )dt
This means that
o
F (x,t) o
is the unique solution of (3 4 ) and hence equal to
Since it is a homeomorphism. and limit of K - q.c.
it is itself K - q.c. 8.12. When F
The quantity
mappings with
and the theorem is proved.
MF undergoes a very simple and predictable change
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
capital letters
~
in an important role we shall now use
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
)
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:
r(x) expressed in
125
Lemma. 4.
We remark first that differentiation of the identi ty X-~ = I
Proof. yie'l.d:s
and hence
D .X-1 = - X-l( D.X )-1 X # J
J
Therefore, differentiation of'
leads to (DrA) ij = - ((DA) - \ DA] ik(foA)k
(38)
+( (DA) -l(DfoA)DA)ij
In the first term on the right -2
where we have used the identity 2
(39)
(DA)TDA = (JA)ii I
.
]26
Differentiation of (39) gives
g
g
(DkDA)TDA + (DA) TDkDA =~ (JA)n (nit log JA)r =~ (JA)n =
~
-1
(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 is a conformal matrix. with values in
Note that
8.13.
VA
More generally, we shall say that a function
s~f transformB like a mixed tensor
has again values in
vex)
if
mf1 .
In order to continue this discussion of the invariance properties
we want to take a good look at the relation (41)
DA
SSf'. cp dx
=-
Jf. S*cp dx
We would like these inner products to be invariant when
f
is replaced
127
by
fA
and
~
transformed by
by
$A - aut this is not so if
(40)
and the integral in
Sf
and
~
are both
_ Indeed, we would have
(41)
would not
~tay
invariant. TO remedy the
situation we regard the integral as an inner product between the function
Sf
~dx,
and the measure
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 regard
S* ~
(41)
makes sense only if we
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·
dx ;;; -
J
(SfoA) (<poA) 1A
I
(x) t n dx =
128
= -
J At (x) -l(SfoA)A' (x)
• tfIA (x)dx
The comparison shows that
Here A1 (x)T == /A'(x)/2 A,(x)-1
and if we were to regard
S~ as a. contravariant
vector, then the transformation rule would read (43)
S *(~A) = ,At (x)
In +2 (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(f ) ;:: (Sf) A A
S {CPA' =
Sf'
,A' (x) In+2 (8' 'cp) A
is a rUxed tensor, "S*~ is a covariant vector density.
Sv or
S*"
8.14.
The fact that
Sf
densities makes the operator
is not a density while
* 8-)(8
S* 'V is defined only for
meaningless except in the euc/lidean case.
To rescue the situation we need an invariant density. such a density is given by the PoincarE! metric. with
We do not define
In hyperbolic space
We use the notation
ds
=
p' dxl
130
2
the density itself being given by
n
p
Now we can pass from tensors and vectors to densities simply multiplication by S*"n ~ Sf
For instance
pnSf
is a coyar i an t vector density, an d
is a mixed tensor density~
p -n-2 S *PnSf
is a contravarian t
To account for these changes it is convenient to define two new operators)
vector. namelY
pn
through
p
=p nSand
P* :;: p -n-2P
They satisfy the relations
and thus also
The basic invariant second order differential operator is this connection we shall say that a vector-valued function
P*Pf=O
(44) or
f
P*P
and in
is harmonic if
It is convenient to split this into two pieces, namely c:p=Ff, P
S *tp = 0
For
*~=O n =2
it turns out that
cp is a holomorphic quadratic
differential. How does ~
*
F P
compare with the Laplace-.Beltrami
applies to scalars and P*P
to vectors.
~.
In the first place
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 whereas
in our termdnology
A differential is harmonic if (do + od)a= a
f f
is harmonic if
1 ) 1 (n - 1 d 60: - -2 8~ - ID = 0 Here
i ID=R. f.dx 04 J
3.
c..
R~ is the Ricci curvature tensor.
where
We sball now study the theory of functions
8.15.
in some detail. LeD1tJ19. 5.
a)
If
S SfijX j
P '*P.r = 0
then
dm(x) = 0
B(r)
r Sfl.J.• x. Xjdlc(x) = 0
..
3.
S(r) c)
r Sf .. X.X.x. d:'Jl{x) =0 l.J 3. J K S(r) J
Proof.
By
Green's formula
and this im,plies (45a).
Similarly~
f
that satisfy P*'Pf = 0
S(r
:x r nSf' x ~ JP i'J" '1 r =
J'
SDj (n P Sf".x. , dx = J pnSf .. 0 .. dY. = 0 ~J 1.
B(r)
I
l.J l.J
B(r)
and
r. nSf'1j JP
xxx. i;i a: r
A_ '-q
r DJ. ( P
I1-.p
=~
~.LijXi '~
) dx
.
B(r) =
J pnSfij(8ijXk +\:jXi)dx
B(r) =
S p~fikxi
dx
=C
by (45a)
B(r) Cor.
:[=*Pf = 0
S Sf •• 1.;]
implies
Y~j (x) 1-
dO' (x) = 0
S(r) In fa::t,
2Sf.k" Sf .k (x) = . J iJ :Vij r n .
x + (n-2) j
r
n+2
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) n Btl')
(46)
Recall that If
c
n
2(n-1) n
c :reO) = n
r
J
S
$
r < 1
.
n
limi ts a. e", the formula ramains true for (46' )
0
ill
has a continuous extension to
f
,
(I + (n-2)Q(x) ):r(x)dm(x)
S(l) r
=1:
,or even if it has radial
-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 (~6 ft )
:r(o)
Sf(x)dm
=2c
n
8.16.
.
S
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
r (I + (n...2)Q,(Ty x)TY• (x)f(x) ITY
cnr(y) = (1- ry( 2)
f
(x)J n-ldm(x)
t,I
S
Recall that
Ty I ex)
'T '(x) Y or
1. -
ly(2
Ix-yJ2
=I. Ty '(x)j
A(x,y)
and
I = 1- Iyf 2 [x,y]2
because
ix' =1
.
We have also shown (see
2.8~
(40») that
lah Ty x =A(X,y)x when
Ixl =1
.
Thus
or
Q(T x) ~(x,y) = ~(x,y)Q(x) y
Taking these simplir1cat1ons into account we end up with Theorem
4..
c fey)
=j
n
S
I [2 n+l (1-;,) A(x,y) (I + (n-2)Q(x) )f(x)dc.o(x)
1x-yI-n
There is again a simplification if .6(x,y) = (I - 2Q(x,y» (I - 2Q,(x»)
Moreover,
is tangential, for then Q(x)f(x) = 0
f
•
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
8.17.
Ix -y I
Evidently we can get a corresponding formula for
erentiating(47) or (l7').
For arbitrary
y
12
( l -Jy ) n+1
2n
J
and
x-y1
f'>J
,1 0'" 1+2n(xy) ( 1 - 2xy )
by diff-
the campu~ation becomes almost
prohibitive, but we can use the device of ccmputing the Ignoring quadratic and higher terms in y
Sf(y)
we have
d~rivRtives
only at
y=O
.
135
Q(X-Y\j
= (Xt -'1i )(:fY}
'" (xi"'j - .xil'j - XjYi){l +2xyl
\](" -yl rv
xixj +2xi Xj(XY):'XiYj-XjYi
so that (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 Xj f j
=0
tan,gentia~
on the boundary the result reduces cnDkfi (0) =
S (2nfi~ +
ca.&;le.
By virtue of
~o
2f x i )din(x) k
S Symmetriz~tion
leads to
n (Dk f i (O)"+Di f'k(O»)=2(n+l)
C
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
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 S
and
j
(fi%k +f'kxi)Cko=
B
(Dkf'i +D :f'k)dx 1
~
B imi laxly
0==
!
fhXhda> =
S
I J)hfh dx 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.
(49)
to
iT -1
Sf(X}
~
before, we applY
is then replaced by
if
(T -1) '(x)-lSf(T- 1 x)(T -1) '(x) Y' y Y For x
=0
factor is
the left and right factors cancel against each other and the middle
Srey)
variab Ie by T x y
In the integral on the right we replace the integration If we observe that
the integral
becomes
S'fy' (x}S:f(X)l'y' (x) -1
.
B
The scalar factors in (50)
Ty'(x)
cngrey) ==2{n+l)
and
T' (x )
-1
y
cancel, and we obtain
r b(Ji:,y)Sf(x)A(y,x)
u
•
B
It is again preferable to
f'OCUB
the attention on
We obtain the reproducing formula for Tkeorem '5.
If'
cp =p nSf
~
in the form:
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.
Izl
<1
~(z)
is complek analYtic ~or
,then
tp{ ,) for a.l1
It
=;
1,1 < 1
(:- i~ I z 12 ) (l-Cz)
SS
2
spe z )
rzl
dxdy
,provided that
S S , cp(~) I (1 -
Jz
r2 ) 2dxdy <~
Izi <1 One checks that
does indeed reduce to
6(Y,X)
1.
1C
when
n =2
Clearly, the matrices
A(x~y)
and
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
J ! 1~(x)11 Cl_lxI2)n
(52)
d.x<,(O
B
8.18.
Because of the invariance properties of P
considerations are easily adaptable to discrete groups.
and P*
all our
be a. discrete subgroup of' M(aD )
r
Let
We introduce the following
definitions: If the vector-valued fUnction
Def.l.
for all
AEr
Def.2.
for a.ll
If
then
f
is said to be au.tomorphic under
A mixed tensor density
A mixed tensor v (x)
AEr f
follow that
is automorphic, then
f
is automorphic under r
(D{x)
if
is a. B ~ltrami differential if
Sf
is aBeltra.mi differential a.nd p nSf (J)~
Conversely, if
Pf
+, In other words,
is trivial, i. e. the components are quadratic polynomial1J. We write t21e period. of
f
under the mapping
The periods satlsf,y the cocycle condition
is an
is automorphic it does not
S(f - t) = a A
is automorphic, but only that
£;. - f =PAf and call PAt'
for
r
.
au;omorphic mixed density.
fA .., f
satisfies
A Er
Det'. 3'.
for all
l
f(x)
A
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 is measurable, if
A mixed tensor
11" ~x) r ,
'Y
(x)
is said to be of class
is (essential1y) boun:led
We can use the same terminology for tensor densities
and vA ~(x)
=\)
Leo(r)
if it
for a.ll "E r
I 12) n
1
when " = cp(x)( - x 2
is in L
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
In particUlar, ~
S "CPJJ
dx <
E
"q:>(x)"
reters to the square norm of the matrix.
Ll(I) ~f
ex)
B
without ary automorphy condition. applies ar..d leads to Theorem 6.
If"
e cp
In this situation the Poincare' e-series
E Ll(r)
q>E LI(I)
then
converges absolute1y a.e. and belongs to Proof.
Let
P
be a fundamental set :for
J II cpr IelY =E SII cpll dx < B
1 L (r)
AEr AP
00
r
By assumption
~40
13ut
p
and CPA {x} = rA· (X), nA t (x) -lcp(Ax) A' (x)
f f q>A (x) , , = 'A' (x) ,n, , tp(~) r ,
From this we conclude that
r
(53)
L;
"qJAfI dx <
00
p
and the convergence of the series follows.
Fina.lly, e cp
The same estimate shows that
is automorphic because
((i)rp)B=L: (CPA)B=~CPAB::::®tp
A
A
It is convenient to use the inner product
8.20.
( cp, ,) =
J cp.
1jr
(1 - f
x, 2 ) ndx
r B/r when
cpEL1Cr)
,
,ELooCr)
the inner Pl":>duct is obvious.
Proof~
(i .. e.
oo \) =.(1- Ixr2)D E L )
I Also, if co EL (I)
,
Let X. be the characteristic function of
•
The invariance of
• E Loo( r)
P
~
then
Then
r:r
Remark.
8.21. and x,
tp
~ EL I
and
nL
Cb
(® qhi.~ ) r ~ (cp, ® ,) r
then
.
We return to Th~orem 4 and 'Poisson's formula (47).
r xl == 1
Let us fix
-' and consider the vector-valued kernel K(x.y)
k
with the
components
There is good reason to believe that K(y)
satisfies
*
P F.K=O
•
Again,
direct computation is almost prohibitive. Let us fIrst drop tte condition
Ixl =1
and !'eplace (54) by
Then (1L~d2)n+1 (l-jx[ 2 )nK.{y)= (1- [xI2)n+1 " ._.
[x,y]2n+2
1.
B.¥ our usual formulas
and Tx' (y) ;:;
Tx '(
1_/xl22 [x,y]
y
)_~_(l-
!x,2) -1
[xtY]
so that
A(Y,x)
2
~,f
u\x,y
)
2 .( (1-l x1 ) "" [x,y] 2
-1 A(x,y) ok 1.
•
142
The right hand side is nothing else than [(1-lyI2)n+lekJT
so that
x Ki(y)
where
ek
=
(1-lxI 2 )-n [(1-ly}2)n+1 ekJT x ~ector
is the unit
with components
0ik.
Almost without computation one obtains
p«1..-ly1 2 )e k ),
= canst.
,
1J
(s
#:
p ( (1- 1 y
#:
(P P (
2 1
(1-1 y I
)
2
ek ) i =
)e k ) i =
(o'ky.+6'kY.J
1.
J
l.
2
1)
0, 'Y ) 1.J k
canst. t>ik
canst. (1-l yI 2)n+2 5
ik
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 tx,y]2n
S
is a harmonic function of
A(x,y)h(x)dx
y
.
8.22. There is a similar fact for Theorem 5 • More expljcitly,
(51) reads
·l4-3
One can show that A(X'Y)ihA(X'Y)jk [x.y]2?l
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 is of the form
.!!!l 'J{x)
'I"hk,
the function
w(y) =
r A(x,Y)\J(x)A(Y,x)dJc 2
.
B
*
satisfies
S L\II(Y) = 0
[x,y] n .
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 is in
vELC»(r)
then the same is true of Lv
so is Lv
LICr)
Proof of 3.
It is clear that
and hence
r
lIILv(y)/ldy ~
P
p
dY
S Ilv(x~ll
dx
.
B [X,lJ
On partitioning B into copies AP one sees that I' .,
p
••
dy
r .. dx= r .. dx r .. dy
tJ
~
oJ
B
p
B
.
.
144 In this
way
But
and we obtain
A weak finiteness theorem.
8.22.
We shall now introduce a rather special class Definition I. II. III.
qJ E Q(r) if
S*cp::: 0 CPA = cp CD
and
cp == Pf
for some
f
wi th
xf ::: 0
on
S
*)
.
.
n LQ)(r)
(In other words,
IV.
the following 113 true:
for all A E r
E L;J,.(r)
Q(r) defined as follows:
Jf
f
B/r
tpl , dx <
0:1
and
q:; has a smooth extension to S no
r f C!>(x) If (1-1 x12)n and this extension
is bounded. )
satisties
More explicitly, cplJ ••x.x. = XiX .CO-4k J It J'o.J on
S\A.
*) It is understood that
f
has a continuous extension to S
.
Condition IV needs quadratic differential.
a
motivation.
For
q> dz
The condition means that When this is so
outside the limit set.
n:=; 2
qJ
2
is a holomorphic
cp dz2
is
real on
can be extended to
1zl
Iz!
>1
=1
by
symmetry.
n the symmetric extension of
For arbitrary
'PJ = C!> where
Jx = x
cp(x*)
Ixl
and for
=
=1
*
~
would be defined by
This condition reads
Ixl2n(I - 2Q(x) hp(x~ (I - 2Q(x) ) 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 Theorem 7space
Q(r)
r
If
is finitely generated, then the dimension of the linear
As already mentioned (see 8.18) if
is a trivial vector function.
quadratic polYnomial in the Suppose that o.f
Q(r)
are not clear.
is finite.
Sketch of proof. fA-f
M(r)
r
Xi
CPA =cp then PAf=
As such each component of PAr is a
determined by a finite number of coefficients.
is generated by Al , ... ,.AN .
There is a linear map
on a l'inite dimensional vector space which takes each cpt:: Q(r)
into
the coefficients of P f"."PA f The kernel. of this homomorphism consists A1 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
For this purpose we study the
~ntegral
t=1U~h
s.
(,fI
must be idfmtjcaJly l7.ero.
~46
which can also be written as
.r Djfi
-.r fiDjqlijdy.+! fi(Oijnjdcr
CPijdX=
P
P
bP
is the outer normal of oP The boundary
oP consists
Plrt of n and points on A . faces.
* .)
of pairwise congruent faces of the polyhedron, Let
A and
A~
be a pair of corresponding
Then
r
.1 AA
foi tn~J.nJ.d a =
. . . -1.
f f.1 (Ax)cplJ •• (Ax)n.J (Ax)' A
•.
I
(x)
In - Ida
A
On the right we substitute
f.(Ax) = (A'(x)f(x»). ::;A'(x). f (x) 1
1
The minus sign is because
at Ax
A maps the outer normal at
x
on the inner normal
.
The integrand
*)
1a a
~ecom~s
lie are temporarily assumming that
Green's formula is applicable.
147
-
f
a
ti) 4> ac (x) n c
(x)
and thus
S'
A6
f.
~
4l • •n . dO' 1J
J
f
=-
A
f. 1
~ . . n . dO' 1)
•
)
The integrals cancel against each other.
Suppose now that
of
A
in
(56)
ap.
belong to
P
is a finite polyhedron and that no points
The remaining part of the boundary integral
is then
S
f . cp •• x. do •
nnP
1
1J
)
Now we make use of Condition IV
~
= 0
2
f.~ .. x.;dw 1. .l..J J
=
f. '. • x . x~ dw .l.. 1.J J K
f.x. 1. ].
=
0
which is zero because and bence
to write
=
by assumption.This proves
($,~)
=
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 00
support on of
ap.
P \
~he
A and satisfies
-
C (P)
for Kleinian groups. Suppose that
A(Ax)
I
= A(X)
A
~
0
I
has compact
at equivalent points
earlier reasoning is sufficient to prove that
148
(57)
= -
(CP , ).
Sf. P
~A
l. aX
•
l.)
j
The theorem will follow if we can find a sequence of
A +1
"
and Let
f.
~
caA vlax.J )+0
boundedly on
P
ponding points
x
and
AX e: ap
equal. To see this observe that
2 2 (1-IAxI )/(1-rxf ) = 1 • Furthermore,
!AX-Ayl =
/x-ylIA' (x)
min (I Ax-Ay I , (Ax-y r) ~
follows that d
~ o(~)
I~IA'
hv(t)
be
0
(y)
[O,l/vJ~
in
(x)
I~=
. At
A
and
corres-
0 (Ax)
IA'{x)1
are
=
y
Ix-y/IA' (y)
r~
IA'(y) I-~ .
rx-y I. J ~
to
and hence
Ix-y/
A
varies over
y
As
and by symmetry
'o(x)-o(y)
is not smooth, but Let
t
o(Ax)
<5
for any
JAX-YI= IAX-AA-lyJ= /x-yl
and hence
x
the distances Jxl=IAX~
V
.
be the euclidean distance from.
<5 (x)
such that
).
o (Ax)
and
linear in
=
o(x}
as asserted.
Jgrad ol~ 1 [1/v,2/v]
it
a.e.
ana
1
for
> 2/v . We shall choose
h [(log log v
Although this function is not smooth it is not hard to see that
(57)
remains in force when
It is evident that
A v
the derivative one finds that
A is replaced by tends boundedly to
(57)
~v
1
as
v-+-co
•
As for
149
lax/ax.1 .v J
<
=
~(x)log
2e
2'Je (
o(x)
2e
2
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
I aA v /8x I
(58)
Let us now extend f
=
J
f
,
2e
< 8e/vo(x)log o(x)
by the symmetry relation
that is to say according to the rule
Because of Condition I
Q(x)f(x)
=
I ISfl I
on
S
IxI 2 f(x*)= (I-2Q(x»f(x). and the extension is
(I-2Q(x»Sf(x) (I-2Q(x»
renains bounded. By Lemma 3 (8.10)
we may therefore
satisfies a near-Lipschitz condition If(x)-f(y) I =
f
conclude that
0
* = Sf(x)
consequently continuous. Moreover,
so that
Rn
to all of
f
•
OClx-yllOglx:yl) . We observe next that for any
A
£
r , f(AO)
2 (1-IAOI ) If(O) x
€
I.
= AI
f
A • In fact,
is identically zero on
(O)f(O) and hence It{AO) «IAI (0)
Choose a sequence of
A • It follows by continuity that
A
such that
AO
I If(O) 1=
tends to
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
f.OA lax.J ~ \.I
tends boundedly to
0
as
(58) \.1700.
was stil+ needed in order to conclude from and Theorem 7 is proved.
this implies that This is precisely what (57)
that
(
,4»
==
0
I
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')
with respect to the
have some rather speci,al p:r;operties, i,or instance ~opology
of
BIT •
