Holomorphic motions and the extension problem

Holomorphic motions and
the extension problem
Hiroshige Shiga
Department of Mathematics
Tokyo Institute of Technology
東京工業大学理学院数学系
1. Introduction
※
郿
_
M
:
ECI
A
0
map
motion
1
)
(
2
)
set
a
:
(
mfd
complex
a
of
4
70.1
M
:
E
E
x
( p。
4
( R
0
(
)
w
,
・
)
i
E
Ah
。
1
0
(
,
Ej
E
→
E
is
4 )
for
:
M
→
d
is
moton
morph c
normalized
)
u
。
、
called
is
for
and
( 3 )
point
a
h 。 10 morphと
if
W
=
cs
,
→
M
over
base
Po
7
0
M
:
に
WE
inject
h
E
x
0
→
1
0
morph
E
年
for
ve
に
for
c
Mj
E
WEE
called
is
if
0
( P
に
PEM
、
0
)
=
0
,
0 (p
,
1
)
=
1
and
0 ( 13
.
i
C
E
で
→
m
0
)
=
g
.
Results
Known
入
・
lemma
一
V
For
I
ヨ
st
h
(
1
。
M
:
0
、
Mani
morph
0
E
x
( P
u
入
Improved
4
M
:
for
⇒
「
0
and
E
x
)
0
:
E
i
h
ヨ
4
1
i
Yp
I
f
is
donor
h
:
E
0 (p
い
a
.
)
:
E
→
E
1
た
⒵
home
adf.is
,
4
"
1 9 8
,
of
6 )
Thurston
Sullivan
E
of
P
phc motion of も
OP
0 p
、
0
(p
(
)
.
,
)
:
も
→
で
conformal
quasi
fz 弐
食
Moreover
E
→
of
E
x
(
neighborhood
a
M
t
( 19 86 )
Royden
-
)
,
三
of
conformal
orphic motion
Ins ion
ex
an
w
,
E
of
quasi
is
over
st
E
→
"
Bars
E
→
E
×
for
)
砦
台
さ
峺
魚
品
m
M
i
19 8 3 )
,
orphic motion
an
E
し
0
1
u
,
→
Up
E
px
。
。
Cp
lemma
-
M
pe
4
E
→
h
=
が
・
E
)
Sullivan
-
motion
c
→
,
Sad
-
is
can
。
MEL
が
・
4
11
、
私
quasi conformal
.
M 11
.
く 1
)
Kowski
Slod
△
4
4
ZE
△
i
0
ヨ
⇒
{
=
9
( p
Groupe
G
0
M
:
4
p
that
G ( E )
た。
orphic
motion
。 m
△
over
E
of
いい」
、
suppose
し
E
E
X
A
over
.
.
ヨ
if
fp
E
=
G
:
、
Mob
→
iso
a
( PE M )
st
fp
of
motion
c
for に
い
,
arrant
equiv
morph
version
E
→
_
h do
( p
motion
orphic
art
and
E
x
G
is
4
quran
,
m
I
=
)
( 19 9 1
.
た。1。
→
)
w
、
Mdba
く
E
E
x
}
く 1
し
→
△
:
st
に
1
E
x
theorem
's
i d
=
。
0
しp
and
.
g ⒵)
,
0
=
9
し
つ
、
) (
01
( し3
2
-
)
for
)
2
-
、
)
t
MXE
Krakmshkd
and
for
.hn
G
出
Earle
Mob
く
4
し
a
△
:
ヨ
)
GE )
,
E
x
I
た
:
over
x
=
E
→
△
Vgt G
E
△
、
E
、
Geqwwi
→
E
Geq
at
hbm
。
5
of E
over
△
www.tholo.motmofd
Question
1
Negative
⇒
Question
a
⇒
2
2 0
0
4
i.e.
.
Do kkdy
Chika
condition
about
a
which is
た。 1 。 morph
NOT
)
"
gives
.
manifold
"
a
necessary
ct on
which
counterexample
月
surface
、
hdomorpL.cm
a
complex
How
、
Riemann
(
about
dimensional
higher
a
How
.
is
simply connected
.
c
moton
simply
and
over
over
connected
sufficient
.
Monodromy of a holomorphic motion
Po
0
X
E
,
0
1
X
:
E
EEC
0
,
Riemann
a
:
x
E
→
surface
closed
a
i
set
En of
holomorph.cm
( normalized )
X
I
。
over
Take
a
a
closed
finite
curve
set
の
ci
す
u
CX
ー
っ
・
.
,
Wn
passing
}
CE
through
and
Po
操
ビビー
三
x
。
.
E
We
if
for
the
兄
that
say
の
any
braid
by
given
p )
ban
and
の
for じん
and
。
.
W
1
2
2
・
毖
に
→
・
・
・
.
・
( Beck
地
K
く
0
:
△
Jiang
{12-1<1}
=
0
K)
-
Mitra
-
・
S
2
.
0
→
:
(
motion
く
of
→
-
△
・
E
K
)
over
a
x
E
( △
-
d
K
-
1。 morph
△
→
AB
and
over
Then
trial
is
12 )
Compact
E
ho
:
E
×
,
・
.
の
}
EE
!!!
・
・
Wn
いう
mon
.
trial
is
月
ど:
W
・
( X
た
E
0
of
monotony
the
K
、
,
a
removable
cm
。
く母
of
E
、
hdom orphic
)
monotony
of
0
is
trial
.
2. Main Results.
Thm
0
I
、
X
:
4
⇒
X
0
E
x
X
:
if
the
Moreover
Ge
,
that
←→
of
ヨ
:
X
×
:
whcl
x
x
E
d
extends
is
.
also
true
for
G
&
component
of
EC
's
motions
→
武E
4
hdomorph.cm
Endomorph
→
principle
over
is
)
c
ota
motion
.
holds for
Riemann
た
surfaces
do morph
.
c
,
PS 4 2 4 )
.
E
8
set
Cine
Oka
く
Connected
is
connected
0
trinal
closed
connected
every
simply
F
E
motion
X
is
version
cd
base pt
a
h 。 1 0 morph c
over
1
0
at
P。
hdomorph.cm otcn
a
E
of
quran
CE
Suppose
⇒
to
monodrama of
県千
1,03
(
E
→
っ
normalised
a
extended
E
x
E
→
be
can
surface
Riemann
a
:
、
Chika
0
X
:
for
E
x
E
→
closed
a
n しの
w
:
、
if
)
wz
,
は
。
w
;
、
なd
、
motion
CX
and
arg ( 0
(
。
for
a
,
)
to
w
2
h 。 10
a
)
=
orphic
m
も
for
0
and
W
、
0(
-
2ms
extended
n
morph
1
。
の
=
:
be
can
h
curve
dai
Chika
0
condition
's
m 。
,
凶
、
,
w
、
W
中
Wz
21て
区
of dover
X
closed
for Kw
(
)) も
た
・
E
ヒ
に
、
の C X
are
EE
)
(
w
、
キ W
2
)
⒳
世
However
Thm
区
E
ヨ
st
to
.
{
=
X
:
0
,
a
1
,
Riemann
0
,
Wo
h
。
し
。
-_-
,
,
surface
satsties
4
、
have
we
,
morph
し*
c
un
}
ヨ
,
)
(
0
1
but
mctm
M
3
X
:
×
)
E
→
と hdoma.pk moton
cannot
it
of
0
E
over
be
X
.
extended

Download Report