Publicacions
Ma ema iques,
Vol
37
(1993),
111-126
.
A
bs ac
INTERSECTIONS
OF
TOTALLY
REAL
AND
HOLOMORPHIC
DISKS
TOM
DUCHAMP
AND
FRANC
FORSTNERIC
I
is
shown
ha
a
holomo phically
embedded open
disk
in
(C2
and
a
o ally
eal
embedded
open
disk
which
ha e
a
common
smoo h
bounda y
ha e
non i ial
in e sec ion
.
1
.
In oduc ion
I is
now
clea ,
h ough
he
wo k
o
G omo
and
o he s,
ha
he e
is
a
s ong
ela ionship
be ween
he
heo y
o
Lag angian imme sions
in o
symplec ic
mani olds
and
he
heo y
o
o ally
eal
imme sions
in o
complex
mani olds
.
The
ela ion
ollows
om
he
ac
ha
he
G ass-
mann
space
o
Lag angian
subspaces
o
R
2
n
is
homo opy
equi alen
o
he
G assmann
space
o o ally eal
subspaces
o
C'
.
The e
a e
su p ising
di e ences
be ween
he
wo
heo ies,
howe e
.
Recall
ha
complex
2-space,
(C
2
,
is
a
symplec ic
mani old wi h
symplec-
ic
o m
w
=
2
(dz
l
A
dz
l
+
dz
2
A
dz
2
)
.
Le
0
:
0
--->
(C
2
be
a
symplec ic
embedding
o
he
open
uni
disk
in
(C,
ha
is,
suppose
ha
he pull-back
O*w
is
a
symplec ic
o m on
A
.
I is
no
di icul
o
p o e
ha
e e y
holomo phic
embedding
o
0
is
symplec ic
.
Assume
u he
ha
~b
ex ends
o
a
smoo h
embedding
o
he
closed
disk
0
.
The e
is
no
Lag angian
embedding
o
0
whose
bounda y
coincides
wi h
0(0A)
.
Fo
suppose
ha
0
is
such
an
embedding
.
Then
O(
0
)
and
0(,i1)
a e
homologous
ela i e
o
he
bounda y
;
hence,
The
second au ho
was
pa ially
suppo ed
by
he
Resea ch
Council
o
he
Republic
o
Slo enia
.
11
2

T
.
DUCHAMP,
F
.
FORSTNERIC
which
is
a
con ac ion
because
he
i s
in eg al
is
0
and
he
second
is
posi i e
.
On
he
o he hand,
i is
easy
o
cons uc
examples
o
a
pai
consis ing
o
a
holomo phically
embedded
disk
and
a o ally eal
disk
which
ha e
a
common
smoo h
bounda y
(see
Sec ion
4
below
o
one
such example)
.
In
his
pape
he
in e sec ion
heo y
o
such
a
pai
is
in es iga ed
.
The
main
esul
is
he
ollowing
.
In e sec ion
Theo em
.
I
a
holomo phically
embedded
disk
and
a
o ally eal
embedded
disk in
(C
Z
ha e
a
common
bounda y
hen
hey
in e sec
in
a
leas
one
in e io
poin
.
Mo e
p ecisely,
le
0
:
0
->
(C2
be
a
smoo h
embedding
o
he elosed
uni
disk,
holomo phic
on
A
;
and
je
0
:
0
~___>
C
2
be
a
smoo h
o ally
eal
embedding
such
ha
0(ao)
=
0(ao)
.
Then
he in e sec ion
0(0)
1
O(0)
is
non-emp y
.
The
p oo
o
he
heo em
is
by
con adic ion
:
Suppose
he
heo em
is
alse
.
Then
he e
a e
embeddings
0
and
0
o
0
wi h
o(0)
1
o(0)
=
0
and
0(8
0
)
=
O(aA)
.
We
will
show
how
o
de o m
he
images
O(
0
)
and
0(
0
)
o
p oduce
wo
o ally
eal
embeddings which
abu
smoo hly
along
he
bounda y
O(
80
)
in
such
a
way
ha
he
union
o
he
images
o
he
de o med
o ally
eal
embeddings
de ines
a
o ally eal
embedding
o
he
wo
sphe e
in o
C2
.
Bishop
[1]
(see also
Wells
[5])
p o ed
ha
he e
is
no
o ally eal
embedding
o
he
wo
sphe e
in o
(C 2
.
Acknowledgmen
.
We
wish
o
hank
P o esso
E
.
L
.
S ou
o
b ing-
ing
his
p oblem
o
ou
a en ion
.
2
.
P ope ies
o
o ally eal
embeddings
Be o e
beginning
he
p oo
i is
necessa y
o
e iew
se e al
condi ions
which
a e equi alen
o
he
o al eali y
condi ion
and
o de ine
an
index
o
o ally
eal
embeddings
o
annuli
.
2
.1
.
Condi ions
o
o al
eali y
.
An
embedding
ip
:
U
->
(C2,
U
C
cC
a
egion,
is
said
o
be
o ally
eal
i
o
all
~
E
U
he
in e sec ion
0,T
C
UnJO
*
T
C
U
is
i ial
.
(He e
TU
deno es
he
eal
angen
bundle
o
U
and
J
he
complex
s uc u e enso
o
C
2
.)
Le
(z,
w)
deno e
complex
coo dina es
on
C
2
and
(u,
)
a bi a y
eal
coo dina es
on
U
.
The
nex
lemma
gi es
se e al
cha ac e iza ions
o
he
o al
eali y
condi ion
.
They
a e
mo e
o
less
well-known
and
easily
e i ied,
so
we
lea e
i
o
he eade
o
check
hem
.
(See
S ou -Zame
[4]
o
a
discussion
o o al
eali y)
.
TOTALLY
REAL
AND
HOLOMORPHIC
DISKS

113
Lemma
1
.
Le
0
:
U
-
(C
2
be
an
embedding,
U
C
(C,
de ined
by
he
unc ions z
=
Z(S),
w=
W«)
.
Then
he
ollowing
a e
equi alen
:
(i)
The
embedding
0
is
o ally eal
.
(ii)
Fo
each (
E
U
he
equali y
(C
-
0,,(T
S
U)
=
TC
2
holds
l
.
az
OZ/a~
(iii)
The
de e minan

Y

ne e
anishes
.
aW/a(
az
az
(i )
The
de e minan

au

a

ne e
anishes
.
aW aW
au a
Rema k
1
.
Condi ion
(ii)
can
be
es a ed
as
ollows
.
Le
X
=
(X
1
,
X
2
)
be a
basis
o
he
angen
space
TAU, (
E
U
.
Then
he
ec o s
~b*
(X1)
and
0*
(X2)
o m acomplex
basis
o
he
complex
ec o
space
T,
G(C)C
2
.
We
in oduce
he
no a ion
A~

1
-
a
<
l~l
<
1}
and
Aá -
{~
:
1_< l~i
~i
<_
1
+
a}
o
a
>
0
.
The
special
case
in
which ~b
is
de ined
on
he
annula
egion
AQ
,
a
>
0
and
0
is
o
he
o m
«)=0
o
~(j=1
is
o
pa icula
in e es o
us
.
Fi s
obse e
ha
condi ion
(iii)
educes
o
a~
o
.
I
will
p o e
use ul
o
w i e
equa ion
(1)
in
pola
coo dina es,
z
=
eie,
(2)

00
-
i Y

0
.
Because
(z)
anishes
o
all
1
z~
=
1, i
ollows
ha
áe
=
0and
o al
eali y
implies
ha -
=,b
0
o
all
z
wi h
Iz1
=
1
.
This
implies
he
ollowing
lemma
.
Lemma
2
.
Le
0
be
a
o ally eal
embedding
o
he
annulus
A
as
gi en
abo e
.
Then
he e
is
an
annula
egion
o
he
o m
1
<_
Iz1
<
1
+
a',
0
<
a'
<
a
<
1,
on
which
can
we
w i en
in
he
o m
.
(z)
=
R(z)e¡8(z)
'I
V
C
W
is
a
eal
subspace
o
he
complex
ec o space
W
hen
(C
-
V
deno es
he
complex
subspace
spanned
by
V
.
11
4

T
.
DUCHAMP,
F
.
FORSTNERIC
whe e
R(z)
is
a
non-nega i e,
smoo h
eal- alued
unc ion
which
anishes
o
~z1
=_1
and
whe e
O(z)
is
smoo hmodulo
27
.
The
inequali y

8
a í,(z)
>
0
is
sa is ied
o
all
z in a
neighbo hood
o
he
uni
ci cle
Iz1
=
1
.
Now
suppose
ha
is
any
smoo h
unc ion
de ined
on
A
o
he
o m
.
(z)
=
R(z)eio(z)
Subs i u ion
o
he
o mulas
_
a

_

aR(z

a0
_
io

a

aR(z)

0"
a
-
(

a

)
+
iR(z)
a
)
e

and
ó0
=
(

00

+
iR(z)
a0
)
ego
in o
he
o al
eali y
condi ion
(2)
and
sepa a ing
eal
and
imagina y
pa s
o
he
coe icien
o
e
io
yields
he
condi ion
'9R

+
R(z)
ae
a0
)
+
i
(R(z)
50
-
a
a z))

0
.
In
pa icula ,
i
he
imagina y
pa
o
he
le
hand
side
is
nega i e,
he
embedding
is
necessa ily
o ally eal
.
Lemma
3
.
Le
0
:
Aá
-->
C2,
0
<
a
<
1,
be
an
embedding
o
he
o m
O(z)
=
(z,
R(z)eio(z)),
whe e
R
and
O
a e
eal- alued
unc ions
wi h
R
smoo h
and
O
smoo hmodulo
27
.
I
he
inequali y
aR(z)
>
R(z)a0
a

á0
is
sa is ied
hen he
embedding
is
o ally
eal
.
2
.2
.
An
index
o
o ally
eal
embeddings
o annuli
.
In
his
sec ion
we
de ine
an
index
o
a
o ally eal
embedding
o
an
annulus
in
(C
2
.
I
is
closely
ela ed
o
he
Maslo
index
and
is
a
special
case
o
an
index
de ined
by
Kambe
and
Tondeu
[3]
.
A
de ailed
p esen a ion,
wi hin
he
con ex
o o ally eal
embeddings
o
su aces
in
C
2
,
is
gi en
in
[2]
.
We
gi e
a
sel -con ained
exposi ion
he e
.
Le
00
:
A
->
(C
2
be
any
o ally
eal
embedding
o
an
annula
egion
A
C
C
.
To
de ine
he
index
o
00
begin
by
choosing
a
complex
aming
2
2
By
a
complex aming
we
mean
a
pai o
complex
ec o
ields
which
a e
poin wise
independen
o e
(C
.
=
( ,,
2)
o
he
holomo phic
angen
bundle
T(
1
,
0
)U,
whe e L
is
any
open
con ac ible
neighhbo hood
o
A
.
Nex
choose
a
eal
aming
3
X
=
(X1,
X2)
o
he
eal
angen
bundle
TA
which
is
compa ible
wi h
he
o ien a ion
o
A
as
a
subse
o
(C
.
By
i ue
o
Rema k
1,
he e
is
a
smoo h
ma ix- alued
unc ion
de ined
by
he
o mula
TOTALLY
REAL
AND
HOLOMORPHIC
DISKS

115
Since
C
is
eal,
so
is
de (C),
hence,
MO
o
:
A
,
GL(2,C)
1
(doo(X1)

dOo(X2))
=
( l

2)
~m2
1
De ini ion
1
.
The
índex
o
he
embedding
0
0
:
A
->
C
2
,
is
he
deg ee
o
he
map
A
-i
80
:

-->
de (M)
de (M)
and
is
deno ed
by
Ind(0o)
E
7G
.
Rema k
2
.
(i)
I
is
easily
e i ied
ha
he
in ege
Ind(oo)
is
inde-
penden
o
he
amings
X
and
.
Fo
suppose
ha
X'
and
'
is
ano he
pai o
amings,
wi h
'
de ined
on
Ll',
a
con ac ible
neighbo hood
o
Oo(A)
.
Then
he e
a e
smoo h
maps
B
:
un
u'
,
GL(2,C)
and
C
:
A
E
->
GL
+
(2,
R)
such
ha
'= -B
and
X'=X
.C
.
I
M'
:
A
-->
GL(2,
C)
is
he
map
de ined
by
he
o mula
d0
o
(X)
=
-M',
a
s aigh o wa d
calcula ion
wi h
ma ices
yields
he
iden i y
M'=B-1-M
.C
.
l
m
.
2
m
2
2
de (M')

_

de (B
--1 )
de (M)
de (C)

_

de (B)j
de (M)
de (M')¡

de (B
-1
)
11
de (M)
11
de (C)
1

de (B)

Ide (M)1
Since
U
and
U'
a e
con ac ible,
each o
he
amings
and
'
a e
is
homo opic
o
he
aming
(
a
a
l ,
a
a
2 )
.
The
map
B
-1
is,
he e o e,
homo-
opic
o
he
iden i y
.
This
ac ,
oge he
wi h
he
obse a ion ha
he
3
By
a
eal
aming
we
mean
a
pai o
eal
ec o
ields
which
a e
poin wise
indepen-
den
o e
R
.

11
6

T
.
DUCHAMP,
F
.
FORSTNERIC
deg ee
depends
only
on
he
homo opy
class
o
he
map,
comple es
he
a gumen
.
(ii)
No e
also
ha
he
abo e
a gumen
shows
ha
i
q>
:
U
,
(C
2
is
a
biholomo phism
on o
an open
se
in
(C
2
hen
Ind(-P
o
0
0 )
=
Ind(0o)
.
(iii)
Finally,
because
he
index
depends
only
on
he
homo opy
class
o
he
map,
00,
i is
de e mined
by
he
image
00 (A)
oge he
wi h
an
o i-
en a ion
.
Thus,
i
A
C
C
2
is
a
o ally
eal,
o ien ed,
embedded
annulus,
he
in ege
Ind(A)
is
well-de ined
.
Lemma
4
.
I
A
C
C
2
is
a
o ally
eal
embedded
annulus
which
is
con ained
in
a
o ally
eal
embedded
disk
DC
C2
hen
Ind(A)
=
0
.
P oo
:
Le
0
:
0
-->
(C
2 a
smoo h
map
such ha
O(0)
=
D
and
choose amings
X
o
Tá
and
o
(C
2
.
Then
le
M(S),
E
0, be
he
GL(2,
C)- alued
ma ix
as
de ined
abo e
.
The
deg ee
o
he
map
(
->
de (M«))/I
de (M(~))1,

E

-I
(A)
is
ze o
because
i
is
homo opy
o
a
cons an
.
3
.
Reduc ion
o
he
case o
eal
analy ic
bounda y
Begin
by assuming
ha he e a e
smoo h
embeddings
0
and
0
o
0
wi h
0
holomo phic
on
0
and
o ally
eal
and
such
ha
he
condi ions
z/~(0)
1
o(0)
=
D
and
O(á0)
_
O(á0)
a e
bo h
sa is ied
.
Wi hou
loss
o
gene ali y
we
may
assume
ha
0
ex ends holomo phi-
cally
o
a
neighbo hood
o
0
.
To
see
his
we
obse e
ha ,
because
he
condi ion
o
o al
eali y
is
an
open
condi ion,
any
C
l
-small
de o ma ion
o
he
map
0
is
also
a
o ally
eal
embedding
.
In
pa icula ,
le
and
in
such
a
manne
ha
0o6
=
{~
:
10
=1-
0,
whe e
S
>
0
is
a
small
cons an
o
be
chosen
la e
.
Then
0
can
be
de o med
o
a
map
0'
so ha
0'(ao)
=
0(aos)
o
(o)
n~b
(oó)
=
0
.
TOTALLY
REAL
AND
HOLOMORPHIC
DISKS

117
Oneway
o
accomplish
such
a
de o ma ion
is
o
le
0'
be
he
composi ion
0
o
ó o

wi h
he
low,
,
o
a
ec o
ield
which
is
angen
o
he
image
o

and
cons uc ed so
ha
ó
(0(OO)
=MO5),
b
>
0
.
The
map
~b'
:
0
-->
CZ
de ined
by
he
equa ion
00= )((1-
b)0
ex ends
holomo phically
o
a
neighbo hood
o
0
C
C
.
Now
eplace
he
pai
0,
0 by
he
pai
0', 0'
.
By
cons uc ion,
O(OO)
is
eal
analy ic
.
whe e
4
.
Holomo phic
disks
a e
ela i ely
iso opic
o
o ally
eal
disks
Re um
now
o
he
p oblem
o
eplacing
0
by
a o ally
eal
embedding
.
Because
0
ex ends
o a
holomo phic
embedding
o
a
neighbo hood
o
0,
he e
is
a
biholomo phism
(D
:
U
-j
C
2
,
de ined
on a
neighbo hood
U
o
~b
(A)
such
ha
he
composi ion
<D
o
0
is
he
map
Conside
he
amily
o
maps
0
E
:0_(C2,
~
H
(z~
w)
=
(S,
E«)),
c
.(~)
=,E
( 1
-
1(I2)
ex(1-I(I2)
~
No e
ha
E
sa is ies
he condi ions
:
o
=
0,
,(~)
=
0 o
I(I
=
1
.
By
i ue
o
equa ion
(1)
and
he
compu a ion
O E(~)
=
E
{(1
_
21(1
2
)
-
(
1
-
I(I
2
)I(I
2
i}
e'('
_1C12)
0
0
he
embedding
de ined
by
0
.is
o ally
eal
o
all
E
>
0
.
Because
o
E
su icien ly
small
he
image
o
0
.
lies
in
he
se
<D(u),
he
map
0E
=P
-1
o
0E
is
well-de ined
.
Lemma
5
.
Fo
e
su icien ly
small,
he
amily 0E
has
he
ollowing
p ope ies
:
( 1
)
O,Iao
=
Ojao
o
all
E
.
(ii)
0E
is
a
o ally eal
embedding
o
0
o E
>
0
.
(üi)
z/~ E
(0)
CU
.
(i )

E(A)
n
o(o)
_
11
8

T
.
DUCHAMP,
F
.
FORSTNERIC
P oo
.
P ope ies
(i)
h ough
(iii)
a e
immedia e om
he
de ini ion
o
,
The
e i ica ion o
(i )
is
based
on
he
implici
unc ion
heo em
.
Fi s
no e
ha
since
O(
0
)
is
o ally eal
and
V)0(0)
is
holomo phic,
Tp(oo(
0
))
n
Tp(o(0))
=
Tp(o(a
0
))
o
all
p
E
0(á
0
)
.
Because
~b E
de-
pends
smoo hly
on
e,
he
condi ion
Tp(,pE(
0
))
n
Tp(O(0))
=
Tp(O(a0))
holds
o
all
e
su icien ly
small
.
By
he
implici
unc ion
heo em
and
compac ness
o
80,
i
ollows
ha
he e
is
a
numbe
S
>
0
such
ha
0E(A)
U
O(A6)
_
0 o
all
su icien ly
small
c
.
Mo oeo e ,
since
0o(A)
n
~(0)
=
0
and
PE
depends smoo hly on
e, i
ollows
om
he
compac ness
o
0(0
Ab
)
ha 0,
(A)
n
O(0
A6)
_
0
o
all
su icien ly
small
e
.
Hence,
o
e
su icien ly
small,
0,
(A)
n
0(0)
=
0
.
5
.
Modi ying
wo
o ally eal
disks
o
abu
smoo hly
Conside
he
small
annula
neighbo hood
Ab
C
0,
0
<
S
<
1,
o
he
bounda y
80
.
We
will
modi y
0(0)
on
O(A6)
so
ha O(
0
)
and
0F(
0
)
abu
in
a
C
l
manne
along
<b(á0)
and
hus
de ine
a
o ally eal
embedding
o
he
2-sphe e
in o
(C 2
.
By
i ue
o
he
equali y
O(ó0)
=
0(
80
),
o
S
su icien ly
small
he
imago
O(A,
)is
con ained
in
he
neighbo hood
U
o
he
p e ious
sec ion
.
Fo
his
eason
he
map
4)
o0
:
A6__+
C
2
is
well-de ined
and,
since
we
will
modi y
0
only
along
A6
,
he
de o ma-
ion
o
0
can
be
ein e p e ed
as
a
de o ma ion
o 0'
.
The
modi ica ion
will
be
done
in
wo
s ages
:
(i)
we
i s
de o m
0' so
ha
O'(A,
)is
he
g aph
o
a
unc ion
g
;
(ii)
hen
we
de o m g
so ha
O(0)
and
0(
0
)
abu
smoo hly
.
5
.1
.
Replacing
O'(A
.
)
by
he
g aph
o
a unc ion
.
Conside
he
neighbo hoods
o
O'(0)
o
he
o m
N
E
,
Q
=
{(z,
w)

:

Iz1
<
1
+
e,
IWI
<
Q}
wi h
e
>
0 and a
>
0
chosen
so
small
ha
he e
is
an
inclusion
N3E,,
C
-D(U)
and
such
ha
he
condi ion
~
_1
(N3e,a)n0(
0
A6)
=
is
sa is ied
.
By
choosing
S'
<
S
su icien ly
small
we
can
insu e
ha
inclusion
0'(A,
-
,)
CV,,
is
sa is ied
.
TOTALLY
REAL
AND
HOLOMORPIIIC
DISKS

119
We
a e
going
o
eplace
0'
by
ano he
o ally eal
embedding
~"
A
6
-~
NzE,Q
which
sa is ies
he
wo
condi ions
:
and
"(Ab
)
1

{(z,
g
(z))
:
z
E
Aó,}

o
o,'
<
su icien ly
small,
whe e
g
is
a
complex- alued
unc ion
de ined
on
Aá,
.
By
cons uc ion,
he
map
4>
-1
o
A
b
_
(C 2
ag ees
wi h
0 on
he
in e io
bounda y
componen
o
he
annulus
and
so
de ines
ano he
o ally
eal
embedding,
01
:
0
-->
(C
2
which
in e sec s
O(0)
along
he
ci cle
O(a0)
.
To
begin
he
cons uc ion
o
0"
obse e
ha
he
image O'(A6
)
is
o
he
o m
whe e
he unc ions
Z
and
W
sa is y
he
condi ions
(4)

Z(~)
=
(
and
W«)
=
0 o
1(1
=
1
.
Hence,
in
pola
coo dina es
(
=
pe
e
",
z
=
e
e
we
can
w i e
wi h
R(1, a)
=
1
and
O(1, a)
=
a
.
Applying
condi ion
(i )
o
Lemma
1
yields
he
inequali y
0
:~
aZ aZ
áp 8a
aw
aw
ap aa
on
he
annula
egio l
Aó
Aó,
z
=
Z(0,

w
=
W(O
,

1
-
S
<
j(j<
1,
Z=
R(p,
a)e2o(P,a)
aZ
aRe
io
áw
ap
aa
0
Thus
áP
is
non-ze o
o
(p,
a)
=
(1,
a)
.

Con inui y
implies
ha
o
S'
>
0
su icien ly
small
OP
does
no
anish
anywhe e on
he
annulus
Ab
.
This
and
equa ion
(4)
show
ha , a e
possibly
dec easing
S'
s ill
u he ,
he
map
(
H
(e
z©
,
W)
is
an
embedding
o
A,,
;
and,
he e o e,
ha
o
any
smoo h
unc ion
R(p, a)
>
0
he
map
0'
,
:
(
-
(Z
(0,
w
(S))
=
(Re
20
,
w)
_
-i
eZa
áPW
when
p
=
1
.
de ines
an
embedding
o
A,
-
,
.
O
cou se,
we
wish
o
choose
R
so ha
he
map
0"
sa is ies
he
condi ions
s a ed
abo e
.
Tha
we
can
do
so
is
implied
by
he
ollowing
lemma
.
12
6

T
.
DUCHAMP,
F
.
FORSTNERIC
Choose
u"
>
0 so
ha
m/M
>C
.
We
claim
ha
he e
is
a
cons an Q"'
<
"
and
a
unc ion
h
such
ha h( )
=
0
o
1
<
<
1
+
o,"'
,
h( )
=
1
o
>
1+o,"/2,
and
h'( )
<
H( )
whe e H( )
=
(Mm)-1
(( /M
-C)
.
Tha
such
a
unc ion
exis s
is
clea
because
he
igh
hand
side o
he
las
inequali y
is
posi i e
o
1
<
<
1
+
u" and
because
he
in eg al
l+,"
H( )
d
di e ges
o
+oo
.
Hence
we
may
se
h( )
=
i
k( )d
whe e
k( )
is
any
unc ion
sa is ying
he
condi ions,
(i)
0
<_
k( )
<
H( ),
(ii)
i
+al
k( )d
=
1,
and
(iii)
k( )
=
0
o
1 <_
<
o,"'
<
u"/2
and
o
>o,"/2
.
1
.

E
.
BISHOP,
Di e en iable
mani olds
in
complex
Euclidean
space,
Duke
Ma h
.
J
.
32
(1965),
1-21
.
2
.

F
.
FORSTNERIC,
Analy ic
disks
wi h
bounda ies
in
a
maximal
eal
submani old
o
(C
2
,
Ann
.
Ins
.
Fou ie
37
(1987),
1-44
.
3
.

F
.
W
.
KAMBER
AND
PH
.
TONDEUR,
Cha ac e is ic
in a ian s
o
olia ed
bundles,
Manusc ip a
Ma hema ica
11
(1974),
51-89
.
4
.

E
.
L
.
STOUT
AND
W
.
ZAME,
To ally
eal
imbeddings
and
he
uni-
e sal
co e ing
spaces
o
domains
o
holomo phy
:
some
examples,
Manusc ip a
Ma hema ica
50
(1985),
29-48
.
5
.

R
.
O
.
WEI,I,S,
Compac
eal
submani olds
o
a complex
mani old
wi h
nondegene a e
holomo phic
angen
bundles,
Ma h
.
Ann
.
179
(1969),123-129
.
6
.
M
.
GROMOV,
"Pa ial
di e en ial ela ions,"
E gebnisse
3
olge
Bd
.9,
Sp inge -Ve lag,
Be lin-Heidelbe g-New
Yo k,
1986
.
Tom
Duchamp
:
Depa men
o
Ma hema ics
Uni e si y
o
Washing on
GN-50
Sea Ie,
WA
98195
U
.S
.A
.
Re e ences
F~anc
Fo s ne ic
:
Depa men
o
Ma hema ics
Uni e si y
o
Wisconsin
Madison,
WI
53706
U
.S
.A
.
P ime a
e sió
ebuda
el
12 de
Ma i
de
1992,
da e a
e sió
ebuda
el
10
de Juny
de 1992