Publicacions
Ma emá iques,
Vol
36
(1992),
131-155
.
Abs ac
FINSLER
METRICS
WITH
PROPERTIES
OF
THE
KOBAYASHI
METRIC
ON
CONVEX
DOMAINS
MYUNG-YULL
PANG
The
s uc u e
o
complex
Finsle
mani olds
is
s udied
when
he
Finsle
me ic
has he
p ope y
o
he
Kobayashi
me ic
on
con-
ex
domains
:
( eal)
geodesics
locally
ex end
o
complex
cu es
(ex emal
disks)
.
l is
shown
ha
his
p ope y
o
he
Finsle
me ic
induces
a
complex
olia ion
o
he
co angen
space
closely
ela ed
o
geodesics
.
Each
geodesic
o
he
me ic
is
hen
shown
o
ha e
a
unique
ex ension
o
a maximal
o ally
geodesic
complex
cu e
E
which
has,p ope ies
o
ex emal
disks
.
Unde
he
addi-
ional
condi ions
ha
he
me ic
is
comple e
and
he
holomo phic
sec ional
cu a u e
is
-4,
E
coincides
wi h
an
ex ema¡
disk
and
a
heo em
o
Fa an
is
eco e ed
:
he
Finsle
me ic
coincides
wi h
he
Kobayashi
me ic
.
1
.
In oduc ion
The
Riemann
mapping
heo em
says ha
all
simply connec ed
do-
mains
in
C,
di e en
om
C
a e
biholomo phically
equi alen
.
I
is
a
well
known
ac
ha
his
heo em
does
no
hold
o
domains
in
T'
o
n
>
1,
and
he
classi ica ion
o
bounded
domains up
o
biholomo phism
has
been an
impo an
p oblem
in
se e al
complex
a iables
.
One
app oach
o
unde s anding he
s uc u e
o
bounded
domains
is
o
s udy
biholo-
mo phically
in a ian
mé ics
such
as
he
Kobayashi
o
Ca a héodo y
me ics
[K]]
[BD]
[L3]
[Pa]
.
In
[L1]
and
[L2],
Lempe
showed
ha
hese me ics
a e
ex emely
well-beha ed
in
he
special
case
when
he
domain
is
s ic ly
linea ly
con ex
and
has
smoo h
bounda y
:
In
his case,
he
wo
me ics
coincide,
and
he
in ini esimal
o m
FK
o
he
Kobayashi
me ic
alls
in o a
special
class
o
smoo h
Finsle
me ics
wi h
cons an
holomo phic
sec ional
cu a u e
K
=
-4
.
Since
he
no ion
o
a
s ic ly
linea ly
con ex
domain
is
no
a
biholomo phically
in a ian
concep ,
i
132

M
.-Y
.
PANG
is
na u al
o
ask
how
a
Lempe 's
esul s
can
be
ex ended
o
a
mo e
gene al (biholomo phically
in a ian )
complex
mani olds
.
One
app oach
o his
p oblem
is
o
s udy
FK
om
a
mo e
in a ian
poin
o
iew
.
The
i s
s ep
is
o
cha ac e ize
he p ope ies
o
an
ab-
s ac
Finsle
me ic
F
on
an
abs ac
complex
mani old
M'
which
a e
necessa y
o
Lempe 's
esul s
o
hold
.
A
second,
and
mo e
di icul ,
s ep
is
o
de e mine
when
he
Kobayashi
me ic
o
a
bounded
domain
in
(U"
has
hese
p ope ies
.
In
[F],
Fa an
analyzed
he
local
s uc u e
o
(complex)
Finsle
mani olds
and
ob ained
a
se
o
local
in a ian s
by
applying Ca an's
me hod
o
equi alen e
.
He
p o ed
ha
anishing
o
ce ain
local
in a ian s
o ces
F
o
coincide
wi h
he
Kobayashi
me ic
o
he
unde lying mani old
M
p o ided ha
F
is
a
comple e
me ic wi h
K
= -4
.
Howe e ,
om
he
complex
p ocess
o
cons uc ing hese
lo-
cal
in a ian s
i
is
no
easy o
see
how
hese
in a ian s
na u ally
a ise
om
he p ope ies
o
Kobayashi
me ics
ob ained
om
Lempe 's
wo k
.
Thus,
one would
like
o
o mula e
a
somewha mo e
di ec
desc ip ion
o
he
local
s uc u e
;
ha
is
in ui i ely
mo e
appealing
.
In
his
pape ,
we
gi e
such
desc ip ion
om
he
poin
o
iew
o
he
calculus
o
a ia ions
by
examining
he
local
p ope ies
o
he
Kobayashi
me ic
on
s ic ly
linea ly
con ex
domains,
and
de i e
equi alen condi ions
o
he
anish-
ing
o
he
Fa an's
in a ian s
om
a
simple
p ope y
o
Kobayashi
me ic
(P ope y
1
.3)
.
In
o de
o
desc ibe
he
local
s uc u e
o
he
Kobayashi
me ic,
we
gi e
b ie
e iew o
Lempe 's
wo k
.
We
de ine
he
in ini esimal
Kobayashi
me ic
FK
on
a
complex
mani old
M
as
ollows
:
Fo
each
E
T
x
M,
x E
M,
le
be
a
holomo phic
map
om
he
uni disk
0
C
C
in o
M
such
ha
(0)
=
xand (0)
=
A
o
A
>
0
.
The
magni ude
FK
( )
o
wi h
espec
o
he
in ini esimal
Kobayashi
me ic
FK
is
de ined
o
be
he
in imum
o

whe e
he
in imum
is
aken
o e
all
such
.
I
ac ually
a ains
he
in imum
(Le
.

FK( )
=
á
),
hen
is
called
ex emall
.
I
can
be
easily
seen
ha
he
me ic
FK
is
in a ian
unde
he
ac ion
o
he
g oup
o
biholomo phisms
o
M
.
Lempe
showed
ha ,
i
M
=
D
C
C
V`
is
a
bounded
s ic ly
linea ly
con ex
domain
wi h
smoo h
bounda y,
hen
FK
is
a
smoo h
complex
Finsle
me ic
[L1], [L2],
Le
.
FK
is
smoo h
ou side
he
ze o
sec ion
o
TD
and
sa is ies
he
ollowing
condi ions
:
(1
.1)
FK( )
>
0

'o

,
:
0,

FK(z
. )
=
Iz1FK( )

o

z
E
C,

and
(1
.2)
FK( i
+
V2)
.<
FK( 1)+FK( 2)

o l,
2
E
T
.D,

x
E
D,
whe e
equali y
in (1
.2)
holds
only
when
l
and 2
a e
colinea
.
Mo eo e ,
FINSLGR
MGTRICS
ANDTHC
KOBAYASHI
MGTRIC

133
he
p o ed
he
ollowing
heo em
:
Theo em
(Lempe )
.
Suppose
ha
D
is
a
bounded
s ic ly
linea ly
con ex
domain
wi h
smoo h
bounda y
.
(1)
The e
is
a
unique
ex emal
map
co esponding
o
each
E
TD
.
(2)
All
he
ex emal
maps
a e p ope
isome ic imbeddings,
and
can
be
smoo hly ex ended
o
he closed
uni
disk
0
.
(3)
The
ex emal
disks
(,~i)
passing h ough
a
poin
xE
D
o m
a
complex
olia ion
o
D
-
{x}
.
(4)
Ex emal
disks
a e
( he
only)
one-dimensional
holomo phic
e-
ac s
o
D
.
One
o
he
key
ideas in
desc ibing
he
geome y
o
D
is
he
cons uc ion
o
he
holomo phic
e ac o
D
on o
he
ex emal
disk
(A)
.
Lempe
p o ed
ha
he
ield
o
holomo phic
angen
planes
o
áD
on
(¿9A)
can
be
holomo phically
ex ended
o
he
in e io
o
he
disk
(
0
),
and
de ines
a
holomo phic
ield
o
complex
hype planes
on
(A)
ha
a e
ans e sal
o
(A)
.
In
o he
wo ds,
he e
is
a
well de ined
(n
-
1)-dimensional
holomo phic
ec o
bundle
p
:
E
,
(A)
o e
he
ex emal
disk
wi h
ibe s
de ined
by
he
hype planes
in
T'
.
The
union
o
he
hype planes
con ains
he
domain
D,
and
he
holomo phic
e ac
is
de ined
by
he
es ic ion
o
D
o
he
p ojec ion
mapp
.
The
exis en e
o
such
holomo phic
e ac s
has
u he implica ions
.
Fo
example,
i
o ces
e e y
(locally
leng h
minimizing, connec ed)
geo-
desic
cu e
o
FK
o
be
con ained
in
an
ex emal
disk
.
The
p ope ies
o
he
Kobayashi
me ic
ha
in e es s
us
a e
he
ollowing
:
Co olla y
.
Le
:
A
-
D
be
a ex emal
map
o
E
T
D
.
(1
.3)

The
ex emal
disk
(A)
coincides
wi h
he
union
o
geodesic cu -
es
h ough
x
angen
o
a
common
complex
line
in
T
x
M
.
(1
.4)

The e
is
a
canonical
spli ing
TD, (o)
=
T(
(0))
®
E
whe e
E
is
an
(n
-
1)-dimensional
holomo phic
subbundle
o
he
es ic ion
TD, (o)
o
he
angen
bundleTD
o
(A)
.
We
wish
o
gene alize
he condi ion
(1
.3)
o
an
abs ac
complex
Finsle
me ic
F
de ined
on
an'n-dimensional
complex
mani old
M
.
Le
exp,
deno e
he
exponen ial
map
om
a
neighbo hood
o
0 E
T,
M
¡ i o
M
de ined
by
he
geodesics
o
F
.
Fo
each
angen
ec o
E
T~M,
he
image exp
(U,)
o
a
small
neighbo hood
U,
o 0 in
he
complex
line
0
de ines
a
su ace
in
M
.
A
easonable
gene aliza ion
o
he
condi ion
(1
.3)
is
he
ollowing
:
(1
.5)

Fo
all
E
TM,
he
su ace
exp
(U,)
is
a
complex
cu e
(1-
dimensional
complex
submani old)
in
M
.
134

M
.-Y
.
PAl c
This
condi ion
was
i s
in oduced
by
Royden
[R],
and
i
is,
in
ac ,
equi alen
o
a
condi ion gi en
by
Fa an
[F]
.
The
pu pose
o
his
pape
is
o
s udy
he
local
s uc u e
on
o
Finsle
me ics
sa is ying
condi ion
(1
.5),
and
u he ,
o
show
ha
many
p op-
e ies
o
he
Kobayashi
me ic o
con ex
domains
ex end
o his
mo e
gene al
class
o
complex
mani olds
.
One
o
ou
majo
esul s
is
a
con-
s uc ion
o a
biholomo phically
in a ian
amily
o
complex
cu es
which
enjoys
many
o
he
p ope ies
o
ex emal
disks in
con ex
domains
.
To
see
how
such
complex
cu es
a e
cons uc ed,
ecall,
om
he
calculus
o
a ia ions,
ha
he
me ic
F
uniquely
de e mines
a
ec o
ield
X
on
he
co angen
space
o
M,
called
he
geodesic
ec o
eeld,
such
ha
he
in e-
g al
cu es o
X
a e
mapped
in o
geodesics
o
F
by
he p ojec ion
map
7
:
T*M
>
M
.
Le
Z
be
he ec o
ield
on Tó
M
=
{
E
T*MI
=~
0}
gene a ed
by
he
ci cle
ac ion
o
unimodula
complex
numbe s
de ined
by
mul iplica ion
on
TO*
M
.
Theo em
A
.
The
ollowing
condi ions
a e
equi alen
(Theo em
4
.7)
:
1
.

The
su ace
exp
(U

)
C
M
is
a
complex
cu e
o
all
E
TM
.
2
.
[X,
JX]
=
KZ
o
some
smoo h
unc ion
,
on
Tó
M
.
3
.
The
dis ibu ion
D
=
e
X
®
(E
Z
C
T(TO
M)
is
in olu i e
.
I
any
o
he
abo e
condi ions
is
sa is ied
hen
each
complex
cu e
exp
(U

)
ex ends
uniquely
o
a
maximal,
o ally
geodesic,
imme sed
com-
plex cu e
E
--~
M
(Theo em
4
.9)
.
The
sígni icance o
he
condi ion
2
and
3
in
he
heo em
is
as
ollows
:
The
condi ion
2
p o ides
a
compu a ional
me hods
o
check
whe he
F
sa is ies
he
p ope y
(1
.5)
.
The
condi ion
3
implies
ha ,
by
F obenius
Theo em,
he
dis ibu ion
D
de ines
a
2-dimensional
complex
olia ion
.Fo
o
Tó
M
.
The
complex
cu e
E
is
cons uc ed
by
p ojec iog
each
lea
o
he
olia ion
Fo
by
he
p ojec ion
map
7
on o
M
.
The
cu es
E
sha e
many
local
p ope ies
in
common
wi h
he
ex emal
disks
desc ibed
in
he
Lempe 's
heo em
.
Fo
example,
by
Theo em
A,
any
eal
geodesic
cu e
o
F
is
con ained
in
one
o
he cu es
E
.
Fu he mo e,
a
gene aliza ion
o
he
p ope y
(3)
in
Lempe 's
heo em
holds
:
he
complex
cu es
E
passing
h ough
a
poin
x o m
a
complex
olia ion
o
some
neighbo hood
o
x
.
A
less
i ial
esul
is
he
ollowing
gene aliza ion
o
p ope y
(1
.4)
:
Theo em
B
.
Fo
each
complex
cu e
E,
he e
is
a canonical
spli ing
TM¡£
=
TE
®P-E,
whe e
T
1
E
is
an
(n
-
1)-dimensional
holomo phic
subbundle
o
TMI£
.
(Theo em
4
.9
.)
FINSLER
METRICS
AND
THE
KOBAYASli1
METRIC

135
The
signi ican e
o
he
unc ion
,
in
he
Theo em
A
is
i s
ela ion
o
he
holomo phic
sec ional
cu a u e
o
F
.
No e
ha
each
complex
cu e
in
M
is
na u ally
equipped
wi h
a
He mi ian
me ic
induced
by
F,
and,
he e o e,
has
an
associa ed
Gaussian
cu a u e
.
Following
he
de ini ion
by
Wong
and Royden
[W]
[R],
we
de ine
he
holomo phic
sec ional
cu a u e
o
F
a
by
he
Gaussian
cu a u e
o
he
cu e
exp (U

)
.
Theo em
C
.
I
F
is
a
complex
Finsle
me ic
sa is ying
he
condi ion
(1
.5)
hen
he
holomo phic
sec ional
cu a u e
o
F
is
de e mined
by
he
unc ion
Finally,
using
he
esul
desc ibed
abo e,
wc
show
ha ,
unde
he
condi ion
ha
F
is
comple e
a,nd
,
=
-4
he
complex
cu es
E
coincide
wi h
he
ex emal
disks
.
This
esul
was
p o ed
ea lie
by
Fa an
[F]
.
No e
ha ,
om
Lempe 's
esul ,
he
Kobayashi
me ic
FK
on a
s ic ly
linea ly
con ex
domain
D
C
en
has
cons an
holomo phic
sec ional
cu -
a u e
-4
.
(This
is
a
di ec
consequence
o
he
ac
ha
e e y ex emal
map
:
A
->
D
is
an
isome y
wi h
espec
o
he
Poinca é
me ic
and
FK
such
ha
(
0
)
is
locally
de ined
by exp (U

) .)
4
.24
Theo em
[F]
.
Suppose
F
is
a
comple e
complex
Finsle
me ic
on
a
complex
mani old,
M
wi h
cons an ,
holomo phic
sec ional
cu a-
u e
-4
sa is ying
h,e
p ope y
(1
.5)
.
Then
F
=
FK,
)hc e
FK
is
he
Kobayashi
me ic
on
M
.
The
pape
is
o ganized
as
ollows
:
In
Sec ion
2,
we
de elop
basic
ools
and
p o e
some
basic
ac s
abou
complex
Finsle
mani olds
.
Sec ion
3
is
an
in oduc ion
o
Legend e
olia ions
and
i s
applica ion
o
complex
Finsle
mani olds
.
In
Sec ion
4,
we
p o e
he
main
heo em
using
he
esul s o
Sec ions
2and
3
.
Th oughou
he pape ,
M
deno es
an
n
dimensional
complex
mani old
and
F
a
complex
Finsle
me ic
;
on
M
.
The
'ollowing
no a ions
a e
used
:
(1)
The
indices
a,
band
c
ango
om
1
h ough
2n,
and
o
.,
(5,
y
ango
om
1
h ough
2n
-
1
.

Summa ion
con en ions
a e in o e
h oughou
.
(2)
(X
I
, . .
.,
xn,
x
n+1
. .
.,
x2n)
deno e he
eal
coo dina es
on
M
ob-
ained
om
a
holomo phic
coo dina es
x,
+
ixn+
,
o
=
1,
. .
.,
n
.
(3)
(xl

X
n
x
n+1

x
2n
u
l

un u
n+l

u2n)
deno e
he
coo -
dina es
on
T*M
induced
by
(xl,
. .
.,
xn, x
n+1
,.
.
.,
x
2n)
.
(4)
Fo
F
E
C°°(T*M),
Fa,
FaL,
. . .
deno e
'
g
"
-
,

9z
"

and
so
on
.
du°
(5)
I
V
is
a
ec o
ield
on
a
mani old
M,
e'
V
:
M-
M
deno es
he
1-pa a le e
amily
o
di
eomo phisms
gene a ed
by
V
.
Thus,
o

136

M
.-Y
.
PANG
Acknowledgmen s
.
I
would
like
o
hank
T
.
Duchamp
o his
help
and
encou agemen ,
and
o
in oducing
me
o his
subjec
.
I also
wish
o
exp ess
my
hanks
o
J
.
Bland
o
con e sa ions
.
In
his
sec ion,
we
p o e
some
gene al
ac s
abou
complex
Finsle
me ics
.
A
complex
Finsle
me ic
on
he
co angen
bundle
o
M
is
a
map
F
:
T*M
-
R
sa is ying
p ope ies
(1
.1)
and
(1
.2)
.
When
M
is
equipped
wi h
acomplex
Finsle me ic,
we
will
call
M
a
complex
Finsle
mani old
.
2
.1
The
Geodesic
Vec o
Field
and
Complex
S uc u e
.
In
o de
o
de ine
he
exponen ial
map,
we
in oduce
he
geodesic
ec o
ield
on
T*M
.
Recall
ha
T*M
is
na u ally
equipped
wi h
a
1- o m
de ined
by
he
equa ion
(2
.2)
and
ha
he
2- o m d(
is
a
symplec ic
2- o m
on
T*M
(Le
.
a
smoo h
closed
2- o m
on
T*M
sa is ying
he
non-degene acy
condi ion
(do)2,,
=~
0)
.
The
geodesic
ec o
ield
X
on T*
M
is
uniquely
de e mined
by
he
condi ion
(2
.3)

XJdS
=
-FdF
.
In
pa icula ,
lle
iden i y
XF
=
0
holds
.
In
e ms
o
coo dina es,
we
Na e
(2
.4)
cach
xE
M,
~--~
e
x
is
an
in eg al
cu e
o
V
s a ing
a
x
(Le
.
e
°
x
=
x and

d
I
e
x
=
Ve

.)
.
2
.
Complex
Finsle
Me ics
X=F
Fa
_ó
_
_
áF
_
8
~
.
áxa

Óx
a
áua
a=1
2n
_
E
ua
dxa,
a-1
To
desc ibe
how
X
is
ela ed
o
he
complex
S uc u e,
conside
co-
o dina e
exp ession
o
he
complex
S uc u e
on
T*M
.
The
complex
S uc u e
J
on
M
is
exp essed
as
-i)
(2
.5)

J
5
=
,In'
~

whe e

(Jb)
_
(1

0

,
(2
.6)
FINSLER
METRICS
AND
THI
;
KOBAYASIII
METRIC

137
and I
is
he
(n
x
n)-iden i y
ma ix
.
The
na u al
complex
s uc u e
on
T*M
is
hen
gi en
by
a

b
a

_a
__

ba
I
gX
a
I
can
be
di ec ly
checked
ha
he
complex
s uc u e
de ined
by
his
is
independen
o
he
choice
o
coo dina es
.
By
abuse
o
no a ion,
we
will
deno e
his
complex
s uc u e
on
T*M
by J
.
F om
he
dc ini ion
abo e,
i is
clea
ha
J
o
7
*
=
7
*
oJ
.
No e
ha
condi ion
(1
.1)
p o ides
a
compa ibili y
condi ion
o
F
and
he
complex
s uc u e,
which
can
be
exp essed
as ollows
:
Le
Y
be
he
adial
ec o
ield
on Tó
M
gene a ed
by
he
,
ac ion
o IR
1
:>y
nul iplica ion
o
é,
E
I1
Z
.
I
can
be
easily
checked
ha
Y
and
Z
sa is y
he
ela ion
Thus,
i
F
sa is ies
he condi ion
(1
.1),
we
ha e
F(e` )
=
F( )
a,nd
F(e' )
=
e'F( ),
and
he e o e
he
ollowing
iden i ies
hold
:
(2
.8)

ZF
=
0,

YF
=
F
.
No e
ha
he
coo dina e
cxp essions
o
Y
and
Z
a e
(2
.9)

Y
=
u°

09
,

Z=
-Ja
U''
a'9
a
.
The e o e, condi ions
(2
.8)
a e equi alen
o
(2
.10)

.1'
71"
F
a
=
0,

F,,
u'
=F
.
2
.11
Lemma
.
The
ollowing
iden i ies
a e
sa is i ied
:
(2
.12)

[Z,
X]
=
-JX,

[Z,
JX]
=
X,
[Y,
X]
=
X,

[Y,
JX]
=
JX,

[Y,
Z]
=
0
.
P oo
:
The
compu a ions
in
he
p oo
o
his
lemma
a e
based
on
he
iden i ies
(2 .8)-(2
.10)
and
he
ollowing basic
iden i ies
de i ed
om
hem
:
_a
_
_a
__

_a
____a

a

b
a
I
Y~
ax
a
]
-
LZ~
Oxa
]

~~

LY~
au
a ]

a7l
.a

[
Z
'
a_ua

_
-
,%a a7/,b
138

M
.-Y
.
PANG
Now
compu e
[Z,
X]
using
coo di la es
:
[Z,
X]
_
{

1
:~
GiX
_G
Z
F

Fa

a

_
aF

a
ax
a

axa aua
a=1
=
F
E
{
(ZFa)
a
axa
_
ax
aF
a~Z
á
.a)
}-
Obse e
ha
(ZFa)

Z
(au
)

a
BuF
)
+
[Z'
aáa,
F-
(Ja
a ,
b
)
F

Ja,Fb
.
The e o e, using
he
skew-symme y
o I
ba,
we
ob ain
=(YF)
)
J
(J
6
Fb
ax
a
+
'Ia
axa
W
,
)
a=1
-
7
{
F
2n
(Fa
_a

-
(9F

o9
)}
=
DF

(ax
a
aU
a
a=1
=-JX
.
To
show
he
iden i y
[Z,
.IX]
=
X,
do e
haa
.Cz,I
=
0,
and
compu e
[Z,
JX]
=
GZ(JX)
=
J(GZX)
=
J(-JX)
=
X
.
To
p o e
he
iden i y
[Y,
X]
=
X,
no e
lla
YF
a
,
=
Y
( O
F
,
)
=
aua
(YF)
+
Y,
aa
y

F

F
a
-
Fa
=
0
.
Using
his iden i y,
compu e
[Y,
X]
as
ollows
:
[Y,
X]
=GyX
=
Gy

F

(Fa
a

-
aF

a
Y!

axa

axa
au
a
a=1
F
a
0

OF
0
)
axa

axa
au
a
a=1
_
F'
Gy

(F
.

a

_
aF

a
)
}
---
axa

ax
a
(gU
a
a=1
_a
__OF_a
(Íxa

ax
a
(gua
a
2n

()

(

)
}
_aF _a _aF
_
l
Y
ax
a
a~
a
+
axaau-
~y
a=l
~(aa
F
) aáa

ááaa}
=X
.
FINSLGR
MCTRICS
ANDTHG
KOBAYASI-II
ML'TRIC

139
The
iden i y
[Y,
Z]
=
0
is
clea
since
he
ac ions
o e
and
e
-
"
commu e,
and
he
iden i y
[Y,
JX]
=
JX
ollows
om
he
compu a ion
:
[Y,
JX]
=
[Y,
-Gzx]
=
-Gz[Y,
X]
+
[GZY,
X]
=
-GzX
=
JX
.
2
.13
Lemma
.
The
ec o
ields
JX
and
[X,
JX]
sa is y
he
iden i-
ies
:
(2
.14)
((JX)
=
dF(JX)
=
0,

and

(([X,
JX])
=
dF([X,
JX])
=
0
.
In
pa icula ,
JX
and
[X,
JX]
a e
angen
o
he
submani old
SFM
C
To
M
.
P oo
::
To
p o e
he
lemma,
ecall
ha
we
ha e
he
iden i ies
Cx(
=
X
-id(
=-F
dF,
X
F=
0
and
C(Z)
=
0
.
Also,
ecall
om
iden i ies
(2
.8)
and
(2
.12)
ha
Z
F
=
0
and
JX
=
[X, Z]
.
Using
hese
iden i ies
and
he
ac
ha
Gx
is
a de i a ion,
compu e
as ollows
:
((JX)
=
(([X,
Z])
=
«£x
Z)
=
£x«(Z»
-
(£x
C)
(Z)
=
FdF(Z)
=
0
dF(JX)
=
dF([X,
Z])
=
XZ
F
-
ZX
F
=
0
Using
hese
iden i ies
again,
we
comple e
he
p oo
o
he
lemma
:
(([X,
JX])=(
(£x(JX))=Gx{((JX)}
-
(Gxo)
(JX)=FdF(
.IX)=0
dF([X,
JX])
=
X(JX)
F
-
(
.IX)X
F=
0
.
2
.15
The
Exponen ial
Map
.
The
exponen ial
map
is
de ined
simi-
la ly
as in
he
case
o
He mi ian
mani olds
.
To
de ine
i ,
we
in oduce
he
dual
complex
Finsle
me ic
F
:
TM
,
R,
sa is ying
condi ions
(1
.1)
and
(1
.2),
and
de ine
geodesics
as
cu es
wi h
locally
leng h
minimizing
p ope y
wi h
espec
o
F
.
We
b ie ly
e iew
some
concep s
o
he
cal-
culus
o
a ia ions
.
Fo
mo e
de ails
abou
he
calculus
o
a ia ions
see
[GF] and
[S]
.
To
de ine
F,
le
ToM
=
{
E
TM1
:~
0},
and
de ine
:
a
bundle
map
xP
:TóM~ToMby
(2
.16)

T
(w)
=
7

(X

,)

o

w
ETó
M
.
In
coo dina es,
we
ha e
7L
(2
.17)

xPx(w)
_

F
(w)
F,,
(w)
8
á
.,,
a=
I
.
.1
14
6

M
.-Y
.
PANG
Bu
om
Le nma
3
.8,
[X,
JX]
does
no
ha e
a
componen
in
he
di ec-
ion
o
X
and
Xl,
and
hus
[X,
JX]

,
E
L

, .
Again,
om
Le nma
3
.8,
i
ollows
ha
S i
(X)
=
0,
and
we
ob ain
(4
.3)

JX
=
XII

and

[X,
JX]
=
Si
Z
<,
.
F om
he
s uc u e
equa ion
(3
.6), i
can
be
easily
shown
ha
he
iden i y
Ex
Z
a
=
X~
+
i '
(X)
Zp
holds
.
This
iden i y
and
he
second
iden i y
o
(4
.3)
gi es
Ex
[X, .JX]
=
Ex
(Si
Z
a
=
(XSi
)
Z x
+
Si
(,ex
Za)
_
{(X
S-)
Z,,+SpZ ~(X)}
Z
<
,,
+Sl
X
'
.
Bu ,
Again,
since
we
p o ed
(Ex
[X,
JX])
,
E
RX

,
(D
1FI(Xl)

,
®
L,D,
componen s
o
he ec o
Ex
[X,
JX]
in
he
di ec ion
o
X
a o
a
=~
0
has
o
anish
.
The e o e,
i
ollows
ha
Si
=
0 o
a
>
1
and
[X,
JX]
_
Si Z,
and
he condi ion
A2
ollows
.
To
p o e
he
con e se,
suppose
ha
[X,
JX]
= Z
holds
o
some
,
E
C°°(S M)
.
We
p o e
ha
he
su ace
de ined
by
C
,( ,
s)
=
7 (e
x e
-
S
z
iu)
is
a
complex
cu e
o
all
Zu
E
T*M
by showing
ha
J
`~

-(-T,
0)
is
angen
o
he
cu e
C

,
o
small
T
<
0
.

No e
ha
Gx
£xZ =
Ex
[X, Z]
= Ex
(JX)
=
Z, and
le
771
=
e-Tx
u)
.
Using
he
de ini ion
o
he
Lic de i a i e,
we
compu e
2
2
(e* xZ(e X ,))

x{
.CXZ}~~ Xw))
=
.e* x
{GXGXZ}(e xm)
=
K
(
e
x
7u )
{e*
x
z
(
,IX 
,)
}
.
The e 'o e,
i
we
lc
W( )
=
e*
xz(
e
x,b)
E
T
z
(S*M),
W( )
sa is ies
a
second
o de
o dina y
di ie en ial
equa ion
W"( )
=
( )W( )
wi h
ini ial
condi ions
W(0)
=
Z,z
and
W'(0)
=
JX,T,
.
Consequen ly,
we
ha e
W( )
E
span{Z,b,
JXú,}
o
small
,
and
in
pa icula ,
W(' )
_
e*Tx
z
(
, x,D)
E
sean{Z,7
JX,J
.
Subs i u lng
7
=
e-Tx7 ,
(4
.4)

W
(T)
=
e
*
'
x
Z

,
E
span{Zw,
JX

,}
.
Using
he
iden i y
1
(T,
0)
=
7
*
X

,
and
7
*
o
J
=J
o
7
*
,
we
ob ain
a
a
w
(T,
0)
=
ás

{
7
(e
'x
esz7
V)
}
s=o
=
7
*
e'
x
Z

,
=
k
7
*
(JX),í,
=
k
J
7 *
(X,D)
=
kJa
a
w
(T,
0)

FINSLER
METRICS
AND
TIiE
KOBAYAS1i1
METI IC

147
o
some
k
E
IR,
and
hence
J
~áe
(T,
0)
is
angen
o
he
su ace de ined
by
C

,
.
(ü)
I
[X,
JX]
=
Z
holds
o
some
c
E
C'
(S*
M),
hen
he
in olu-
i i y
o
D
ollows
om
he
iden i ies
(4
.5)

[Z,
X]
=
-JX,

[Z,
JX]
=
X

and

[X,
JX]
=
KZ
.
The
con e se
o his
is
an
immedia e
conséquence
o
he he
iden i y
(3 .10)
.
4
.6
Rema k
.
(1)
No e
ha ,
in
he
p oo
o
Theo em
4
.2,
he
wo
equa ions
S i
(X)
_
0
and
SQ
=
0
o
/3
>
1
a e equi alen
o
he
single
condi ion
[X,
JX]
=
Z
wi h
c
=
Si
.

The
condi ion
S i
(X)
=
0
can
be
in e p e ed
as a
compa ibili y condi ion
o
F
and
he
complex
s uc u e
.
Fo
example,
i
F
is
he
induced
no m
o
a
Kaehle
mani old,
his
condi ion
is
sa is ied
.
In
his
case,
he condi ion
Sp
=
0
o
,(j
>
1
pu s
es ic ions
on
he
cu a u e
o
he
Kaehle
me ic
.
One
special
case
o
his
is
when
M
is
a
Kaehle
mani old
wi h
cons an
holomo phic
sec ional
cu a u e
.
In
his
case,
i
can
be
e i ied
using
esul s
in
[P]
ha
Sá
=
cb
í
3
o
some
cons an
e
.
(2)
Condi ions
equi alen
o
Al-A3
o
Theo em
4
.2
we e
in oduced
by
Royden
[R]
and
Fa an
[F]
.
The
condi ions
Al-A3
in
Theo em
4
.2
can
be
equi alen ly
s a ed
as
condi ions
on
Tó
M
.
4
.7
Theo em
.
The
ollowing condi ions
a e
equi alen
:
B1
.

exp(U

)
is
a
complex
cu e o
all
E
TOM
.
132
.
[X,
JX]
=
Z
onTó
M
o
some
e
E
C°°(TO
M)
.
133
.
The
dis ibu ion
CX
®
0Z
C
T(TO
M)
is
in olu i e
.
Mo eo e ,
hese condi ions
a e
equi alen
o
condi ions
Al-A3
in
Theo-
em
4
.2
.
P oo
:
No e
ha
he condi ions
Al
and Bl
a e clea ly
equi alen
.
To
p o e
he
heo em,
we
show
(i)
ha
A2
implicas
B2,
and
(ii)
ha
A3
implies
B3
.
The
con e ses
o
hese
a e
i ial
o
p o e
.
(i)
Suppose
ha
[X,
JX]
= c
Z
holds
on
SFM
o
some
c
E
C°°
(S M)
.
Ex end
K
o
Tó
M
by
c( w)
=
z
K(W)
o
all
>
0
and
w
E
SI*M
.
We
claim
ha
[X,
JX]
=
c
Z
on Tó
M
.
To
show
his,
we
show
ha
bo h
W
=
[X,
JX]
and
W
=
n
Z
on
Tá
M
mus
sa is y
he
o dina y
di e en ial
equa ion
GyW
=2W
.
No e
ha
he
in eg al
cu es o
Y
14
8

M
.-Y
.
PANG
a e
he
adial
li ios
in
Tó
M
.
Thus,
i
bo h
[X,
.IX]
and
c
Z
sa is y
he
equa ion,
hey
nus
coincide
because
he
iden i y
[X,
JX]
=
Z
gi es
he
same
ini ial
condi ion
a
poin s
on
SFM
.
To show
ha
he
ec o
ield
[X,
JX]
sa is ies
he
di e en ial
equa ion,
ecall,
o
Lemma
2
.11,
ha
G
y
X
=X
and
GyJX=JX
.
Using
hese
iden i ies
and
he
ac
ha
Gy
is
a,
de i a io ,
we
compu e
Gy
[X,
JX]=[GyX,
IX]
+[X,
GyJX]=[X,
JX]+[X,
JX]=2[X,
JX]
.
To
show
ha
he
ec o
ield
,
Z
sa is ies
he
di e en ial
equa ion,
ecall
ha
he
ec o
ield
Y
on
Tó
M
is
gene a ed
by
he ac ion
o
II3,
by
mul iplica ion
o
e'
.
Using
homogenei y
o
K,
we
ob ain
he
ollowing
iden i y
:
Fo
w
E TO
M,
d
~~
d ,
=o
(Le
.
GyZ
=
[X,
Y]
=
0),
we
compu e
d

{e2
(w)
}
=
2~
;( ~)
.=o
Using
his iden i y
and
he
ac
ha
he
ec o
ields
Y
and
Z
commu e
{
.Cy
(
Z)}
=
(Y ,)
Z=
2(K
Z)
.
(ii)
The
;
p oo
ha
he
condi ion
A3
i nplics
he
condi ion
B3
im ne-
dia c
;ly
ollows
o
hc
;
iden i ies
in
Lc n na
3
.8
.
4
.8
To ally
Geodesic
Complex
Cu es
.
We
call
a complex
cu e
E
o allly
geodesic
i ,
o
any
angen
ec o
o
he
complex
cu e
E
and
geodesic
segn cn
y
:
(-e,
e)
-
M
such
ha
^c
(0)
=
,
y,
( )
is
co ai ed
in
he
complex
cu e
o
small
.
The
main
esul
o
his
see io
is
l a ,
unde
condi ion
(1
.5),
he
geodesics
o
F
can be
uniquely
ex ended
o
i nme sed
complex
cu es
ha
a e
o ally
geodesic
subman-
i olds
o
M
.
In
ac ,
l ese
cu es
a e
p ecisely
hc
ones
de i ed
by
he
complex
cu es
exp (U

)'s
in
condi ion
(1
.5)
.
Theo em
4
.9
.
I
he
condi ion
(1
.5)
holds,
he
complex
cu e
exp
(U

)
can
be
uniquely
ex ended
o
a
maximal
o ally
geodesic
complex
cu e
:
E
-
M
imme sed
in
M
.
Mo eo e ,
he e
is
a
canonical
(n-1)-
dimensional
holomo phic
ec o
subbundle
T'E
o
*(TM)
ans e sal
o
E
.
P oo
::
Recall
om
Theo em
4
.2
ha
D=
span{X,
JX,
Z}
is
an
in-
olu i e dis ibu ion
.
By
he
F obenius
heo e ,
his
i plies
ha
S*
M
is
olia ed
by
3-di ne sional
maximal
in eg al
sub nani olds
o
D
.
Le
(4
.10)
FINSLGRMETRICS
AND
THC
KOBAYASII1
MGTRIC

149
E
be a
lea o
his
olia ion
.-D,
hen
he e
is
a
well
de ined
S
I
C
e
ac ion
on
E
since
Z
is
angen
o
E
.
Le
us
deno e
he
quo ien
space
E/S
I
by
E
.
I
is
no
di ñcul
o see
ha
E
is
a
complex
cu e
wi h
he
complex
s uc u e
induced
om
he
complex
s uc u e
o
E
C
T*
M,
and
ha
he e
is
a
holomo phic
imme sion
such
ha
he ollowing
diag am
commu es
:
inclusion
E~S m
E
J
M
Recall
ha
he
complex
cu e
exp
(U

)
is
he
su ace de ined
by
(4
.11)

( ,,
s)
-
exp
( esw)
=
n
(e Xe-
.yi4,( ))
.
F om
his,
i is
clea
ha
he
complex
cu e
:
E
-
M
is
locally
de ined
by exp (U
)
since
e'
x
c
-9z
i
E
E
.
F om
his,
i
clea ly
ollows
ha
:
E
-
M
is
o ally
geodesic
.
To
de ine
he
ans e sal
holomo phic
subbundle
T
-
L
E
o
*
(TM),
no e
ha
7 z
E
is
a
p incipal
ci cle
bundle
.
De ine
he
ibe
T,LE
o T
l
Ea xEEbyT~E={ E *(TM)Jw(
*
)=0
o
al¡
wEE
-
,,}
.
I is
clea
ha
T
l
E
is
a
holomo phic
ec o
bundle
wi h dimension
n-1
.
To
show
ha
T~E
is
ans e sal
o E,
suppose
E
TLE
n
TE
.
No e
ha
xP(w)
is
angen
o
E
o
all
w
E
E
y
since
:
xP( )
=
7
*
X

,
and
X
is
angen
o
E
.
Mo eo e ,
'Y(ui)
:,¿
0
because
w
(xP(w))
=
w(7
*
X

,)
=
FF
a
ua
=F
2
=
1 z~
0,
whe e
uJ
=Ea
-l
, a
dx
a
.
Since
,
*
Y'
(w)
E
T,
;
E,
we
ha e
*
=
zkP(w)
o
some
z
E
(E
.
Bu ,
ecall
ha
E
T
L
E,
and
he e o e,
)( )
=
0
.
This
implies
ha
=
0
because
71)
(V)
=w
(zXP(w))
=
z
{w
(xP(w))}
=
z
.
4
.12
The
Holomo phic
Sec ional
Cu a u e
.
I
F
sa is ies
he
p ope y
(1
.5),
he e
is
a
na u al
way
o de ine
holomo phic
sec ional
cu a u e
K
o
F
.
In his
sec io'n,
we
show
ha
K
is
de e mi ied
by
he
smoo h
unc ion
K
in
he condi ion
A2
o
Theo em
4
.2
.
Holomo phic
sec ional
cu a u e
K
o
a complex
Fi isle
ne ic
F
has
been
s udied
by
Wong
and
Royden
[W]
[R]
.
To
de ine
K( )
o
a
uni
ec o
E
T,
M
(Le
.
F( )
=
1),
no e
ha
each
complex
cu e
U
C
M
angen
o
has
a
canonical
complex
Finsle
ne ic,
de ined
by
he
es ic ion
F,1
,
u
:
TU
->
IR
.
In
ac ,
because
U
is
o
complex
15
0

M
.-Y
.
PANG
dimension
one,
i
can
be
easily
seen ha
he
me ic
F1
h
u
is
a
no m
induced
by
a
He mi ian
me ic
g
on
U
:
I
F( )
=
1,
F
((a
+
i/l) )
=
l
a
+
i,31
F( )
=

a2
-+Q2
.
In
[W],
Wong
de ined
he
holomo phic
sec ional
cu a u a
K( )
as
he
sup emum
o
he
Gaussian
cu a u a
o
g a
x
E
U,
whe e
sup emum
is
aken
o e
all
complex
cu es
angen
o
.
In
he
special
case
when
F
is
he
no m
induced
by a
He mi ian
me ic,
his
de ines
he
usual
holomo phic
sec ional
cu a u a
o
He mi ian
me ic
.
Obse e
ha ,
i
F
sa is ies
he
p ope y
(1
.5),
hen
by
Theo em
4
.2,
he e
is
a
na u ally
de ined
o ally
geodesic
complex
cu e
o
he
o m
exp
(U

)
angen
o
.
In
[R],
Royden
showed
ha
he
Gaussian
cu a u a
o
he
induced
me ic
g
a
x on
his
complex
cu e
a ains
he
g ea es
alue
and,
hence,
i
de ines
he
holomo phic
sec ional
cu a ú e
K( )
.
Thus,
he
ollowing
heo em
holds
:
4
.13
Theo em
.
Le ,
N
be
a
complex
submani old
o
M,
and
le
K'
be
he
holomo phic
sec ional
cu a u e
o
he
induced me ic
FITN
on
N
.
Then
he
inequali y
K'( )
<
K
( )
holds e e y
uni
ec o
angen
o
N
.
In
pa icula ,
i
U
C
M
is
a
complex
cu e
angen
o
a
uni
ec o
E
TM,
he
Gaussian
cu a u a o
he
induced,
me ic g
de ined
by
FITU
is
bounded
om
abo e
by
K( )
.
The
unc ion
K
can
be
ega ded
as
a
unc ion
on
he
quo ien
space
S*
MIS'
.
Recall
om
(2
.12)
and(4
.5)
he
iden i ies
[Z,
X]
=
-JX,
[Z,
JX]
=
X
and
KZ=
[X,
,IX],
and
compu e
(ZK)Z
=G
Z(KZ)
=
£z
[x,
JX]
-
[L
ZX,
Jx]
+
[x,
L
Z
(Jx)]
=
[-Jx, Jx]
+
[x,
x]
=
0
.
F om
his
ide i i y,
we
conclude
ZK=
0
.
This
implies
ha
he
unc ion
K
is
in a ian
unde
he
ci cle
ac ion
o
unimodula
complex
numbe s
de ined
by
mul iplica ion
on TO
M,
and
hence
he
unc ion
K
can
be
ega ded
as a
unc ion
on
S*M/S
1
.
4
.14
Theo em
.
I
K
is
he
holomo phic
sec ional
cu a u a
o
F,
o
e e y
uni
ec o
E
TM
.
K
( )
=
K
o
-D( )
Be o e
we
hegin
he
p oo ,
no e
ha
by
Theo em
4
.9
he
complex
cu e
U
C
M
has
a
unique
ex ension
o
a
maximal
o ally
geodesic
complex
cu e
:
E
-+
M,
and
ha
he e
is
a
ci cle
bundle
:
É
,
E
o a
E
.
FINSLP
;I1
METRICS
ANll TIIG
KO 3AYAS1ll
METRIC

151
4
.15
Lemma
.

The
be s
o
he
ci cle
bundle
7 z
:
E
-
E
de ines
a
Legend e
olia ion
.
:L
wi h
espec
o
a
na u al
con ac ,
1- o m
de ned
by
he
pull-back
o
77
o
E
.
The
s uc u e
equa ions
o
his
Legend e
olia ion
a e he
pull
baks
o
E
o
he
equa ions
:
(4
.16)

dB'
=-
7A~',

d =0
1
n
;',

d~'
=K17A0
1
P oo
.
:
The
s uc u e
equa ions
(4
.16)
a e
ob ained
by
pulling
back
he
equa ion
(3
.4)
o
É
.
No e
ha ,
since
JX=
XI
and
Z
=
ZI,
we
lla e
Ba
(JX)
=
~'
(Z)
=
0
o
cx
>
1
.
The e o e,
on
E,
we
lla e
dB'
=
-77
A
;'
+
G'110
1
A
~'
d i=0
1
A~'
d
l
=SigAB
1
+Q
1
10
1
A~
l
.
Bu ,
om
hc
;
iden i ies
(4
.5), i
easily ollows
ha
G' 1
=
Q
11
=
0 a d
Si
=
,
.
Hence,
we
ob ain
he
s uc u e
equa ions
(4
.16)
:
F om
he equa ions,
i is
clea
ha
he
pulí
back
o
É
o
l
is
a
con ac
o m
since
)
A
d
=
A
B'
A

'

0on
E
.
P oo
..
To
p o e
i e
heo e n,
choose
E
T,,M
and
le
U
C
M
be a
o ally
geodesic
complex
cu e
such
ha
x
E
U
a d
E
T,;U
.
Also,
le
:
E
,
M
be
he
unique
ex ension
o
U
desc ibed
in
Theo em
4
.9,
and
le
Éu
deno e
he
es ic ion
o
he
ci cle
bundle
7
:
É
-~
E
o
U
.
Obse e
ha
U
is
also a
submani old
o
S
M/S'
since
U
C
E=
E/S'
C
S M/S'
.
We
claim
ha
he
Gaussian
cu a u a
o
g
o
U
is
Klu
E
C_
(U),
wl c e
K
is
ega ded
as a
unc ion
o
S¡,M/S'
.
The
heo e n
ollows
om
he
claim
.
To
sea
his,
p o eed
as ollows
:
No e
ha
by
commu a i i y
o
he
diag am
(4
.10),
7
=
o
7 z
.
Hence,
i
w
E
Eu,
hen
xP(u )
=
7 ,X
w
=
*
o
(7 ),X

, .
The e o e, he
ap
S M
,
SpM
sends
Eu
in o
he
zeni
a ge
bundle
S
.U
o
g
.
Since
T
is
a
bundle
map
o a
U
such
ha
T(es'u»
=
e`T(w),
kP
maps
u
di
eomo phically
on o
S
.U,
o
equi alen ly,
we
lla e
a
bundle
ap
1
Is,u
=
<PIs,u
:
SU
-
Eu
o a
U
.
Thus,
since
,
is
co s an
along
ibe s
o
Eu
a d
E
T,,U,
we
lla e
,
o
<D( )
=
K(x)
.
B,y hc
;
clai ,
K(x)
is
he
Gaussian
cu a u a
o
g
a
x
E
U,
and
he
iden i y
K( )
=
Koq>( )
ollows
.
To
p o e
he
clai n,
we
deno e
i e
Gaussian
cu a u a
o g
on
U
by
ic
E
C'(U),
and
show
K
=
k
.
Recall
ha ,
' o
P oposi io
3
.5,
hc
;
Legend e
olia ion
on
Sg*U
i as
he
s uc u e
equa ions
(4
.17)

dé'
=-~A
:
',

di7=8'A~',

d~'=k~AB',

15
2

M
.-Y
.
PANC
whe c
:
{H1,
~,
`1}
is
he
in a ian
co ame
on
S*
U
.
The
;
p oo
o
he
claim
is
done by
es
S
ablishing
he
equi alen e
o
he
Legend e
olia ion
on S
.
.U
wi h
,Fz
.
This
is
p o ed
by
showing
ha
he e
is
a
di
eomo phism
0
:
Eu
~
S9
U
such
ha
(4
.18)

0*~= 7,

and

7
o0=7
.
I
ollows
ha
O*Hl
=
0
1
and
?P*~ 1
=
~1,
and
in
pa icula ,
he pull-back
by
0
o
he
equa ions
(4
.17)
a e he
s uc u e
equa ions
(4
.16)
o
.Fz
.
The
iden i y
k
o
0=
,
ollows
.
We
de ine
O(w)
o
w
E
Eu
C
T*M
as
he
by
pull-back
o
w
o
he
angen
space
o
U
.
In
coo dina es,
we
ha e
(4
.19)
l

n+1

1

n
n+1,
. .
.,
2 a
(
:L
)
0,
.
.,O,X'

,0,
.
.,0¡U
, . .
.,26

,4L

2L

~(x
l
xn+1
u1
,9xn+1)
whe e
(x
1
, . .
.,
.xn,
xn+1
,. .
.,
x2,)
is
aken
so
ha
U
C
M
is
locally
de ined
by
. ,°'
=
0
o
a
=~
1,
n
+
1
.
To
comple e
he
p oo
;
i
emains
o
show
ha
(1)

naps
Eu
di eo no pllically
o o
S
.*U,
and
(2)
"

~
=
71
(1)
:
To
show
ha
0
is
a
di eo no phism
on o
Sg*U,
ecall
ha
T
maps
Eu
on o
SU
.
Hence,
i
w
E
Eu,
hen
he
ec o s
xP(w)
and
T(Jw)
_
-JT(w)
'o m
an
o hono mal ame
o T,
;U
o
some
x
E
U
.
The
ol-
lowing
compu a ion
shows
ha
he
co ec o s
0(7u)
and
0(Jul)
=
,Iz/1(w)
o o
he
dual
co ame
o
{xP(w),T(Jul)}
:
Using
he
ide l i ies
(2
.10),
compu e
;
{1h(1u)}
(IP(w))
=
w(7
*
X
w
)
=
FF
<l
U,
a
=
F
2
=
1
{O(Jw)}
(

0%711))
=
{Jo(w)}
(-JT(w))
=
{Y'(
1
V)}
(XP(lu))
=
1
{O(IV)}
(IpG%w))
=
{O(lu)}
(-JP(ul))
=
{-J~J(7U)}
(`P(1 ))
=
{-Jw}(7 *X
,)
=
uaJ,"
F
F,
=
0
.
Hence
z/>(Elx)
C
S
.*U
.
Since
V) is
a
bundle
map
p ese ing
he
ci cle
ac ion,
i
casily
ollows
ha
0
is
a
di
eo lo pllis n
.
(2)
:
To
p o e
lle
iden i y z)*~
=
TI,
ecall
' onl
(2
.2)
ha
l
=
ul
dx
1
+
un
-
"
dx
n+1
.
Tlle e ó e,
onl
(4
.19),
' %1
*
=
ul
dx
1-}-
u
"+1
.
dxn+l
.
On
he
o he
hand,
he
con ac
1- o o
77
on
E
is
de ined
by
he pull-back
o
71
_
1
:2n
.=

o
E
.
Bu ,
since
dx°
=
0
o
a
=,~
1,
n+
1
on
E,
we
ha e
77
=
7x
1
d
x
1
+
u
n+1dxn+1,
Hence
he
iden i y
0*~
=
17
ollows
.
a
The
ollowing
co olla y
is
a
consequence
o
he
p oo
o
Theo em
4
.14
:
FINSI,I
;R
MI
.
:TI ICS
AND
'CII1
;
KOBAYASIII
MI TILIC

153
4
.20
Co olla y
.

The
Gaussian
cu a u a
o
he
induced
me ic
.q
on
E
ás
,,É
.
4
.21
Rela ion
o
he
Kobayashi
Me ic
.
In
his
sec ion,
we
p o e
a
e sion
o a
heo em
o
Fa an
which
s a es
ha
anishing
o
ce ain
local
in a ian s
o es
F
o
be
he
Kobayashi
me ic
o
M,
p o ided
ha
F
is
comple e
and
sa is ies
he
condi ion
K
=
-4
(sea
In oduc ion)
.
In
he
e sion
o
he
heo em
p esen ed
he e,
he condi ion
o
anishing
o
in a ian s
is
eplaced
by
he
equi alen
condi ion
(1
.5)
.
Recall
o i
Lempe 's
esul
desc ibed
in
he
in oduc ion
ha ,
i
D
C
T'
is
a
bounded
s ic ly
linea ly
con ex
domain
wi h
smoo h
bounda y,
hen
c e y
ex emal
disk
:
A
->
D
is
an
iso ne ic
imbedding
(Le
.
*FK
coincides
wi h
he
Poinca é
no m
en
,),
and
ha
(A)
is
a
maximal
o ally
geodesia
co nplex
cu e
;
in
D
.
Since
hc
;
Poinca é
nie ie
has
Gaussian
cu a u a
-4, he
holo no phic
;
sec ional
cu a u c
o hc
;
Kobayashi
nc ic
FK
is
-4
.
O i
he
o he hand,
i
F
is
a
ny
co nplex
Finsle
me ic
:
on
a
co nplex
mani old
M,
he
condi ion
o
cons an
holo no phic
;
sec ional
cu a u a
K
=-4
imposes
a
es ic ion
on
he me ic
F
.
In
aca,
we
show
ha ,
i
F
is
any
comple e
co nplex
Finsle
me ic
;
wi h
he p ope ies
K
= -4
and
(1
.5),
hen
F
nus
coincide
wi h
he
Kobayashi
me ic
.
To
show
his,
we
need
hc
;
ollowing
le ima
due
o
Ahl o s
[A] [K]
:
4
.22
Gene alized
Schwa z
Lemma
.
Le (N,
g)
be
a,
1-dimensional
.
He mi ian
mani old
such
ha
he
Gaussian
cu, ' a u 'e
is
bounded,
abo e
by
a
nega i e
cons an
-C
.
Fo '
any
holo no phic
mal)
:
A
,
N,
he
inequali y
(4 .23)
Il
. * ll
.~
<_
c
Il ll
holds
o
allll
E
TA,
)hc e
II
Ils
is
he
no m
on
N
induced
by
y
and
II
II
deno es
he
no ' n
de ined
by
he
Poinca é
ne ic
on
A
.
We
call
a
co nplex
Finsle
me ic
:
F
comple e
i
he
geodesic
:
ec o
ield
X
is
comple e
;
(o equi alen ly,
i
e e y
geodesic
can
be
;
ex ended
o
a
geodesic
de ine(¡
en
all
o
IR)
.
4
.24
Theo em
[F]
.
Suppose
F
is
a comple e
complex
Finsle me ic
on
a
complex
mani old
M
wi h
cons an e
holomo phic
sec ionall
cu a u a
K
=
-4
sa is yinq
he
p ope y
(1
.5)
.
The
dual
me ic
F
coincides
wi h
he
Kobayashi
me ic
FI<
o
M
.
P oo
.-
We
e
; i y
he
equali y
FK
=F
by
e i ying
hc
;
inequali ies
FK<FandF<FK
.
15
4

M
.-Y
.
PANG
(i)
To show
ha
he
inequali y
FI<
<
F
holds,
le
E
TM
and
ecall
o n
Tlico en
4
.9
ha
he e
is
an
i n ne sed o ally
geodesic
co iplex
cu e
:
E
,
M
angen
o
.
Sinee
F
is
a
comple e
me ic wi h
K
=
-4
and
E
is
o ally
geodesic,
he
induced
me ic
g on
E
de ined
by
*F
is
a
comple e
Kaehle
me ic
wi h Gaussian
cu a u e
-4
(see
Theo em
4
.14
and
Co olla y
4
.20)
.
The e o e, he e
is
a
holo no phic
co e ing
nap
2
:
A
-
E
which
¡s
'a
local
isome y
be ween
he
Poinca é
me ic
and
9
(see
chap e
IX
o
[KN])
.
By
composing
and
2,
we
ob ain
a
holo no phic
nap
o
2
:
A
-
M
ha
is
an
isome ic
imme sion
wi h
espec
o
he
Poinca é me ic
and
F
(Le
.
{( o 2)
*
F}(w)
=
11wil)
.
Recall
ha
he
Kobayasl i
me ic
FK( )
is
de ined
as
he
in i num
o
Ji
.
*
il
o e
all
complex
cu e
:
A
-
M
angen
o
.
Hence, he
inequali y
ollows
:
FK( )
<
11
(
l
o
2)* jj
=
F( )

o

E
TM
.
(ü)
To
p o e
he
inequali y
F( )
<
FK( ),
no e
ha ,
o
each
complex
cu e
:
:
A
->
M
angen
o
E
TM,
he e
is
a
He mi ian
me ic me ic
g on
0
de ined
by
*F
.
By
Theo em
4
.13,
l e
Gaussian
cu a u e
n
is
bounded
abo e
by
-4
.
Applying
he
Genc alized
Schwa z
Lem na
4 .22 o
hc
iden i y
mal)
id
:
->
(0,
g
),
we
ob ain
he
inequali y
{ *F}( »
<_ 11wII,
o
equi alen ly,
F(
*
w)
<_
jjwjj
o
all
uj
E
TA
.
In
pa icula ,
his
implies
ha
F( )
<
li
*
il
o
any
complex
cu e
angen
o
.
Since
FK
( )
is
he
in i num
o
11
*
11
o e
all
such
,
he
inequali y
F( )
<
FK( )
ollows
.
Re e en es
[A]
L
.V
.
AnLFORS,
An
ex ension
o
Schwa z's
lemma,
T ans
.
o
Ame
.
Ma h
.
Soc
.
43
(1938),
359-364
.
[BD]
J
.
BLAND,
T
.
DUCHAMP,
Moduli
o
Poin ed
Con ex
Do nains

In en
.
Ma h
.,
( o
appea )
.
[F]
J
.J
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FARAN,
He mi ian
Finsle
Me ics
and
he
Kobayashi
Me ic,
Jou nal
o Di e en ial
Geome y
31,
no
.
3
(1990),
601-625
.
[GF]
I
.M
.
GEAYAND,
S
.V
.
FOMIN,
"Calculas o
Va ia ions,"
P en-
ice-Hall,
Inc
.,
Englewood
Cli s,
1963
.
[K]
S
.
KOl3AYAS11l,
"Hype bolic
Mani olds
and
Holo no phic
Mappings,"
Ma cel
Dekke ,
Inc
.,
New
Yo k,
1970
.
[KN]
S
.
KOBAYASM,
K
.
NOMizu,
"Founda ions
o Di e en ial
Geom-
e y,"
Vol
.
1
a d
2,
John Wiley
&
Sons,
Inc
.,
New
Yo k,
1963
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FINSL R
METRICS
AND
TI-I
KOBAYASIII
MI
TRIC

155
[L1]
L
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L mP R
, ,
La
me ique
de Kobayashi
e
la
ep esen a ion
des
domains
su
la
boule,
Bull
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Soc
.
Ma h
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F ance
109
(1981),
427-474
.
[L2]
L
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LEMPERT,
In insic
Dis an es
and Holomo phic
Re ac s,
Complex
Analysis
and
Applica ions
81
(1984),
341-364
.
[L3]
L
.
LEMPERT,
Holomo phic
in a ian s,
no mal
o ms,
and
he
moduli
space
o
con ex
domains,
Annals
o
Ma h
.
128
(1988),
43-78
.
[P]
M
.
PANG,
The
S uc u e
o
Legend e
Folia ions,
T ans
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o
Ame
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Ma h
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Soc
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320,
no
.
2
(1990),
417-455
.
[Pa]
G
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PATR1zIO,
Disques
ex emaux
de Kobayashi
e
equa ion
de
Monge-Ampe e
complexe,
C
.
R
.
A ad
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Sci
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Pa i
s
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Se ie
1
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[R]
H
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ROYDGN,
Complex
Finsle
Me ics,
Con empo a y
Ma he-
ma ics,
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o
Summe
Resea ch
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Aug
.
12-18,
Ame
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Ma h
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.,
1984,
pp
.
119-124
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[S]
S
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"Lec u es
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Geome y,"
P en ice
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Englwood
Cli s,
1964
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[W]
B
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WONG,
On
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Cu a u e
o
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Me -
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1 (1977),
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Depa men
o
Ma hema ies
Box
1146
Washing on
Uni e si y
S
.
Louis,
MO
63130-4899
U
.S
.A
.
Rebu
el
12 de
Ma e,
de
1991