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SPINc GEOMETRY OF KAHLER MANIFOLDS AND THE HODGE LAPLACIAN ON MINIMAL

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Niveau: Supérieur, Doctorat, Bac+8
SPINc GEOMETRY OF KAHLER MANIFOLDS AND THE HODGE LAPLACIAN ON MINIMAL LAGRANGIAN SUBMANIFOLDS O. HIJAZI, S. MONTIEL, AND F. URBANO Abstract. From the existence of parallel spinor fields on Calabi- Yau, hyper-Kahler or complex flat manifolds, we deduce the ex- istence of harmonic differential forms of different degrees on their minimal Lagrangian submanifolds. In particular, when the sub- manifolds are compact, we obtain sharp estimates on their Betti numbers. When the ambient manifold is Kahler-Einstein with pos- itive scalar curvature, and especially if it is a complex contact manifold or the complex projective space, we prove the existence of Kahlerian Killing spinor fields for some particular spinc struc- tures. Using these fields, we construct eigenforms for the Hodge Laplacian on certain minimal Lagrangian submanifolds and give some estimates for their spectra. Applications on the Morse index of minimal Lagrangian submanifolds are obtained. 1. Introduction Recently, connections between the spectrum of the classical Dirac operator on submanifolds of a spin Riemannian manifold and its ge- ometry were investigated. Even when the submanifold is spin, many problems appear. In fact, it is known that the restriction of the spin bundle of a spin manifoldM to a spin submanifold is a Hermitian bun- dle given by the tensorial product of the intrinsic spin bundle of the submanifold and certain bundle associated with the normal bundle of the immersion ([2, 3, 6]).

  • vector bundles

  • dirac operator

  • spin bundle

  • trivial spin

  • kahler manifold should

  • parallel spinor

  • bundle associated

  • manifold

  • complex manifolds


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INSTITUT NATIONAL DE LA STATISTIQUE ET DES ETUDES ECONOMIQUES
Série des Documents de Travail du CREST
(Centre de Recherche en Economie et Statistique)








n° 2005-24

Econometrics of Individual
Labor Market Transitions

1D. FOUGERE
2T. KAMIONKA




























Les documents de travail ne reflètent pas la position de l'INSEE et n'engagent que
leurs auteurs.

Working papers do not reflect the position of INSEE but only the views of the authors.

1 CNRS and CREST-INSEE, CEPR and IZA. fougere@ensae.fr
2nd CREST-INSEE, Paris. kamionka@ensae.fr ECONOMETRICS
OF
INDIVIDUAL
k
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and
LABOR
er
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a
TRANSITIONS
P
Denis
a@ensae.fr
F
fougere@ensae.fr
OUGERE
Kamionk
CNRS
CNRS
and
CREST-INSEE,
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aris.
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n

)
1
where
;
k
:
8
:
>
:;
<

>
n
:
;

x
;


n
)
g
k
(3)
0
where

n
;
is

the
)
n
1
um
k
b
n
er

of
+1
transitions,
0)
i.e.
k
the
n
n
1
um
us
b
a
er
ell
of
a
mo
erio
dications,
of
of
delimited
the
y
studied
w
pro


tran-
during
The
the
of
p
pro
erio
is
d

[
v

y
0
=
;
u

;
e


)
.
where
This
k
ev
the
en
of
t-history
ell

and
b

e
is
equiv
state
alen

tly
y
dened
indi-
as:
at
!

=
.
n
The

treatmen
0
of
;
sp
u
has
1
een
;
b
x


and
0
(1984),
+
h
u
and
1
a
;
42.1.2
Distributions
of
S
j
u
sp
dened
ell
;
durations.
y
Supp
y
ose
`
no
`
w
1
that
;
the
0
pro



en
z
ters
:;
state
or
x
u

follo
`
;
1
(12)
(
u
x
=

;
`
:
1
u
2
is
f
journ
1
;
;
j
:
(
:
)
:;
`
K
j
g
=
)
j
at
d
time
y

u
`
z
1
y
(
`
`
(8)
=
;
1
z
;
(
:
:;
:
)
:;
y
n
1
+
the
1)
;
.
z
Let
exists
us
function
examine
`
the
u
probabilit
y
y
=
distribution
:
of
;
the
0
so
;
journ
log
duration
:
in
;
state
as
x
:

;
`
#
1
<u
en
u
tered
y
after
rom
the
S
(
:
`

1)
t
-th
y
transition
dt
of
;
the
z
pro
:;

;
F
)
or
f
that
y
purp
:
ose,
1
w

e
du
assume
j
that
:
this
`
so
;
journ
d
duration
u
is
;
generated
y
b
z
y
(9)
a
f

y
probabilit
:
y
1
distribution

P
then
giv
func-
en
hazar
the
the
ev
in
en
sp
t-history
h
(
y
y
:
0
1
;

:
(
:
0
:;
:;
y
;
`
)
1
j
)
:
and
`
a
;
v
d
ector
(
of
0
exogenous
:;
v
;
ariables
)
z
alen
,
(
dened
0
b
:;
y
;
the
)

d
ulativ
Pr
e
U
distribution
d
function
`
F
y
(
:
u
1
j
(11)
y
,
0
that:
;
u
:
;
:
y
:;
z
y
=
`
h
1
y
;
:
z
1
;


H
)
y
=
:
Pr
1
[

U
:
`
y

1
u
z
j

y
dt
0
and
;
(
:
j
:
0
:;
:
y
:;
`
`
1
;
;
;
z
)
;
d

F

u
=
y
1
;
S
:
(
y
u
1
j
z
y

0
=
;
du
:
(
:
j
:;
0
y
:
`
:;
1
`
;
;
z
;
;
)

If
)
function
(7)
(
where
j

0
is
:
a
:;
v
`
ector
;
of
;
unkno
)
wn

parameters.
there
Here
a
U
tion,
`
the
denotes
d
the
of
random
so
v
duration
ariable
the

th
onding
ell,
to
as
the
(
duration
j
of
0
the
:
`
:;
th
`
sp
;
ell
;
of
)
the
f
pro
u

y
starting
;
with
:
its
y
(
1
`
z
1)

th
S
transition.
u
S
y
(
;
u
:
j
y
y
1
0
z
;

:
=
:
du
:;
S
y
u
`
y
1
;
;
:
z
y
;
1

z
)

is
(10)
the
equiv
sur-
tly
vivor
h
function
u
of
y
the
;
so
:
journ
y
duration
1
in
z
the

`
du
th
lim
sp
u
ell.
0
If
[
the

probabilit
`
y
+
distribution
u
P
U
admits

a
;
densit
0
y
:
f
:;
with
`
resp

ect
u
to
F
the
(9)
Leb
it
esgue
ws
measure,
log
then:
(
F
j
(
0
u
:
j
:;
y
`
0
;
;
;
:
)
:
R
:;
0
y
(
`
j
1
0
;
:
z
:;
;
`

;
)
;
=
)
Z
=
u
(
0
j
f
0
(
:
t
:;
j
`
y
;
0
;
;
)
:
5The
function
H
`
history
;
z
(
u
u
1
j
function
y
b
0
(
;
0
:
where
:
;
:;
to
y
k
`
x
1
the
)
)
is
y

Z
the
`

;
inte

gr
z
ate
x
d
ell
haz-

ar
y
d
the
function
to
of
y
the
e
so
:;
journ
0
in
k
the
(15)
`
;
th
y
sp
(16)
ell,
;
giv
k
en
y
the
=
history
u
of
y
the

pro
journ

b
up
the
to

time


Let
`
;
1
z
.
densit

journ

b
mo
and
dels
;
of
z
lab
hazard
our-mark
the
et
0
transitions
;

u;
b
:;
e
)
view
;
ed
z
as
u;
extensions
:;
of
)

k
eting
:
risks

duration
:;
mo
;
dels
)
or
(
m
y
ulti-states
:
m
1
ulti-sp

ells
(
duration
;`
mo
y
dels.
:
These
1

is
will
surviv
no
the
w
in
b
`
e
a
sp
k
ecied.
th
2.1.3
the
Comp
en
eting
the
risks
to
duration
1
mo
(
dels
y
Let
:
us
1
assume

that
the
the
function
n
t
um
in
b
`
er
a
of
k
states
k
K
y
is
:

1
greater

than

2
Then
(
v
K
h
>
j
2)
:
and
`
that,
;
for
g

j
h
:
sp
`
ell,
;
there
(
exists
y
(
:
K
1
1)

indep
S
enden
j
t
:
laten
`
t
;
random
exp
v
0
ariables,
t
denoted
;
U
y
?
z
k
dt
;`
:
(
y
k
1
6
z
=

x
(14)

S
`
u;
1
j
;
0
k
:
2
:;
E
`
).
;

;
h
)
random
Pr
v
U
ariable
k
U


j
k
0
;`
:
represen
:;
ts
`
the
;
laten
)
t
the
so
ditional
journ
al
duration
of
in
so
state
duration
x
state


`
1
1
efore
b
transition
efore
state
a
during
transition
`
to
sp
state
of
k
pro
(
giv
k
the
6
of
=
pro
x
up

time
`
`
1
.
)
g
during
u;
the
j
`
0
th
:
sp
:;
ell
`
of
;
the
;
pro
)

e
The

observ
y
ed
of
so
laten
journ
so
duration
duration
u
state
`

is
1
the
efore
minim
transition
um
state
of
,
these
h
(
(
K
j
1)
0
laten
:
t
:;
durations:
`
u
;
`
;
=
)
inf
asso
k

6
function.
=
w
x
ha

e
`
relations:
1
k
n
u
u
y

;
k
:
;`
y
o
1
(13)
z
Then,

for
=
an
(
y
k

y
`
;
1
:
2
y
!
1
:
z
S

(
S
u
u;
j
j
y
0
0
:
;
:;
:
`
:
;
:;
;
y
)
`
and
1
(
;
k
z
y
;
;

:
)
y
=
1
K
z
Y

k
=
=1

k
u
6
h
=
(
j
j
S
0
(
:
u;
:;
k
`
j
;
y
;
0
)
;

:
6Let
us
remark
(14)
(
;
ulti-states
and
1
(16)
z
imply:
`
S
y
(
L
u
righ
j
y
y

0
del,
;
;
:
the
:
Z
:;
;
y
:
`
sp
1
`
;
1
z
the
;
the

In
)

=
=1
exp
z
0
:;
B
en

Y
Z
ector
u
=1
0
:
X

k

6
the
=
for
x
=


`
:
1
2.1.4
h
These
k
preceding
(
del,
t
single
j
with
y
ells
0
lik
;
follo
:
n
:
`
:;
y
y
(19)
`
0
1
;
;
of
z
y
;
;

y
)

dt
tion
1
=
C
`
A
j
(17)
`
Th
S
us
j
the
n

last
densit

y
the
function
is
of
1
the

observ
(
ed

so
1
journ
;
duration
y
in
z
state
ells
j
mo
during
dels
the
of
`
enden
th
risks
sp
h
ell
of
of
ell
the
sp
pro
ultiple

m
giv
ulti-states
en
t
that
o
this
has
sp
form:
ell
)
starts
Y
at
(
time
y

:
`
1
1

and
f
ends
j
at
:
time
`

)
`
densit
1
`
+
0
u
;
b
y
y
:
a
1
transition
1
to
z
state
a
k
parameters.
,
implies
is:

f
Y
(
(
u;
`
k

j
0
y
:;
0
;
;
)
:
+1
:

:
0
y
:;
`
z
1
(20)
;
of
z
side
;
(20)

tribution
)
observ
=
whic
h

k
general
(
;
u
;
j
)
y
Pr
0
U
;
>
:
e
:
`
:;
j
y
0
`
:
1
:;
;
`
z
;
;
)

Multi-sp
)
m
;
duration

dels
exp
mo

are
Z
extension
u
the
0
indep
K
t
X
eting
k
mo
0
whic
=1
treats
k

0
a
6
sp
=
(the
x
th

ell)
`
m
1
destinations.
h
the
k
ulti-sp
0
m
(
mo
t
the
j
ypical
y
eliho
0
d
;
tribution
:
the
:
wing
:;
L
y

`
=
1
+1
;
`
z
f
;
y

j
)
0
dt
:

:;
(18)
`
This
;
is
;
the
)
lik
where
eliho
(
o
`
d
y

;
tribution
:
of
y
the
1
`

th
is
sp

ell
y
when
Y
this
giv
sp
Y
ell
=
is
0
not
Y
righ
=

1
(i.e.
:
when
:;

`
`
=
=
`

;
`
=
1
and
+
is
u
v

of

Deni-
e
(18)
).
that:
When
(
the
)
`
n
th
`
sp
f
ell

lasts

more
1
than
x

`
e
y

;
`
:
1
y
,
1
the
z


tribution

of
n
this
(
sp
e
ell
n
to
y
the
;
lik
:
eliho
y
o
;
d
;
function
)
is:
The
S
term
(
the

t-hand
e
pro

in
`
is
1

j
of
y
last
0
ed
;
ell,
:
h
:
righ
:;
References
y
a
`
7