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Category Category CONFORMAL HARMONIC FORMS BRANSON GOVER OPERATORS AND DIRICHLET PROBLEM AT INFINITY

CONFORMAL HARMONIC FORMS BRANSON GOVER OPERATORS AND DIRICHLET PROBLEM AT INFINITY

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CONFORMAL HARMONIC FORMS, BRANSON-GOVER OPERATORS AND DIRICHLET PROBLEM AT INFINITY. ERWANN AUBRY AND COLIN GUILLARMOU Abstract. For odd dimensional Poincare-Einstein manifolds (Xn+1, g), we study the set of harmonic k-forms (for k < n2 ) which are C m (with m ? N) on the conformal compactification X¯ of X. This is infinite dimensional for small m but it becomes finite dimensional if m is large enough, and in one-to-one correspondence with the direct sum of the relative cohomology Hk(X¯, ∂X¯) and the kernel of the Branson- Gover [3] differential operators (Lk, Gk) on the conformal infinity (∂X¯, [h0]). In a second time we relate the set of Cn?2k+1(?k(X¯)) forms in the kernel of d + ?g to the conformal harmonics on the boundary in the sense of [3], providing some sort of long exact sequence adapted to this setting. This study also provides another construction of Branson-Gover differential operators, including a parallel construction of the generalization of Q curvature for forms. 1. Introduction Let (M, [h0]) be an n-dimensional compact manifold equipped with a conformal class [h0]. The k-th cohomology group Hk(M) can be identified with ker(d+?h) for any h ? [h0] by usual Hodge-De Rham Theory.

  • invariant spaces

  • n0 ?

  • infinite order

  • sequence

  • natural cohomology maps

  • relative cohomology

  • near ∂x¯

  • dimensional kernel

  • also remark

  • conformal harmonic


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Nombre de lectures

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English

Branson

CONFORMALHARMONICFORMS,BRANSON-GOVEROPERATORS
ANDDIRICHLETPROBLEMATINFINITY.

ERWANNAUBRYANDCOLINGUILLARMOU
Abstract.
ForodddimensionalPoincare´-Einsteinmanifolds(
X
n
+1
,g
),westudythe
setofharmonic
k
-forms(for
k<
2
n
)whichare
C
m
(with
m
∈
N
)ontheconformal
compactification
X
¯of
X
.Thisisinfinitedimensionalforsmall
m
butitbecomes
finitedimensionalif
m
islargeenough,andinone-to-onecorrespondencewiththe
directsumoftherelativecohomology
H
k
(
X
¯
,∂X
¯)andthekerneloftheBranson-
Gover[3]differentialoperators(
L
k
,G
k
)ontheconformalinfinity(
∂X
¯
,
[
h
0
]).Ina
secondtimewerelatethesetof
C
n
−
2
k
+1
(Λ
k
(
X
¯))formsinthekernelof
d
+
δ
g
to
theconformalharmonicsontheboundaryinthesenseof[3],providingsomesort
oflongexactsequenceadaptedtothissetting.Thisstudyalsoprovidesanother
constructionofBranson-Goverdifferentialoperators,includingaparallelconstruction
ofthegeneralizationof
Q
curvatureforforms.

1.
Introduction
Let(
M,
[
h
0
])beann-dimensionalcompactmanifoldequippedwithaconformalclass
[
h
0
].The
k
-thcohomologygroup
H
k
(
M
)canbeidentifiedwithker(
d
+
δ
h
)forany
h
∈
[
h
0
]
byusualHodge-DeRhamTheory.However,thechoiceofharmonicrepresentativesin
H
k
(
M
)isnotconformallyinvariantwithrespectto[
h
0
],exceptwhen
n
isevenand
k
=
2
n
.
Recently,BransonandGover[3]definednewcomplexes,newconformallyinvariantspaces
offormsandnewoperatorstosomehowgeneralizethis
k
=
2
n
case.Moreprecisely,they
introduceconformallycovariantdifferentialoperators
L
k
BG
,`
oforder2
`
onthebundle
Λ
k
(
M
)of
k
-forms,for
`
∈
N
(resp.
`
∈{
1
,...,
2
n
}
)if
n
isodd(resp.
n
iseven).A
particularlyinterestingcaseisthecriticaloneinevendimension,thisis
n(1.1)
L
BG
:=
L
BG
,
2
−
k
.
kkThemainfeaturesofthisoperatorarethatitfactorizesundertheform
L
k
BG
=
G
k
B+G1
d
for
someoperator
(1.2)
G
k
B+G1
:
C
∞
(
M,
Λ
k
+1
(
M
))
→
C
∞
(
M,
Λ
k
(
M
))
andthat
G
k
BG
factorizesundertheform
G
k
BG
=
δ
h
0
Q
k
BG
forsomedifferentialoperator
(1.3)
Q
k
BG
:
C
∞
(
M,
Λ
k
(
M
))
∩
ker
d
→
C
∞
(
M,
Λ
k
(
M
))
where
δ
h
0
istheadjointof
d
withrespectto
h
0
.Thisgivesrisetoanellipticcomplex
GB...
−
d
→
Λ
k
−
1
(
M
)
−
d
→
Λ
k
(
M
)
L
−
k
−→
Λ
k
(
M
)
−
δ
−
h
0
→
Λ
k
−
1
(
M
)
−
δ
−
h
0
→
...
namedthe
detourcomplex
,whosecohomologyisconformallyinvariant.Moreover,the
pairs(
L
k
BG
,G
k
BG
)and(
d,G
k
BG
)onΛ
k
(
M
)
⊕
Λ
k
(
M
)aregradedinjectivelyellipticinthe
sensethat
δ
h
0
d
+
dG
k
BG
and
L
k
BG
+
dG
k
BG
areelliptic.Theirfinitedimensionalkernel
(1.4)
H
Lk
(
M
):=ker(
L
k
BG
,G
k
BG
)
,
H
k
(
M
):=ker(
d,G
k
BG
)
areconformallyinvariant,theelementsof
H
k
(
M
)arenamed
conformalharmonics
,provid-
ingatypeofHodgetheoryforconformalstructure.Theoperator
Q
k
BG
abovegeneralizes
Branson
Q
-curvatureinthesensethatitsatisfies,asoperatorsonclosed
k
-forms,
Q
ˆ
BG
=
e
µ
(2
k
−
n
)
(
Q
BG
+
L
BG
µ
)
kkk1

2ERWANNAUBRYANDCOLINGUILLARMOU
if
h
ˆ
0
=
e
2
µ
h
0
isanotherconformalrepresentative.
ThegeneralapproachofFefferman-Graham[4]fordealingwithconformalinvariants
isrelatedtoPoincare´-Einsteinmanifolds,roughlyspeakingitprovidesacorrespondence
betweenRiemannianinvariantsinthebulk(
X,g
)andconformalinvariantsontheconfor-
malinfinity(
∂X
¯
,
[
h
0
])of(
X,g
),inspiredbytheidentificationoftheconformalgroupof
thesphere
S
n
withtheisometrygroupofthehyperbolicspace
H
n
+1
.AsmoothRiemann-
ianmanifold(
X,g
)issaidtobea
Poincare´-Einsteinmanifold
withconformalinfinity
(
M,
[
h
0
])ifthespace
X
compactifiessmoothlyto
X
¯withboundary
∂X
¯=
M
,andifthere
isaboundarydefiningfunctionof
X
¯andsomecollarneighbourhood(0
,
)
x
×
∂X
¯ofthe
boundarysuchthat
dx
2
+
h
x
(1.5)
g
=
2
x(1.6)Ric(
g
)=
−
ng
+
O
(
x
∞
)
where
h
x
isaone-parameterfamilyofsmoothmetricson
∂X
¯suchthatthereexistsome
familyofsmoothtensors
h
jx
(
j
∈
N
0
)on
∂X
¯,dependingsmoothlyon
x
∈
[0
,
)with
Ph
x
∼
j
∞
=0
h
jx
(
x
n
log
x
)
j
as
x
→
0if
n
+1isodd
)7.1(h
x
issmoothin
x
∈
[0
,
)if
n
+1iseven
(1.8)
h
x
|
x
=0
∈
[
h
0
]
.
Thetensor
h
01
iscalled
obstructiontensor
of
h
0
,itisdefinedin[4]andstudiedfurther
in[9].Weshallsaythat(
X,g
)isasmoothPoincare´-Einsteinmanifoldif
x
2
g
extends
smoothlyon
X
¯,i.e.eitherif
n
+1isevenor
n
+1isoddand
h
jx
=0forall
j>
0.Itis
provedin[6]that
h
01
=0impliesthat(
X,g
)isasmoothPoincare´-Einsteinmanifold.
Theboundary
∂X
¯=
{
x
=0
}
inheritsnaturallyfrom
g
theconformalclass[
h
0
]of
h
x
|
x
=0
sincetheboundarydefiningfunction
x
satisfyingsuchconditionsarenotunique.
AfundamentalresultofFefferman-Graham[4],whichwedonotstateinfullgenerality,is
thatforany(
M,
[
h
0
])compactthatcanberealizedastheboundaryofsmoothcompact
manifoldwithboundary
X
¯,thereisaPoincare´-Einsteinmanifold(
X,g
)for(
M,
[
h
0
]),and
h
x
in(1.7)isuniquelydeterminedby
h
0
uptoorder
O
(
x
n
)anduptodiffeomorphism
whichrestrictstotheIdentityon
M
.Themostbasicexampleisthehyperbolicspace
H
n
+1
whichisasmoothPoincare´-Einsteinmanifoldforthecanonicalconformalstructure
ofthesphere
S
n
,aswellasquotientsof
H
n
+1
byconvexco-compactgroupsofisometries.
IthasbeenprovedbyMazzeo[16]that
1
foraPoincare´-Einsteinmanifold(
X,g
),the
relativecohomology
H
k
(
X
¯
,∂X
¯)iscanonicallyisomorphictothe
L
2
kernelker
L
2
(Δ
k
)of
theLaplacianΔ
k
=(
d
+
δ
g
)
2
withrespecttothemetric
g
,actingonthebundleΛ
k
(
X
¯)
of
k
-formsif
k<
2
n
.Inothertermstherelativecohomologyhasabasisof
L
2
harmonic
representatives.Inthiswork,wegiveaninterpretationofthespaces
H
k
,
H
Lk
intermsof
harmonicformsonthebulk
X
withacertainregularityonthecompactification
X
¯.
Theorem1.1.
Let
(
X
n
+1
,g
)
beanodddimensionalPoincare´-Einsteinmanifoldwith
conformalinfinity
(
M,
[
h
0
])
andlet
Δ
k
=(
d
+
δ
g

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