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TT-EIGENTENSORS FOR THE LICHNEROWICZ
LAPLACIAN ON SOME ASYMPTOTICALLY HYPERBOLIC
MANIFOLDS WITH WARPED PRODUCTS METRICS
ERWANN DELAY
Abstract. Let (M =]0,∞[×N,g) be an asymptotically hyperbolic
manifold equipped with a warped product metrics. We show that there
2existnoTTL -eigentensors with eigenvalue in the essentialspectrumof
the Lichnerowicz Laplacian Δ . If (M,g) is the real hyperbolic space,L
2there is no symmetric L -eigentensors of Δ .L
Keywords : Asymptotically hyperbolic manifold, warped product, Lich-
nerowicz Laplacian, symmetric 2-tensor, TT-tensor, essential spectrum,
asymptotic behavior.
2000 MSC : 35P15, 58J50, 47A53.
1. Introduction
The study of Laplacians acting on symmetric 2-tensors like the Lich-
nerowicz Laplacian Δ is very important to the understanding of geometricL
problems involving deformations of metrics and their Ricci tensor. These
pr appear in a Riemannian context [2] (see also [11], [4], [5] ), in
general Relativity and string theory (see [15], [20], [14], [13] for instance).
Usually, the study of (positivity of) Δ on TT-tensors plays an importantL
role [4] [15].
A natural geometric problem is to ﬁnd a metric with prescribed Ricci
curvature [11], and the inﬁnitesimal version of that problem is to invert the
Lichnerowicz Laplacian on symmetric 2-tensors. In [6], [7] it was shown
that the Ricci curvature can be arbitrarily prescribed in the neighborhood
of the hyperbolic metric on the real hyperbolic space when the dimension is
strictly larger than 9 (see also [9], [10] for some generalizations). In [21], [8]
itisprovedthattheessentialspectrumoftheLichnerowiczLaplacianacting
on trace free symmetric 2-tensors on asymptotically hyperbolic manifolds of
dimension n+1 is the ray
n(n−8)
,+∞ ,
4
with no eigenvalues below if the manifold is the hyperbolic space. The goal
of this article is to study more precisely this essential spectrum for some
particular warped product manifolds. We prove
Date: March 13, 2006.
12 E. DELAY
Theorem. For n ≥ 2, let us consider (N,gb) be an n-dimensionnal
compact Einstein manifold. Let M =]0,+∞[×N equipped with an asymp-
2 2totically hyperbolic metric g =dr +f (r)gb(as in section 2). Then there are
2no L TT-eigentensors of the Lichnerowicz Laplacian Δ with eigenvalueL
embedded in the essential spectrum. For the real hyperbolic space, there are
2no L eigentensors of Δ .L
Asthisresultconcernonlyembeddedeigenvaluesinanessentialspectrum
and it is well know that the essential spectrum is characterized at inﬁnity,
the manifold can be replaced by any manifold which is of the form given
here only outside a compact set.
This theorem is a consequence of the corollary 3.6 and the theorem 6.2.
Note that this result can easily be adapted to Laplacians on symmetric
∗covariant2-tensorsoftheform∇ ∇+curvaturesterms. Thecompleteproof
is in fact given for the rough Laplacian.
n+1Our proof, as the one used by Donnelly [12] for forms on H , consist
to use a Hodge type decomposition theorem for symmetric 2-tensors on a
compact Einstein manifold N (see also [19] for a similar decomposition on
nAdS where N = S ). This decomposition gives a separation of variable
technique to the equations studied here. The problem is then reduced to
asymptotic integration for ordinary diﬀerential equations of the form
00 2 −εr −εry (r)+(α +O(e ))y(r)=O(e ), α≥0, ε>0.
2The asymptotic behaviour of the solutions to these equations and the L
condition will give the result.
As discussed with R. Mazzeo, it is certainly possible to generalize the
result to certain conformally compact manifolds by carefully constructing a
parametrix for the Lichnerowicz Laplacian as he did in [23] for the Hodge
Laplacian. The proof given here has the merit to be purely geometric and
easily tractable.
When the dimension n+1 is less than or equal to 9, 0 is in the essential
2spectrumofΔ . TheresultgivenhereprovesthatΔ staysinjective(inL )L L
2when n+1≥ 3, so its image is not closed (but dense) in L . In particular
Δ is not surjective. This suggests it might be possible to ﬁnd some (atL
least inﬁnitesimally) prohibited directions to prescribe the Ricci curvature
near this space.
The action of the Lichnerowicz Laplacian in the conformal direction cor-
responds to the action of the Laplacian on functions. Since we know that
spectrum (see [23] for instance), we are only interested here about the trace
free direction.
This article is organized as follows : ﬁrst in section 2, we will introduce
all the objects we need along the paper. In section 3 we will see that on the
2real hyperbolic space any L eigentensor must be a TT-tensor. In section 4
we compute the components of the Laplacian of a TT-tensor for a warped
product metric. In section 5, we detail the Hodge type decomposition that
we will need in the proof of the main theorem. In section 6 we give the
proof of the main theorem. The appendix, section 7, recall some results of
asymptotic integration for ordinary diﬀerential equations.TT-EIGENTENSORS ON SOME AH MANIFOLDS 3
Acknowledgments. I am grateful to G. Carron, P. T. Chru´sciel, B. Colbois
and R. Mazzeo for useful conversations, to Ph. Delano¨e and F. Gautero for their
comments on the original manuscript.
2. Definitions, notations and conventions
Let n≥ 2 and let (N,gb) be an n dimensional riemannian manifold. The
riemannianmanifoldswearestudyingthroughoutthepaperafterthesection
3 are of the form M =]0,+∞[×N endowed with a warped product metric :
2 2 2 2 2g =dr +f (r)gb=dr +f (r)dθ ,
where f is a smooth positive function on ]0,∞[. We will say that (M,g) is
an asymptotically hyperbolic manifold if there exist ε > 0 such that, when
r goes to inﬁnity,
10 00 r −εrf(r),f (r),f (r)=e +O(e ) .
2
The basic example studied here is when f =sinh and (N,gb) is the standard
n-dimensional sphere endowed with its canonical metric, so (M,g) is the
real hyperbolic space (minus a point). There are also interesting case when
1 r −r(N,gb) is Einstein with Ric(gb) = κ(n− 1)gb and f(r) = (e − κe ) so
2
Ric(g)=−ng.
We denote by ∇ the Levi-Civita connexion of g and by Riem(g), Ric(g)
respectively the Riemannian sectional and the Ricci curvature of g.
We denote by T the set of rank p covariant tensors. When p = 2, wep
˚denotebyS thesubsetofsymmetrictensorwhichsplitsinG⊕S whereG is2 2
˚the set of g-conformal tensors andS the set of trace-free tensor (relatively2
to g). We observe the summation convention, and we use g and its inverseij
ijg to lower or raise indices.
The Laplacian is deﬁned as
2 ∗4=−tr∇ =∇ ∇,
∗ 2where∇ is the L formal adjoint of∇. The Lichnerowicz Laplacian acting
on symmetric covariant 2-tensors is
4 =4+2(Ric−Riem),L
where
1 k k(Ric u) = [Ric(g) u +Ric(g) u ],ij ik jkj i2
and
kl(Riem u) =Riem(g) u .ij ikjl
For u a covariant 2-tensorﬁeld on M we deﬁne the divergence of u by
j(divu) =−∇ u .i ji
For ω, a one form on M , we deﬁne its divergence
∗ id ω =−∇ ω ,i
the symmetric part of its covariant derivative :
1
(Lω) = (∇ ω +∇ ω ),ij i j j i
24 E. DELAY
∗(note thatL =div) and the trace free part of that last tensor :
1 1 ∗˚(Lω) = (∇ ω +∇ ω )+ d ωg .ij i j j i ij
2 n+1
The well known [2] weitzenb¨ock formula for the Hodge-De Rham Laplacian
on 1-forms reads
∗ kΔ ω =∇ ∇ω +Ric(g) ω .H i i ik
All quantities relative to gbwill have a hat or will be indexed by gb(∇ , div ,g g
bd , Δ ,...)Lg
A TT-tensor (Transverse Traceless tensor) is by deﬁnition a symmetric
divergence free and trace free covariant 2-tensor .
2L denotestheusualHilbertspaceoffunctionsortensorswiththeproduct
(resp. norm)
Z Z
12
2hu,vi 2 = hu,vidμ (resp. |u| 2 =( |u| dμ ) ),g gL L
M M
where hu,vi (resp. |u|) is the usual product (resp. norm) of functions or
tensors relative to g, and the measure dμ is the usual measure relative tog
g (we will omit the term dμ ).g
3. Commutators of some natural operators
We will study the commutator of the Laplacian with the divergence op-
erator in order to apply the result for an eigentensor. We also study the
commutator of the Laplacian with the Killing operatorL needed in section
5. For further references, we will be as general as possible, in particular the
manifold M is not necessarily a product. We ﬁrst begin with the lemma:
Lemma3.1. On a Riemannian manifold (M,g) with Levi-Civita connexion
∇, for all symmetric 2-tensor ﬁeld u, we have
ki ii k k i il l k p k p∇∇ ∇ u = ∇ ∇ ∇ u +R ∇u +2R ∇ u +∇ R u +∇ R uij ij ij pj ipk k l k ilj k jk
1k i il lki p ip= ∇ ∇ ∇ u +R ∇u −2R ∇ u + ∇ Ru +(∇ R −∇ R )u .k ij l ij j k il pj p ij j ip
2
Proof. For a covariant 3-tensor ﬁeld T on M, we have
l l l∇ ∇ T −∇ ∇ T =R T +R T +R T .p q q pkij kij lij iqp klj jqp kilkqp
ip qkContracting this equality with g g , we obtain
kii k k i l(3.1) ∇∇ T −∇ ∇ T =R T .kij kij kilj
For a covariant 2-tensor ﬁeld u on M, we have
p p∇ ∇ u −∇ ∇ u =R u +R u .m k ij k m ij pj ipikm jkm
imContracting this equality with g , we obtain
p p ii i∇∇ u −∇ ∇ u =R u +R u .k ij k ij pj ipk jk
From this last equality, we obtain
(3.2)
p p i p p ik i k i k k k k∇ ∇∇ u −∇ ∇ ∇ u =∇ R u +∇ R u +R ∇ u +R ∇ u .k ij k ij pj ip pj ipk jk k jk
bbbTT-EIGENTENSORS ON SOME AH MANIFOLDS 5
Taking T = ∇ u in equation 3.1 and combining with equation 3.2, wekij k ij
get the annonced equality. The equality in the lemma is due to Bianchi
identities.
Remark. The formula in lemma 3.1 is much simpler when the curvature
kis harmonic (∇ R = 0) or equivalently when Ric(g) is a Codazzi tensorkijl
(∇ R −∇ R = 0) and (or) the tensor u satisfy an equality of the formk ij i kj
∇ u −∇ u +f∇ u =0 (f is any real function).ij i jk kj ik
Corollary 3.2. On a Riemannian manifold (M,g) with Levi-Civita con-
nexion ∇, we have
div ◦Δ =Δ ◦div −DRic,L H
ip ikwhere (DRicu) =(∇ R −∇ R −∇ R )u =(∇ R )u .j p ij j ip i jp j ik
Proof. From lemma 3.1, we have
i k k k kl k i l i∇ (∇ ∇ u −R u −R u +2R u ) = ∇ ∇ ∇ u −R ∇ uk ij ik jk jlik k ij ilj i j
ip+(∇ R −∇ R −∇ R )up ij j ip i jp
k i l i= ∇ ∇ ∇ u −R ∇ uk ij ilj
ip−(∇ R )u .j ip
Also, we have the
Lemma3.3. On a Riemannian manifold with Levi-Civita connexion∇, we
have
Δ ◦L=L◦Δ +DRic,L H
pwhere (DRicξ) =(∇ R −∇ R −∇ R )ξ .ij p ij j ip i jp
Proof. A straightforward computation gives
k k k k l k∇ ∇ ∇ ξ =∇∇ ∇ ξ +R ∇ ξ −2R ∇ ξ −(∇ R −∇ R )ξ .k i j i k j ik j ikjl k ij j ik
TheresultfollowsfromthedeﬁnitionsoftheHodgeandLichnerowiczLapla-
cians.
In particular the Corollary 3.2 and Lemma 3.3 together give a theorem
of Lichnerowicz ( [22] page 29)
Theorem 3.4. On a Riemannian manifold (M,g) with parallel Ricci cur-
vature, we have
div ◦Δ =Δ ◦div,L H
and
Δ ◦L=L◦Δ .L H
Remark. The condition ∇ R −∇ R −∇ R = 0 is equivalent to thep ij j ip i jp
parallel Ricci curvature condition.
Corollary3.5. On a (n+1)-dimensional Riemannian manifold (M,g) with
Levi-Civita connexion ∇ and constant sectional curvature K, for all trace
free symmetric 2-tensor ﬁelds u, we have
i k k i i∇∇ ∇ u =∇ ∇ ∇ u +(n+2)K∇ u ,k ij k ij ij6 E. DELAY
or equivalently
div◦Δ=[Δ−(n+2)K]◦div.
Corollary3.6. On the (n+1)-dimensional real hyperbolic space with n≥2,
2any L symmetric trace free eigentensor ﬁeld is divergence free, and so is a
TT-tensor.
2Proof. If u is an L symmetric trace free eigentensor ﬁeld on the real hyper-
2bolic space, then divu is an L eigenform, but from [12] (see also [23] for a
generalization) this form must be trivial.
4. Components of the divergence and the Laplacian of a
2-tensor
We will compute the components of the divergence and Laplacian for a
symmetric 2-tensor and adapt the result for the case of trace free tensor. In
this section the riemannian manifold (N,gb) is not assumed to be Einstein.
On M =]0,+∞[×N, we consider the metric
2 2g =dr +f (r)gb.
The computation is given in local coordinates adapted to the character of
1 nM. We will use a local coordinate (x ,...,x ) = (x ) on the manifold NA
0 1 n 1 nand the coordinate (x ,x ,...,x )=(r,x ,...,x )=(x ) on M. The indicesi
with big letters are relative to the coordinate on N, for example
2g =1, g =0, g =f gb .00 0A AB AB
The prime in the calculation will denote derivative relative to r.
The Christoﬀel’s symbol of g are
0 C 0Γ =Γ =Γ =0,00 00 A0
0 0 C −1 0 A C CbΓ =−ff gb , Γ =f f δ , Γ =Γ .ABAB A0 C AB AB
If h is a covariant symmetric 2-tensor, we decompose
2 b(4.1) h=udr +ξ⊗dr+dr⊗ξ+h,
bwhere h is the orthogonal projection of h on {r}×N, and ξ is a one form
on{r}×N. In particular we have
bh =u, h =ξ , h =h .00 0A A AB AB
The covariant derivatives of h are
0∇ h = ∂ h =u,0 00 0 00
−1 0 0 −1 0 0∇ h = ∂ h −f f h =ξ −f f ξ ,0 A0 0 A0 A0 A A
−1 0 0 −1 0b b∇ h = ∂ h −2f f h =h −2f f h ,0 AB 0 AB AB ABAB
−1 0 −1 0∇ h = ∂ h −2f f h =d u−2f f ξ ,C 00 C 00 C0 C C
0 −1 0 0 −1 0bb b∇ h = ∇ h +ff gb h −f f h =∇ ξ +ff gb u−f f h ,C A0 C A0 AC 00 AC C A AC AC
0 0b b b∇ h = ∇ h +ff (gb h +gb h )=∇ h +ff (gb ξ +gb ξ ).C AB C AB AC 0B CB 0A C AB AC B CB A
In particular, we can compute the components of minus the divergence of h:
i 0 −1 0 −3 0 A −2 Ab∇ h = u +nf f u−f f h +f ∇ ξ ,i0 AA
i 0 −1 0 −2 Bb b∇ h = ξ +nf f ξ +f ∇ h .iA A BAATT-EIGENTENSORS ON SOME AH MANIFOLDS 7
−2 A −2 bIf Tr h=h +f h =u+f Tr h=0, this yieldg 00 gA
i 0 −1 0 −2 Ab∇ h = u +(n+1)f f u+f ∇ ξ ,i0 A
i 0 −1 0 −2 Bb b∇ h = ξ +nf f ξ +f ∇ h .iA A BAA
We thus obtain the
Lemma 4.1. If h is a symmetric covariant 2-tensor decomposed as in 4.1
then the components of its divergence are
0 −1 0 −3 0 −2 ∗b(divh) = −(u +nf f u−f f Tr h−f d ξ),0 g g
0 −1 0 −2 b(divh) = −(ξ +nf f ξ−f div h) .A Ag
If moreover h is trace free, the ﬁrst component becomes
0 −1 0 −2 ∗(divh) = −(u +(n+1)f f u−f d ξ).0 g
Inasimilarway, wecomputethecomponentsoftheLaplacianofh. After
a straightforward calculation we obtain the following Lemma
Lemma4.2. Let h be a trace free covariant symmetric 2-tensor decomposed
as is 4.1. The components of minus the Laplacian of h are
k 00 −1 0 0 −2 0 2 −2 A −3 0 Ab b b∇ ∇ h = u +nf f u −2(n+1)f (f ) u+f ∇ ∇ u−4f f ∇ ξ ,k 00 A A
k 00 −1 0 0 −2 0 2 −1 00∇ ∇ h = ξ +(n−2)f f ξ −[(2n+1)f (f ) +f f ]ξk A0 AA A
−2 D −3 0 D −1 0b b b+f ∇ ∇ ξ −2f f ∇ h +2f f d u,D A AD A
k 00 −1 0 0 −2 0 2 −1 00b b b∇ ∇ h = h +(n−4)f f h −[(2n−4)f (f ) +2f f ]hk AB ABAB AB
−2 D −1 0 0 2bb b b b+f ∇ ∇ h +2f f (∇ ξ +∇ ξ )+(f ) gb u.D AB A B B A AB
In particular, if h is divergence free,
k 00 −1 0 0 −2 0 2 −2 Ab b∇ ∇ h = u +(n+4)f f u +2(n+1)f (f ) u+f ∇ ∇ u,k 00 A
k 00 −1 0 0 −2 0 2 −1 00∇ ∇ h = ξ +(n)f f ξ −[f (f ) +f f ]ξk A0 AA A
−2 D −1 0b b+f ∇ ∇ ξ +2f f d u,D A A
k 00 −1 0 0 −2 0 2 −1 00b b b∇ ∇ h = h +(n−4)f f h −[(2n−4)f (f ) +2f f ]hAB ABk AB AB
−2 D −1 0 0 2b b b b b+f ∇ ∇ h +2f f (∇ ξ +∇ ξ )+(f ) gb u.D AB A B B A AB
In order to study the action of the zero order terms of the Lichnerowicz
Laplacian, we compute the curvatures of g. A straightforward computation
shows that the non trivial components of the Riemann Christoﬀel curvature
are
A −1 00 AR = −f f δ ,0B0 B
0 00R = −ff gb ,ABA0B
A A 0 2 A AbR = R −(f ) (gb δ −gb δ ).BD BCBCD BCD C D
Thus the non trivial components of the Ricci curvature are
−1 00 00 0 2bR =−nf f , R =R −[ff +(n−1)(f ) ]gb .00 AB AB AB
bbbbb8 E. DELAY
Then, the zero order terms that appear in the Lichnerowicz Laplacian are
−3 00 −1 00b2(Riemh) = −2f f Tr h=2f f u,00 g
−1 002(Rich) = −2nf f u,00
−1 002(Riemh) = 2f f ξ ,0A A
−2 C −1 00 −2 0 2b2(Rich) = f R ξ −((n+1)f f +(n−1)f (f ) )ξ ,0A C AA
−2 AB −2 0 2 00b b b b2(Riemh) = 2f R h +2f (f ) (h −Tr hgb )−2ff ugb ,AB ACBD AB g AB AB
−2 AB −2 0 2b b b= 2f R h +2f (f ) h ,ACBD AB
−2 −1 00 −2 0 2b b2(Rich) = 2f (Ric h) −2(f f +(n−1)f (f ) )h .AB AB ABg
Lemma4.3. Let h be a trace free covariant symmetric 2-tensor decomposed
as is 4.1. The components of the Lichnerowicz Laplacian of h are
00 −1 0 0 −2 A −3 0 Ab b bΔ h = −u −nf f u −f ∇ ∇ u+4f f ∇ ξ ,L 00 A A
00 −1 0 0 −2 0 2 −1 00Δ h = −ξ −(n−2)f f ξ +[(3n+4)f (f ) +nf f ]ξL A0 AA A
−2 D C −3 0 D −1 0b b b b−f (∇ ∇ ξ −R ξ )+2f f ∇ h −2f f d u,D A C AD AA
00 −1 0 0 −2 0 2b b bΔ h = −h −(n−4)f f h −4f (f ) hL AB ABAB AB
−2 −1 0 0 2b b b b+f Δ h −2f f (∇ ξ +∇ ξ )−(f ) gb u.L AB A B B A AB
In particular, if h is divergence free, we obtain
00 −1 0 0 −2 0 2 −1 00 −2 Ab bΔ h = −u −(n+4)f f u −2(n+1)(f (f ) +f f )u−f ∇ ∇ u,L 00 A
00 −1 0 0 −2 0 2 −1 00Δ h = −ξ −nf f ξ −[(n+2)f (f ) +(n−2)f f ]ξL A0 AA A
−2 D C −1 0b b b−f (∇ ∇ ξ −R ξ )−2f f d u,D A C AA
00 −1 0 0 −2 0 2b b bΔ h = −h −(n−4)f f h −4f (f ) hL AB ABAB AB
−2 −1 0 0 2b b b b+f Δ h −2f f (∇ ξ +∇ ξ )−(f ) gb u.L AB A B B A AB
5. Hodge type decomposition for trace free symmetric
2-tensors
2We recall here how to construct an L orthonormal eigenbasis for trace
free symmetric covariants 2-tensors on a compact Einstein manifold N with
Ric(gb)=κ(n−1)gb.
2WerecalltheclassicalL -orthogonalHodge-deRahmdecompositionforone
forms on N :
∞ ∗(5.1) C (N,T )=Imd ⊕Kerd .1 g g
2We also recall the L -orthogonal decomposition for trace free symmetric
2-tensors on N [1] :
∞ ˚ ˚(5.2) C (N,S )=ImL ⊕Kerdiv ,2 g g
1 ∗˚ ˚where, we recall that for a one form ξ,L ξ =L ξ+ d ξgbis the gb-trace freeg g n g
part ofL ξ.g
2Let (Ψ ) be an orthonormal basis of L (N) of eigenfunctions, withi i∈N
∗∇ ∇ Ψ =λ Ψ , 0=λ <λ ≤λ ≤...g i i i 0 1 2g
bbbbbbbbbbbbbTT-EIGENTENSORS ON SOME AH MANIFOLDS 9
For all i∈N\{0}
∗bΔ dΨ =λ dΨ ⇐⇒∇ ∇ dΨ =[λ −(n−1)κ]dΨ ,H i i i i i igg
and Z Z Z
∗hdΨ ,dΨ i dθ = Ψ ∇ ∇Ψ dθ =λ Ψ Ψ dθ.i j g j i i i j
N N N
From the Hodge decomposition theorem (see in particular equation 5.1),
12 √the L (N,T ) orthonormal family { dΨ} can be completed with1 i i∈N\{0}λi
2an L (N,T ) orthonormal family{ν } with1 j j∈N
∗ ∗∇ ∇ ν =μ ν , d ν =0g j j j jg g
2to obtain an orthonormal basis {η } of L (N,T ) composed of eigen-k k∈N 1
forms :
∗∇ ∇ η =α η .g k k kg
1Thank’s to some easy integrations by parts, we get:
Z Z
1 n−2 ∗ ∗˚ ˚hLη ,Lη i dθ = [hΔ η ,η i −(n−1)κhη ,η i + d η d η ]dθk q g g k q g k q g k qg g2 nN N Z
α −(n−1)κ k ∗ ∗ hη ,η i dθ if d η or d η =0,k q g k q g g2 N Z
n−1 2= √ √ [ λ λ −κ(λ +λ )] Ψ Ψ dθi j i j i j n2 λ λ i j N 1 1 √ √if η = dΨ and η = dΨ ,k i q jλi λj
1 ˚so the family{ Lη } is orthonormal. If the sectional curvature isk k∈N˚kLη kk 2L
constant, we also have
∗ ˚ ˚∇ ∇ L η = [α −κ(n+2)]L η .g g k k g kg
In this case from the decomposition of trace free symmetric 2-tensors (see
12 ˚ ˚equation 5.2), the L (N,S ) orthonormal family { Lη } can be2 k k∈N˚kLη kk 2L
2 ˚completed with an L (N,S ) orthonormal family{σ} such that2 l l∈N
∗∇ ∇ σ =βσ, div σ =0,l l l lg gg
2 ˚to obtain an orthonormal basis {Υ } of L (N,S ) composed of eigen-m m∈N 2
tensors, with
∗∇ ∇ Υ =γ Υ .g m m mg
In the more general case where N is only Einstein, then we have
b ˚ ˚Δ L η = [α +κ(n−1)]L η ,L k k kg g
1 ˚So the orthonormal family { Lη } can be completed with ank k∈N˚kLη kk 2L
2 ˚L (N,S ) orthonormal family{π} such that2 l l∈N
bΔ π =βπ, div π =0,L l l l g l
1This is at this step we really need (N,g) to be Einstein because the Laplacian Δ −g
Ric(g) that appear in the calculation is not the Hodge Laplacian due to the fact we work
on symmetric 2-tensors and not on (skew symmetric) 2-forms
bbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbb10 E. DELAY
2 ˚to obtain an orthonormal basis{Φ } of L (N,S ) of eigentensors, withm m∈N 2
bΔ Φ =ε Φ .L m m m
n+1Remark. When (N,gb) is the standard unit sphere ofR or the real pro-
jective space, the spectrum of the Lichnerowicz Laplacian is given explicitly
in [3] (see also [17] and [18]).
6. Conclusion
In this section we work near the inﬁnity of asymptotically hyperbolic
warped product manifold ]0,+∞[×N equipped with the metric
2 2g =dr +f (r)gb,
where gb is for the moment assumed to be of constant sectional curvature.
2nTheorem6.1. For any λ≥ +2, there is no non-trivial trace free, diver-4
2gence free, L covariant symmetric 2-tensor on (]0,+∞[×N,g) satisfying
∗∇ ∇h=λh.
Proof. For h a symmetric 2-tensor ﬁeld on ]0,+∞[×N, we decompose
2 bh=udr +ξ⊗dr+dr⊗ξ+h,
bwhere h is the orthogonal projection of h on {r}×N and ξ is a one form
2on {r}×N. We assume ﬁrst that h is a trace free L eigentensor for an
2neigenvalue λ> +2 so
4
∗∇ ∇h=λh.
We assume also that h is divergence free (see lemma 4.1):
0 −1 0 −2 ∗b(6.1) u +(n+1)f f u−f d ξ =0,
0 −1 0 −2 b(6.2) ξ +nf f ξ−f div h=0.g
From Lemma 4.2, we have for the components of h:
00 −1 0 0 −2 0 2 −2 ∗(6.3) u +(n+4)f f u +[2(n+1)f (f ) +λ]u−f ∇ ∇ u=0,gg
00 −1 0 0 −2 0 2 −1 00 −2 ∗ −1 0(6.4) ξ +nf f ξ −[f (f ) +f f −λ]ξ−f ∇ ∇ ξ+2f f d u=0,g gg
00 −1 0 0 −2 0 2 −1 00b b bh +(n−4)f f h −[(2n−4)f (f ) +2f f −λ]h
(6.5) −2 ∗ −1 0 0 2b−f ∇ ∇ h+4f f L ξ+(f ) ugb=0.g gg
The gb-trace free part of equation 6.5 gives
00 −1 0 0 −2 0 2 −1 00˚ ˚ ˚h +(n−4)f f h −[(2n−4)f (f ) +2f f −λ]h
(6.6) 1−2 ∗ −1 0 ∗˚ b−f ∇ ∇ h+4[f f L ξ+ d ξgb]=0,g gg n
˚ bwhere h is the gb-trace free part of h.
n+4 n n−4− − −˚2 2 2With the change of variables u = f v, ξ = f ζ and h = f b,
equations 6.1 and 6.2 become respectively
n n−1 0 −1 ∗
2 2(6.7) (f v) =f d ζg
n n 10 −2
2 2(6.8) (f ζ) =f (div b+ f d v),g gn
bbbbbbbbbbbbbbb
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