TT-EIGENTENSORS FOR THE LICHNEROWICZ LAPLACIAN ON ...

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
exist no TT L 2 -eigentensors with eigenvalue in the essential spectrum of
the Lichnerowicz Laplacian ∆ L. If (M, g) is the real hyperbolic space,
there is no symmetric L 2 -eigentensors of ∆ L.
Keywords : Asymptotically hyperbolic manifold, warped product, Lichnerowicz
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 Lichnerowicz
Laplacian ∆ L is very important to the understanding of geometric
problems involving deformations of metrics and their Ricci tensor. These
problems 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) ∆ L on TT-tensors plays an important
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]
it is proved that the essential spectrum of the Lichnerowicz Laplacian acting
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.
1

2 E. DELAY
Theorem. For n ≥ 2, let us consider (N, ĝ) be an n-dimensionnal
compact Einstein manifold. Let M =]0, +∞[×N equipped with an asymptotically
hyperbolic metric g = dr 2 +f 2 (r)ĝ (as in section 2). Then there are
no L 2 TT-eigentensors of the Lichnerowicz Laplacian ∆ L with eigenvalue
embedded in the essential spectrum. For the real hyperbolic space, there are
no L 2 eigentensors of ∆ L .
As this result concern only embedded eigenvalues in an essential spectrum
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
covariant 2-tensors of the form ∇ ∗ ∇+ curvatures terms. The complete proof
is in fact given for the rough Laplacian.
Our proof, as the one used by Donnelly [12] for forms on H n+1 , 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
AdS where N = S n ). 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
y ′′ (r) + (α 2 + O(e −εr ))y(r) = O(e −εr ), α ≥ 0, ε > 0.
The asymptotic behaviour of the solutions to these equations and the L 2
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
spectrum of ∆ L . The result given here proves that ∆ L stays injective (in L 2 )
when n + 1 ≥ 3, so its image is not closed (but dense) in L 2 . In particular
∆ L is not surjective. This suggests it might be possible to find some (at
least infinitesimally) prohibited directions to prescribe the Ricci curvature
near this space.
The action of the Lichnerowicz Laplacian in the conformal direction corresponds
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
real hyperbolic space any L 2 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. Chruściel, B. Colbois
and R. Mazzeo for useful conversations, to Ph. Delanoë and F. Gautero for their
comments on the original manuscript.
2. Definitions, notations and conventions
Let n ≥ 2 and let (N, ĝ) be an n dimensional riemannian manifold. The
riemannian manifolds we are studying throughout the paper after the section
3 are of the form M =]0, +∞[×N endowed with a warped product metric :
g = dr 2 + f 2 (r)ĝ = dr 2 + f 2 (r)dθ 2 ,
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,
( )
1
f(r), f ′ (r), f ′′ (r) = e r 2 + O(e−εr ) .
The basic example studied here is when f = sinh and (N, ĝ) 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
(N, ĝ) is Einstein with Ric(ĝ) = κ(n − 1)ĝ and f(r) = 1 2 (er − κe −r ) so
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 p the set of rank p covariant tensors. When p = 2, we
denote by S 2 the subset of symmetric tensor which splits in G ⊕ ˚S 2 where G is
the set of g-conformal tensors and ˚S 2 the set of trace-free tensor (relatively
to g). We observe the summation convention, and we use g ij and its inverse
g ij to lower or raise indices.
The Laplacian is defined as
△ = −tr∇ 2 = ∇ ∗ ∇,
where ∇ ∗ is the L 2 formal adjoint of ∇. The Lichnerowicz Laplacian acting
on symmetric covariant 2-tensors is
where
△ L = △ + 2(Ric − Riem),
(Ric u) ij = 1 2 [Ric(g) iku k j + Ric(g) jk u k i ],
and
(Riem u) ij = Riem(g) ikjl u kl .
For u a covariant 2-tensorfield on M we define the divergence of u by
(divu) i = −∇ j u ji .
For ω, a one form on M , we define its divergence
d ∗ ω = −∇ i ω i ,
the symmetric part of its covariant derivative :
(Lω) ij = 1 2 (∇ iω j + ∇ j ω i ),

4 E. DELAY
(note that L ∗ = div) and the trace free part of that last tensor :
(˚Lω) ij = 1 2 (∇ iω j + ∇ j ω i ) + 1
n + 1 d∗ ωg ij .
The well known [2] weitzenböck formula for the Hodge-De Rham Laplacian
on 1-forms reads
∆ H ω i = ∇ ∗ ∇ω i + Ric(g) ik ω k .
All quantities relative to ĝ will have a hat or will be indexed by ĝ (∇ bg , div bg ,
d bg , ̂∆ L ,...)
A TT-tensor (Transverse Traceless tensor) is by definition a symmetric
divergence free and trace free covariant 2-tensor .
L 2 denotes the usual Hilbert space of functions or tensors with the product
(resp. norm)
∫
∫
〈u, v〉 L 2 = 〈u, v〉dµ g (resp. |u| L 2 = ( |u| 2 dµ g ) 1 2 ),
M
M
where 〈u, v〉 (resp. |u|) is the usual product (resp. norm) of functions or
tensors relative to g, and the measure dµ g is the usual measure relative to
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 operator
in order to apply the result for an eigentensor. We also study the
commutator of the Laplacian with the Killing operator L 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:
Lemma 3.1. On a Riemannian manifold (M, g) with Levi-Civita connexion
∇, for all symmetric 2-tensor field u, we have
∇ i ∇ k ∇ k u ij = ∇ k ∇ k ∇ i u ij + R il ∇ l u ij + 2Rj
l ki
∇k u il + ∇ k R p k u pj + ∇ k R p i jk
u ip
= ∇ k ∇ k ∇ i u ij + R il ∇ l u ij − 2R j lki ∇ k u il + 1 2 ∇p Ru pj + (∇ p R ij − ∇ j R ip )u ip .
Proof. For a covariant 3-tensor field T on M, we have
∇ p ∇ q T kij − ∇ q ∇ p T kij = R l kqp T lij + R l iqpT klj + R l jqpT kil .
Contracting this equality with g ip g qk , we obtain
(3.1) ∇ i ∇ k T kij − ∇ k ∇ i T kij = R l jki
Tkil .
For a covariant 2-tensor field u on M, we have
∇ m ∇ k u ij − ∇ k ∇ m u ij = R p ikm u pj + R p jkm u ip.
Contracting this equality with g im , we obtain
∇ i ∇ k u ij − ∇ k ∇ i u ij = R p k u pj + R p jki u ip .
From this last equality, we obtain
(3.2)
∇ k ∇ i ∇ k u ij − ∇ k ∇ k ∇ i u ij = ∇ k R p k u pj + ∇ k R p i jk
u ip + R p k ∇k u pj + R p i jk
∇ k u ip .

TT-EIGENTENSORS ON SOME AH MANIFOLDS 5
Taking T kij = ∇ k u ij in equation 3.1 and combining with equation 3.2, we
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
is harmonic (∇ k R kijl = 0) or equivalently when Ric(g) is a Codazzi tensor
(∇ k R ij − ∇ i R kj = 0) and (or) the tensor u satisfy an equality of the form
∇ k u ij − ∇ i u kj + f∇ j u ik = 0 (f is any real function).
Corollary 3.2. On a Riemannian manifold (M, g) with Levi-Civita connexion
∇, we have
div ◦ ∆ L = ∆ H ◦ div − D Ric,
where (D Ric u) j = (∇ p R ij − ∇ j R ip − ∇ i R jp )u ip = (∇ j R ik )u ik .
Proof. From lemma 3.1, we have
∇ i (∇ k ∇ k u ij − R ik u k j − R jk u k i + 2R jlik u kl ) = ∇ k ∇ k ∇ i u ij − R l j∇ i u il
Also, we have the
+(∇ p R ij − ∇ j R ip − ∇ i R jp )u ip
= ∇ k ∇ k ∇ i u ij − R l j∇ i u il
−(∇ j R ip )u ip .
Lemma 3.3. On a Riemannian manifold with Levi-Civita connexion ∇, we
have
∆ L ◦ L = L ◦ ∆ H + D Ric,
where (D Ric ξ) ij = (∇ p R ij − ∇ j R ip − ∇ i R jp )ξ p .
Proof. A straightforward computation gives
∇ k ∇ k ∇ i ξ j = ∇ i ∇ k ∇ k ξ j + R ik ∇ j ξ k − 2R ikjl ∇ k ξ l − (∇ k R ij − ∇ j R ik )ξ k .
The result follows from the definitions of the Hodge and Lichnerowicz Laplacians.
□
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 curvature,
we have
div ◦ ∆ L = ∆ H ◦ div ,
and
∆ L ◦ L = L ◦ ∆ H .
Remark. The condition ∇ p R ij − ∇ j R ip − ∇ i R jp = 0 is equivalent to the
parallel Ricci curvature condition.
Corollary 3.5. 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 u ij = ∇ k ∇ k ∇ i u ij + (n + 2)K∇ i u ij ,
□

6 E. DELAY
or equivalently
div ◦ ∆ = [∆ − (n + 2)K] ◦ div.
Corollary 3.6. On the (n+1)-dimensional real hyperbolic space with n ≥ 2,
any L 2 symmetric trace free eigentensor field is divergence free, and so is a
TT-tensor.
Proof. If u is an L 2 symmetric trace free eigentensor field on the real hyperbolic
space, then div u is an L 2 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, ĝ) is not assumed to be Einstein.
On M =]0, +∞[×N, we consider the metric
g = dr 2 + f 2 (r)ĝ.
The computation is given in local coordinates adapted to the character of
M. We will use a local coordinate (x 1 , ..., x n ) = (x A ) on the manifold N
and the coordinate (x 0 , x 1 , ..., x n ) = (r, x 1 , ..., x n ) = (x i ) on M. The indices
with big letters are relative to the coordinate on N, for example
g 00 = 1, g 0A = 0, g AB = f 2 ĝ AB .
The prime in the calculation will denote derivative relative to r.
The Christoffel’s symbol of g are
Γ 0 00 = Γ C 00 = Γ 0 A0 = 0,
Γ 0 AB = −ff ′ ĝ AB , Γ C A0 = f −1 f ′ δ A C , Γ C AB = ̂Γ C AB.
If h is a covariant symmetric 2-tensor, we decompose
(4.1) h = udr 2 + ξ ⊗ dr + dr ⊗ ξ + ĥ,
where ĥ is the orthogonal projection of h on {r} × N, and ξ is a one form
on {r} × N. In particular we have
The covariant derivatives of h are
∇ 0 h 00 = ∂ 0 h 00 = u ′ ,
h 00 = u, h 0A = ξ A , h AB = ĥAB.
∇ 0 h A0 = ∂ 0 h A0 − f −1 f ′ h A0 = ξ ′ A − f −1 f ′ ξ ′ A,
∇ 0 h AB = ∂ 0 h AB − 2f −1 f ′ h AB = ĥ′ AB − 2f −1 f ′ ĥ AB ,
∇ C h 00 = ∂ C h 00 − 2f −1 f ′ h C0 = d C u − 2f −1 f ′ ξ C ,
∇ C h A0 = ̂∇ C h A0 + ff ′ ĝ AC h 00 − f −1 f ′ h AC = ̂∇ C ξ A + ff ′ ĝ AC u − f −1 f ′ ĥ AC ,
∇ C h AB = ̂∇ C h AB + ff ′ (ĝ AC h 0B + ĝ CB h 0A ) = ̂∇ C ĥ AB + ff ′ (ĝ AC ξ B + ĝ CB ξ A ).
In particular, we can compute the components of minus the divergence of h:
∇ i h i0 = u ′ + nf −1 f ′ u − f −3 f ′ h A A + f −2 ̂∇A ξ A ,
∇ i h iA = ξ ′ A + nf −1 f ′ ξ A + f −2 ̂∇B ĥ BA .

TT-EIGENTENSORS ON SOME AH MANIFOLDS 7
If Tr g h = h 00 + f −2 h A A = u + f −2 Tr bg ĥ = 0, this yield
We thus obtain the
∇ i h i0 = u ′ + (n + 1)f −1 f ′ u + f −2 ̂∇A ξ A ,
∇ i h iA = ξ ′ A + nf −1 f ′ ξ A + f −2 ̂∇B ĥ BA .
Lemma 4.1. If h is a symmetric covariant 2-tensor decomposed as in 4.1
then the components of its divergence are
(divh) 0 = −(u ′ + nf −1 f ′ u − f −3 f ′ Tr bg ĥ − f −2 d ∗ bg ξ),
(divh) A = −(ξ ′ + nf −1 f ′ ξ − f −2 div bg ĥ) A .
If moreover h is trace free, the first component becomes
(divh) 0 = −(u ′ + (n + 1)f −1 f ′ u − f −2 d ∗ bg ξ).
In a similar way, we compute the components of the Laplacian of h. After
a straightforward calculation we obtain the following Lemma
Lemma 4.2. 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 ∇ k h 00 = u ′′ + nf −1 f ′ u ′ − 2(n + 1)f −2 (f ′ ) 2 u + f −2 ̂∇A ̂∇A u − 4f −3 f ′ ̂∇A ξ A ,
∇ k ∇ k h A0 = ξ ′′ A + (n − 2)f −1 f ′ ξ ′ A − [(2n + 1)f −2 (f ′ ) 2 + f −1 f ′′ ]ξ A
+f −2 ̂∇D ̂∇D ξ A − 2f −3 f ′ ̂∇D h AD + 2f −1 f ′ d A u,
∇ k ∇ k h AB = ĥ′′ AB + (n − 4)f −1 f ′ ĥ ′ AB − [(2n − 4)f −2 (f ′ ) 2 + 2f −1 f ′′ ]ĥAB
+f −2 ̂∇D ̂∇D ĥ AB + 2f −1 f ′ ( ̂∇ A ξ B + ̂∇ B ξ A ) + (f ′ ) 2 ĝ AB u.
In particular, if h is divergence free,
∇ k ∇ k h 00 = u ′′ + (n + 4)f −1 f ′ u ′ + 2(n + 1)f −2 (f ′ ) 2 u + f −2 ̂∇A ̂∇A u,
∇ k ∇ k h A0 = ξ ′′ A + (n)f −1 f ′ ξ ′ A − [f −2 (f ′ ) 2 + f −1 f ′′ ]ξ A
+f −2 ̂∇D ̂∇D ξ A + 2f −1 f ′ d A u,
∇ k ∇ k h AB = ĥ′′ AB + (n − 4)f −1 f ′ ĥ ′ AB − [(2n − 4)f −2 (f ′ ) 2 + 2f −1 f ′′ ]ĥAB
+f −2 ̂∇D ̂∇D ĥ AB + 2f −1 f ′ ( ̂∇ A ξ B + ̂∇ B ξ A ) + (f ′ ) 2 ĝ AB u.
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
R A 0B0 = −f −1 f ′′ δ A B,
R 0 A0B = −ff ′′ ĝ AB ,
R A BCD = ̂R A BCD − (f ′ ) 2 (ĝ BD δ A C − ĝ BC δ A D).
Thus the non trivial components of the Ricci curvature are
R 00 = −nf −1 f ′′ , R AB = ̂R AB − [ff ′′ + (n − 1)(f ′ ) 2 ]ĝ AB .

8 E. DELAY
Then, the zero order terms that appear in the Lichnerowicz Laplacian are
2(Riem h) 00 = −2f −3 f ′′ Tr bg ĥ = 2f −1 f ′′ u,
2(Ric h) 00 = −2nf −1 f ′′ u,
2(Riem h) 0A = 2f −1 f ′′ ξ A ,
2(Ric h) 0A = f −2 ̂RC A ξ C − ((n + 1)f −1 f ′′ + (n − 1)f −2 (f ′ ) 2 )ξ A ,
2(Riem h) AB = 2f −2 ̂RACBD ĥ AB + 2f −2 (f ′ ) 2 (ĥAB − Tr bg ĥ ĝ AB ) − 2ff ′′ uĝ AB ,
= 2f −2 ̂RACBD ĥ AB + 2f −2 (f ′ ) 2 ĥ AB ,
2(Ric h) AB = 2f −2 (Ric bg ĥ) AB − 2(f −1 f ′′ + (n − 1)f −2 (f ′ ) 2 )ĥAB.
Lemma 4.3. Let h be a trace free covariant symmetric 2-tensor decomposed
as is 4.1. The components of the Lichnerowicz Laplacian of h are
∆ L h 00 = −u ′′ − nf −1 f ′ u ′ − f −2 ̂∇A ̂∇A u + 4f −3 f ′ ̂∇A ξ A ,
∆ L h A0 = −ξ ′′ A − (n − 2)f −1 f ′ ξ ′ A + [(3n + 4)f −2 (f ′ ) 2 + nf −1 f ′′ ]ξ A
−f −2 ( ̂∇ D ̂∇D ξ A − ̂R C Aξ C ) + 2f −3 f ′ ̂∇D h AD − 2f −1 f ′ d A u,
∆ L h AB = −ĥ′′ AB − (n − 4)f −1 f ′ ĥ ′ AB − 4f −2 (f ′ ) 2 ĥ AB
+f −2 ̂∆L ĥ AB − 2f −1 f ′ ( ̂∇ A ξ B + ̂∇ B ξ A ) − (f ′ ) 2 ĝ AB u.
In particular, if h is divergence free, we obtain
∆ L h 00 = −u ′′ − (n + 4)f −1 f ′ u ′ − 2(n + 1)(f −2 (f ′ ) 2 + f −1 f ′′ )u − f −2 ̂∇A ̂∇A u,
∆ L h A0 = −ξ ′′ A − nf −1 f ′ ξ ′ A − [(n + 2)f −2 (f ′ ) 2 + (n − 2)f −1 f ′′ ]ξ A
−f −2 ( ̂∇ D ̂∇D ξ A − ̂R C Aξ C ) − 2f −1 f ′ d A u,
∆ L h AB = −ĥ′′ AB − (n − 4)f −1 f ′ ĥ ′ AB − 4f −2 (f ′ ) 2 ĥ AB
+f −2 ̂∆L ĥ AB − 2f −1 f ′ ( ̂∇ A ξ B + ̂∇ B ξ A ) − (f ′ ) 2 ĝ AB u.
5. Hodge type decomposition for trace free symmetric
2-tensors
We recall here how to construct an L 2 orthonormal eigenbasis for trace
free symmetric covariants 2-tensors on a compact Einstein manifold N with
Ric(ĝ) = κ(n − 1)ĝ.
We recall the classical L 2 -orthogonal Hodge-de Rahm decomposition for one
forms on N :
(5.1) C ∞ (N, T 1 ) = Im d bg ⊕ Ker d ∗ bg .
We also recall the L 2 -orthogonal decomposition for trace free symmetric
2-tensors on N [1] :
(5.2) C ∞ (N, ˚S 2 ) = Im ˚L bg ⊕ Ker div bg ,
where, we recall that for a one form ξ, ˚L bg ξ = ˚L bg ξ + 1 n d∗ bg
ξĝ is the ĝ-trace free
part of L bg ξ.
Let (Ψ i ) i∈N be an orthonormal basis of L 2 (N) of eigenfunctions, with
∇ ∗ bg ∇ bgΨ i = λ i Ψ i , 0 = λ 0 < λ 1 ≤ λ 2 ≤ ...

For all i ∈ N\{0}
and
TT-EIGENTENSORS ON SOME AH MANIFOLDS 9
̂∆ H dΨ i = λ i dΨ i ⇐⇒ ∇ ∗ bg ∇ bgdΨ i = [λ i − (n − 1)κ]dΨ i ,
∫
N
∫
〈dΨ i , dΨ j 〉 bg dθ =
N
Ψ j ∇ ∗ ∇Ψ i dθ = λ i
∫
N
Ψ i Ψ j dθ.
From the Hodge decomposition theorem (see in particular equation 5.1),
the L 2 (N, T 1 ) orthonormal family { 1 √ λi
dΨ i } i∈N\{0} can be completed with
an L 2 (N, T 1 ) orthonormal family {ν j } j∈N with
∇ ∗ bg ∇ bgν j = µ j ν j , d ∗ bg ν j = 0
to obtain an orthonormal basis {η k } k∈N of L 2 (N, T 1 ) composed of eigenforms
:
∇ ∗ bg ∇ bgη k = α k η k .
Thank’s to some easy integrations by parts, we get: 1
∫
k ,
N〈˚Lη ˚Lη q 〉 bg dθ = 1 ∫
[〈∆ bg η k , η q 〉 bg − (n − 1)κ〈η k , η q 〉 bg + n − 2
2 N
n
d∗ bg η kd ∗ bg η q]dθ
⎧ ∫
α k −(n−1)κ
2
〈η k , η q 〉 bg dθ if d
⎪⎨
∗ bg η k or d ∗ bg η q = 0,
N
∫
= n−1
2 √ √ [ 2 λ i λj n λ iλ j − κ(λ i + λ j )] Ψ i Ψ j dθ
N
⎪⎩ if η k = √ 1
λi
dΨ i and η q = √ 1 dΨ j ,
λj
1
so the family { ˚Lη
‖˚Lη k ‖ k } k∈N is orthonormal. If the sectional curvature is
L 2
constant, we also have
∇ ∗ bg ∇ bg ˚L bg η k = [α k − κ(n + 2)]˚L bg η k .
In this case from the decomposition of trace free symmetric 2-tensors (see
equation 5.2), the L 2 (N, ˚S 1
2 ) orthonormal family { ˚Lη
‖˚Lη k ‖ k } k∈N can be
L 2
completed with an L 2 (N, ˚S 2 ) orthonormal family {σ l } l∈N such that
∇ ∗ bg ∇ bgσ l = β l σ l , div bg σ l = 0,
to obtain an orthonormal basis {Υ m } m∈N of L 2 (N, ˚S 2 ) composed of eigentensors,
with
∇ ∗ bg ∇ bgΥ m = γ m Υ m .
In the more general case where N is only Einstein, then we have
̂∆ L ˚Lbg η k = [α k + κ(n − 1)]˚L bg η k ,
1
So the orthonormal family { ˚Lη
‖˚Lη k ‖ k } k∈N can be completed with an
L 2
L 2 (N, ˚S 2 ) orthonormal family {π l } l∈N such that
̂∆ L π l = β l π l , div bg π l = 0,
1 This is at this step we really need (N, bg) to be Einstein because the Laplacian ∆ bg −
Ric(bg) 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

10 E. DELAY
to obtain an orthonormal basis {Φ m } m∈N of L 2 (N, ˚S 2 ) of eigentensors, with
̂∆ L Φ m = ε m Φ m .
Remark. When (N, ĝ) is the standard unit sphere of R n+1 or the real projective
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
g = dr 2 + f 2 (r)ĝ,
where ĝ is for the moment assumed to be of constant sectional curvature.
Theorem 6.1. For any λ ≥ n2
4
+ 2, there is no non-trivial trace free, divergence
free, L 2 covariant symmetric 2-tensor on (]0, +∞[×N, g) satisfying
∇ ∗ ∇h = λh.
Proof. For h a symmetric 2-tensor field on ]0, +∞[×N, we decompose
h = udr 2 + ξ ⊗ dr + dr ⊗ ξ + ĥ,
where ĥ is the orthogonal projection of h on {r} × N and ξ is a one form
on {r} × N. We assume first that h is a trace free L 2 eigentensor for an
eigenvalue λ > n2
4 + 2 so ∇ ∗ ∇h = λh.
We assume also that h is divergence free (see lemma 4.1):
(6.1) u ′ + (n + 1)f −1 f ′ u − f −2 ̂d∗ ξ = 0,
(6.2) ξ ′ + nf −1 f ′ ξ − f −2 div bg ĥ = 0.
From Lemma 4.2, we have for the components of h:
(6.3) u ′′ + (n + 4)f −1 f ′ u ′ + [2(n + 1)f −2 (f ′ ) 2 + λ]u − f −2 ∇ ∗ bg ∇ bgu = 0,
(6.4) ξ ′′ +nf −1 f ′ ξ ′ −[f −2 (f ′ ) 2 +f −1 f ′′ −λ]ξ −f −2 ∇ ∗ bg ∇ bgξ +2f −1 f ′ d bg u = 0,
(6.5)
ĥ ′′ + (n − 4)f −1 f ′ ĥ ′ − [(2n − 4)f −2 (f ′ ) 2 + 2f −1 f ′′ − λ]ĥ
−f −2 ∇ ∗ bg ∇ bgĥ + 4f −1 f ′ L bg ξ + (f ′ ) 2 uĝ = 0.
The ĝ-trace free part of equation 6.5 gives
(6.6)
˚h′′ + (n − 4)f −1 f ′˚h′ − [(2n − 4)f −2 (f ′ ) 2 + 2f −1 f ′′ − λ]˚h
−f −2 ∇ ∗ bg ∇ bg˚h + 4[f −1 f ′ L bg ξ + 1 n ̂d ∗ ξĝ] = 0,
where ˚h is the ĝ-trace free part of ĥ.
With the change of variables u = f − n+4
2 v, ξ = f − n 2 ζ and ˚h = f − n−4
2 b,
equations 6.1 and 6.2 become respectively
(6.7) (f n 2 −1 v) ′ = f n 2 −1 d ∗ bg ζ
(6.8) (f n 2 ζ) ′ = f n 2 (div bg b + 1 n f −2 d bg v),

TT-EIGENTENSORS ON SOME AH MANIFOLDS 11
while equation 6.3, 6.4 and 6.6 become respectivily
(6.9) v ′′ + [λ −
(6.10)
n(n − 2)
4
f −2 (f ′ ) 2 − n + 4 f −1 f ′′ ]v − f −2 ∇ ∗ bg
2
∇ bgv = 0,
ζ ′′ +[λ− n2 − 2n + 4
f −2 (f ′ ) 2 − n + 2 f −1 f ′′ ]ζ−f −2 ∇ ∗ bg
4
2
∇ bgζ = −2f −2 (f −1 f ′ d bg v),
(6.11)
b ′′ + [λ − n2 −2n+8
4
f −2 (f ′ ) 2 − n 2 f −1 f ′′ ]b − f −2 ∇ ∗ bg ∇ bgb
= −4f −2 [f −1 f ′ L bg ζ + 1 ̂d n ∗ ζĝ].
We now decompose each component of h according to the L 2 basis described
in section 5. This yield
v(r, θ) = ∑ i∈N
x i (r)Ψ i (θ),
where x i (r) = ∫ N u(r, θ)Ψ i(θ)dθ is smooth and the same is true for
ζ(r, θ) = ∑ k∈N
y k (r)η k (θ),
b(r, θ) = ∑ m∈N
z m (r)Υ m (θ).
From equation 6.9, each component x i of v satisfies
(6.12) x ′′ n(n − 2)
+ [λ − f −2 (f ′ ) 2 − n + 4 f −1 f ′′ − f −2 λ i ]x = 0.
4
2
This equation can be rewritten
(6.13) x ′′ + [α 2 + O(e −εr )]x = 0,
√
where α = λ − n2
4
− 2. An asymptotic integration (see theorem 7.1 of
appendix) gives
(6.14) x = (A 1 + O(e −εr ))e αir + (A 2 + O(e −εr ))e −αir ,
and
(6.15) x ≡ 0 ⇐⇒ A 1 = A 2 = 0.
We have
d bg v = ∑ i∈N
x i d bg Ψ,
with each of the x i ’s having the same asymptotic as x in 6.14. From equation
6.10, each components y k of ζ satisfies
(6.16)
y ′′ +[λ− n2 − 2n + 4
4
f −2 (f ′ ) 2 − n + 2 f −1 f ′′ −f −2 µ k ]y =
2
which can be written
⎧
⎨ O(e −εr )
(6.17) y ′′ + [α 2 + O(e −εr )]y = or
⎩
0
⎧
⎨
⎩
.
−2f −2 (f −1 f ′ x)
or
0
,

12 E. DELAY
From asymptotic integration (see corollary 7.2 of appendix), we thus have
(6.18) y = (B 1 + O(e −εr ))e αir + (B 2 + O(e −εr ))e −αir ,
and if x = 0,
(6.19) y ≡ 0 ⇐⇒ B 1 = B 2 = 0.
In a similar way since
each components z m of b satisfies
(6.20)
L ζ ĝ + 1 n d∗ bg ζĝ = ∑ k∈N
y k (L ηk ĝ + 1 n d∗ bg η kĝ),
z ′′ + [λ − n2 − 2n + 8
f −2 (f ′ ) 2 − n 4
2 f −1 f ′′ − f −2 α m ]z =
which can also be written
⎧
⎨ O(e −εr )
(6.21) z ′′ + [α 2 + O(e −εr )]z = or
⎩
0
⎧
⎨
.
⎩
−4f −2 f −1 f ′ y
or
0
From asymptotic integration (see corollary 7.2 in appendix), once again
(6.22) z = (C 1 + O(e −εr ))e αir + (C 2 + O(e −εr ))e −αir ,
and if y = 0,
(6.23) z ≡ 0 ⇐⇒ C 1 = C 2 = 0.
Now, the squared norm of h is
|h| 2 = u 2 + 2f −2 |ξ| 2 bg + f −4 |ĥ|2 bg
= u 2 (1 + 1 n ) + 2f −2 |ξ| 2 bg + f −4 |˚h| 2 bg
= f −n [f −4 v 2 (1 + 1 n ) + 2f −2 |ζ| 2 bg + |b|2 bg ],
and the mesure relative to g is dµ = f n drdθ.
But as h ∈ L 2 , the only possibility is C 1 = C 2 = 0 for all components
z m of b. If we include this fact in equation 6.8, we obtain that all the
components y k of ζ are of order O(e −εr ) thus B 1 = B 2 = 0 for all of them.
Now we insert this last fact in equation 6.7 and we obtain that the x i ’s are
also of order O(e −εr ). Finally from 6.15, for all i ∈ N, x i = 0, from 6.19, for
all k ∈ N, y k = 0, and from 6.23 all the z m ’s are equals to zero. We thus
conclude that h = 0 and this complete the proof when λ > n2
4 + 2.
Assume now that λ = n2
4
+ 2 and repeat the preceding arguments step
by step. Here we have α = 0 and from Theorem 7.3 and Corollary 7.4 the
equations 6.14, 6.18, 6.22 are substituted by
(6.24) x = A 1 + (A 2 + O(r 2 e −εr )r,
(6.25) y = B 1 + (B 2 + O(r 2 e −εr )r,
,

TT-EIGENTENSORS ON SOME AH MANIFOLDS 13
(6.26) z = C 1 + (C 2 + O(r 2 e −εr )r.
The end of the proof proceeds as before.
The adaptation of the proof above for the Lichnerowicz Laplacian in place
of the rough Laplacian is achieved with some minor modifications. This
proceed using Lemma 4.3 and the end of section 5 only assuming here that
N is compact and Einstein. We finally obtain:
Theorem 6.2. For all λ ≥ n(n−8)
4
, there is no non-trivial trace free, divergence
free, L 2 covariant symmetric 2-tensor on (]0, +∞[×N, g) satisfying
∆ L h = λh.
7. Appendix : asymptotic integration for ordinary differential
equations
In this section, we recall some results of asymptotic integration which are
certainly well known since a long time. They can be found partly in [16]. We
detail some parts of the proof given there to obtain a more precise version
in the particular case which is of interest here.
Theorem 7.1. Let a be a continuous function on ]0, ∞[ such that for r
going to infinity, a(r) = O(e −εr ) for some ε > 0. If α is a positive real then
any solution to the equation
(7.1) y ′′ + (α 2 + a)y = 0,
satisfies
(7.2) y = [A 1 + O(e −εr )]e iαr + [A 2 + O(e −εr )]e −iαr ,
(7.3) y ′ = [A 1 iα + O(e −εr )]e iαr − [A 2 iα + O(e −εr )]e −iαr ,
for some constants A 1 and A 2 , and
y ≡ 0 iff A 1 = A 2 = 0.
Moreover, for any given A 1 and A 2 there exists at least one solution to 7.1
with the asymptotics 7.2 7.3.
Proof. From [16], Theorem XI.8.1 page 370, the theorem given here is true
with the O(e −εr ) there replaced with o(1). So we only have to prove this
behavior at infinity. Let us denote by u 0 (r) = e iαr and v 0 (r) = e −iαr . We
define the matrics
A =
(
0 1
−α 2 0
) (
0 0
, B =
−a 0
Since Z ′ = AZ, the usual change of variable
( ) ( )
y ξ1
y ′ = Z = Zξ,
ξ 2
gives
In particular
) (
u0 v
, Z =
0
u ′ 0 v 0
′
ξ ′ (r) = F (r, ξ(r)), where F (r, ξ) = Z −1 BZξ.
‖F (r, ξ)‖ ≤ C|a|‖ξ‖,
)
.
□

14 E. DELAY
where C is a constant, ( so)
from [16], Theorems X.1.1 and X.1.2 page 274,
A1
lim r→∞ ξ = ξ ∞ =: exists. Moreover (see [16] equation (1.11) page
A 2
275),
∫ ∞
‖ξ(r) − ξ ∞ ‖ ≤ C 1 |a(s)|ds,
where C 1 is a constant. As a(r) = O(e −εr ), the integral on the right-hand
side of this last equation is of order O(e −εr ). This concludes the proof. □
Corollary 7.2. Let a, b be two continuous functions on ]0, ∞[ such that for
r going to infinity, a(r), b(r) = O(e −εr ) for some ε > 0. If α is a positive
real, then any solution to the equation
(7.4) y ′′ + (α 2 + a)y = b,
satisfies
(7.5) y = [A 1 + O(e −εr )]e iαr + [A 2 + O(e −εr )]e −iαr ,
(7.6) y ′ = [A 1 iα + O(e −εr )]e iαr − [A 2 iα + O(e −εr )]e −iαr ,
for some constants A 1 and A 2 .
Proof. From Theorem 7.1, there exist two solutions u 1 , v 1 to the equation
7.1 such that
u 1 = [1+O(e −εr )]e iαr +O(e −εr )e −iαr , u ′ 1 = [iα+O(e −εr )]e iαr +O(e −εr )e −iαr ,
v 1 = O(e −εr )e iαr +[1+O(e −εr )]e −iαr , v 1 ′ = O(e −εr )e iαr −[iα+O(e −εr )]e −iαr .
Let us consider the matrics
(
) ( )
0 1
u1 v
A =
−α 2 , Z
− a 0 1 =
1
u ′ 1 v 1
′ .
Once again Z 1 ′ = AZ 1, so the usual change of variable
( ) ( )
y η1
y ′ = Z 1 = Z
η 1 η,
2
gives
( )
η ′ (r) = G(r), where G(r) = Z1
−1 0
.
b
In particular from the asymptotic behavior of a and b we have
‖G(r)‖ ≤ Ce −εr ,
where C is a constant. This shows that ∫ ( )
∞ η ′ converges absolutely so that
A1
lim r→∞ η = η ∞ = exists. Moreover
A 2
‖η(r) − η ∞ ‖ = ‖
∫ ∞
We finally obtain η =
r
η ′ (s)ds‖ ≤
(
A1
∫ ∞
r
r
‖η ′ (s)‖ds ≤ C
∫ ∞
r
e −εs ds = C ε e−εr .
A 2
)
+ O(e −εr ), which concludes the proof. □
Similarly to theorem 7.1 and to corollary 7.2:

TT-EIGENTENSORS ON SOME AH MANIFOLDS 15
Theorem 7.3. Let a be a continuous function on ]0, ∞[ such that for r
going to infinity, a(r) = O(e −εr ) for some ε > 0. Then any solution to the
equation
(7.7) y ′′ + ay = 0,
satisfies
(7.8) y = A 1 + [A 2 + O(r 2 e −εr )]r,
(7.9) y ′ = A 2 + O(r 2 e −εr ),
for some constants A 1 and A 2 , and
y ≡ 0 iff A 1 = A 2 = 0.
Moreover, for any given A 1 and A 2 there exists at least one solution to 7.7
with the asymptotics 7.8 7.9.
Proof. The proof proceeds as for theorem 7.1 with α = 0, u 0 (r) = 1 and
v 0 (r) = r. Here we will only have
for r ≥ r 0 . Then
‖F (r, ξ)‖ ≤ C|a|r 2 ‖ξ‖,
∫ ∞
‖ξ(r) − ξ ∞ ‖ ≤ C 1 |a(s)|s 2 ds,
where C 1 is a constant. As a(r) = O(e −εr ), the integral on the right-hand
side of this last equation is of order r 2 e −εr and this complete the proof. □
Corollary 7.4. Let a, b be two continuous functions on ]0, ∞[ such that for
r going to infinity, a(r), b(r) = O(e −εr ) for some ε > 0. Then any solution
to the equation
(7.10) y ′′ + ay = b,
satisfies
(7.11) y = A 1 + [A 2 + O(r 2 e −εr )]r,
r
(7.12) y ′ = A 2 + O(r 2 e −εr ),
for some constants A 1 and A 2 .
Proof. The proof proceeds as in Corollary 7.2, using Theorem 7.3.
□
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Erwann Delay, Laboratoire d’analyse non linéaire et géométrie, Faculté
des Sciences, 33 rue Louis Pasteur, 84000 Avignon, France
E-mail address: Erwann.Delay@univ-avignon.fr
URL: http://www.math.univ-avignon.fr/Delay