TT EIGENTENSORS FOR THE LICHNEROWICZ LAPLACIAN ON SOME ASYMPTOTICALLY HYPERBOLIC

Description

Niveau: Supérieur, Doctorat, Bac+8
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 L2-eigentensors with eigenvalue in the essential spectrum of the Lichnerowicz Laplacian ∆L. If (M, g) is the real hyperbolic space, there is no symmetric L2-eigentensors of ∆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 ∆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.

• trace free


• tensor


• riemannian manifold


• lichnerowicz laplacian


• hyperbolic metric


• dimensional riemannian


• manifold


• ∆l


• tt-eigentensors

Sujets

• Tensor
• Riemannian manifold
• Laplace operators in differential geometry
• Manifold

Informations

• Publié par: profil-zyak-2012
• Nombre de lectures: 40
• Langue: English

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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