HILBERT MANIFOLD STRUCTURE FOR ASYMPTOTICALLY HYPERBOLIC RELATIVISTIC INITIAL DATA ERWANN DELAY AND JÉRÉMIE FOUGEIROL

Keywords: Hilbert manifold, asymptotically hyperbolic, elliptic operators, general relativity, constraint equations, weak regularity

We provide a Hilbert manifold structure à la Bartnik for the space of asymptotically hyperbolic initial data for the vacuum constraint equations. The adaptation led us to prove new weighted Poincaré and Korn type inequalities for AH manifolds with inner boundary and weakly regular metric.

Introduction

This work follows on from a paper of 2005 [START_REF] Bartnik | Phase space for the Einstein equations[END_REF] in which R. Bartnik described a Hilbert manifold structure for the space of asymptotically flat solutions of the Einstein equations (see also [START_REF] Mccormick | The phase space for the Einstein-Yang-Mills equations and the first law of black hole thermodynamics[END_REF], [START_REF] Mccormick | The Hilbert manifold of asymptotically flat metric extensions[END_REF], [START_REF] Rai | Hilbert manifold structure of the set of solutions of constraint equations for coupled Einstein and scalar fields[END_REF]). The work is done with rather weak regularity assumptions concerning the metric involved (curvature constant only modulo weighted L 2 terms) and so can be related to the context of the bounded L 2 curvature conjecture of KRS [START_REF] Sergiu Klainerman | Overview of the proof of the bounded L 2 curvature conjecture[END_REF]. Actually R. Bartnik showed in [START_REF] Bartnik | The mass of an asymptotically flat manifold[END_REF] that these assumptions on the regularity are the weakest possible to define the ADM mass of the manifold, explaining why we chose the same regularity conditions. In an undergoing work [START_REF] Fougeirol | Etude des équations de contraintes sur une variété Asymptotiquement hyperbolique[END_REF], we are showing that these very assumptions allow us to properly define the mass of an asymptotically hyperbolic manifold compatible with previous definitions of the mass (see [START_REF] Chrusciel | The mass of asymptotically hyperbolic Riemannian manifold[END_REF] , [START_REF] Herzlich | The AdS/CFT correspondence (Einstein metrics and conformal geometry)[END_REF] or [START_REF] Dahl | Asymptotically hyperbolic manifolds with small mass[END_REF]) and with the Hilbert manifold structure exposed here. In order to overcome difficulties arising in the asymptotically hyperbolic case, we had to create a Hessian-type operator T and a differential operator of order two, called Ů , built up with the first derivatives of the Killing operator S. In particular, we obtain Poincaré and Korn-type estimates of second order on an asymptotically hyperbolic manifold with boundary. These estimates are the key to prove triviality of the adjoint's kernel.

Acknowledgement : Jérémie Fougeirol thanks Marc Herzlich for some helpful remarks.

Notations and conventions

Let (M, g) be a Riemannian manifold. We define T r m (M) to be the bundle of tensor covariant of rank m and contravariant of rank r . For all u ∈ T r m (M), |u| g will denote the norm of u with respect to the metric g and notation |u| g,x allows us to precise the point of the manifold we consider. dµ(g) is the Riemannian measure determined by g. Riem g, Ric g and R(g) are respectively the Riemann tensor, the Ricci tensor and the scalar curvature of the metric g. For a Riemannian metric g with connection ∇, we set the following notations concerning the Hessian and Laplacian of a function u.

∇ 2 ij u = ∇ i ∇ j u ∆u = tr g ∇ 2 u = g ij ∇ 2 ij u w.r.t is the abbreviation of "with respect to". The work presented in this article is done on a n-dimensional manifold as often as possible and, otherwise explicitely stated, the results presented here are valid in any dimension n. Nevertheless, Sobolev inequalities strongly constrain the dimension to n = 3 in several proofs ; this is clearly specified when this is the case. We chose to leave n and specify n = 3 in the concerned results rather than replace n by its specific value since it helps to understand where the dimension plays a role. Concerning constants in norm inequalities, the constant c will design in general a constant depending on the background metric g and the decaying rate δ and its expression may change from line to line in a proof. The nature of the dependence of the constant C will be systematically specified because it will depend on other parameters.

Conformally compact manifold.

Let (M,g) be a C ∞ n-dimensional complete non compact Riemannian manifold. The manifold (M,g) is conformally compact if there exists a Riemannian metric h such that g = ρ -2 h , where (M, h) is a C ∞ compact Riemannian manifold with boundary ∂ ∞ M and ρ is a function on M, called defining function on

∂ ∞ M: • ρ ∈ C ∞ ( M) • ρ 0 on M • ∂ ∞ M = {x ∈ M : ρ(x) = 0}
• dρ ∂∞M never vanishes.

We call asymptotically hyperbolic metric on M every Riemannian metric g verifying:

• g is conformally compact.

• |dρ| 2 h,∂∞M = 1. In some inequalities, we will allow M to have an inner boundary ∂M. In that case, M will the disjoint union of M, ∂M and ∂ ∞ M.

The existence of these two metrics on M can be source of confusion thereafter, so we have to fix some notations. ∇, ∆, Γ k ij , |u| g (resp. ( h) ∇, ∆ h, ( h) Γ k ij , |u| h) will respectively denote the connection, Laplace operator, Christoffel symbols and tensor norm w.r.t g (resp. h). There are some correspondances between these quantities. For volume measures, in dimension n, clearly dµ(g) = ρ -n dµ( h). For Christoffel symbols,

Γ k ij = ( h) Γ k i j -1 ρ δ k j ∂ i ρ + δ k i ∂ j ρ -hkl hij ∂ l ρ .
In particular, for the Hessian of ρ (e.g. equation (C.12) from [START_REF] Chrusciel | On mapping properties of the general relativistic constraints operator in weighted functions space, with applications[END_REF])

∇2 ij ρ = ( h) ∇2 ij ρ + ρ 2 ∂ i ρ ρ ∂ j ρ ρ -gij |dρ| 2 h . (1)
Taking the trace of (1), we obtain the expression of the Laplace operator of ρ

∆ρ = ρ 2 ∆ hρ -(n -2)ρ|dρ| 2 h. (2)
More generally,

∀u ∈ C ∞ c (M) , ∆u = ρ 2 ∆ hu - n -2 ρ dρ . hdu , (3)
where . h is the scalar product w.r.t the metric h.

For tensor norms, for all u ∈ T r m (M) , |u| g = ρ m-r |u| h.

Definition 1. A cut-off function: (M,g) is a conformally compact manifold with defining function ρ. Let χ : IR → IR be a smooth cut-off function such that:

• χ(IR) ⊂ [0, 1] • supp χ ⊂ (-∞, 2] • χ (-∞,1] = 1
then for R large enough, we can define a cut-off function on M by χ R (x) = χ(-ln(ρ(x))/R).

Setting Ω R = {x ∈ M : ρ(x) > e -2R } , then χ R verifies

χ R = 1 on Ω R 2 0 on M \ Ω R
In other words, χ R is a cut-off function near the boundary at infinity of M. The Lebesgue weighted space L p δ is now defined as the space of measurable functions of L p loc whose norm mentioned above is finite. The Sobolev weighted space W k,p δ is then the space of measurable functions of W k,p loc whose following norm is finite:

||u|| k,p,δ = |α| k || ∇α u|| p,δ ,
where α is a multi-index of size n and ∇α u = ∇α 1 i 1 . . . ∇αn in u , α = (α 1 , . . . , α n ) and |α| = n i=1 α i .

NB: for δ = 0, we get back to norms of the classic Lebesgue and Sobolev spaces.

W k,p δ (T r m M) will refer to Sobolev spaces of sections of the (r, m)-tensor bundle over M. We will speak of W k,p δ -norm to indicate the norm of a tensor field u ∈ W k,p δ (T r m M). For a domain U ⊂ M, ||u|| p,δ;U will be the restriction to U of the W k,p δ -norm of u. The Hölder weighted space C s,α δ (M, g) , with 0 < α < 1 is endowed with the norm

||u|| C s,α δ = max |k| s || ∇k u|| C 0,α δ with ||u|| C 0,α δ = sup x∈M ρ δ |u| g + sup x∈M ρ δ sup d g (x,y) 1 | u(x) -u(y)| g d g(x, y) α ,
where u and g represent tensors u and g in an appropriate orthonormal basis.

Elliptic operators.

Here we recall classic results on elliptic operators that we may find in [START_REF] Andersson | Elliptic systems on manifolds with asymptotically negative curvature[END_REF] for example. Let B 1 and B 2 be two tensor bundles over a conformally compact manifold (M,g) with defining function ρ and A :

C ∞ (B 1 ) → C ∞ (B 2 ) be a partial differential linear operator of order m define d by A = |α| m a α ∇α (4)
Set s ∈ N. We say that the operator A of the form (4) has symbol in

OP m s if a α ∈ C sα -|α| L(B 1 , B 2 ), with s α = max(s, |α| -m + 1
). We say that A is an elliptic operator if

• For all α such that |α| = m , for all ξ α = ξ α 1 1 . . . ξ αn n = 0, a α ξ α : B 1 → B 2 is a tensor bundles isomorphism.

• For all ξ α , there exists two constants c 1 and c 2 such that

||a α ξ α || g < c 1 |ξ α | g and ||(a α ξ α ) -1 || g < c 2 |ξ α | g.
Lemma 1. Set s 0 ∈ N. For every elliptic operator A with symbol in OP m s 0 , there exists a positive constant c = c (g, δ) such that the following inequality is valid for all s s 0 :

||u|| 2,s+m,δ c (||Au|| 2,s,δ + ||u|| 2,0,δ ) . ( 5
)
Theorem 1. For all R > 0, let Ω R be as in Definition 1. Given δ ∈ IR and A an elliptic operator with symbol in OP m 0 . Suppose there exists R large enough so that

A verifies ∀u ∈ C ∞ c (M \ Ω R ) , ||u|| 2,δ C ||Au|| 2,δ , (6)
where C depends on A.

Then we can choose R large enough so that the following inequality

||u|| m,2,δ C (||Au|| 2,δ + ||u|| 2,δ;Ω R ) (7
)

is valid for all u ∈ C ∞ c (M). In particular, A : W m,2 δ → L 2 δ is semi-Fredholm, i.e.
A has finite dimensional kernel and closed range.

Preliminary analysis

In this section, we introduce some useful inequalities (see [START_REF] Andersson | Elliptic systems on manifolds with asymptotically negative curvature[END_REF] , th 2.3 for example): Proposition 1. Weighted Hölder inequalities (in any dimension):

• Set δ ∈ IR so that δ = δ 1 + δ 2 , let p, q, r ∈ N be such that 1 p q r ∞ and

1 p = 1 q + 1 r , then ||uv|| p,δ ||u|| q,δ 1 ||v|| r,δ 2 . (8)
• Set δ ∈ IR and λ ∈ [0, 1] , let p, q, r ∈ N be such that 1 p q r ∞ and

1 p = λ q + 1 -λ r , then ||u|| p,δ ||u|| λ q,δ ||u|| 1-λ r,δ . (9)
Theorem 2. Let (M, g) be a conformally compact n-dimensional manifold.

Weighted Sobolev inclusion:

For all 1 p q < ∞ , for all k k ′ , if δ δ ′ -n p (q-p) q , then W k,q δ ⊂ W k ′ ,p δ ′
, and there exists a positive constant c = c (g, δ, δ ′ , k, k ′ , n, p, q) such that

||u|| k ′ ,p,δ ′ c ||u|| k,q,δ . If q = ∞, for all 1 p ∞; for all k k ′ , if δ δ ′ -n p , then W k,∞ δ ⊂ W k ′ ,p δ ′
, and there exists a positive constant c = c (g, δ, δ ′ , k, k ′ , n, p) such that

||u|| k ′ ,p,δ ′ c ||u|| k,∞,δ .
Weighted Hölder inclusion: Set k ∈ N, n = 3 and p = 2, then for all 0 < α 1 2 , for all δ δ ′ , there exists a

positive constant c = c (g, δ, δ ′ , k, k ′ , α) such that ∀u ∈ W 2,2 δ (M) , ||u|| C 0,α δ ′ c ||u|| 2,2,δ . (10)
Weighted Sobolev inequalities: Set 1 p < ∞ and let k, j be integers.

In each of the following cases, there exists a positive constant c = c (g, δ, k, n, j, p, q) such that for all u ∈ W j+k,p δ (M),

• If pk < n, ||u|| j,q,δ c ||u|| j+k,p,δ , ∀ p q np n-kp . • If pk = n, ||u|| j,q,δ c ||u|| j+k,p,δ , ∀ p q < ∞. • If pk > n, ||u|| j,q,δ c ||u|| j+k,p,δ , ∀ p q ∞.

Ehrling inequality:

For all ε > 0 , for all integers j, k such that 0 < j < k , there exists a positive constant

C(ε) such that ∀u ∈ W k,p δ , ||u|| j,p,δ ε||u|| k,p,δ + C(ε)||u|| p,δ . (11)
Rellich Theorem:

For all k > k ′ and δ < δ ′ , the inclusion W k,2 δ ⊂ W k ′ ,2 δ ′ is compact.
A consequence of the Sobolev inequalities (cf. [START_REF] Chrusciel | On mapping properties of the general relativistic constraints operator in weighted functions space, with applications[END_REF] for example) is

Proposition 2. In dimension n = 3, for all k > 3 2 , u ∈ W k,2 δ ⇒ u = o(ρ -δ ). ( 12
)
Lemma 2. In dimension 3, for all δ ∈ IR, there exists a positive constant c = c (g,

δ 1 , δ 2 ) such that ||uv|| 2,δ c ||u|| 1,2,δ 1 ||v|| 1,2,δ 2 , with δ 1 + δ 2 = δ. (13)
Remark: In the particular case where δ 1 , δ 2 and δ are non positive, then δ 1 and δ 2 are both greater than δ. The weighted Sobolev inclusion leads to

||uv|| 2,δ c ||u|| 1,2,δ ||v|| 1,2,δ . (14)
Lemma 3. For each of the following inequalities, in dimension 3, there exists a positive constant c = c (g, δ) such that

∀u ∈ W 2,2 δ (M) , ||u|| ∞,δ ε ||u|| 2,2,δ + cε -3 ||u|| 1,2,δ . ( 15
) ∀u ∈ W 1,2 δ (M) , ||u|| 3,δ ε ||u|| 1,2,δ + cε -1 ||u|| 2,δ . (16)
Lemma 4. In all dimension n, we define for R real large enough,

E R := M \ Ω R = {ρ e -2R }. Then ∀u ∈ L p δ (E R ), ∀ δ ∈ IR, ||u|| p,η;E R e 2R(δ-η) ||u|| p,δ;E R , ∀ δ η , ∀ 1 p ∞. (17)

The Hessian type operator T

Here we give some preliminary results concerning the operator T defined for a function N by:

T = T (N) := ∇2 N -Ng. ( 18
)
Lemma 5. For all δ ∈ IR, there exists a positive constant c > 0 depending on g such that for all

N ∈ W 2,2 -δ (M), || T (N)|| 2,-δ || ∇2 N|| 2,-δ -c ||N|| 2,-δ . ( 19
)
Proof: The result stems from the definition of T and the Triangle inequality.

Lemma 6. For all δ ∈] -

(n + 1)/2 , 0], there exists a positive constant c = c (g, δ) such that for all N ∈ W 2,2 -δ (M), ||N|| 1,2,-δ c || T (N)|| 2,-δ . ( 20
)
Proof: By density, we can suppose N ∈ C ∞ c (M). We use the proof of Proposition 4 which establishes (20) if the support of N is in a neighborhood of the boundary at infinity. Here we ignore the δ restriction due to positivity of the interior boundary term since N is compactly supported. We obtain the result near the boundary for δ ∈] -(n + 1)/2 ; 0] and conclude with kernel triviality of T for -δ < n+1 2 (see [START_REF] Chrusciel | Exotic hyperbolic gluings[END_REF]) thanks to a proof similar to the one of Theorem 1.

Combination of Lemmas 5 and 6 gives

Proposition 3. For all δ ∈] -(n + 1)/2 , 0] , there exists a positive constant c = c (g, δ)

such that ||N|| 2,2,-δ c || T (N)|| 2,-δ . ( 21
)
We will need the next three lemmas which give general equalities on (M,g), a ndimensional asymptotically hyperbolic Riemannian manifold with g = ρ -2 h. Here we allow an possible inner boundary ∂M. From now on, dσ(g) will be the measure induced by g on ∂M and η is the exterior unit normal to ∂M. The term o(1) will tend to zero when approaching ∂ ∞ M. Lemma 7. Let (M,g) be a n-dimensional asymptotically hyperbolic manifold and

N ∈ C ∞ (M) with a compact support on M. ∀δ ∈ IR , M 2N dN, dρ ρ g ρ 2δ dµ(g) = - M [2δ+1-n+o(1)]N 2 ρ 2δ dµ(g) + ∂M N 2 dρ ρ , η g ρ 2δ dσ(g) .
Proof: Integration by parts gives

M ∇i (-N 2 ∇i (ρ -1 )ρ 2δ+1 ) dµ(g) = M 2N dN, dρ ρ g ρ 2δ dµ(g) - M N 2 ∆(ρ -1 )ρ 2δ+1 dµ(g) + M (2δ + 1)N 2 |dρ| 2 hρ 2δ dµ(g). (22)
Let us compute (see [START_REF] Chrusciel | On mapping properties of the general relativistic constraints operator in weighted functions space, with applications[END_REF], (D.4) for instance)

∇2 (ρ -1 ) = ρ -1 |dρ| 2 hg -ρ -2 ( h) ∇ 2 ρ. ( 23
)
Taking the g-trace, ∆(ρ -1 ) = nρ -1 |dρ| 2 h -∆ hρ. The metric h being defined (and so bounded) until ∂ ∞ M and ρ being a smooth function on M , ∆ hρ is a smooth function bounded on M and so, we can write

∆ hρ = O(1) = o(ρ -1 ) near ∂ ∞ M. We obtain ∆(ρ -1 ) = ρ -1 n|dρ| 2 h + o(1) . ( 24
)
According to |dρ| 2 h = 1 + o(1) near the boundary at infinity on an asymptotically hyperbolic manifold, (22) become

M ∇i (-N 2 ∇i (ρ -1 )ρ 2δ+1 ) dµ(g) = M 2N dN, dρ ρ g ρ 2δ dµ(g) + M [2δ + 1 -n + o(1)]N 2 ρ 2δ dµ(g). ( 25
)
From the Divergence theorem,

M ∇i (-N 2 ∇i (ρ -1 )ρ 2δ+1 ) dµ(g) = ∂M N 2 dρ ρ , η g ρ 2δ dσ(g). (26)
We end the proof replacing the left-hand side of (25) by its expression (26).

Lemma 8. Let (M,g) be a n-dimensional asymptotically hyperbolic manifold and

N ∈ C ∞ (M) with a compact support on M. ∀δ ∈ IR , -2 M T (dN, dρ ρ )ρ 2δ dµ(g) = M {2δ + 1 -n + o(1)}|dN| 2 gρ 2δ dµ(g) - M [2δ + 1 -n + o(1)]N 2 ρ 2δ dµ(g) + ∂M (N 2 -|dN| 2 g) dρ ρ , η g ρ 2δ dσ(g), (27)
Proof: We integrate by parts the term ∇i (|dN| 2 g ∇i (ρ -1 ) ρ 2δ+1 ) and the result follows on from the Divergence theorem and Lemma 7.

Lemma 9. Let (M,g) be a n-dimensional asymptotically hyperbolic manifold and N ∈ C ∞ (M) with a compact support on M. ∀δ ∈ IR ,

- M tr g T N ρ 2δ dµ(g) = - M [ δ(2δ + 1 -n) -n + o(1)]N 2 ρ 2δ dµ(g) + M |dN| 2 gρ 2δ dµ(g) - ∂M N dN, η g ρ 2δ dσ(g) + ∂M δN 2 dN, η g ρ 2δ dσ(g), (28)
Proof: We integrate by parts the term ∇i (N ∇i Nρ 2δ ) and the result follows on from the Divergence theorem and Lemma 7.

The next proposition stems from the two previous lemmas and will play an important role to prove the adjoint kernel triviality: Proposition 4. For every ε > 0 , for all δ ∈] -(n + 1)/2, -1[, there exists R ε,δ > 0 such that for all R > R ε,δ , there exists a positive constant c such that

∀N ∈ C ∞ c (E R ) , ||N|| 2,2,-δ;E R c || T || 2,-δ;E R . (29)
Proof: From T and tr g T expressions and ∇n N (resp. ∇T N) being the component of dN normal (resp. tangential) to ∂ ∞ M, ∇n N := dN, η g and

|dN| 2 g = | ∇n N| 2 g + | ∇T N| 2 g. ( 28
) -1 2 (27) give M T (dN, dρ ρ )ρ 2δ dµ(g) - M Ntr g T ρ 2δ dµ(g) = M { n+1 2 -δ + o(1)}|dN| 2 gρ 2δ dµ(g) + M [-2δ 2 + nδ + n+1 2 + o(1)]N 2 ρ 2δ dµ(g) + ∂M {(δ -1 2 )N 2 + 1 2 |dN| 2 g} dρ ρ , η gρ 2δ dσ(g) - ∂M N ∇n N ρ 2δ dσ(g). (30)
Application on E R : E R possesses two disjoint boundary components. A boundary at infinity, noted ∂E ∞ = ∂ ∞ M, and an inner boundary

∂Ω R = {ρ = e -2R }. Since N ∈ C ∞ c (E R )
, N is surely null near ∂E ∞ but not necessarily on ∂Ω R and that's the reason why boundary terms in (30

) will only concern ∂Ω R . If η R is the normal to ∂Ω R exterior to E R and considering that when R → +∞ , η R -dρ ρ → 0 , so that dρ ρ , η R g = |dρ| 2 g ρ 2 + o(1) = 1 + o(1). E R T (dN, dρ ρ )ρ 2δ dµ(g) - E R N tr g T ρ 2δ dµ(g) = E R { n+1 2 -δ + o(1)}|dN| 2 gρ 2δ dµ(g) + E R [-2δ 2 + nδ + n+1 2 + o(1)]N 2 ρ 2δ dµ(g) + ∂E R {(δ -1 2 + o(1))N 2 + 1 2 |dN| 2 g -N ∇n N + o(1)}ρ 2δ dσ(g).
According to the following inequalities

     |tr g T | 2 g n | T | 2 g. T (dN, dρ ρ ) a 2 | T | 2 g + 1 2a |dN| 2 g|dρ| 2 h , ∀a > 0 , a >> 1. -N tr g T b 2 |tr g T | 2 g + 1 2b N 2 |dρ| 2 h , ∀b > 0 , b >> 1. ∀ε > 0, ∃R ε > 0 such that ∀R > R ε , a + bn 2 E R | T | 2 gρ 2δ dµ(g) E R { n+1 2 -δ -ε}|dN| 2 gρ 2δ dµ(g) + E R [-2δ 2 + nδ + n+1 2 -ε]N 2 ρ 2δ dµ(g) + ∂E R (δ -1 -ε)N 2 ρ 2δ dσ(g) + ∂E R 1 2 | ∇T N | 2 g + 1 2 (N -∇n N) 2 ρ 2δ dσ(g).
• The N 2 interior term is non negative if δ ∈] -1 2 ; n+1 2 [.

• The |dN| 2 g interior term is non negative if δ < (n + 1)/2.

• The boundary term is non negative if δ > 1. Moreover, a quick calculation shows that on the interval

[0 ; n+1 2 [ , n+1 2 -δ -2δ 2 + nδ + n+1 2 . Consequently, for δ ∈]1 ; n+1 2 [, ∀ε > 0, ∃R ε > 0 such that ∀R > R ε , a + bn 2 E R | T | 2 gρ 2δ dµ(g) E R { n+1 2 -δ -ε} N 2 + |dN| 2 g ρ 2δ dµ(g). Namely ||N|| 1,2,δ;E R c || T || 2,δ;E R .
Combining this inequality with Lemma 5 valid in particular on E R , we end the proof of (29).

The Killing operator S

Let S be the Killing operator defined on 1-forms by

S(Y ) ij = 1 2 ( ∇i Y j + ∇j Y i ) = ∇(i Y j) . (31)
The trace of this operator is

tr g S(Y ) = gij S(Y ) ij = ∇i Y i =: divY. ( 32
)
The next three lemmas are respectively versions of Lemma D.1 and Propositions D.2 and D.3 from [START_REF] Chrusciel | On mapping properties of the general relativistic constraints operator in weighted functions space, with applications[END_REF] with inner boundary: Lemma 10. Let V be a vector field and Y a 1-form both compactly supported on M. Then,

M ( S(Y ) + 1 2 tr g( S(Y ))g)(Y, V ) dµ(g) = - 1 2 M ∇V (Y, Y ) + 1 2 divV |Y | 2 g dµ(g) + 1 2 ∂M Y, V g Y, η g dσ(g) + 1 4 ∂M |Y | 2 g V, η g dσ(g),
Lemma 11. Let u be a function, V a vector field and Y a 1-form all compactly supported on M. Then,

M e 2u ( S(Y ) + 1 2 tr g( S(Y ))g)(Y, V ) dµ(g) = - 1 2 M e 2u ∇V (Y, Y ) + 1 2 divV |Y | 2 g dµ(g) + 1 2 ∂M e 2u Y, V g Y, η g dσ(g) - 1 2 M e 2u 2 du, Y g Y, V g + du, V g|Y | 2 g dµ(g) + 1 4 ∂M e 2u |Y | 2 g V, η g dσ(g), Lemma 12. Let Y be a 1-form compactly supported M and u, v ∈ C ∞ (M) two functions defined in a neighborhood of the support of Y . Then, -2 M v e 2u S(Y )(dv, dv) dv, Y g dµ(g) = M e 2u dv, Y g dv, Y g |dv| 2 g + v ∆v + 2v dv, du g + 2v ∇2 v(Y, dv) dµ(g) - ∂M v e 2u dv, Y 2 g dv, η g dσ(g),
Respective versions of Corollaries D.4 and D.5 from [START_REF] Chrusciel | On mapping properties of the general relativistic constraints operator in weighted functions space, with applications[END_REF] with possible inner boundary come out from these previous lemmas: Corollary 1. Let (M,g) be an asymptotically hyperbolic manifold and Y a 1-form compactly supported on M. ∀δ ∈ IR ,

2 M ρ 2δ ( S(Y ) + 1 2 tr g( S(Y ))g)(Y, dρ ρ ) dµ(g) = M ρ 2δ ( n+1 2 -δ + o(1))|Y | 2 g -(2δ + 1) dρ ρ , Y 2 g dµ(g) + 1 2 ∂M ρ 2δ |Y | 2 g dρ ρ , η g dσ(g) + ∂M ρ 2δ Y, dρ ρ g Y, η g dσ(g), (33)
Proof: We apply lemma 11 with

V = d(ρ -1 ) = -ρ -2 dρ u = (δ + 1 2 ) ln ρ and ∇V = ∇2 (ρ -1 ) = ρ -1 |dρ| 2 hg -ρ -2 ( h) ∇ 2 ρ from (23) divV = ∆(ρ -1 ) = ρ -1 (n|dρ| 2 h + o(1)) from (24) . du = (δ + 1 2 )
dρ ρ and e 2u = ρ 2δ+1 .

The metric h being defined (and so bounded) until ∂ ∞ M and ρ being a smooth function on M , ( h) ∇ 2 ρ is a smooth function bounded on M and so, we can write

( h) ∇ 2 ρ = o(ρ -1 ) near ∂ ∞ M. Thus, ∇V (Y, Y ) = ρ -1 (|dρ| 2 h|Y | 2 g + o(1)|Y | 2 g) divV |Y | 2 g = ρ -1 (n|dρ| 2 h + o(1))|Y | 2 g .
We obtain

M ρ 2δ ( S(Y ) + 1 2 tr g( S(Y ))g)(Y, dρ ρ ) dµ(g) = 1 2 M ρ 2δ 1 -(δ + 1 2 ) + n 2 |dρ| 2 h|Y | 2 g -(2δ + 1) dρ ρ , Y 2 g + o(1)|Y | 2 g dµ(g) + 1 4 ∂M ρ 2δ |Y | 2 g dρ ρ , η g dσ(g) + 1 2 ∂M ρ 2δ Y, dρ ρ g Y, η g dσ(g).
We end the proof with |dρ| 2 h = 1 + o(1) on an asymptotically hyperbolic manifold.

Corollary 2. Let (M,g) be an asymptotically hyperbolic manifold and Y a 1-form compactly supported on M. Then,

2 M ρ 2δ S(Y )( dρ ρ , dρ ρ ) dρ ρ , Y g dµ(g) = M ρ 2δ (n -1 -2δ + o(1)) dρ ρ , Y 2 g dµ(g) + ∂M ρ 2δ dρ ρ , Y 2 g dρ ρ , η g dσ(g), (34)
Proof: We apply lemma 12 with

v = ρ -1 u = (δ + 2) ln ρ , dv = d(ρ -1 ) = -ρ -1 dρ ρ du = (δ + 2) dρ ρ and e 2u = ρ 2δ+4 |dv| 2 g = ρ -2 |dρ| 2 h together with    ∇2 v = ∇2 (ρ -1 ) = ρ -1 |dρ| 2 hg -ρ -1 ( h) ∇ 2 ρ from (23) ∆v = ∆(ρ -1 ) = ρ -1 n|dρ| 2 h + o(1) from (24)
. 1) on an asymptotically hyperbolic manifold, we end up with

Since |dρ| 2 h = 1 + o(
2 M ρ 2δ S(Y )( dρ ρ , dρ ρ ) dρ ρ , Y g dµ(g) = M ρ 2δ n + 3 -2(δ + 2) + o(1) dρ ρ , Y 2 g dµ(g) + ∂M ρ 2δ dρ ρ , Y 2 g dρ ρ , η g dσ(g).
The following lemma establishes a Korn-type inequality for the Killing operator S:

Lemma 13. Assume M has no inner boundary. Then for all δ > -(n + 1)/2 and δ = -(n -1)/2, there exists a positive constant c = c (g, δ) such that for all 1-form

Y ∈ W 1,2 -δ (T * M), ||Y || 1,2,-δ c || S(Y )|| 2,-δ .
Proof: Here again we base ourselves on the proof of Theorem 1 but for the operator S. Lemma 2.8 from [START_REF] Chrusciel | On mapping properties of the general relativistic constraints operator in weighted functions space, with applications[END_REF] (for g with N = 0) replaces Lemma 1, in order to get a Korn-type inequality

||Y || 1,2,-δ c (|| S(Y ) -tr g(S(Y ))g|| 2,-δ + ||Y || 2,-δ ) c (|| S(Y )|| 2,-δ + ||Y || 2,-δ ) ,
where c = c (g, δ) is a positive constant. We now use Proposition D12 from [START_REF] Chrusciel | On mapping properties of the general relativistic constraints operator in weighted functions space, with applications[END_REF]: Let (M,g) be a conformally compact manifold with g = ρ -2 h . For all δ = -(n + 1)/2 and δ = -(n -1)/2, there exists two constants c δ > 0 and ρ ε,δ > 0 such that for all differentiable vector field Y compactly supported in

{ρ < ρ ε,δ } : ||Y || 2,-δ c δ || S(Y )|| 2,-δ . ( 35
)
Defining R such that ρ ε,δ = e -2R , we set Ω R as in Definition 1 and we have (35) on M\Ω R , as in hypothesis of Theorem 1. The rest of the proof is analogous to the one of Theorem 1 and we can choose R large enough so that for all

Y ∈ C ∞ c (T * M) , ||Y || 1,2,-δ c || S(Y )|| 2,-δ + ||Y || 2,-δ;Ω R .
Hence, the operator S :

W 1,2 -δ (T * M) → L 2 -δ (T * M) has a finite dimensional kernel. So we can write W 1,2
-δ (T * M) = ker S ⊕ (ker S) ⊥ . From then, there exists a positive constant c such that for all Y ∈ (ker S) ⊥ ,

||Y || 1,2,-δ c || S(Y )|| 2,-δ . It remains to show that ker S ∩ W 1,2 -δ = {0} , ∀δ > -(n + 1)/2.
For that matter, we use the coordinate system (x 1 = ρ, x 2 , . . . , x n ) = (ρ, θ) on a neighborhood of the boundary [0, ε] × ∂ ∞ M that we may find in [START_REF] Chrusciel | On mapping properties of the general relativistic constraints operator in weighted functions space, with applications[END_REF]. From the expression of the defining function ρ, the metric g can be written:

g = ρ -2 h = ρ -2 (dρ 2 + ĝ(ρ)) with ĝ(ρ)(∂ ρ , .) = 0.
The same conventions and notations as in the paper are in order: the index ρ will be the radial coordinate one whereas indices relative to tangential coordinates will be designated by latin capital indices. Finally lower case latin indices designate any components. Christoffel symbols, in this coordinate system, are given in [START_REF] Chrusciel | On mapping properties of the general relativistic constraints operator in weighted functions space, with applications[END_REF], just as the equation ∇i X j + ∇j X i = 0 (36) which becomes the system:

∂ ρ X ρ + ρ -1 X ρ = 0 (37) ∂ ρ X A + ∂ A X ρ + 2ρ -1 X A -ĝCD (ρ)ĝ ′ DA (ρ)X C = 0 (38) ∂ A X B + ∂ B X A -2 Γ C A B (ρ)X C + (ĝ ′ AB (ρ) -2ρ -1 ĝAB (ρ))X ρ = 0 where f ′ := ∂ ρ f . Solving equation (37) give X ρ = ρ -1 K(θ).
As the metric g is polyhomogeneous , ĝ can be written as a development of powers of ρ and ln ρ, and the first terms only contain powers of ρ. We set ĝCD ĝ′ DA := T C A , where T is and order two tensor whose development is

T C A (ρ, θ) = ρ 1 T C A (θ) + o(1). Andersson and Chruściel have shown in [2] that the solution X of (∆ -Ricg)X = 0 is also polyhomogeneous X A (ρ, θ) = ρ s Z A (θ) + o(ρ s ). Replacing in equation (38), we find s = -2 and X A = ρ -2 Z A (θ) + o(ρ -2 ).
We obtain the form of the solution of (36) near the boundary

X = ρ -1 K(θ) dρ + [ρ -2 Z A (θ) + o(ρ -2 )] dx A . Moreover, X ∈ W 1,2 -δ , leading to Z = 0. Indeed, suppose Z = 0. The g-norm of X is |X| 2 g ≃ ρ -2 |K| 2 g + ρ -4 |Z| 2 g, with |K| 2 g ≃ |Z| 2 g ≃ ρ 2 O(1). |X| 2 g ≃ ρ -2 O(1). Consequently ||X|| 2,-δ < ∞ ⇔ ε 0 |X| 2 gρ -2δ ρ -n dρ < ∞ ⇔ ε 0 ρ -2 ρ -2δ ρ -n dρ < ∞ ⇔ -2δ -n -2 > -1 ⇔ δ < -(n + 1)/2. Given that δ > -(n + 1)/2 , then X / ∈ W 1,2 -δ .
Hence, Z is necessarily null near the boundary. Analysing the X development coefficients, we realize Z = 0 lead to |X| g = O(ρ ∞ ). We conclude the proof using the unique continuation theorem from [START_REF] Mazzeo | Unique continuation at infinity and embedded eigenvalues for asymptotic hyperbolic manifolds[END_REF].

The contraint operator Φ

Let M be a n-dimensional connected non compact oriented manifold. We consider M as a spacelike hypersurface of a (n + 1)-dimensional Lorentzian manifold (N , γ), from now on refered to as spacetime. We will distinguish the two manifolds by different indices: Latin indices will take values from 1 to n and are spatial indices whereas Greek indices will take values from 0 to n and are spacetime indices. K is the second fundamental form of M in N defined by

K(X, Y ) = γ(X, (γ) ∇ Y n), (39)
where (γ) ∇ is the spacetime connection on T N , X, Y ∈ T M and n is the future-directed unit normal to M in N . It is convenient to consider the conjugate momentum π as a reparametrisation of K

π ij = π ij √ g with π ij = K ij -tr g Kg ij . ( 40
)
where √ g is the volume measure of the metric g.

√ g = det(g) det(g) dµ(g).
Remark: ∇( √ g) = 0 since the covariant derivative of the volume form is null.

π is a section of the bundle S 2 T M whereas π is a section of the bundle S = S 2 T M ⊗ Λ 3 T * M. We consider a smooth and polyhomogeneous asymptotically hyperbolic metric g asymptotically hyperbolic on M as model. We ask g to satisfy the following integrability condition Riemg ijklgil gjk + gik gjl ∈ L 2 δ .

(41) In particular,

Ricg jl + (n -1)g jl ∈ L 2 δ . (42)
For any sufficiently regular Riemannian metric g on M, we define the constraint operator Φ = (Φ 0 , Φ i ) = Φ(g, π) as follows:

Φ 0 (g, π) := R(g) -2Λ -|K| 2 g + (tr g K) 2 √ g = (R(g) -2Λ) √ g -|π| 2 g -1 n-1 (tr g π) 2 / √ g. ( 43
) Φ i (g, π) := 2(∇ j K ij -∇ i (tr g K)) √ g = 2g ij ∇ k π jk = 2g ij ∇ k π jk √ g. (44)
We set K = τg , (45) where τ is a real parameter. The cosmological constant Λ is normalized here in dimension n by

2Λ = n(n -1)(τ 2 -1), (46)
so that Φ(g, K) = 0 at infinity. Taking the g-trace of (42) considering (46), we end up with the integrability condition

R(g) -2Λ + n(n -1)τ 2 ∈ L 2 δ . (47)
The conjugate momentum π is then

πij = ( Kij -tr g Kg ij ) dµ(g) = τ (1 -n)g ij dµ(g). (48)
Remark : ∇π = ∇ K = 0.

If the spacetime satisfies Einstein's equations, the normalisation chosen insures that the constraint operator and the energy-momentum tensor are related by

Φ α = 16πGT nα √ g ,
where G is Newton's gravitationnal constant. ξ = (N, X i ) is the lapse-shift associated to the spacetime foliation. We study the constraint operator Φ for Riemannian metrics of the form g = g + h. g is asymptotic to g , i.e. |g -g| g = |h| g -→ ∞ 0.

S := S 2 T * M is the bundle of symmetric bilinear forms on M. S := S 2 T M ⊗ Λ 3 T * M is the bundle of symmetric 2-tensors-valued densities (3-forms) on M. T := T N is the spacetime tangent bundle. The following spaces will be of particular interest in the sequel:

G := W 2,2 δ (S). K := {π : π -π ∈ W 1,2 δ ( S)}. G + := {g : g -g ∈ G, g > 0}. G + λ := {g ∈ G + : λg < g < λ -1 g} , 0 < λ < 1. L * := L 2 δ (T * ⊗ Λ 3 T * M) is the dual space of L := L 2 -δ (T )
. From [START_REF] Fougeirol | Etude des équations de contraintes sur une variété Asymptotiquement hyperbolique[END_REF], tensors in G are Hölder-continuous and thus, matrices inequalities in spaces G + and G + λ are satisfied pointwise. In particular, for all metric g ∈ G + λ , metrics g and g are equivalent in the following sense

λg ij (x) v i v j < g ij (x) v i v j < λ -1 gij (x) v i v j , ∀x ∈ M , ∀v ∈ T M (49) So |g| g ≃ c |g| g ≃ c |g| g ≃ √ n.
F = G + × K will be the phase space of the contraint operator Φ. We will use (g, π) as well as (g, K) to express coordinates on F .

Let Γ and ∇ (resp. Γ and ∇) be the Christoffel symbols and the Levi-Civita connection for g (resp. g). We define

A k i j = Γ k i j -Γ k i j (50)
Remark: A is a symmetric tensor, arising from symmetry of the Christoffel symbols. We easily show

A k i j = g kl A ilj = 1 2 g kl ( ∇i g jl + ∇j g il -∇l g ij ) (51)
The scalar curvature of g can be express with ∇ and A j i k (see eq. ( 21) of [START_REF] Bartnik | Phase space for the Einstein equations[END_REF]):

Lemma 14. R(g) = g jk Ricg jk + g jk ( ∇i A i j k -∇j A i i k + A l j k A i i l -A i j l A l k i ) = g jk Ricg jk + Q(g -1 , ∇g) + g ik g jl ( ∇2 ij g kl -∇2 ik g jl ) ( 52
)
where Q is a sum of quadratic terms in g -1 , ∇g.

This result relies on the following fact:

Lemma 15. Ric g jk -Ricg jk = ∇i A i j k -∇j A i i k + A µ j k A i i µ -A i j µ A µ k i (53)
Proof:

Ric g jk -Ricg jk = ∂ i A i j k -∂ j A i i k + [Γ i l i Γ l j k -Γ i l i Γ l j k ] -[Γ l k i Γ i j l -Γ l k i Γ i j l ] = ∂ i A i j k -∂ j A i i k + [A i l i A l j k + Γ i l i A l j k + Γ l j k A i l i ] -[A l k i A i j l + Γ l k i A i j l + Γ i j l A l k i ]
We end the proof adding and substracting Γ l j i A i l k .

Here we show Φ is a well-defined mapping between the Hilbert spaces F and L * .

Proposition 5. Set (g, π) ∈ G + λ × K , with 0 < λ < 1.
Then in dimension n = 3, for all δ 0, there exists a positive constant c = c (λ,g, δ) such that

||Φ 0 (g, π)|| 2,δ c 1 + ||g -g|| 2 2,2,δ + ||π -π|| 2 1,2,δ (54)
||Φ i (g, π)|| 2,δ c || ∇(π -π)|| 2,δ + || ∇g|| 1,2,δ (1 + ||π -π|| 1,2,δ ) (55)
Proof: From R(g) expression (52), we can write

Φ 0 (g, π) = (R(g) -2Λ) √ g -(|π| 2 g -1 n-1 (tr g π) 2 ]/ √ g = [R(g)-R(g) + R(g) -2Λ+n(n -1)τ 2 -n(n -1)τ 2 ] √ g -[|π -π| 2 g -|π| 2 g + 2(π -π) ij πij + 2 |π| 2 g ]/ √ g + 1 n-1 [(tr g (π -π)) 2 + (tr g π) 2 + 2 tr g (π -π) tr g π]/ √ g = [R(g)-R(g) + R(g) -2Λ+n(n -1)τ 2 -n(n -1)τ 2 ] √ g -[|π -π| 2 g + 2(π -π) ij πij + |π| 2 g -1 n-1 (tr gπ) 2 ]/ √ g + 1 n-1 [(tr g (π -π)) 2 + ((g -g) ij πij ) 2 + 2(g -g) ij πij tr gπ + 2 tr g (π -π) tr g π]/ √ g = [R(g) -R(g) + R(g) -2Λ + n(n -1)τ 2 ] √ g -[|π -π| 2 g + 2(π -π) ij πij ]/ √ g + 1 n-1 [(tr g (π -π)) 2 + ((g -g) ij πij ) 2 + 2(g -g) ij πij tr gπ + 2 tr g (π -π) tr g π]/ √ g.
Since g ∈ G + λ , we can use (49) and from Cauchy-Schwarz inequality and 2ab

a 2 + b 2 |Φ 0 (g, π)| g [|R(g) -R(g)| g + |R(g) -2Λ + n(n -1)τ 2 | g] √ g +c [1 + |π -π| 2 g + |g -g| 2 g ]/ √ g. From (53), Ric g -Ricg ≃ ∇A + A 2 ≃ ( ∇g) 2 + g ∇2 g + g -2 ( ∇g) 2 .
Using [START_REF] Hille | Functional analysis and semi-groups[END_REF],

||Ric g -Ricg|| 2,δ (c || ∇g|| 2 1,2,δ + c || ∇2 g|| 2,δ ) c ||g -g|| 2,2,-δ .
In particular, we have the following integrability conditions

Ric g -Ricg ∈ L 2 δ . (56) R(g) -R(g) ∈ L 2 δ . (57)
Thanks to (57),( 47) and ( 14),

||Φ 0 (g, π)|| 2,δ c 1 + ||(π -π) 2 || 2,δ + ||(g -g) 2 || 2,δ c 1 + ||π -π|| 2 1,2,δ + ||g -g|| 2 2,2,δ . Hence, Φ 0 (g, π) ∈ L * . Regarding Φ i (g, π) , using (50), Φ i (g, π) = 2g ij ( ∇k (π -π) jk + A j k l (π -π) kl ) + A j k l πkl ).
Considering (51), Φ i (g, π) is of the form

Φ i (g, π) ≃ g( ∇(π -π) + g -1 ∇g (π -π) + g -1 ∇g π). ( 58
) ||Φ i (g, π)|| 2,δ c (|| ∇(π -π)|| 2,δ + || ∇g (π -π)|| 2,δ + || ∇g π|| 2,δ ) c (|| ∇(π -π)|| 2,δ + || ∇g|| 1,2,δ ||π -π|| 1,2,δ + || ∇g|| 2,δ ||π|| ∞,0 ) c || ∇(π -π)|| 2,δ + || ∇g|| 1,2,δ (1 + ||π -π|| 1,2,δ ) .
Proposition 6. Let (g, π) ∈ F . Then in dimension n = 3, for all δ 0, Φ : F → L * is a smooth map between Hilbert spaces.

Proof: We recall the proof of [START_REF] Bartnik | Phase space for the Einstein equations[END_REF] for completeness. From Proposition 5,

||Φ(g, π)|| L * c(1 + ||g -g|| 2 G + ||π -π|| 2 K ) , i.e.
Φ is locally bounded on F . The polynomial structure of the constraint operator allows us to show Φ is smooth, i.e. indefinitely differentiable in a Fréchet sense. From the expression (52) of scalar curvature and given (58) , Φ can be expressed as

Φ(g, π) = F (g, g -1 , √ g, 1/ √ g, ∇g, ∇2 g, π, ∇π) ,
where F = F (a 1 , . . . , a 8 ) is a polynomial function quadratic in a 5 and a 7 and linear in the remaining parameters. The map g → (g, g -1 , √ g, 1/ √ g) is analytic on the space of positive definite matrices and the maps g → ∇g , g → ∇2 g and π → ∇π are bounded linear, thus smooth, from F to L * , which are Hilbert spaces. A result from Hille [START_REF] Hille | Functional analysis and semi-groups[END_REF] on locally bounded polynomial functionals shows Φ admit continuous Fréchet-derivatives of all orders.

The set C = {(g, π) ∈ G + × K : Φ(g, π) = 0} := Φ -1 ({0}) ⊂ F is the set of initial data for the vacuum Einstein's equations. To prove that C is a submanifold of F , we show that 0 is a regular value of Φ , so we are interested in the surjectivity of the differential of Φ.

Expressions of the linearization of Φ and its adjoint

In this section we recall the expression of the linearization of Φ and its adjoint that we may find in [START_REF] Bartnik | Phase space for the Einstein equations[END_REF] or [START_REF] Fischer | Topics in the dynamics general relativity[END_REF] for example.

Proposition 7. DΦ 0 (g, π).(h, p) = (∇ i ∇ j h ij -∆ g tr g h) √ g -h ij [R ij -1 2 (R(g) -2Λ)g ij ] √ g +h ij 2 n-1 tr g ππ ij -2π i k π kj + 1 2 |π| 2 g g ij -1 2(n-1) (tr g π) 2 g ij / √ g +p ij ( 2 n-1 tr g πg ij -2π ij )/ √ g. ( 59
) DΦ i (g, π).(h, p) = π jk (2∇ k h ij -∇ i h jk ) + 2h ij ∇ k π jk + 2g ik ∇ j p jk . ( 60
)
Using notations of [START_REF] Bartnik | Phase space for the Einstein equations[END_REF] ,

δ g δ g h = ∇ i ∇ j h ij . E ij = R ij -1 2 (R(g) -2Λ)g ij . Π ij = 2 n-1 tr g ππ ij -2π i k π kj + 1 2 |π| 2 g g ij -1 2(n-1) (tr g π) 2 g ij /( √ g) 2 .
We can express DΦ in the matricial following form

DΦ(g, π).(h, p) = √ g(δ g δ g -∆ g tr g + Π -E) -2K π∇ + 2δ g π 2δ g h p , (61)
with π∇h = πjkl i ∇ j h kl = (π jk δ l i + π jl δ k i -π kl δ j i )∇ j h kl .
To prove surjectivity of the differential of Φ , we investigate injectivity of the adjoint operator. Integrating by parts and ignoring boundary terms leads (cf. [START_REF] Fischer | Topics in the dynamics general relativity[END_REF] for example) to the expression of the formal L 2 (dµ(g))-adjoint of DΦ(g, π) :

M DΦ(g, π).(h, p) (N, X i ) = M (h, p) • DΦ(g, π) * (N, X i ). Proposition 8. (h, p) • DΦ 0 (g, π) * N = h ij [∇ i ∇ j N -g ij ∆ g N -[R ij -1 2 (R(g) -2Λ)g ij ]N] √ g +Nh ij 2 n-1 tr g ππ ij -2π i k π kj + 1 2 |π| 2 g g ij -1 2(n-1) (tr g π) 2 g ij / √ g +Np ij ( 2 n-1 tr g πg ij -2π ij )/ √ g. (h, p) • DΦ i (g, π) * X i = h ij (X k ∇ k π ij + ∇ k X k π ij -2∇ k X (i π j)k ) -2p ij ∇ (i X j) .
Then we can put DΦ * in the matricial form

DΦ(g, π) * .(N, X) = √ g(∇ 2 -g∆ g + Π -E) ∇π -π∇ -2K -L g N X , (62)
with

(∇π -π∇)X = L X π = ∇ X π ij -πkij l ∇ k X l . L g (X) = L X g = 2 ∇ (i X j) = 2 S(X).

DΦ(g, π) *

1 .ξ and DΦ(g, π) * 2 .ξ will denote the two components of DΦ(g, π) * in (62).

L 2 δ ξ (resp. W 1,2 δ ∇ξ) is the set of terms of the form u ξ (resp. u ∇ξ) such that ||u|| 2,δ C (resp. ||u|| 1,2,δ C ), where C is a constant depending on g, δ and ||(g, π)|| F . DΦ(g, π) * 1 .ξ = [∇ i ∇ j N -g ij ∆ g N + (Π ij -E ij )]N] √ g + (∇π -π∇)X = DΦ(g, 0) * (N, 0) + Π ij N √ g + (∇π -π∇)X (63) (∇π -π∇)X = X k ∇ k π ij -(π k i δ lj + π k j δ li -π ij δ k l )∇ k X l = L 2 δ X + W 1,2 δ ∇X + (n -1)τ (2 S(X) -g tr g S(X)). Π(g, π)N = L 2 δ N + Π(g,π)N = L 2 δ N + -1 2 (n -1)(n -4)τ 2 gN. So we have the integrability condition Π(g, π) + 1 2 (n -1)(n -4)τ 2 g ∈ L 2 δ . (64)
Taking into account (42) and ( 47),

E + (n -1)g -1 2 n(n -1)τ 2 g ∈ L 2 δ . (65)
On one hand,

DΦ(g, π) * 1 .ξ/ √ g = ∇ 2 N -g∆ g N + (n -1)g + [Π + 1 2 (n -1)(n -4)τ 2 g]N +(n -1)τ (2 S(X) -g tr g S(X)) -[E + (n -1)g -1 2 n(n -1)τ 2 g]N] -(n -1)(n -2)τ 2 gN + L 2 δ ξ + W 1,2 δ ∇X = ∇ 2 N -g∆ g N + (n -1)g + (n -1)τ (2 S(X) -g tr g S(X)) -(n -1)(n -2)τ 2 gN + L 2 δ ξ + W 1,2 δ ∇X. ( 66
)
On the other hand,

DΦ(g, π) * 2 .ξ = -2KN -2S(X) = -2( S(X) + τgN) + W 1,2 δ ξ.
From the definition of the operator T = ∇ 2 N -Ng and the expression of S as a function of DΦ(g, π) * 2 .ξ, we obtain DΦ(g, π) * 1 .ξ/ √ g = Tg tr g T + (n -1)τ (2 S(X)g tr g S(X))

-(n -1)(n -2)τ 2 gN + L 2 δ ξ + W 1,2 δ ∇X. ( 67
) DΦ(g, π) * 2 .ξ = -2( S(X) + τgN) + W 1,2 δ ξ. (68)
It is useful to restructure DΦ * into the operator P * defined by

P * (ξ) = P * (g,π) (ξ) = g 1/4 ∇ i ∇ j N -δ i j ∆ g N + (Π i j -E i j )N + g -1/4 L X π i j -2g -1/4 ∇ l (K i j N + S(X) i j ) = ζ • 1 0 0 ∇ • DΦ(g, π) * ξ , (69)
where g 1/4 = (det(g)/det(g)) 1/4 dµ(g) is a density of weight 1 2 and

ζ = ζ(g) = g -1/4 g jk 0 0 g 1/4 g ik . (70)
Finally, we can put P * (g,π) (ξ) into the form

P * (g,π) (ξ) = g -1/4 DΦ(g, π) * 1 .ξ g 1/4 ∇DΦ(g, π) * 2 .ξ . (71)
Expression (69) of P * allows us to rewrite the L 2 (dµ(g))-adjoint of P * as follows

P (g,π) = DΦ(g, π) • 1 0 0 -δ g • ζ , (72)
with δ g q = ∇ l (q ij l ) so that P (f i j , q j li ) = DΦ(f ij , q ij l ) and so the composition P P * is well defined.

Elliptic estimates relative to the adjoint

In this section, we gather elliptic estimates satisfied by the adjoint operator DΦ * . Proposition 9. Set δ ∈] -(n + 1)/2 , 0], with n = 3, and δ = -(n -1)/2. There exists a positive constant C = C (g, λ, δ, ||g|| F ) such that the following elliptic estimate is satisfied:

∀ξ ∈ W 2,2 -δ (T ) , ||ξ|| 2,2,-δ c ||DΦ(g, π) * 1 .ξ|| 2,-δ + ||DΦ(g, π) * 2 .ξ|| 1,2,-δ + C ||ξ|| 1,2,-2δ , (73)
Proof: Considering expression (68) of S as a function of DΦ(g, π

) * 2 .ξ, T -g tr g T = DΦ(g, π) * 1 .ξ/ √ g + (n -1)τ DΦ(g, π) * 2 .ξ -1 2 g tr g DΦ(g, π) * 2 .ξ +L 2 δ ξ + W 1,2 δ ∇X (74)
From Proposition 3 and (8),

||N|| 2,2,-δ c ||DΦ(g, π) * 1 .ξ|| 2,-δ + (n -1)τ (1 + n 2 4 ) 1 2 ||DΦ(g, π) * 2 .ξ|| 2,-δ +C (||ξ|| ∞,-2δ + || ∇ξ|| 3,-2δ ).
Using [START_REF] Mazzeo | Unique continuation at infinity and embedded eigenvalues for asymptotic hyperbolic manifolds[END_REF], [START_REF] Mccormick | The phase space for the Einstein-Yang-Mills equations and the first law of black hole thermodynamics[END_REF] and Sobolev inclusion (δ 0), there exists a positive constant

C = C (g, λ, δ, ||g|| F ) such that ||N|| 2,2,-δ c ||DΦ(g, π) * 1 .ξ|| 2,-δ + (n -1)τ (1 + n 2 4 ) 1 2 ||DΦ(g, π) * 2 .ξ|| 2,-δ +ε ||ξ|| 2,2,-δ + C ||ξ|| 1,2,-2δ . (75)
Besides, for every sufficiently regular 1-form X on M, we have the following identity for the metric g (cf equation (29) of [START_REF] Bartnik | Phase space for the Einstein equations[END_REF] for example)

∇2 kj X i := ∇k ∇j X i = Riemg ijkl X l + ∇k S(X) ij + ∇j S(X) ik -∇i S(X) jk , (76)
Hence

|| ∇2 X|| 2,-δ ||Riemg X|| 2,-δ + c || ∇S (X)|| 2,-δ c ||X|| 2,-δ + c || ∇S (X)|| 2,-δ . (77)
A consequence of Lemma 13 is

||X|| 2,-δ ||X|| 1,2,-δ c || S(X)|| 2,-δ , (78)
which imply, with (77

) || ∇2 X|| 2,-δ c || S(X)|| 1,2,-δ , (79)
∈ W 2,2 -δ (T ) , ||ξ|| 2,2,-δ c ||DΦ(g, π) * 1 .ξ|| 2,-δ + ||DΦ(g, π) * 2 .ξ|| 1,2,-δ + C ||ξ|| 2,-2δ , (82)
where C depends on g, λ, δ and ||(g, π)|| F .

The next lemma will be very useful during the proof of proposition 10 since it is the Time-symmetric version.

Lemma 16. In dimension n = 3, let δ 0 , then the operator DΦ(g, 0) * (., 0) : W 2,2

-δ (M) -→ L 2 -δ ( S) is bounded and depends on g in a Lipschitz way, DΦ(g, 0) * -DΦ(g, 0) * (N, 0)

2,-δ C||g -g|| F ||N|| 2,2,-δ , (83)
where constant C depends on g, δ, ||g|| F and ||g|| F .

Proof: Let us recall the expression of DΦ(g, 0)

* DΦ(g, 0) * .(N, 0) = [∇ i ∇ j N -g ij ∆ g N -[R ij -1 2 (R(g) -2Λ)g ij ]N] √ g. (84)
Let us begin by showing DΦ(g, 0) * is bounded. In order to do this, we introduce the operator O acting on functions

O(N) = ∇ 2 N -g ∆ g N (85)
and we notice that O(N) = L(∇ 2 N) where L is a linear invertible operator, so

||O|| 2,-δ c ||∇ 2 N || 2,-δ = c || ∇2 N|| 2,-δ + ||A dN|| 2,-δ C ||N|| 2,2,-δ .
Indeed, A dN ≃ g -1 ∇g dN. Using Hölder inequality ( 8) , [START_REF] Hille | Functional analysis and semi-groups[END_REF] and Sobolev inclusion,

||A dN|| 2,-δ ||g -1 || ∞,0 || ∇g dN || 2,-δ c || ∇g|| 1,2,δ ||dN|| 1,2,-2δ C ||N|| 2,2,-δ . ( 86
) ||DΦ(g, 0) * .(N, 0) √ g|| 2,-δ ||O|| 2,-δ + ||(Ric g -Ricg) N || 2,-δ + ||(n -1)gN || 2,-δ +||[Ricg + (n -1)g] N|| 2,-δ + || 1 2 n(n -1)τ 2 g N || 2,-δ + 1 2 R(g) -2Λ + n(n -1)τ 2 g N 2,-δ . Considering (56), (42), ||(Ric g -Ricg)N|| 2,-δ C ||N|| 2,2,-δ ||[Ricg + (n -1)g]N || 2,-δ c ||N|| 2,2,-δ .
For the scalar curvature term, using Hölder inequality [START_REF] Dahl | Asymptotically hyperbolic manifolds with small mass[END_REF] and Sobolev inclusion together with (47) and (57),

||(R(g) -2Λ + n(n -1)τ 2 ) g N || 2,-δ ||(R(g) -2Λ + n(n -1)τ 2 ) g N|| 2,-δ +||(R(g) -R(g)) g N || 2,-δ C ||N|| 2,2,-δ . ||(n -1)gN || 2,-δ (n -1)||g|| ∞,0 ||N|| 2,-δ c ||N|| 2,2,-δ .
Similarly,

|| 1 2 n(n -1)τ 2 gN|| 2,-δ c ||N|| 2,2,-δ .
We end up with ||DΦ(g, 0) * .(N, 0) √ g|| 2,-δ C ||N|| 2,2,-δ and finally

||DΦ(g, 0) * .(N, 0)|| 2,-δ C || √ g|| ∞,0 ||N|| 2,2,-δ C ||N|| 2,2,-δ , ( 87
)
where C is a constant depending upon g, δ and ||g|| F .

Proof of (83): ∇ , ∆ , Ric(g) and R(g) will denote The Levi-Civita connection, the Laplacian, the Ricci tensor and the scalar curvature of the Riemannian metric g. In order to lighten notations, we set DΦ 0 (g) * N := DΦ(g, 0) * (N, 0) and DΦ 0 (g) * N := DΦ(g, 0) * (N, 0)

[DΦ 0 (g) * -DΦ 0 (g) * ]N = ( √ g -g) DΦ 0 (g) * N √ g + g DΦ 0 (g) * N √ g - DΦ 0 (g) * N √ g . [DΦ 0 (g) * -DΦ 0 (g) * ]N 2,-δ ||g -g|| F DΦ 0 (g) * N √ g 2,-2δ +c DΦ 0 (g) * N √ g - DΦ 0 (g) * N √ g 2,-δ . ( 88
) DΦ 0 (g) * N √ g - DΦ 0 (g) * N √ g (∇ -∇) dN + g ∆ g N -g ∆N -[Ric(g) -Ric(g)]N + 1 2 [(R(g) -2Λ)g -(R(g) -2Λ)g] N. DΦ 0 (g) * N √ g - DΦ 0 (g) * N √ g 2,-δ ||(∇ -∇) dN|| 2,-δ + ||g ∆ g N -g ∆N || 2,-δ + 1 2 [(R(g) -2Λ)g -(R(g) -2Λ)g] N 2,-δ -||[Ric(g) -Ric(g)]N || 2,-δ .
• For the Hessian:

∇ -∇ = (g -1 -g-1 ) ∇g + g-1 ∇(g -g). (89)
Using Hölder and Sobolev weighted inequalities, Sobolev inclusion (δ 0) and (13),

||(∇ -∇) dN|| 2,-δ C ||g -g|| 2,2,δ ||N|| 2,2,-δ .
(90) • • For the Laplacian:

g ∆ g N -g ∆N = g ∆ g N -g ∆ g N + g ∆ g N -g ∆N = (g -g)∆ g N + g(∆ g N -∆N) = (g -g)g -1 ∇dN + g(g -1 -g-1 )∇dN + gg -1 (∇ -∇)dN.
Using Hölder inequality [START_REF] Dahl | Asymptotically hyperbolic manifolds with small mass[END_REF], Sobolev inclusion (δ 0) and Sobolev inequality,

||g ∆ g N -g ∆N|| 2,-δ c ||g -g|| 2,2,δ ||∇dN|| 2,-δ + c ||(∇ -∇)dN|| 2,-δ .

Considering (86) and given that

∇ ≃ A + ∇ , ||∇dN|| 2,-δ C ||N|| 2,2,-δ , (91)
Using (90) and (91) , we get

||g ∆ g N -g ∆N|| 2,δ C ||g -g|| 2,2,δ ||N|| 2,2,-δ .
• For the Ricci tensor:

We define

A k i j = Γ k i j -Γ k i j . Set T := ∇ -∇ = g-1 ∇g -g -1 ∇g = (g -1 -g-1 ) ∇g + g-1 ∇(g -g).
Using Hölder inequality [START_REF] Dahl | Asymptotically hyperbolic manifolds with small mass[END_REF], Sobolev inclusion (δ 0) and Sobolev inequality,

||T || 1,2,δ C||g -g|| 2,2,δ . (92)
We can show, adding and substracting Ric(g) and using (53), that

[Ric(g) -Ric(g)]N ≃ ( ∇T + AT + T 2 )N , which leads to ||[Ric(g) -Ric(g)]N|| 2,-δ || ∇T N|| 2,-δ + || AT N || 2,-δ + ||T 2 N || 2,-δ . ( 93
)
Using Hölder and Sobolev inequalities along with Sobolev inclusion 0) and ( 92)

|| ∇T N|| 2,-δ C ||g -g|| 2,2,δ ||N|| 2,2,-δ
The same method for the term AT N gives , considering ( 13)

|| AT N|| 2,-δ C ||g -g|| 2,2,δ ||N|| 2,2,-δ
Using (8) , ( 14) , Sobolev inclusion (δ 0) and Sobolev inequality together with (92) ,

||T 2 N || 2,-δ C ||g -g|| 2 2,2,δ ||N|| 2,2,-δ . Replacing in (93), we obtain ||[Ric(g) -Ric(g)]N || 2,-δ C ||g -g|| 2,2,δ ||N|| 2,2,-δ . ( 94
)
• For the scalar curvature,

(R(g) -2Λ)g -(R(g) -2Λ)g = (g -g)(R(g) -2Λ) + gg -1 (Ric g -Ric g) +g(g -1 -g-1 )Ric g = (g -g) R(g) -2Λ + n(n -1)τ 2 -n(n -1)τ 2 (g -g) + gg -1 (Ric g -Ric g) +g(g -1 -g-1 ) Ric g -Ricg .
Hölder and Sobolev inequalities as well as Sobolev inclusion (δ 0) and (94) yield

||[(R(g) -2Λ)g -(R(g) -2Λ)g]N|| 2,-δ C ||g -g|| 2,2,δ ||N|| 2,2,δ , given that ∀u ∈ L ∞ 0 , ∀v ∈ L 2 δ such that ||vN|| 2,-δ C ||N|| 2,2,-δ , ||(g -g) u v N || 2,-δ ||g -g|| ∞,0 ||u|| ∞,0 ||vN|| 2,-δ C ||g -g|| 2,2,δ ||v|| 2,δ ||N|| 2,2,-δ ,
where C is a positive constant depending on g, δ and ||g|| F . Putting the pieces all together in (88) and taking (87) into account lead to

||[DΦ 0 (g) * -DΦ 0 (g) * ]N|| 2,-δ C ||g -g|| 2,2,δ ||N|| 2,2,δ .
The dependence in (g, π) of P * is controled as follows :

Proposition 10. Let δ 0 , then in dimension 3, the operator

P * : W 2,2 -δ (T ) -→ L 2 -δ
is bounded and satisfies

||ξ|| 2,2,-δ c ||P * ξ|| 2,-δ + C ||ξ|| 1,2,-2δ , ( 95
)
where C depends on g, δ and ||(g, π)|| F . Moreover, P * (g,π) depends on (g, π) ∈ F in a Lipschitz way,

||(P * (g,π) -P * (g,π) ) ξ|| 2,-δ C 1 ||(g -g, π -π)|| F ||ξ|| 2,2,-δ , ( 96
)
where constant C 1 depends on g, δ, ||(g, π)|| F and ||(g, π)|| F .

Proof: Let us begin by showing P * is bounded, i.e.

||P * ξ|| 2,-δ C ||ξ|| 2,2,-δ . ( 97
) Set      P * = P * (g,π) DΦ * 1 = DΦ(g, π) * 1 DΦ * 2 = DΦ(g, π) * 2 .
From (71),

||P * ξ|| 2,-δ c (||DΦ * 1 .ξ|| 2,-δ + ||∇DΦ * 2 .ξ|| 2,-δ ) c (||DΦ * 1 .ξ|| 2,-δ + || ∇DΦ * 2 .ξ|| 2,-δ + ||ADΦ * 2 .ξ|| 2,-δ ). (98)
From ( 67),( 8), ( 13), Sobolev inequality and inclusion (δ 0)

||DΦ * 1 .ξ|| 2,-δ c ||T || 2,-δ + || S(X)|| 2,-δ + ||N|| 2,-δ + C ||ξ|| ∞,-2δ + || ∇X|| 1,2,-2δ c ||N|| 2,2,-δ + ||X|| 1,2,-δ + ||N|| 2,-δ + C ||ξ|| 2,-δ + || ∇X|| 1,2,-δ C ||ξ|| 2,2,-δ . ( 99
)
From (68) along with ( 8), ( 13), Sobolev inequality and inclusion (δ 0) 8), Sobolev inequality and inclusion (δ 0)

||DΦ * 2 .ξ|| 2,-δ c || S(X)|| 2,-δ + ||N|| 2,-δ + C ||ξ|| 1,2,-2δ . ( 100
) ||ADΦ * 2 .ξ|| 2,-δ c ||A S(X)|| 2,-δ + ||AN|| 2,-δ + ||ξ|| ∞,-2δ C || S(X)|| 1,2,-δ + ||N|| 1,2,-δ + ||ξ|| 2,2,-2δ || ∇DΦ * 2 .ξ|| 2,-δ c || ∇S (X)|| 2,-δ + || ∇N || 2,-δ + ||ξ|| ∞,-2δ C ||ξ|| 2,
D Φ * 1 = DΦ(g, π) * 1 D Φ * 2 = DΦ(g, π) * 2 . (P * -P * ) ξ = g -1/4 DΦ * 1 .ξ -g-1/4 D Φ * 1 .ξ g 1/4 ∇DΦ * 2 .ξ -g1/4 ∇D Φ * 2 .ξ =: E F . So ||(P * -P * ) ξ|| 2,-δ ||E|| 2,-δ + ||F || 2,-δ . ( 102
) E = g -1/4 DΦ * 1 .ξ -g-1/4 D Φ * 1 .ξ = (g -1/4 -g-1/4 ) DΦ * 1 .ξ + g-1/4 (DΦ * 1 .ξ -D Φ * 1 .ξ). Using (
||E|| 2,-δ ||(g -1/4 -g-1/4 ) DΦ * 1 .ξ|| 2,-δ + ||g -1/4 (DΦ * 1 .ξ -D Φ * 1 .ξ)|| 2,-δ c ||g -g|| F ||DΦ * 1 .ξ|| 2,-δ + c ||DΦ * 1 .ξ -D Φ * 1 .ξ|| 2,-δ . From (63), DΦ * 1 .ξ -D Φ * 1 .ξ = [DΦ(g, 0) * -DΦ(g, 0) * ] (N, 0) + (Π √ g -Π g)N + X ∇(π -π) +(π -π) ∇X + AX(π -π).
Using ( 8), ( 13), Sobolev inequality and inclusion (δ 0)

||(π -π) ∇X|| 2,-δ + ||X ∇(π -π)|| 2,-δ ||π -π|| 1,2,δ || ∇X|| 1,2,-2δ +|| ∇(π -π)|| 2,δ ||X|| ∞,-2δ c ||π -π|| 1,2,δ ||X|| 2,2,-δ . ||AX(π -π)|| 2,-δ ||A(π -π)|| 2,δ ||X|| ∞,-2δ C ||π -π|| 1,2,δ ||X|| 2,2,-δ . Π √ g -Π g ∼ 1 √ g g -1 π 2 - 1 √ g g-1 π2 ∼ 1 √ g (g -1 -g-1 )π 2 + 1 √ g g-1 (π 2 -π2 ) + ( 1 √ g - 1 √ g )g -1 π2 . ||(Π √ g -Π g)N|| 2,-δ C ||(g -g, π -π)|| F ||N|| 2,2,-δ .
Given (83),

||DΦ * 1 .ξ -D Φ * 1 .ξ|| 2,-δ C ||(g -g, π -π)|| F ||ξ|| 2,2,-δ and taking (99) into account, ||E|| 2,-δ C ||(g -g, π -π)|| F ||ξ|| 2,2,-δ . ( 103
) F = g 1/4 ∇DΦ * 2 .ξ -g1/4 ∇D Φ * 2 .ξ = g 1/4 (∇ -∇)DΦ * 2 .ξ + (g 1/4 -g1/4 ) ∇D Φ * 2 .ξ + g 1/4 ∇(DΦ * 2 .ξ -D Φ * 2 .ξ).
Using (89),( 8), ( 13), Sobolev inequality and inclusion (δ 0) 100) and ( 101),

||F || 2,-δ c ||∇ -∇|| 1,2,δ ||DΦ * 2 .ξ|| 1,2,-2δ + ||g 1/4 -g1/4 || ∞,-δ || ∇D Φ * 2 .ξ|| 2,-2δ +c || ∇(DΦ * 2 .ξ -D Φ * 2 .ξ)|| 2,-δ + c ||A(DΦ * 2 .ξ -D Φ * 2 .ξ)|| 2,-δ C ||g -g|| F ||DΦ * 2 .ξ|| 1,2,-2δ + c ||g -g|| F || ∇D Φ * 2 .ξ|| 2,-2δ +c || ∇(DΦ * 2 .ξ -D Φ * 2 .ξ)|| 2,-δ + c ||A(DΦ * 2 .ξ -D Φ * 2 .ξ)|| 2,-δ . Considering (
||F || 2,-δ C ||g -g|| F ||ξ|| 2,2,-δ + c || ∇(DΦ * 2 .ξ -D Φ * 2 .ξ)|| 2,-δ (104) +c ||A(DΦ * 2 .ξ -D Φ * 2 .ξ)|| 2,-δ . DΦ * 2 .ξ -D Φ * 2 .ξ ∼ (K -K)N + (A -Ã)X ∼ (π -π)N + (∇ -∇)X.
Using (89),( 8), ( 13), Sobolev inequality and inclusion (δ 0)

|| ∇(DΦ * 2 .ξ -D Φ * 2 .ξ)|| 2,-δ c || ∇(π -π)|| 2,δ ||N|| ∞,-2δ + c ||π -π|| 1,2,δ || ∇N || 1,2,-2δ +|| ∇(∇ -∇)|| 2,δ ||X|| ∞,-2δ + ||∇ -∇|| 1,2,δ || ∇X|| 1,2,-2δ c ||π -π|| 1,2,δ ||N|| 2,2,-δ + c ||∇ -∇|| 1,2,δ ||X|| 2,2,-δ C ||(g -g, π -π)|| F ||ξ|| 2,2,-δ .
In the same way,

||A(DΦ * 2 .ξ -D Φ * 2 .ξ)|| 2,-δ c ||A(π -π)|| 2,δ ||N|| ∞,-2δ + ||A(∇ -∇)|| 2,δ ||X|| ∞,-2δ C ||(g -g, π -π)|| F ||ξ|| 2,2,-δ .
We deduce from (104)

||F || 2,-δ C ||(g -g, π -π)|| F ||ξ|| 2,2,-δ . (105)
and the estimate (96) arises from (102), considering (103) and (105).

We show in the following proposition that the estimate (82) of Corollary 3 is also verified by weak solutions ξ only in L 2 -δ (T ). We say that ξ ∈ L is a weak solution of

DΦ(g, π) * ξ = (f 1 , f 2 ) , with (f 1 , f 2 ) ∈ L 2 -δ ( S) × W 1,2 -δ (S) when M ξ, DΦ(g, π).(h, p) g = M (f 1 , f 2 ), (h, p) g , ∀(h, p) ∈ G × K. It suffices to test with (h, p) ∈ C ∞ c (S × S) since this place is dense in G × K. Proposition 11. Let δ ∈] -(n + 1)/2 , 0] \ {-(n -1)/2} with n = 3 , (g, π) ∈ G + × K , (f 1 , f 2 ) ∈ L 2 -δ ( S) × W 1,2 -δ (S). Let ξ ∈ L be a weak solution of DΦ(g, π) * ξ = (f 1 , f 2 ). Then ξ ∈ W 2,2
-δ (T ) is a strong solution and satisfies (82). Proof: In [START_REF] Bartnik | Phase space for the Einstein equations[END_REF], Bartnik shows that ξ ∈ W 2,2 loc . We can find a cut-off function χ R as in Definition 1 such that

• χ R ∈ C ∞ c (Ω R ). • χ R = 1 on Ω R/2 .
In particular, χ R ξ ∈ W 2,2 -δ (T ) and from Proposition 9 , we can write:

||χ R ξ|| 2,2,-δ c ||DΦ(g, π) * 1 .(χ R ξ)|| 2,-δ + ||DΦ(g, π) * 2 .(χ R ξ)|| 1,2,-δ + C ||ξ|| 2,-δ , (106)
using Sobolev inclusion (δ 0) and considering that χ R ξ converge to ξ in L 2 -δ . We have to show that χ R ξ is uniformly bounded in W 2,2 -δ , i.e. bounded independently of R. In order to do so, we adapt S. McCormick's method found in [START_REF] Mccormick | The Hilbert manifold of asymptotically flat metric extensions[END_REF]. From (66) and (68),

DΦ(g, π) * 1 .(χ R ξ) ≃ χ R (DΦ(g, π) * 1 .(ξ)) + N ∇ 2 χ R + dN ∇χ R + X ∇χ R + ξ W 1,2 δ ∇χ R . DΦ(g, π) * 2 .(χ R ξ) ≃ χ R (DΦ(g, π) * 2 .(ξ)) + X ∇χ R . As derivatives of χ R are supported in A R := Ω R \Ω R/2
, we can use ( 8), ( 15), ( 16), Sobolev inclusion (δ 0) and Ehrling inequality to obtain

||DΦ(g, π) * 1 .(χ R ξ)|| 2,-δ c ||χ R (DΦ(g, π) * 1 .(ξ))|| 2,-δ + ||N ∇ 2 χ R || 2,-δ +||dN ∇χ R || 2,-δ + ||X ∇χ R || 2,-δ + ||ξ W 1,2 δ ∇χ R || 2,-δ c ||DΦ(g, π) * 1 .(ξ)|| 2,-δ + C ||ξ|| 2,-δ;A R + ε|| ∇2 ξ|| 2,-δ;A R , (107)
where C is a constant depending on g, λ, ε, δ and ||(g, π)|| F . Likewise,

||DΦ(g, π) * 2 .(χ R ξ)|| 1,2,-δ c ||χ R (DΦ(g, π) * 2 .(ξ))|| 1,2,-δ + ||X ∇χ R || 1,2,-δ c ||DΦ(g, π) * 2 .(ξ)|| 1,2,-δ + C ||X|| 2,-δ;A R + ε|| ∇2 X|| 2,-δ;A R . (108)
Plugging ( 107) and ( 108) into (106) yields

||χ R ξ|| 2,2,-δ c ||DΦ(g, π) * 1 .(ξ)|| 2,-δ + ||DΦ(g, π) * 2 .(ξ)|| 1,2,-δ + C ||ξ|| 2,-δ +ε|| ∇2 ξ|| 2,-δ;A R . (109)
Equation ( 85) allows us to determine the following link between operators O and T :

O = T -g tr g T -(n -1)gN
and we deduce from (74)

O = DΦ(g, π) * 1 .ξ/ √ g + (n -1)τ DΦ(g, π) * 2 .ξ -1 2 g tr gDΦ(g, π) * 2 .ξ +L 2 δ ξ + W 1,2 δ ∇X -(n -1)gN. (110)
Since O(N) = L(∇ 2 N) with L a linear invertible operator, we also get

||∇ 2 N|| 2,-δ c ||O|| 2,-δ , leading to || ∇2 N || 2,-δ c (||O|| 2,-δ + ||A dN|| 2,-δ ) . (111)
From ( 111), (110) and using ( 8), ( 15), ( 16), Sobolev inclusion (δ 0) and Ehrling inequality, there exists a constant C depending on g, λ, ε, δ and ||(g, π)|| F such that

|| ∇2 N || 2,-δ c ||DΦ(g, π) * 1 .ξ|| 2,-δ + ||DΦ(g, π) * 2 .ξ|| 2,-δ +ε || ∇2 ξ|| 2,-δ + C ||ξ|| 2,-δ . (112)
Moreover, combining (79) and (80) and using Sobolev inclusion (δ 0) and Ehrling inequality, we get

|| ∇2 X|| 2,-δ c ||DΦ(g, π) * 2 .ξ|| 1,2,-δ + ε|| ∇2 ξ|| 2,-δ + C ||ξ|| 2,-δ . (113)
Now combination of ( 112) and (113) gives

|| ∇2 ξ|| 2,-δ c ||DΦ(g, π) * 1 .ξ|| 2,-δ + ||DΦ(g, π) * 2 .ξ|| 1,2,-δ + C ||ξ|| 2,-δ . (114)
Given that ξ ∈ W 2,2 -δ (A R ) and that all the inequalities used to obtain (114) are valid in particular on every compact set, we have the following local estimate on

A R || ∇2 ξ|| 2,-δ;A R c ||DΦ(g, π) * 1 .ξ|| 2,-δ;A R + ||DΦ(g, π) * 2 .ξ|| 1,2,-δ;A R + C ||ξ|| 2,-δ;A R c ||DΦ(g, π) * 1 .ξ|| 2,-δ + ||DΦ(g, π) * 2 .ξ|| 1,2,-δ + C ||ξ|| 2,-δ . (115)
Finally, equation (109) leads to

||χ R ξ|| 2,2,-δ c ||DΦ(g, π) * 1 .(ξ)|| 2,-δ + ||DΦ(g, π) * 2 .(ξ)|| 1,2,-δ + C ||ξ|| 2,-δ c ||f 1 || 2,-δ + ||f 2 || 1,2,-δ + C ||ξ|| 2,-δ . χ R ξ is then a Cauchy sequence in W 2,2
-δ , and so converge in

W 2,2 -δ . As χ R ξ converge to ξ in L 2
-δ , uniqueness of the limit implies that χ R ξ converge to ξ in W 2,2 -δ (T ). So ξ ∈ W 2,2 -δ (T ) and consequently ξ verifies (82).

The operator Ů

Here we introduce an operator Ů inspired by the formula (76). It will allow us to control the W 2,2 δ -norm of X with the L 2 δ -norms of S and Ů , in other words with the W 1,2 δ -norm of S. The key estimate will arise from a succession of lemmas. Let Ů be the operator defined on 1-forms by

Ůkji (X) = ∇2 kj X i -gjk X i + gik X j . (116)
This readily implies

|| ∇2 X|| 2,-δ -c ||X|| 2,-δ || Ů(X)|| 2,-δ . (117)
The next four lemmas are established on an asymptotically hyperbolic manifold (M,g) , with g = ρ -2 h and |dρ| 2 h = 1 + o(1) near the boundary at infinity.

Proof: From (116), Ůkji (X) ∇j X i ∇k ρ ρ = ∇2 kj X i ∇j X i ∇k ρ ρ -∇X( dρ ρ , X) + ∇X(X, dρ ρ ).
We integrate the term ∇k (| ∇X| 2 g ∇k ρ ρ ρ 2δ ) and the lemma stems from the Divergence theorem and Lemma 17.

Considering also the equality

2 | S(X)| 2 g = | ∇X| 2 g + ∇j X k ∇k X j , we get
Lemma 20. Let (M,g) be an asymptotically hyperbolic manifold and

X ∈ C ∞ (T * M) compactly supported on M. ∀δ ∈ IR , M ρ 2δ Ůkji (X)g ik X j + 2 | S(X)| 2 g -2δ divX X, dρ ρ dµ(g) = M ρ 2δ [n -1 -2δ + o(1)]|X| 2 g dµ(g) + M ρ 2δ | ∇X| 2 g dµ(g) (122)
+ M ρ 2δ 2δ(2δ + 1) X, dρ ρ 2 g dµ(g) + ∂M ρ 2δ ∇X(X, η) -2δ X, dρ ρ g X, η g dσ(g) , Proof: From (116), Ůkji (X)g ik X j = ∇2 kj X k X j + (n -1)|X| 2 g.
We integrate by parts the term ∇k ( ∇j X k X j ρ 2δ ) and the result follows on from the Divergence theorem along with (119) and the equality

2 | S(X)| 2 g = | ∇X| 2 g + ∇j X k ∇k X j .
We can now prove the following proposition, crucial in the demonstration of the adjoint kernel triviality.

Proposition 12. Set E R := M \ Ω R , for an open set Ω R . For all ε > 0 , for all δ ∈] -2 , -1[ , there exists R ε,δ > 0 such that for all R > R ε,δ , there exists a constant c > 0 such that

∀X ∈ C ∞ c (E R ) , ||X|| 1,2,-δ;E R c (|| Ů(X)|| 2,-δ;E R + || S(X)|| 2,-δ;E R ). (123)
Proof: The linear combination (121) -1 2 (120) + 1 2 (122) + (33) -1 2 (34)

yields M ρ 2δ Ůkji (X) ∇j X i ∇k ρ ρ -1 2 gkj X i + 1 2 gik X j dµ(g) + M ρ 2δ | S(X)| 2 g -S(X)( dρ ρ , dρ ρ ) X, dρ ρ g + (2 -δ)divX X, dρ ρ dµ(g) = M {-δ 2 + ( n-3 2 )δ + n + 1 + o(1)}|X| 2 gρ 2δ dµ(g) + M ρ 2δ 2δ 2 -2δ -( n+3 2 ) + o(1) X, dρ ρ 2 g dµ(g) + M n+1 2 -δ + o(1) | ∇X| 2 gρ 2δ dµ(g) + ∂M ρ 2δ 1 2 (δ -1)|X| 2 g dρ ρ , η g dσ(g) + ∂M 1 2 | ∇X| 2 g dρ ρ , η g + 1 2 |X| 2 g dρ ρ , η g + 1 2 ∇X(X, dρ ρ ) -1 2 ∇X( dρ ρ , X) dσ(g) + ∂M ρ 2δ (2 -δ) X, dρ ρ g X, η g -1 2 X, dρ ρ 2 g dρ ρ , η g dσ(g). (124)
Application on E R : With the same notations as in Proposition 4, since X ∈ C ∞ c (E R ), boundary terms in (30) will only concern ∂Ω R . X n (resp. X T ) being the component of X normal (resp. tangential) to ∂ ∞ M, X n := X, η g and

|X| 2 g = X 2 n + X 2 T . E R ρ 2δ Ůkji (X) ∇j X i ∇k ρ ρ -1 2 gkj X i + 1 2 gik X j dµ(g) + E R ρ 2δ | S(X)| 2 g -S(X)(η R , η R ) X n + (2 -δ)divXX n dµ(g) = E R ρ 2δ {-δ 2 + ( n-3 2 )δ + n + 1 + o(1)}X 2 T dµ(g) + E R ρ 2δ n+1 2 -δ + o(1) | ∇X| 2 g + δ 2 + ( n-7 2 )δ + n-1 2 + o(1) X 2 n dµ(g) + ∂E R ρ 2δ 1 2 (δ -1) X 2 T + o(1) dσ(g) + ∂E R ρ 2δ 1 2 (2 -δ) X 2 n + o(1) dσ(g) + ∂E R 1 2 | ∇X| 2 g|η| 2 g + 1 2 |X| 2 g + 1 2 ∇X(X, η) -1 2 ∇X(η, X) dσ(g).
Given the following equalities

0 1 4 | ∇i X j η i -X j | 2 g = 1 4 | ∇X| 2 g|η| 2 g + 1 4 |X| 2 g -1 2 ∇X(η, X) 0 1 4 | ∇i X j η j + X i | 2 g = 1 4 | ∇X| 2 g|η| 2 g + 1 4 |X| 2 g + 1 2 ∇X(X, η)
we get that for all ε > 0 , there exists R ε > 0 and c ε >> 1 such that ∀R > R ε ,

c ε E R ρ 2δ | Ů (X)| 2 g + | S(X)| 2 g dµ(g) E R ρ 2δ {-δ 2 + ( n-3 2 )δ + n + 1 -ε}X 2 T dµ(g) E R ρ 2δ [ n+1 2 -δ -ε]| ∇X| 2 g dµ(g) + E R ρ 2δ {δ 2 + ( n-7 2 )δ + n-1 2 -ε}X 2 n dµ(g) + 1 2 ∂E R ρ 2δ (2 -δ -ε)X 2 n dσ(g) + 1 2 ∂E R ρ 2δ (δ -1 -ε)X 2 T dσ(g). (125)
• The X 2 T term is non negative if δ ∈] -2 ; n+1 2 [.

• The X 2 n term is non negative ∀n 3 , ∀δ ∈] n-1 2 ; n+1 2 [.

• The | ∇X| 2 g term is non negative if δ < (n + 1)/2.
• The boundary term is non negative if δ ∈]1, 2[.

Hence, ∀δ ∈]1 ; 2[, ∀ε > 0, ∃R ε > 0 and c ε >> 1 such that ∀R > R ε , c ε E R ρ 2δ | Ů(X)| 2 g + | S(X)| 2 g dµ(g) E R ρ 2δ [ n+1 2 -δ -ε]| ∇X| 2 g dµ(g) + E R ρ 2δ {-δ 2 + ( n-3 2 )δ + n + 1 -ε}X 2 T + {δ 2 + ( n- 7
2 )δ + n-1 2 -ε}X 2 n dµ(g) dµ(g).

The adjoint kernel triviality

In this section we show that the kernel of DΦ(g, π) * is trivial. We will need the following lemmas and propositions in the proof of Theorem 3.

Lemma 21. Let δ ∈ IR and ξ ∈ W 2,2
-δ (T ) be a solution of DΦ 0 (g, π) * ξ = 0. Then ξ = (N, X) satisfies a system of the form

T (N) = b 0 ξ + b 1 ∇ξ S(X) = -gτ N + b 2 ξ , ( 126
)
with b 0 ∈ L 2 δ and b 1 , b 2 ∈ W 1,2 δ .
Proof: From ( 74) and (68) , DΦ 0 (g, π) * ξ = 0 leading to

T -(tr g T )g = L 2 δ ξ + W 1,2 δ ∇X. (127) S(X) = -gτ N + W 1,2
δ ξ. Taking the trace of (127),

T (N) = L 2 δ ξ + W 1,2 δ ∇ξ. (128) S(X) = -gτ N + W 1,2 δ ξ.
Theorem 3. Triviality of ker DΦ 0 (g, π) * .

Let Ω ⊂ M be a connected open set such that E R ⊂ Ω. We fix (g, π) ∈ F . Set δ ∈] -2 , -1[ and n = 3. Suppose ξ ∈ L 2 -δ (M) verifies DΦ 0 (g, π) * ξ = 0 on Ω. Then ξ ≡ 0 on Ω.

Proof: From Proposition 11 , ξ ∈ W 2,2
-δ (M). According to the Lemma 21 and since ξ is a solution of DΦ 0 (g) * ξ = 0 , ξ satisfies

T (N) = b 0 ξ + b 1 ∇ξ S(X) = -gτ N + b 2 ξ , with b 0 ∈ L 2 δ and b 1 , b 2 ∈ W 1,2 δ . From (41), Ůkji (X) = ∇2 kj X i -Riemg ijkl + L 2 δ X. (129) Regarding (76) Ů(X) = c 1 ∇S (X) + L 2 δ X = -c 1 τg ∇N + L 2 δ ξ + W 1,2 δ ∇ξ. (130)
We must show that a solution ξ of (126) such that ξ = o(ρ δ ) (from ( 12)) vanishes. Before pursuing the proof of the theorem, let us recall Proposition 3.9 of [START_REF] Bartnik | Phase space for the Einstein equations[END_REF] :

Proposition 13. In dimension n = 3, set δ 0 and (g, π) ∈ F . Let Ω be a connected subset of M. Let ξ satisfy DΦ(g, π) * ξ = 0 on Ω. If in addition ξ ≡ 0 on an open set U ⊂ Ω , then ξ ≡ 0 on Ω.

Given the previous proposition, it remains to show ξ vanishes near infinity. As for (117), we have

|| ∇2 X|| 2,-δ;E R -c ||X|| 2,-δ;E R || Ů(X)|| 2,-δ;E R . (131)
Combine to (123), together with (130), Sobolev inequality and ( 8), we obtain

||X|| 2,2,-δ;E R c (|| Ů(X)|| 2,-δ;E R + || S(X)|| 2,-δ;E R ) c ||N|| 1,2,-δ;E R + C (||ξ|| ∞,-2δ;E R + || ∇ξ|| 3,-2δ;E R ). (132)
Using Proposition 4 , there exists ε 0 << 1 such that (29) + ε 0 (132) give

||ξ|| 2,2,-δ;E R c || T || 2,-δ;E R + C (||ξ|| ∞,-2δ;E R + || ∇ξ|| 3,-2δ;E R ).
Considering Lemma 21 along with Sobolev inequality and ( 17) with δ 0,

||ξ|| 2,2,-δ;E R C ||ξ|| 2,2,-2δ;E R C e 4Rδ
ξ ∈ W 2,2 -δ (T ), ||ξ|| 2,2,-δ C ||P * ξ|| 2,-δ . (133)
Proof: In order to show that the kernel of P * is finite dimensional, we apply Riesz theorem showing every bounded subset of ker P * is || || 2,2,-δcompact. Let {ξ k } be a sequence of ker P * such that ||ξ k || 2,2,-δ = 1. Rellich theorem tells us we can extract from {ξ k } a sub-sequence, also noted {ξ k }, converging in W 1,2 -2δ to a limit ξ. Hence,

{ξ k } is a Cauchy sequence in W 1,2 -2δ . From (73), considering that {ξ k } ∈ kerP * , {ξ k } is a Cauchy sequence in W 2,2
-δ , and so converges to ξ in W 2,2 -δ , from the limit uniqueness. This ends the proof of the finite dimension of ker P * . ker P * is thus a closed subspace of the Hilbert vector space W 2,2 -δ . Being a finite dimensional closed subspace of a normed vector space , it splits and if we set W to be the closed complement of ker P * , W 2,2 -δ = kerP * ⊕ W.

From the same argument as in the proof of Theorem 1 , there exists a constant C > 0 depending on ||(g, π)|| F such that for all ξ ∈ W ,

||ξ|| 2,2,-δ C ||P * ξ|| 2,-δ (134)
We conclude thanks to the triviality of ker P * from Theorem 3.

11. The submanifold structure Lemma 22. Let X, Y be two Banach spaces and T a linear operator with closed range.

T : X → Y T * : Y * → X * then (CokerT ) * ≃ ker T * , where CokerT = Y Im T Proof: We define ψ : ker T * → (Coker T ) * = L Y Im T , IR ρ → (λ : y + T X → ρ(y))
The map λ is well defined because ∀x ∈ X , ρ(T x) = T * (ρ)(x) = 0. The map ψ is invertible and In order to prove the Theorem 4, we will use the implicit function theorem, so we have to show:

• ker DΦ(g, π) splits.

• • DΦ(g, π) is surjective.

Given that the kernel of DΦ(g, π) is finite dimensional, we show that ker DΦ(g, π) is closed and hence splits. DΦ(g, π) being a bounded operator, its kernel is closed by continuity. The triviality of ker DΦ(g, π) * , established in Theorem 3, leads to

(kerDΦ(g, π) * ) ⊥ = L * .
Using the classic relation

(kerDΦ(g, π) * ) ⊥ = ImDΦ(g, π), we get ImDΦ(g, π) = L * .
Thus, in order to have the surjectivity of DΦ 0 (g) , it suffices to prove DΦ(g, π) has closed range. To do so, we consider particular variations (h, p) of (g, π) of the form

h ij = 2 y g ij p ij = 2S(Y ) ij -g ij tr g S(Y ) -(n -1)(n -2)τ y g ij √ g , (135)
determined from fields (y, Y i ). We define the operator

F (y, Y i ) = [F 0 (y, Y i ), F i (y, Y i )] = [DΦ 0 (g, π)(h, p), DΦ i (g, π)(h, p)]. (136)
From ( 59) and (60),

     F 0 (y, Y i ) = 2(n -1) √ g [-∆y + ny] + (4 -n) Φ 0 (g, π) y + 2(n -2)τ divY √ g +L 2 δ [y + Y ] + W 1,2 δ ∇Y. F i (y, Y i ) = -2 √ g [-∆Y i + (n -1)Y i ] + 2 Φ i (g, π) y + W 1,2 δ ∇y + L 2 δ [y + Y ].
Definition 2. Operator asymptotic to ∆.

We say an operator P of the form

P u = a ij (x) ∇2 ij u + b i (x)∂ i u + c(x)
u is asymptotic to ∆ with a decaying rate τ if there exists n < q < ∞ , τ 0 and two positive constants C 1 , λ such that

λ|ξ| 2 g a ij (x)ξ i ξ j λ -1 |ξ| 2 g, ∀x ∈ M , ξ ∈ T M. ||a ij -gij || 1,q,τ + ||b i || q,τ + ||c|| q/2,τ C 1 .
Proposition 14. Let g ∈ G + with δ 0. Then ∆ is asymptotic to ∆ with a decaying rate δ.

Proof:

∆ = g ij ∇ 2 ij = g ij ∇2 ij + g ij (∇ i -∇i ) ∇j = g ij ∇2 ij -g ij A k i j ∇k . (137)
The metrics g and g being equivalent, equation (49) directly gives

λ|ξ| 2 g g ij (x)ξ i ξ j λ -1 |ξ| 2 g, ∀x ∈ M , ξ ∈ T M. Setting b k = g ij A k i j , then b ∈ W 1,2
δ from (51). Given the Sobolev inequality, there exists a constant C 1 > 0 such that

||g ij -gij || 1,6,δ + ||b k || 6,δ c (||g ij -gij || 2,2,δ + ||b k || 1,2,δ ) C 1 .
We will justify later on the terminology "asymptotic" used in this definition. The operator A = -∆ + n , acting on functions, will be of great interest. Proof: By elliptic regularity, u ∈ W 2,2 loc and the estimate arises from interior estimates (see [START_REF] Gilbarg | Elliptic Partial Differential Equations of Second Order[END_REF] for example) and scaling.

Theorem 5. Let g ∈ G + with δ 0 so that ∆ is asymptotic to ∆.

Set A = -∆ + n , with n = 3. Let |s| < (n + 1)/2. Then A : W 2,2 s (M) → L 2 s (T * ⊗ Λ 3 T * M) is bounded. Moreover, it satisfies the following elliptic estimate ||u|| 2,2,s C (||Au|| 2,s + ||u|| 2,s;Ω R ) . (138)
In particular, A is a semi-Fredholm operator , i.e. A has finite dimensional kernel and closed range.

Proof: We define the following operator norm:

||∆ -∆|| op = sup M {||(∆ -∆)u|| 2,s : u ∈ W 2,2 s , ||u|| 2,2,s = 1} and || • || op,R denotes the same norm restricted to functions supported in E R = M \ Ω R . If supp(u) ⊂ E R , then from the expression (137) of ∆ , ||(∆ -∆)u|| 2,s ||(g ij -gij ) ∇2 ij u|| 2,s + ||b k ∇k u|| 2,s sup E R {g ij -gij }|| ∇2 ij u|| 2,s + ||b k ∇k u|| 2,s c ||g -g|| ∞,0;E R || ∇2 u|| 2,s + ||b ∇u|| 2,s .
Using [START_REF] Dahl | Asymptotically hyperbolic manifolds with small mass[END_REF], Sobolev inequality and inclusion (δ 0),

||(∆ -∆)u|| 2,s c ||g -g|| 2,2,δ;E R + ||b|| 1,2,δ;E R ||u|| 2,2,s . Recalling that ||g -g|| 2,2,δ + ||b|| 1,2,δ is bounded because g ∈ G + , ||∆ -∆|| op,R = o(1) when R → +∞. ( 139
χ R = 1 on Ω R 2 0 on M \ Ω R Then we can decompose u = u 0 + u ∞ , with u ∞ = (1 -χ R )u.
We look into the operator Å = -∆ + n acting on functions. Using Corollary 3.13 of [START_REF] Andersson | Elliptic systems on manifolds with asymptotically negative curvature[END_REF] with λ = n , we obtain that ∀|s| < (n + 1)/2 , Å : W From the expression (137) of ∆, Let B = -∆ + n -1 be an operator acting on 1-forms. (143)

χ R Au -Au 0 = -uAχ R + nχ R u + 2g ij ∂ i u∂ j χ R = 2g ij ∂ i u ∂ j χ R + (g ij ∇2 ij χ R + b i ∂ i χ R )
In particular, B is a semi-Fredholm operator , i.e. B has finite dimensional kernel and closed range.

Proof: From Proposition E of [START_REF] Lee | Fredholm operators and Einstein metrics on conformally compact manifolds[END_REF], the indicial radius of B is (n+1)/2 and by Theorem C of [START_REF] Lee | Fredholm operators and Einstein metrics on conformally compact manifolds[END_REF], ∀|s| < (n + 1)/2 , B : W (145)

In particular, F is a semi Fredholm operator, i.e. F has finite dimensional kernel and closed range.

Proof: Starting from the definition of F , the Triangle inequality together with (54) and the Sobolev inclusion (with δ 0) directly yield

||F (y, Y )|| 2,δ C ||(y, Y )|| 2,2,δ ,
where C is a constant depending on g and ||g|| F . Hence F is a bounded (continuous) operator. Plugging the expression of F 0 (y, Y i ) in (138) and using Hölder inequality ( 8) , [START_REF] Mazzeo | Unique continuation at infinity and embedded eigenvalues for asymptotic hyperbolic manifolds[END_REF] ,Ehrling inequality [START_REF] Gilbarg | Elliptic Partial Differential Equations of Second Order[END_REF]

Finally, combination of (146) and (147) gives (145). For all δ ∈]-(n+1)/2, 0] , the estimate (145) verified by F is analoguous to the one of Theorem 1 and by a similar proof, we show F is semi-Fredholm, i.e. F has finite dimensional kernel and closed range. Now F and its adjoint F * have similar structure (F is formally self-adjoint)

F * : L 2 -δ (T ) → W -2,2
-δ (T * ⊗ Λ 3 T * M). Let F * be the restriction of F * defined as follows Plugging the expression of F * 0 (y, Y i ) (formally identical to F 0 (y, Y i )) in (138) and using Hölder inequality [START_REF] Dahl | Asymptotically hyperbolic manifolds with small mass[END_REF] , [START_REF] Mazzeo | Unique continuation at infinity and embedded eigenvalues for asymptotic hyperbolic manifolds[END_REF] ,Ehrling inequality [START_REF] Gilbarg | Elliptic Partial Differential Equations of Second Order[END_REF] along with Sobolev inclusion (with δ 0) and Φ 0 (g, π) ∈ L Plugging the expression of F * i (y, Y i ) (formally identical to F i (y, Y i )) in (143) and using Hölder inequality ( 8) , [START_REF] Mazzeo | Unique continuation at infinity and embedded eigenvalues for asymptotic hyperbolic manifolds[END_REF] ,Ehrling inequality [START_REF] Gilbarg | Elliptic Partial Differential Equations of Second Order[END_REF]

Finally, combination of ( 149) and ( 150) gives (148). Similarly to F , for all δ ∈] -(n + 1)/2, 0] , F * is a semi-Fredholm operator.

We are now in possession of all the tools necessary to finish the proof of Theorem 4. By elliptic regularity, kerF * = ker F * is also finite dimensional. If we apply Lemma 22 to F , we get (CokerF ) * ≃ kerF * . So (CokerF ) * is finite dimensional so (CokerF ) * ≃ CokerF . Thus we have the isomorphism CokerF = L * ImF ≃ kerF * . The operator F satisfies ImF ⊂ ImDΦ(g, π) ⊂ L * . Let π be the canonic projection: π : L * → L * ImF . π(ImDφ) is closed, being the subspace of a finite dimensional vector space. Im(Dφ) is closed, being the inverse image of a closed set by a continuous map. This ends the proof of the manifold structure of C, as a smooth submanifold of F . In fact, all level sets of Φ(g, π) are smooth submanifolds of F .

2. 2 .

 2 Sobolev and Hölder weighted spaces. Thanks to the background metric g , we can define the following norm ∀ 1 p < ∞ and δ ∈ IR : ||u|| p,δ = M |u| p g ρ pδ dµ(g) 1/p For p = ∞ and δ ∈ IR : ||u|| ∞,δ = sup M ρ δ |u| g

1 4 1 4

 11 and considering (78), ||X|| 2,2,-δ c 1 || S(X)|| 1,2,-δ . From (68) and (15), || S(X)|| 1,2,-δ ||DΦ(g, π) * 2 .ξ|| 1,2,-δ + nτ ||N|| 1,2,-δ + ε||ξ|| 2,2,-δ + C ||ξ|| 1,2,-2δ . (80) Thus there exists a constant C depending on g, λ, ε, δ and ||(g, π)|| F such that ||X|| 2,2,-δnc 1 τ ||N|| 1,2,-δ c ||DΦ(g, π) * 2 .ξ|| 1,2,-δ + ε||ξ|| 2,2,-δ + C ||ξ|| 1,2,-2δ . (81) We can choose ε 0 << 1 so that (75) + ε 0 (81) combine to yield (73). Remark: it may be possible to extend this result to δ = -(n -1)/2 using operator Ů introduced later on. Combining Proposition 9 and Ehrling inequality (11), we get Corollary 3. Let δ ∈] -(n + 1)/2 , 0], with n = 3 and δ = -(n -1)/2. Then the following estimate is verified: ∀ξ

Corollary 4 .

 4 ||ξ|| 2,2,-δ;E R . We end up with ∀ δ ∈] -(n + 1)/2 , -1], ||ξ|| 2,2,-δ;E R C e 4Rδ ||ξ|| 2,2,-δ;E R . For R large enough, ||ξ|| 2,2,-δ;E R = 0 and thanks to Sobolev inequality, ξ vanishes on E R , for R >> 1. From Prop.13, ξ ≡ 0 on Ω, because Ω is connected by assumption. This ends the proof of Theorem 3. Set δ ∈] -(n + 1)/2 , 0] \ {-(n -1)/2}, with n = 3. There exists a constant C > 0 depending on ||(g, π)|| F such that for

ψ - 1 :Theorem 4 .

 14 (Coker T ) * → ker T * λ → ρwhere ρ(y) := λ(y + T X).Note that ρ ∈ ker T * because T * (ρ)(x) = ρ(T x) = λ( 0) = 0. Remark: The closed range of T imply CokerT is a Banach space. Let Φ : F → L * be the constraint operator in dimension n = 3.For every ε ∈ L * , for all δ ∈] -2 , -1[ , the set of solutions of the constraint equationsC(ε) := {(g, π) ∈ F : Φ(g, π) = ε}is a submanifold of F . In particular, the space of solutions of the vacuum constraint equations C = C(0) has a Hilbert submanifold structure.

Proposition 15 .

 15 Let g ∈ G + with δ 0 and A = -∆ + n. Let s ∈ IR. There exists a constant C = C(n, p, q, s, δ, C 1 , λ) such that if u ∈ L 2 s and Au ∈ L 2 s , then u ∈ W 2,2 s and ||u|| 2,2,s C (||Au|| 2,s + ||u|| 2,s ) .

  ) This justifies a posteriori the terminology used in Definition 2.Let χ R be a cut-off function as in Definition 1

Theorem 6 .

 6 Let δ 0 and g ∈ G + . Setting B = -∆ + n -1 and |s| < (n + 1)/2. Then B : W 2,2 s (T * M) → L 2 s (T * ⊗ Λ 3 T * M) is bounded. Furthermore, it satisfies ||Y || 2,2,s C (||BY || 2,s + ||Y || 2,s;Ω R ) .

2 , 2 s → L 2 s

 222 is a Fredholm operator. By Corollary 3.13 of[START_REF] Andersson | Elliptic systems on manifolds with asymptotically negative curvature[END_REF] , B is an isomorphism for the same span of s. So there exists a positive constantC = C(n, s) such that ||Y || 2,2,s C || BY || 2,s .(144)The proof is nearly identical to the one of Theorem 5 with (144) replacing (140).

Theorem 7 .

 7 Let δ ∈] -(n + 1)/2, 0] with n = 3. Then the operatorF : W 2,2 δ (T ) → L 2 δ (T * ⊗ Λ 3 T * M) := L * is bounded. Furthermore, it verifies ||(y, Y )|| 2,2,δ C (||F (y, Y )|| 2,δ + ||(y, Y )|| 2,0 + ||(y, Y )|| 2,δ;Ω R ) .

FTheorem 8 .

 8 * : W 2,2 -δ (T ) → L 2-δ (T * ⊗ Λ 3 T * M). We can apply Theorem 7 to F * : Let δ ∈] -(n + 1)/2, 0] with n = 3. Then the operatorF * : W 2,2 -δ (T ) → L 2 -δ (T * ⊗ Λ 3 T * M) is bounded. Furthermore, it satisfies ||(y, Y )|| 2,2,-δ C || F * (y, Y )|| 2,-δ + ||(y, Y )|| 2,-2δ + ||(y, Y )|| 2,-δ;Ω R .(148)In particular, F * is a semi-Fredholm operator , i.e. F * has finite dimensional kernel and closed range. Proof: From the definition of F * , the Triangle inequality together with (54) and the Sobolev inclusion with δ 0 directly yield || F * (y, Y )|| 2,-δ C ||(y, Y )|| 2,2,-δ .

  2 δ , ||(y, Y )|| 2,2,-δ C (||-∆y + ny|| 2,-δ + ||N|| 2,-δ;Ω R ) C || F * 0 (y, Y )|| 2,δ + ||(y, Y )|| 2,-2δ + ||Y || 2,2,-δ + ||y|| 2,-δ;Ω R . (149)

  2,-δ .

	Consequently,
	||DΦ * 2 .ξ|| 2,-δ ||DΦ * 2 .ξ|| 1,2,-δ C ||ξ|| 2,2,-δ . Every term of (98) is controled by ||ξ|| 2,2,-2δ leading to (97). Estimate (95) verified by P * directly comes from (73). We now look into the Lipschitz (101)
	behaviour of P * :
	Set	   P
		 

* = P * (g,π)

  2,2 s → L 2 s is a Fredholm operator and an isomorphism. So there exists a positive constant C = C(n, s) such that ||u|| 2,2,s C || Åu|| 2,s .(140)Applying (140) to u ∞ , ||u ∞ || 2,2,s C || Åu ∞ || 2,s C ||Au ∞ || 2,s + ||∆ -∆|| op,R ||u ∞ || 2,2,s .(141)YetAu ∞ = Au -Au 0 = Au-χ R Au+χ R Au -Au 0 . Thus, ||Au ∞ || 2,s ||Au|| 2,s + ||χ R Au|| 2,s + ||χ R Au -Au 0 || 2,s C ||Au|| 2,s + ||χ R Au -Au 0 || 2,s;Ω R .

  u , leading to ||χ R Au -Au 0 || 2,s;Ω R c ||u|| 1,2,s;Ω R . Finally ||Au ∞ || 2,s C (||Au|| 2,s + ||u|| 1,2,s;Ω R ) . Replacing (141) and considering (139), we obtain for R large enough ||u ∞ || 2,δ ||u ∞ || 2,2,s C (||Au|| 2,s + ||u|| 1,2,s;Ω R ) . (142) Using (142) and the fact that on Ω R , |u 0 | g |u| g ||u|| 2,s ||u ∞ || 2,s + ||u 0 || 2,s C (||Au|| 2,s + ||u|| 1,2,s;Ω R ) + ||u|| 2,s;Ω R . C (||Au|| 2,s + ||u|| 2,s;Ω R ) + ε||u|| 2,2,s;Ω R C (||Au|| 2,s + ||u|| 2,s;Ω R ) + ε||u|| 2,2,s .

	Thanks to Ehrling inequality (11),
	||u|| 2,s

and we conclude with Proposition 15.

  along with Sobolev inclusion (with δ 0) andΦ 0 (g, π) ∈ L 2 δ , ||y|| 2,2,δ C (||-∆y + ny|| 2,δ + ||y|| 2,δ;Ω R ) C (||F 0 (y, Y )|| 2,δ + ||(y, Y )|| 2,0 + ||Y || 2,2,δ + ||y|| 2,δ;Ω R ) . (146)Plugging the expression of F i (y, Y i ) in (143) and using Hölder inequality (8) ,[START_REF] Mazzeo | Unique continuation at infinity and embedded eigenvalues for asymptotic hyperbolic manifolds[END_REF] ,Ehrling inequality[START_REF] Gilbarg | Elliptic Partial Differential Equations of Second Order[END_REF] along with Sobolev inclusion (with δ 0) andΦ i (g, π) ∈ L 2 δ , ||Y || 2,2,δ C (||-∆Y + (n -1)Y || 2,δ + ||Y || 2,δ;Ω R ) C (||F i (y, Y )||2,δ + ||(y, Y )|| 2,0 + ||Y || 2,δ;Ω R ) .

  along with the Sobolev inclusion (with δ 0) andΦ i (g, π) ∈ L 2 δ , ||Y || 2,2,-δ C (||-∆Y + (n -1)Y || 2,-δ + ||Y || 2,-δ;Ω R ) C || F * i (y, Y )|| 2,-δ + ||(y, Y )|| 2,-2δ + ||Y || 2,-δ;Ω R .

 Lemma 17. Let (M,g) be an asymptotically hyperbolic manifold and X ∈ C ∞ (T * M) compactly supported on M. ∀δ ∈ IR ,

Proof: For (118), we integrate by parts the term ∇i (|X| 2 g ∇i ρ ρ 2δ-1 ) and the result follows from the Divergence theorem , along with the definition (116) of Ů and (1). For (119), we integrate by parts the term ∇i X i X, dρ ρ 2δ-1 and the result follows on from the Divergence theorem , along with the definition (116).

Lemma 18. Let (M,g) be an asymptotically hyperbolic manifold and

We integrate the term ∇k ( ∇k X i X i ρ 2δ ) and the lemma stems from the Divergence theorem and Lemma 17.

Lemma 19. Let (M,g) be an asymptotically hyperbolic manifold and


Armelle Gardeisen

Katerina Trantalidou

Andreas Darlas

come


Ludwik Dobrzynski

S Baffioni

C Charlot

F Ferri

N Godinovic

P Meridiani

I Puljak

R Salerno

Y Sirois

D Futyan

D Fortin

D Giordano

CMS status

Keywords: CMS, Assembly, Magnet, Tracker, Calorimeter, Muon

teaching and research institutions in France or abroad, or from public or private research centers.

CMS detector and collaboration

The Compact Muon Solenoid (CMS) is a general-purpose proton-proton detector designed to run at the highest luminosity at the LHC [1]. The detector (Fig. 1) is built around a long (13 m) and large bore (φ = 5.9 m) highfield (4T) super-conducting solenoid leading to a compact design for the muon spectrometer. The magnetic flux is returned through ~1.5 m of saturated iron yoke (1.8 T) instrumented with muon chambers.

Figure 1. Overview of the CMS detector From the beginning CMS was designed to be accessible and maintainable. The barrel yoke is sectioned in five wheels (YB-2, YB-1, YB0, YB+1 and YB+2) movable on rails with air pads. The central wheel (YB0, ~2000 tons) supports the vacuum tank containing the super-conducting coil, the vacuum tank in turn supports the barrel calorimeters (900 tons), which support the full tracker. The endcap yokes are made of three iron disks each (YE1, YE2 and YE3), which can be opened to access the endcap muon chambers. The first endcap disks (YE+1 and YE-1) support the 300 tons endcap calorimeters. The CMS Collaboration consists currently of 2030 physicists and engineers from 176 institutes in 36 countries.

CMS detector assembly overview

The CMS experimental cavern (Fig. 2) is now delivered and ready to receive detector elements as soon as the magnet commissioning is achieved. Heavy lifting operations to transfer CMS underground will start immediately after the magnet test is finished. The system (Figs. 3) has been validated and is ready for operation. The heaviest element (YB0) will be in the cavern by Christmas 2006. All detector elements will be in place by early February 2007.

Installation and commissioning of the muon chambers on the magnet yoke is largely made. Figure 4 presents detector's equipped heavy pieces, during this Summer process of detector closing procedure on the surface. 3 Towards a working detector: the Cosmic Challenge

CMS solenoid achievement

The Cosmic Challenge

The magnet commissioning procedure spread over about 6 weeks will produce for the first time in the surface hall the 4T field. This operation time is used to record cosmic data in the installed detector elements. The muons chambers are used to define the trigger and to start the data acquisition to record the HCAL, ECAL and Tracker data. Figure 6 shows the installed elements inside the magnet.

Detector response to muons

For the first time end of July 2006, CMS has seen in the drift tubes (Fig. 7 right) tracks curved by the magnetic field.

Figure 7. Muon data recorded in CMS DT chambers without magnetic field (left) and with field on (right)

Figure 8 shows muons data as recored by the HCAL and the Forward Tracker (TEC) during their validation phases. All ECAL SuperModules are also tested with cosmic muons providing the first crystal inter-calibration constants. 4 CMS detector construction status.

Muon chambers

The CMS muon system involves 3 types of chambers: Drift Tubes (DT) in the barrel, Cathode Strip Chambers (CCS) in the endcaps and RPCs both in the barrel and in the endcaps used for the trigger.

Centrally produced muons are such measured three times, in the inner tracker and in the return flux. They are then identified and measured in four Muon Barrel stations (MB) inserted in the five barrel yoke wheels. Special care has been taken to avoid pointing cracks and to maximize the geometric acceptance. Each muon station contains two superlayers (SL) of four planes of aluminum drift tubes oriented along the z-axis designed to give a muon vector in r-φ space, with 100 μm precision in position and better than 1 mrad in direction. The first 3 stations also contain one superlayer of four planes of orthogonal drift tubes measuring the θ coordinate. The 250 MB chambers have been built in four production sites: CIEMAT, Aachen, Legnaro and Torino. Dubna and IHEP, Protvino have produced the plates and I-beam electrodes.

The endcap muon system also consists of four Muon Endcap stations (ME). Each station consists of six planes of Cathode Strip Chambers (CSCs). The chambers are arranged such that all muon tracks traverse four stations at all rapidities, including the transition region between the barrel and the endcaps. The complete endcap muon system consists of 540 ME chambers. The 108 ME4 chambers of the fourth station have been staged. In total, the 432 ME chambers have been delivered by FNAL, US, PNPI-St Petersburg, Russia, IHEP-Beijing, China and Dubna, Russia.

The four muon stations include RPC triggering planes that also identify the bunch crossing and enable a cut on the muon transverse momentum at the first trigger level. A new conceptual design of the RPC trigger algorithm confirmed that a noise below 10 Hz/cm 2 is needed for safe triggering in the barrel region. All chambers have been produced by General Tecnica, Italy. The production of endcap RPCs has been done in Korea.

The full CMS muon system is now installed and commissioned. Figure 9 shows two endcap disks fully equipped with CCS and RPCs. Figure 10 shows the foreseen precisions performance of the CMS muon system.

Electromagnetic calorimeter : ECAL

The high precision electromagnetic calorimeter (ECAL) of CMS is made of dense lead tungstate (PbWO4) crystals (61200 for the barrel and 7324 for each endcap). It fits well the compact design of CMS. A total of 30 SM will be installed into the HCAL before these detectors goes down to the cavern to be inserted into the coil. Figure 12 shows two SM placed inside the HCAL for the Cosmic Challenge. Figure 13 gives the achieved design performance of the ECAL.

Figure 13. Performance of the ECAL. On the left, the achieved electronics noise (40 MeV/channel) and the achieved resolution for a 3x3 crystal matrix. On the right the corrected energy measurement for a widely illuminated SM area. The achieved resolution for 120 GeV electrons was measured to be 0.5%.

Tracker

The volume available for tracking is a cylinder of 6m long and 1.2m in radius. The whole volume is filled with 210 m 2 of silicon strip sensors, comprising 10M channels, and ~40M silicon pixels close to the interaction re- The baseline pixel system consists of 3 barrel layers and 2 forward disks. Figure 14 (right) shows a picture of the design principle of the pixel detector. The mechanics of the pixel system have been designed to allow independent insertion with the beam pipe in place. The insertion procedure has been tested on a full-scale mechanical mockup.

Figure 15 and 16 show respectively silicon tracker and pixel achieved elements.

CMS trigger and DAQ schemes

The trigger and data acquisition consists of four parts: the front-end detector electronics, the calorimeter and muon first level trigger processors, the readout network and an on-line event filter system. The first two parts are synchronous and pipelined with a pipeline depth corresponding to ≈ 3μs. The latter two are asynchronous and based on industry standard data communication components and commercial PCs. The resources that would have been required for a hardware second level trigger processors are invested in a high bandwidth ( ≈500 Gbit/ s) readout network and in the event filter processing power (10 6 -10 7 MIPs), both of which are more suitable for upgrading as commercially available technology develops. The CMS Level-1 trigger decision is based upon the presence of physics objects such as muons, photons, electrons, and jets, as well as global sums of Et and missing Et (to find neutrinos). The Level-1 Trigger as described in Technical Design Report [2] is outlined in Fig. 18 (left).

Figure 18. The DAQ system (right) is made of 8 independent systems (slices ). Each slice consists of 64 RUs, a 64x64 EVB and 64 BUs and associated filter units. A slice can read up to 12.5 kHz.

The DAQ system (Fig. 18 right) has to assemble the data from the triggered event, contained in about 500 frontend buffers (readout units, RUs), into a single processor in a "farm" for executing physics algorithms (so-called High Level Trigger or HLT algorithms) so that the input rate of 100 kHz is reduced to the 100 Hz of sustainable physics at high luminosity. The size of a raw data event in CMS is 1 Mbyte. The DAQ system consists of 8 independent slices able to handle 12.5 kHz of L1 rate each. This modularity allows to start with a reduced system at low luminosity and to progressively add more slices as the luminosity increases. CMS is following an assembly sequence that allows a complete detector (except for ME4/2 and 3 rd forward pixel layer) to be ready for the first physics run. Presently the 4T solenoid is in its commissioning phase. Figure 19 shows an event taken in August with a field of 3.95 Tesla during the Cosmic Challenge and showing active muon, HCAL, ECAL and Tracker detectors.

The muon system as well as the HCAL are delivered and used in the present Cosmic Challenge combined with the magnet reception tests. First curved tracks have been observed in the involved detector elements (Muons, HCAL,ECAL and Tracker) placed in the magnetic field. The ECAL barrel will be in place before the first LHC collisions as well as the Tracker which after its commissioning phase starting this Autumn, will be inserted inside the calorimeters in April 2007. The pixel detector will be installed during the 2008 shut down as well as the EndCap ECAL and the preshower.

The proposed detector will be able to exploit the physics potential of the first physics run at medium luminosity ( ~2.10 33 cm -2 s -1 ) for which an integrated luminosity of the order of 10 fb -1 could be accumulated during the first Year of operation (30% efficiency). With 10 fb -1 most of the standard model Higgs can be probed. SUSY squarks and gluino with masses up to about 2 TeV can be discovered in the jet plus missing Et channel. For these discoveries full acceptance must be maintained. The installation of a complete barrel and endcap ECAL for the first physics run is vital to discovering a Higgs below a mass of 140 GeV. References.

Figure 2 .

 2 Figure 2. Main CMS cavern, ready to receive the detector

Figure 5

 5 Figure5shows the insertion of the 4 Tesla coil into the vacuum tank. The coil is made of 5 modules each of length 2.5m. To reach 4 Tesla four layers of reinforced superconducting cable are necessary. For the nominal current of 20 kA the radial magnetic pressure reaches 64 atmospheres and the stored energy is 2.7 GJ! The five coil modules have been wound at Ansaldo in Genova, Italy. The 5 modules have been assembled vertically as a single coil in SX5 on a swiveling platform, which was used to turn the coil in the horizontal position and insert it in the outer shell of the vacuum tank supported by YB0. During early spring 2006 the coil was cooled down up to its working temperature without any problem. Figure5(right) gives the coil cool down curves as recorded during the 4 weeks of the operation. The magnet is now in its commissioning procedure to supply the 20 kA and deliver the nominal 4T field.

Figure 5 .

 5 Figure 5. The coil insert (left) in the vacuum tank supported by YB0 and the coil cooled down curve (right).

Figure 6 .

 6 Figure 6. HCAL, ECAL and Tracker elements installed inside the magnet for the Cosmic Challenge.

Figure 8 .

 8 Figure 8. Muon event in the HCAL (left) and in the TEC tracker (right)

Figure 9 .

 9 Figure 9. Two endcap disks equipped with muon chambers

Figure 11 (

 11 Figure 11 (left) shows the 18 assembled wedges making a half-barrel at it insert into the coil. Figure 11 (right) gives the foreseen jet Et and jet-jet Mass resolutions.

Figure 11 .

 11 Figure 11. Half barrel HCAL insertion into the magnet coil (left). Jet Et and Mass jet-jet resolutions (right)

Figure 12 .

 12 Figure 12. First two ECAL SM inserted into the HCAL for the Cosmic Challenge.

  gion. The whole tracker volume will be maintained at -10 o C in dry atmosphere to minimize the effects of radiation damage of the silicon detectors. Stereo information is provided by back-to-back microstrip detectors with strips at a small angle. The Silicon Strip Tracker (SST) is made of 3 independent mechanical structures: the Tracker Outer Barrel (TOB), the Tracker Inner Barrel (TIB) and Inner Disks (TID) and the Tracker End-Caps (TEC) Fig.14 (left). The SST has silicon detectors of two different thicknesses: 320 μm in the inner region (r<600mm) and 500 μm in the outer region. A highly automated procedure was needed to assemble the sensors plus hybrids into modules. A total of 16,000 silicon modules have been assembled. System tests of small parts of the tracker (TEC petal, TOB rod, and TIB shell) have been carried out and have shown that the signals are clean (low spurious noise). The rapidity coverage of the tracker is -2.5 < |η| < 2.5. All high pt charged particles, produced in the central rapidity region, are reconstructed with a momentum precision of ∆pt /pt ≈ 0.005 + 0.15 pt (pt in TeV). High track finding efficiencies are achieved for high pt tracks even inside jets.

Figure 14 .

 14 Figure 14. Silicon tracker design (left). On the right the barrel mechanics for the pixel detector.

Figure 15 .

 15 Figure 15. The completed Tracker Inner Barrel (TIB) on the left. A TEC petal on the right

Figure 17 .

 17 Figure 17. Higgs discovery potential with the CMS detectors

Figure 19 .

 19 Figure 19. An event display taken in August with a field of 3.95 Tesla during the Cosmic Challenge and showing active muon, HCAL ECAL and Tracker detectors.

Project, Technical Design Report, CERN/LHCC 97-31 CMS, The Muon Project, Technical Design Report, CERN/LHCC 97-32 CMS, The Electromagnetic Calorimeter, Technical Design Report, CERN/LHCC 97-33 CMS, The Tracker Project, Technical Design Report, CERN/LHCC 98-6 CMS, Addendum to CMS Tracker TDR, CERN/LHCC 2000-016 CMS, The Trigger and Data Acquisition project, Volume 1, The Level-1 Trigger, Technical Design Report, CERN /LHCC 2000-038 2. CMS, The Trigger and Data Acquisition project, Volume 2, Data Acquisition and High Level Trigger, Technical Design Report, CERN /LHCC 2002-026 3. CMS Detector Performance and Software CERN/LHCC 2006-001 CMS Physics Performance CERN/LHCC