Inverse problems for Einstein manifolds

We show that the Dirichlet-to-Neumann operator of the Laplacian on an open subset of the boundary of a connected compact Einstein manifold with boundary determines the manifold up to isometries. Similarly, for connected conformally compact Einstein manifolds of even dimension $n+1$, we prove that the scattering matrix at energy $n$ on an open subset of its boundary determines the manifold up to isometries.


Introduction
The purpose of this note is to prove two results: first that compact connected Einstein manifolds with boundary are determined modulo isometries from the Dirichletto-Neumann map on an open subset of its boundary. Secondly, that a conformally compact connected Einstein manifolds of even dimension n + 1 is determined, modulo isometries, by the scattering matrix on an open subset of the boundary.
The Dirichlet-to-Neumann (DN in short) map N : C ∞ (∂X) → C ∞ (∂X) for the Laplacian on a Riemannian manifold with boundary (X, g) is defined by solving the Dirichlet problem where f ∈ C ∞ (∂X) is given, then Nf := −∂ n u| M where ∂ n is the interior pointing normal vector field to the boundary for the metric g. It is an elliptic pseudodifferential operator of order 1 on the boundary, see for example [16]. Mathematically, it is of interest to know what this map determines about the geometry of the manifold, but N can also be interpreted as a boundary measurement of current flux in terms of voltage in electrical impedance tomography. We refer to [24] for a survey in the field, and to [14,15,16,22,23] for significant results about that problem.
Our first result answers a conjecture of Lassas and Uhlmann [15] Theorem 1.1. Let (X 1 , g 1 ) and (X 2 , g 2 ) be two smooth connected compact manifolds with respective boundaries ∂X 1 and ∂X 2 . We suppose that g 1 and g 2 are Einstein with the same constant λ ∈ R, i.e. Ric(g i ) = λg i for i = 1, 2. Assume that ∂X 1 and ∂X 2 contain a common open set Γ such that the identity map Id : Γ ⊂X 1 → Γ ⊂X 2 is a smooth diffeomorphism. If the Dirichlet-to-Neumann map N i of ∆ gi onX i for i = 1, 2 satisfy (N 1 f )| Γ = (N 2 f )| Γ for any f ∈ C ∞ 0 (Γ), then there exists a diffeomorphism J :X 1 →X 2 , such that J * g 2 = g 1 .
Then we consider a class of non-compact complete Einstein manifolds, but conformal to a compact manifold. In this case we say that (X, g) is Einstein, with dim X = n + 1, if Ric(g) = −ng.
We say that a Riemannian manifold (X, g) is conformally compact if X compactifies into a smooth manifold with boundaryX and for any smooth boundary defining function ρ ofX,ḡ := ρ 2 g extends toX as a smooth metric. Such a metric g is necessarily complete on X and its sectional curvatures are pinched negatively outside a compact set of X. If in addition the sectional curvatures of g tends to −1 at the boundary, we say that (X, g) is asymptotically hyperbolic. It has been shown in [9,10] that if (X, g) is asymptotically hyperbolic, or in particular if (X, g) is Einstein, then there exists a family of boundary defining functions ρ (i.e. ∂X = {ρ = 0} and dρ| ∂X does not vanish) such that |dρ| ρ 2 g = 1 near the boundary. These will be called geodesic boundary defining functions. Note that, in this case, a DN map can not be defined as in (1.1) since ∆ g is not an elliptic operator at the boundary. The natural analogue of the DN map on a conformally compact Einstein manifold (X, g) is related to scattering theory, at least in the point of view of Melrose [21]. We consider an n + 1-dimensional conformally compact Einstein manifold (X, g) with n + 1 even. Following [11,13], the scattering matrix or scattering map in this case, and more generally for asymptotically hyperbolic manifolds, is an operator S : C ∞ (∂X) → C ∞ (∂X), constructed by solving a Dirichlet problem in a way similar to (1.1). This will be discussed in details in section 4. We show that for all f ∈ C ∞ (∂X), there exists a unique function u ∈ C ∞ (X) such that Since there is no canonical normal vector field at the boundary defined from g (recall that g blows-up at the boundary), we can considerḡ := ρ 2 g for some geodesic boundary defining function and take the unit normal vector field forḡ, that is ∇ḡρ, which we denote by ∂ ρ . It turns out that (∂ k ρ u| ∂X ) k=1,...,n−1 are locally determined by u| ∂X = f and the first term in the Taylor expansion of u which is global is the n-th ∂ n ρ u| ∂X . We thus define Sf ∈ C ∞ (∂X) by Notice that S a priori depends on the choice of ρ, we shall say that it is associated to ρ. It can be checked that ifρ = e ω ρ is another geodesic boundary defining function with ω ∈ C ∞ (X), then the scattering mapŜ associated toρ satisfyŜ = e −nω0 S where ω 0 = ω| ∂X , see [11] and Subsection 4.1 below. We also remark that the fact that u ∈ C ∞ (X) strongly depends on the fact that the manifold under consideration is Einstein and has even dimensions. For more general asymptotically hyperbolic manifolds, the solution u to (1.2) possibly has a logarithmic singularity as shown in [11]. Our second result is the following Theorem 1.2. Let (X 1 , g 1 ) and (X 2 , g 2 ) be connected, C ∞ , (n + 1)-dimensional conformally compact manifolds, with n+1 even. Suppose that g 1 and g 2 are Einstein and that ∂X 1 and ∂X 2 contain a common open set Γ such that the identity map Id : Γ ⊂X 1 → Γ ⊂X 2 is a smooth diffeomorphism. If for i = 1, 2, there exist boundary defining functions ρ i of ∂X i such that the scattering maps S i of ∆ gi associated to ρ i satisfy ( The proofs are based on the results of Lassas and Uhlmann [15], and Lassas, Taylor and Uhlmann [14], and suitable unique continuation theorems for Einstein equation. It is shown in [15] that a connected compact manifold with boundary (X = X ∪ ∂X, g), is determined by the Dirichlet-to-Neumann if the interior (X, g) is real analytic, and if there exists an open set Γ of the boundary ∂X which is real analytic with g real analytic up to Γ. In [14] Lassas, Taylor and Uhlmann prove the analogue of this result for complete manifolds.
A theorem of De Turck and Kazdan, Theorem 5.2 of [6], says that if (X, g) is a connected Einstein manifold with boundary then the collection of harmonic coordinates give X, the interior ofX, a real analytic structure which is compatible with its C ∞ structure, and moreover g is real analytic in those coordinates. The principle is that Einstein's equation becomes a non-linear elliptic system with real analytic coefficient in these coordinates, thus the real analyticity of the metric. But since the harmonic coordinates satisfy the Laplace equation, they are analytic as well.
However this construction is not necessarily valid at the boundary. Therefore one cannot guarantee that (X, g) is real analytic at the boundary, and hence one cannot directly apply the results of [14].
To prove Theorems 1.1 and 1.2, we first show that the DN map (or the scattering map) determines the metric in a small neighbourhood U of a point p ∈ Γ ⊂ ∂X, then we shall prove that this determines the Green's function in U × U. However one of the results of [14] says that this determines the whole Riemannian manifold, provided it is real-analytic, but as mentioned above, this is the case of the interior of an Einstein manifold.
The essential part in this paper is the reconstruction near the boundary. This will be done using the ellipticity of Einstein equation in harmonic coordinates, and by applying a unique continuation theorem for the Cauchy problem for elliptic systems with diagonal principal part. The unique continuation result we need in the compact case was essentially proved by Calderón [4,5]. The conformally compact case is more involved since the system is only elliptic in the uniformly degenerate sense of [19,18,20,17], see also [1]. When the first version of this paper was completed we learned that O. Biquard [3] proved a unique continuation result for Einstein manifolds without using the DN map for functions, which was a problem that was part of the program of M. Anderson [2]. Under our assumptions, it seems somehow natural to use harmonic coordinates for Einstein equation, and we notice that our approach is self-contained and does not require the result of [3].
Throughout this paper when we refer to the real analyticity of the metric, we mean that it is real analytic with respect to the real analytic structure defined from harmonic coordinates corresponding to the Einstein metric g.

Acknowledgments
The work of both authors was funded by the NSF under grant DMS-0500788. Guillarmou acknowledges support of french ANR grants JC05-52556 and JC0546063 and thanks the MSI at ANU, Canberra, where part of this work was achieved. We thank Erwann Delay, Robin Graham and Gunther Uhlmann for helpful conversations. Finally, we are grateful to the anonymous referee for a very careful reading, for many useful comments, and especially for uncovering an error in the first submitted version of the paper

Inverse problem for Einstein manifolds with boundary
The result of De Turck and Kazdan concerning the analyticity of the metric does not apply to Einstein manifolds with boundary (X = X ∪ ∂X, g). Their argument breaks down since the boundary can have low regularity even though g has constant Ricci curvature. This means that the open incomplete manifold (X, g) is real-analytic with respect to the analytic structure defined by harmonic coordinates, but a priori (X, g) does not satisfy this property. We will use the Dirichlet-to-Neumann map to overcome this difficulty.
3.1. The Dirichlet-to-Neumann map. As in section 1, is defined by solving the Dirichlet problem (1.1) with f ∈ C ∞ (∂X), and setting Nf := −∂ n u| M where ∂ n is the interior pointing normal vector to the boundary for the metric g. Its Schwartz kernel is related to the Green function G(z, z ′ ) of the Laplacian ∆ g with Dirichlet condition on ∂X by the following identity Lemma 3.1. The Schwartz kernel N(y, y ′ ) of N is given for y, y ′ ∈ ∂X, y = y ′ , by where ∂ n , ∂ n ′ are respectively the inward pointing normal vector fields to the boundary in variable z and z ′ .
Proof : Let x be the distance function to the boundary inX, it is smooth in a neighbourhood of ∂X and the normal vector field to the boundary is the gradient defined by φ(t, y) := e t∂n (y) and we have x(φ(t, y)) = t. This induces natural coordinates z = (x, y) near the boundary, these are normal geodesic coordinates. The function u in (1.1) can be obtained by taking Now taking φ with support disjoint to the support of f, thus φf = 0, and differentiating (3.1) in x, we see, using the fact that Green's function G(z, z ′ ) is smooth outside the diagonal, that which proves the claim.

The Ricci tensor in harmonic coordinates and unique continuation.
Let us take coordinates x = (x 0 , x 1 , . . . , x n ) near a point p ∈ ∂X, with x 0 a boundary defining function of ∂X, then Ric(g) is given by definition by Lemma 1.1 of [6] shows that ∆ g x k = i,j g ij Γ k ij , so Einstein equation Ric(g) = λg for some λ ∈ R can be written as the system (see also Lemma 4.1 in [6]) is smooth and polynomial of degree two in B, where A, B denote vectors (g kl ) k,l ∈ R (n+1) 2 and (∂ xm g kl ) k,l,m ∈ R (n+1) 3 . From this discussion we deduce the following Now we may use a uniqueness theorem for the Cauchy problem of such elliptic systems.
Proof : The system is elliptic and the leading symbol is a scalar times the identity, the result could then be proved using Carleman estimates. For instance, uniqueness properties are proved by Calderon [4,5] for elliptic systems when the characteristics of the system are non-multiple, but in our case they are multiple. However, since the leading symbol is scalar and this scalar symbol has only non-multiple characteristics, the technics used in Calderon could be applied like in the case of a single equation with non-multiple characteristics. Since we did not find references that we can cite directly, we prefer to use Proposition 4.3 which is a consequence of a uniqueness result of Mazzeo [19]. Indeed, first it is straightforward to notice, by using boundary normal coordinates, that two solutions of (3.5) with same Cauchy data agree to infinite order at the boundary, therefore we may multiply (3.5) by x 2 0 and (3.5) becomes of the form (4.7) thus Proposition 4.3 below proves uniqueness.

3.3.
Reconstruction near the boundary. Throughout this section we assume that (X 1 , g 1 ), (X 2 , g 2 ) are C ∞ connected Einstein manifolds with boundary M j = ∂X j , j = 1, 2, such that M 1 and M 2 contain a common open set Γ, and that the identity map Id : Γ ⊂ ∂X 1 −→ Γ ⊂ ∂X 2 is a C ∞ diffeomorphism. Moreover we assume that for every f ∈ C ∞ 0 (Γ), the Dirichlet-to-Neumann maps satisfy We first prove Proof : For i = 1, 2, let t i = dist(., ∂X i ) be the distance to the boundary inX i , then the flow e t∇ g i ti of the gradient ∇ gi t i induces a diffeomorphism and we have the decomposition near the boundary (φ i ) * g i = dt 2 +h i (t) for some oneparameter family of metrics h i (t) on ∂X i . Lee-Uhlmann [15] proved that Let us now consider H i := φ i * g i on the collar [0, ǫ) t × Γ. Let p ∈ Γ be a point of the boundary and (y 1 , . . . , y n ) be a set of local coordinates in a neighbourhood of p in Γ, and extend each y j to [0, ǫ) × Γ by the function (t, m) → y j (m). Notice that φ 2 • (φ 1 ) −1 is a smooth diffeomorphism from a neighbourhood of p inX 1 to a neighbourhood of p inX 2 , this is a consequence of the fact that Id : Γ ⊂ X 1 → Γ ⊂X 2 is a diffeomorphism. Using z := (t, y 1 , . . . , y n ) as coordinates on and we can always assume U ∩ {t = 0} = Γ. Let y 0 ∈ C ∞ 0 (Γ) with y 0 = 0 on U ∩ {t = 0} but y 0 not identically 0, and by cutting off far from p me may assume that y j ∈ C ∞ 0 (Γ) for j = 1, . . . , n. Now let (x 1 0 , x 1 1 , . . . , x 1 n ) and (x 2 0 , x 2 1 , . . . , x 2 n ) be harmonic functions near p in [0, ǫ) × Γ for respectively H 1 and H 2 such that x 1 j = x 2 j = y j on {t = 0}. These functions are constructed by solving the Dirichlet problem ∆ gi w i j = 0 onX i with boundary data w i j | Mi = y j , i = 1, 2, and j = 0, . . . , n, and by setting We in U with u vanishing at order 2 at the boundary t = 0, a standard Taylor expansion argument shows that The metrics g = H 1 and g = ψ * H 2 both satisfy Einstein equation Ric(g) = λg in U . Moreover in coordinates (x 1 0 , . . . , x 1 n ) this correspond to the system (3.5) and since the coordinates are harmonic with respect to g, the system is elliptic and diagonal to leading order. From the unique continuation result in Proposition 3.3, we conclude that there exists a unique solution to this system in U 1 with given initial data g| U∩{t=0} and ∂ x 1 0 g| U∩{t=0} . In view of (3.7), this proves that H 1 = ψ * H 2 in U . Although it is not relevant for the proof, we remark that ψ is actually the Identity on U since ψ| U∩Γ = Id and it pulls back one metric in geodesic normal coordinates to the other. Now it suffices to go back toX 1 andX 2 through φ 1 , φ 2 and we have proved the Lemma by setting U i := φ i (U ) and Remark that F is analytic from U 1 ∩ {t 1 = 0} to U 2 ∩ {t 2 = 0} since the harmonic functions w i j define the analytic structure in U i ∩ {t i = 0} for all j = 0, . . . , n and F is the map that identify w 1 j to w 2 j for all j.
Next we prove where F was defined in (3.8) Proof : First we remark that g 1 is Einstein and thus real analytic in U 1 \ (U 1 ∩ M ), so is any harmonic function in this open set. Let ∂ n , ∂ n ′ be the normal vector fields to the boundary in the first and second variables in U 1 × U 1 respectively, as defined in Lemma 3.1. We see from the proof of Lemma 3.4 that F * ∂ n and F * ∂ n ′ are the normal vector fields to the boundary in the first and second variable in .) is the Schwartz kernel of N 2 . Using again that F | U1∩Γ = Id, we deduce that ∂ n T 2 | U1∩Γ\{z ′ } = N 2 (., z ′ ). By our assumption N 1 | Γ = N 2 | Γ , we conclude that T 1 and T 2 solve the same Cauchy problem near U 1 ∩ Γ \ {z ′ }, so first by unique continuation near the boundary and then real analyticity in U 1 \ (U 1 ∩Γ), we obtain T 1 = T 2 there. Now we can use again similar arguments to prove that by what we proved above. Thus unique continuation for Cauchy problem and real analyticity allow us to conclude that 3.4. Proof of Theorem 1.1. To conclude the proof of 1.1, we use the following Proposition which is implicitly proved by Lassas-Taylor-Uhlmann [14] Proposition 3.6. For i = 1, 2, let (X i , g i ) be C ∞ connected Riemannian manifolds with boundary, assume that its interior X i has a real-analytic structure compatible with the smooth structure and such that the metric g i is real analytic on X i . Let G i (z, z ′ ) be the Green function of the Laplacian ∆ gi with Dirichlet condition at ∂X i , and assume there exists an open set U 1 ⊂ X 1 and an analytic diffeomorphism Then there exists a diffeomorphism J : X 1 → X 2 such that J * g 2 = g 1 and J| U1 = F .
The proof is entirely done in section 3 of [14], although not explicitly written under that form. The principle is to define maps where H s (U 1 ) is the s-Sobolev space of U 1 for some s < 1 − (n + 1)/2, then prove that G j are embeddings with G 1 (X 1 ) = G 2 (X 2 ), and finally show that J := Proposition 3.6 and Corollary 3.5 imply Theorem 1.1, after noticing that an isometry ψ : (X 1 , g 1 ) → (X 2 , g 2 ) extends smoothly to the manifold with boundary (X 1 , g 1 ) by smoothness of the metrics g i up to the boundaries ∂X i .

Inverse scattering for conformally compact Einstein manifolds
Consider an n+ 1 dimensional connected conformally compact Einstein manifold (X, g) with n + 1 even, and let ρ be a geodesic boundary defining function and g := ρ 2 g. Using the flow φ t (y) of the gradient ∇ ρ 2 g ρ, one has a diffeomorphism φ : [0, ǫ) t × ∂X → φ([0, ǫ) × ∂X) ⊂X defined by φ(t, y) := φ t (y), and the metric pulls back to for some smooth one-parameter family of metrics h(t) on the boundary ∂X. Note that here φ * ρ = t.

4.1.
The scattering map. The scattering map S, or scattering matrix, defined in the introduction is really S = S(n), where S(λ) for λ ∈ C is defined in [13,11]. Let us construct S by solving the boundary value problem ∆ g u = 0 with u ∈ C ∞ (X) and u| ∂X = f where f ∈ C ∞ (∂X) is given. This follows the construction in section 4.1 of [11]. Writing ∆ g in the collar [0, ǫ) t × ∂X through the diffeomorphism φ, we have and for any f j ∈ C ∞ (∂X) and j ∈ N 0 Now recall that since g is Einstein and even dimensional, we have ∂ 2j+1 t h(0) = 0 for j ∈ N 0 such that 2j + 1 < n, see for instance Section 2 of [9]. Consequently, H(n − j)f j is an even function of t modulo O(t n ) for j = 0, and modulo O(t n+2 ) when j = 0. Since H(n − j)f j also vanishes at t = 0, we can construct by induction a Taylor series using (4.2) for j < n such that ∆ g F j = O(t j+1 ). Note that, since H(n − j)f j has even powers of t modulo O(t n ), we get f 2j+1 = 0 for 2j + 1 < n. For j = n, the construction of F n seems to fail but actually we can remark that ∆ g F n−1 = O(t n+1 ) instead of O(t n ) thanks to the fact that t 2j H(n − 2j)f 2j has even Taylor expansion at t = 0 modulo O(t 2j+n+2 ) by the discussion above. So we can set F n := F n−1 and then continue to define F j for j > n using (4.3). Using Borel's Lemma, one can construct Green operator, i.e. such that ∆ g G = Id, recalling that ker L 2 ∆ g = 0 by [17]. From the analysis of G in [17], one has that G mapsĊ ∞ (X) : . This proves that u ∈ C ∞ (X) and has an asymptotic expansion In particular the first odd power is of order t n and its coefficient is given by the smooth function [t −n φ * (G∆ g F ∞ )] t=0 of C ∞ (∂X). Notice that the f 2j in the construction are local with respect to f , more precisely f 2j = p 2j f for some differential operator p 2j on the boundary. Note that we used strongly that the Taylor expansion of the metric t 2 φ * g at t = 0 is even to order t n , which comes from the fact that X is Einstein and has even dimensions. Indeed for a general asymptotically hyperbolic manifold, u has logarithmic singularities, see [11,12].
Since φ * ∇ ρ 2 g ρ = ∂ t , the definition of Sf in the Introduction is equivalent to Sf = 1 n! ∂ n t φ * u| t=0 , i.e. the n-th Taylor coefficient of the expansion of φ * u at t = 0, in other words From the analysis of Mazzeo-Melrose [17], one can describe the behaviour of the Green kernel G(z, z ′ ) near the boundary and outside the diagonal diagX ×X : We can show easily that the kernel of S is the boundary value of (4.5) at the corner ∂X × ∂X: Lemma 4.1. The Schwartz kernel S(y, y ′ ) of the scattering map S is, for y = y ′ , where G(z, z ′ ) is the Green kernel for ∆ g .
Proof : Consider (G∆ g F ∞ )(z) for z ∈ X fixed and use Green formula on the compact where ∂ n ′ is the unit normal interior pointing vector field of ∂U ǫ (in the right variable z ′ ) and dν ǫ the measure induced by g there. Consider the part ρ(z ′ ) = ǫ in the variables as in (4.1) using the diffeomorphism φ, i.e. φ(t ′ , y ′ ) = z ′ , then . Using (4.5) and F ∞ = f + O(ρ 2 ) by the construction of F ∞ above the Lemma, we see that the integral on ρ ′ = ǫ converges to as ǫ → 0. As for the part on dist(z ′ , z) = ǫ, by another application of Green formula and ∆ g (z ′ )G(z, z ′ ) = δ(z − z ′ ), this converges to F ∞ (z) as ǫ → 0. We deduce that the solution u of ∆ g u with u| ∂X = f is given by Let us write dy for dv h0 (y). So given y ∈ ∂X, let f be supported in a neighborhood of y and take ψ ∈ C ∞ (∂X) with ψf = 0 and consider the pairing ∂X φ * u(t, y)ψ(y)dy.
The Taylor expansion of u at t = 0 and the structure of G(z, z ′ ) given by (4.5) show that which proves the claim.
Remark: A more general relation between the kernel of the resolvent of ∆ g , (∆ g − λ(n − λ)) −1 , and the kernel of the scattering operator S(λ) holds, as proved in [13]. But since the proof of Lemma 4.1 is rather elementary, we included it to make the paper essentially self-contained.

4.2.
Einstein equation for g. We shall analyze Einstein equation in a good system of coordinates, actually constructed from harmonic coordinates for ∆ g . First choose coordinates (y 1 , . . . , y n ) in a neighbourhood V ⊂ ∂X of p ∈ ∂X. Take an open set W ⊂ ∂X which contains V , we may assume that y i ∈ C ∞ 0 (W ). Let φ be the diffeomorpism as in (4.1). From the properties of the solution of the equation ∆ g u = 0, as in subsection 4.1 (which follows Graham-Zworski [11]), there exists n smooth functions (x 1 , . . . , x n ) onX such that where p k are differential operators on ∂X determined by the (∂ k t h(0)) k=0,...,n−1 at the boundary (using the form (4.1)). Similarly let y 0 ∈ C ∞ 0 (W ) be a non zero smooth function such that y 0 = 0 in V , then by Subsection 4.1 there exists v ∈ C ∞ (X) such that so v would vanish to infinite order at an open set of V and by Mazzeo's unique continuation result [19], it would vanish identically inX. Thus, possibly by changing p to another point (still denoted p for convenience), there exists v ∈ C ∞ (X) such that v is harmonic for ∆ g and v = ρ n (w + O(ρ)) with w > 0 near p, the function x 0 := v 1/n then defines a boundary defining function of ∂X near p, it can be written as ρe f for some smooth f . Then (x 0 , x 1 , . . . , x n ) defines a system of coordinates near p.
Let us now consider Einstein equations in these coordinates. Again like (3.4), the principal part of Ric(g) is given by But all functions x r are harmonic, except x 0 , and the latter satisfies But this involves only terms of order 0 in the metric g orḡ := x 2 0 g so the principal part of Ric(g) in these coordinates is This is a non-linear system of order 2, elliptic in the uniformly degenerate sense of [18,20,17] and diagonal at leading order. We state the following unique continuation result for this system: Proposition 4.3. Assumeḡ 1 andḡ 2 are two smooth solutions of the system (4.7) withḡ 1 =ḡ 2 + O(x ∞ 0 ) near p. Thenḡ 1 =ḡ 2 near p. Proof : This is an application of Mazzeo's unique continuation result [19]. We work in a small neighbourhood U of p and set w = (ḡ 1 −ḡ 2 ) near p. For h metric near p and ℓ symmetric tensor near p, let where Q := (Q ij ) i,j=0,...,n . Note that G is smooth in all its components. We have from (4.7) Let g 1 := x −2 0ḡ 1 and let ∇ be the connection on symmetric 2 tensors on U induced by g 1 , then ∇ * ∇w = µ,ν g µν 1 ∇ ∂x ν ∇ ∂x µ w and in coordinates it is easy to check that x 0 (∇ ∂x µ − ∂ xµ ) is a zeroth order operator with smooth coefficients up to the boundary, using (3.3) for instance. Therefore one obtains, using (4.8), for some C depending onḡ 1 ,ḡ 2 . It then suffices to apply Corollary 11 of [19], this proves that w = 0 and we are done.

4.3.
Reconstruction near the boundary and proof of Theorem 1.2. The proof of Theorem 1.2 is fairly close to that of Theorem 1.1. Let (X 1 , g 1 ) and (X 2 , g 2 ) be conformally compact Einstein manifolds with geodesic boundary defining functions ρ 1 and ρ 2 . Let S i be the scattering map for g i defined by (1.3) using the boundary defining function ρ i , assume that ∂X 1 and ∂X 2 contain a common open set Γ such that the identity map which identifies the copies of Γ is a diffeomorphism, and that S 1 f | Γ = S 2 f | Γ for all f ∈ C ∞ 0 (Γ). Using the geodesic boundary defining function ρ i for g i ,i = 1, 2, there is a diffeomorphism is a family of metric on ∂X i . We first show the Proof : For a compact manifold M, let us denote Ψ z (M ) the set of classical pseudodifferential operators of order z ∈ R on M . Since S i is the scattering operator S i (λ) at energy λ = n for ∆ gi as defined in [13], we can use [13, Th.1.1], then we have that S i ∈ Ψ n (∂X i ) for i = 1, 2, with principal symbol σ i n (y, ξ) = 2 −n Γ(− n 2 )/Γ( n 2 )|ξ| hi(0) , thus h 1 (0) = h 2 (0) on Γ and χ(S 1 − S 2 )χ ∈ Ψ n+1 (Γ) for all χ ∈ C ∞ 0 (Γ). Now we use Einstein equation, for instance the results of [7,8] (see also [9,Sec. 2]) show, using only Taylor expansion of Ric(g) = −ng at the boundary, that Then we use Theorem 1.2 of [13] which computes the principal symbol of S 1 − S 2 . Since this result is entirely local, we can rephrase it on the piece Γ of the boundary: if there exists a symmetric 2-tensor L on Γ such that h 1 (t) = h 2 (t)+t k L+O(t k+1 ) on [0, ǫ) t × Γ for some k ∈ N, then for any χ ∈ C ∞ 0 (Γ) we have χ(S 1 − S 2 )χ ∈ Ψ n−k (Γ) and the principal symbol of this operator at (y, ξ) ∈ T * Γ is 1 (4.10) where h 0 := h 1 (0)| Γ = h 2 (0)| Γ , ξ * := h −1 0 ξ ∈ T y Γ is the dual of ξ through h 0 , and A i (k, λ) are the meromorphic functions of λ ∈ C defined by where T l (k, λ) is defined, when the integral converges, by and M (λ) ∈ C is a constant not explicitly computed in [13]. However at λ = n the constant M (n) is defined in [13,Sec. 4] such that u(z)/n = M (n)f + O(ρ(z)) where u(z) is the function of (4.6), so M (n) = n by (4.4). Since we are interested in the case k = n, only the term with A 1 (n, n) appears, and setting λ = n in A 1 (n, λ), with the explicit formulae above and the fact that T 1 (n, n) > 0 converges by Lemma 5.2 of [13], we see easily that A 1 (n, n) = 0 if n > 2. Since we assumed χS 1 χ = χS 2 χ, this implies that L = 0 and h 1 − h 2 = O(t n+1 ) near Γ. We finally use again [7] (see [8,Sect. 4] for full proofs), where it is proved that if g 1 = g 2 +O(ρ n−1 ) with g 1 , g 2 conformally compact Einstein and n odd, then g 1 = g 2 + O(ρ ∞ ). Notice that their arguments are entirely local near any point of the boundary, so we can apply it near the piece Γ of the boundary. Lemma 4.5. For i = 1, 2, there exist p ∈ Γ, neighbourhoods U i of p inX i and a diffeomorphism F : U 1 → U 2 , F | U1∩X1 analytic, such that F * g 2 = g 1 and F | U1∩Γ = Id.
We finish by the following Corollary, similar to Corollary 3.5.