Limiting Carleman weights and anisotropic inverse problems
aa r X i v : . [ m a t h . A P ] M a r LIMITING CARLEMAN WEIGHTS ANDANISOTROPIC INVERSE PROBLEMS
DAVID DOS SANTOS FERREIRA, CARLOS E. KENIG, MIKKO SALO,GUNTHER UHLMANN
Abstract.
In this article we consider the anisotropic Calder´onproblem and related inverse problems. The approach is based onlimiting Carleman weights, introduced in [13] in the Euclidean case.We characterize those Riemannian manifolds which admit limitingCarleman weights, and give a complex geometrical optics construc-tion for a class of such manifolds. This is used to prove uniquenessresults for anisotropic inverse problems, via the attenuated geo-desic X-ray transform. Earlier results in dimension n ≥ Contents
1. Introduction and main results 12. Limiting Carleman weights 93. The Euclidean case 164. Carleman estimates 255. Complex geometrical optics 306. Uniqueness results 337. Attenuated ray transform 388. Boundary determination 45Appendix A. Riemannian geometry 51References 561.
Introduction and main results
Introduction.
In this paper we consider the Calder´on problemin the anisotropic case. This inverse method, also called ElectricalImpedance Tomography (EIT), consists in determining the conductiv-ity of a medium by making voltage and current measurements at theboundary. Applications range from geophysical prospection to medicalimaging.
Anisotropic conductivities depend on direction. Muscle tissue in thehuman body is an important example of an anisotropic conductor. Forinstance cardiac muscle has a conductivity of 2.3 mho in the transversedirection and 6.3 mho in the longitudinal direction [4]. The conductiv-ity in this case is represented by a positive definite, smooth, symmetricmatrix γ = ( γ jk ( x )) in a domain Ω in Euclidean space.If there are no sources or sinks of current in Ω, the potential u in Ω,given a voltage potential f on ∂ Ω, solves the Dirichlet problem ∂∂x j (cid:18) γ jk ∂u∂x k (cid:19) = 0 in Ω ,u = f on ∂ Ω . Here and throughout this article we are using Einstein’s summationconvention: repeated indices in lower and upper position are summed.The boundary measurements are given by the Dirichlet-to-Neumannmap (DN map), defined byΛ γ f = γ jk ∂u∂x j ν k (cid:12)(cid:12)(cid:12) ∂ Ω where ν = ( ν , . . . , ν n ) denotes the unit outer normal to ∂ Ω and u isthe solution of the Dirichlet problem. The inverse problem is whetherone can determine γ by knowing Λ γ .Unfortunately, Λ γ doesn’t determine γ uniquely. This observation isdue to L. Tartar (see [15] for an account). Let ψ : Ω → Ω be a C ∞ diffeomorphism with ψ | ∂ Ω = Id where Id is the identity map. ThenΛ e γ = Λ γ where e γ = (cid:18) t ψ ′ · γ · ψ ′ | det ψ ′ | (cid:19) ◦ ψ − . Here ψ ′ denotes the (matrix) differential of ψ , t ψ ′ its transpose, andthe dot · represents multiplication of matrices.We have then a large number of conductivities with the same DNmap: any change of variables of Ω that leaves the boundary fixed givesrise to a new conductivity with the same boundary measurements. Thequestion is whether this is the only obstruction to unique identifiabilityof the conductivity. It is known that this is the case in two dimensions.The anisotropic problem can be reduced to the isotropic one by usingisothermal coordinates (Sylvester [31]), and combining this with theresult of Nachman [21] for isotropic conductivities gives the result foranisotropic conductivities with two derivatives. The regularity wasimproved by Sun and Uhlmann [30] to Lipschitz conductivities using IMITING CARLEMAN WEIGHTS 3 the techniques of Brown and Uhlmann [5], and to L ∞ conductivities byAstala-Lassas-P¨aiv¨arinta [2] using the work of Astala-P¨aiv¨arinta [3].In the case of dimension n ≥
3, as was pointed out in [18], this isa problem of geometrical nature. In this article we will focus on thegeometric problem.Let (
M, g ) be a compact Riemannian manifold with boundary. Allmanifolds will be assumed smooth (which means C ∞ ) and oriented.The Laplace-Beltrami operator associated to the metric g is given inlocal coordinates by∆ g u = | g | − / ∂∂x j (cid:18) | g | / g jk ∂u∂x k (cid:19) where as usual ( g jk ) is the matrix inverse of ( g jk ), and | g | = det( g jk ).Let us consider the Dirichlet problem∆ g u = 0 in M, u | ∂M = f. The DN map in this case is defined as the normal derivativeΛ g f = ∂ ν u | ∂M = g jk ∂u∂x j ν k (cid:12)(cid:12)(cid:12) ∂M where ν = ν l ∂ x l denotes the unit outer normal to ∂M , and ν k = g kl ν l is the conormal. The inverse problem is to recover g from Λ g .There is a similar obstruction to uniqueness as for the conductivity.We have(1.1) Λ ψ ∗ g = Λ g where ψ is a C ∞ diffeomorphism of M which is the identity on theboundary. As usual ψ ∗ g denotes the pull back of the metric g by thediffeomorphism ψ .In the two dimensional case there is an additional obstruction sincethe Laplace-Beltrami operator is conformally invariant. More precisely∆ cg = 1 c ∆ g for any function c , c = 0. Therefore we have that for n = 2(1.2) Λ c ( ψ ∗ g ) = Λ g for any smooth function c = 0 so that c | ∂M = 1.Lassas and Uhlmann [17] proved that (1.1) is the only obstructionto unique identifiability of the metric for real-analytic manifolds indimension n ≥
3. In the two dimensional case they showed that (1.2)is the only obstruction to unique identifiability for smooth Riemanniansurfaces. Moreover these results assume that Λ g is measured only onan open subset of the boundary. LIMITING CARLEMAN WEIGHTS
Notice that these two results don’t assume any condition on thetopology of the manifold except for connectedness. An earlier result ofLee-Uhlmann [18] assumed that (
M, g ) was strongly convex and sim-ply connected. The result of [17] in dimension n ≥ n ≥ M proceeds by analytic continuation, using the knowledgeof Taylor series of g at the boundary. Thus, these results do not giveinformation from the interior of the manifold.On the other hand, in the isotropic case where g is a conformalmultiple of the Euclidean metric, many results are available even fornonsmooth coefficients. These results are based on special complexgeometrical optics solutions to elliptic equations, introduced in [32].These have the form u = e − h h ζ,x i (1 + r ) , where ζ ∈ C n is a complex vector satisfying ζ · ζ = 0, and r is small as h tends to 0. However, complex geometrical optics solutions have notbeen available in the anisotropic case, which has been a major difficultyin the study of that problem.One of the main contributions of this paper is a complex geomet-rical optics construction for a class of Riemannian manifolds. Thisis based on the work of Kenig-Sj¨ostrand-Uhlmann [13], where moregeneral complex geometrical optics solutions, of the form u = e − h ( ϕ + iψ ) ( a + r ) , were constructed in Euclidean space. Here ϕ is a limiting Carlemanweight.In this paper we characterize those Riemannian manifolds which ad-mit limiting Carleman weights, and also characterize all such weightsin Euclidean space. We give a construction of complex geometricaloptics solutions on a class of Riemannian manifolds, and we use thesesolutions to prove uniqueness results in inverse problems. The inverseproblems considered are the recovery of an electric potential and a mag-netic field from boundary measurements on an admissible Riemannianmanifold, and the determination of an admissible metric within a con-formal class from the DN map. Let us now state the precise results ofthis article. IMITING CARLEMAN WEIGHTS 5
Statement of results.
We first recall the definition of limitingCarleman weights. Let h > P = − h ∆ g . If ϕ is a smoothreal-valued function on M , consider the conjugated operator P ,ϕ = e ϕ/h P e − ϕ/h . (1.3)Here it is natural to work with open manifolds (i.e. manifolds withoutboundary such that no component is compact). Definition 1.1.
A real-valued smooth function ϕ in an open manifold ( M, g ) is said to be a limiting Carleman weight if it has non-vanishingdifferential, and if it satisfies on T ∗ M the Poisson bracket condition { p ϕ , p ϕ } = 0 when p ϕ = 0 , (1.4) where p ϕ is the principal symbol, in semiclassical Weyl quantization, ofthe conjugated Laplace-Beltrami operator ( ) . Our first result is a characterization of those Riemannian manifoldswhich admit limiting Carleman weights.
Theorem 1. If ( M, g ) is an open manifold having a limiting Carlemanweight, then some conformal multiple of the metric g admits a parallelunit vector field. For simply connected manifolds, the converse is alsotrue. Locally, a manifold admits a parallel unit vector field if and only ifit is isometric to the product of an Euclidean interval and another Rie-mannian manifold. This is an instance of the de Rham decomposition[24], or is easy to prove directly (see Lemma A.4). Thus, if (
M, g ) has alimiting weight ϕ , one can choose local coordinates in such a way that ϕ ( x ) = x and g ( x , x ′ ) = c ( x ) (cid:18) g ( x ′ ) (cid:19) , where c is a positive conformal factor. Conversely, any metric of thisform admits ϕ ( x ) = x as a limiting weight.In the case n = 2, limiting Carleman weights in ( M, g ) are exactlythe harmonic functions with non-vanishing differential (see Section 2).The case n ≥ Theorem 2.
Let Ω be an open subset of R n , n ≥ , and let e be theEuclidean metric. The limiting Carleman weights in (Ω , e ) are locallyof the form ϕ ( x ) = aϕ ( x − x ) + b LIMITING CARLEMAN WEIGHTS where a ∈ R \ { } and ϕ is one of the following functions: h x, ξ i , arg h x, ω + iω i , log | x | , h x, ξ i| x | , arg (cid:0) e iθ ( x + iξ ) (cid:1) , log | x + ξ | | x − ξ | with ω , ω orthogonal unit vectors, θ ∈ [0 , π ) and ξ ∈ R n \ { } . We use the following definition for the argument functionarg z = 2 arctan Im z | z | + Re z , z ∈ C \ R − . Remark 1.2.
The possible weights are all real analytic functions.Some comments are made at the end of section 3 about the globalaspect of Theorem 2.Let us now introduce the class of manifolds which admit limitingCarleman weights and for which we can prove uniqueness results ininverse problems. For this we need the notion of simple manifolds [27].
Definition 1.3.
A manifold ( M, g ) with boundary is simple if ∂M isstrictly convex , and for any point x ∈ M the exponential map exp x isa diffeomorphism from some closed neighborhood of in T x M onto M . Definition 1.4.
A compact manifold with boundary ( M, g ) , of dimen-sion n ≥ , is admissible if it is conformal to a submanifold withboundary of R × ( M , g ) where ( M , g ) is a compact simple ( n − -dimensional manifold. Examples of admissible manifolds include the following:1. Bounded domains in Euclidean space, in the sphere minus apoint, or in hyperbolic space. In the last two cases, the manifoldis conformal to a domain in Euclidean space via stereographicprojection.2. More generally, any domain in a locally conformally flat man-ifold is admissible, provided that the domain is appropriatelysmall. Such manifolds include locally symmetric 3-dimensionalspaces, which have parallel curvature tensor so their Cottontensor vanishes (see the Appendix).3. Any bounded domain M in R n , endowed with a metric whichin some coordinates has the form g ( x , x ′ ) = c ( x ) (cid:18) g ( x ′ ) (cid:19) , with c > g simple, is admissible. cf. Definition A.7. IMITING CARLEMAN WEIGHTS 7
4. The class of admissible metrics is stable under C -small pertur-bations of g .The first inverse problem involves the Schr¨odinger operator L g,q = − ∆ g + q, where q is a smooth complex valued function on ( M, g ). We make thestanding assumption that 0 is not a Dirichlet eigenvalue of L g,q in M .Then the Dirichlet problem (cid:26) L g,q u = 0 in M,u = f on ∂M has a unique solution for any f ∈ H / ( ∂M ), and we may define theDN map Λ g,q : f ∂ ν u | ∂M . Given a fixed admissible metric, one can determine the potential q fromboundary measurements. Theorem 3.
Let ( M, g ) be admissible, and let q and q be two smoothfunctions on M . If Λ g,q = Λ g,q , then q = q . This result was known previously in dimensions n ≥ A be a smooth complex valued1-form on M (the magnetic potential), and denote L g,A,q = d ¯ A ∗ d A + q, where d A = d + iA ∧ : C ∞ ( M ) → Ω ( M ) and d A ∗ is the formal adjointof d A (for the sesquilinear inner product induced by the Hodge dual onthe exterior form algebra). This reads in local coordinates L g,A,q u = −| g | − / (cid:0) ∂ x j + iA j (cid:1)(cid:0) | g | / g jk (cid:0) ∂ x k + iA k (cid:1) u (cid:1) if A = A j dx j .As before, we assume throughout that 0 is not a Dirichlet eigenvalueof L g,A,q in M , and consider the Dirichlet problem (cid:26) L g,A,q u = 0 in M,u = f on ∂M. We can define the DN map as the magnetic normal derivativeΛ g,A,q : f d A u ( ν ) | ∂M . LIMITING CARLEMAN WEIGHTS
This map is invariant under gauge transformations of the magneticpotential: we have Λ g,A + dψ,q = Λ g,A,q for any smooth function ψ which vanishes on the boundary. Thus, it isnatural that one recovers the magnetic field dA and electric potential q from the map Λ g,A,q . Theorem 4.
Let ( M, g ) be admissible, let A , A be two smooth 1-formson M and let q , q be two smooth functions on M . If Λ g,A ,q = Λ g,A ,q ,then dA = dA and q = q . This result was proved in [22] for the Euclidean metric. Our proofis closer to [7] which considers partial boundary measurements. See[25] for further references on the inverse problem for the magneticSchr¨odinger operator in the Euclidean case.The next result considers the anisotropic Calder´on problem. Underthe additional condition that the metrics are in the same conformalclass, one expects uniqueness since the only diffeomorphism that leavesa conformal class invariant is the identity. In dimensions n ≥ Theorem 5.
Let ( M, g ) and ( M, g ) be two admissible Riemannianmanifolds in the same conformal class. If Λ g = Λ g , then g = g . This article is organized as follows. In the next two sections, we studylimiting Carleman weights and prove in particular Theorems 1 and 2.In Sections 4 and 5, we prove Carleman estimates and construct com-plex geometrical optics solutions to the Schr¨odinger equation. Section6 deals with the proofs of Theorems 3, 4, and 5. The last two sectionsare devoted to two results needed in the resolution of the anisotropicinverse problems. The first is the injectivity of an attenuated geodesicX-ray transform on simple manifolds, and the second states that theDN map determines the Taylor expansion of the different quantities in-volved at the boundary. Finally, there is an appendix containing basicdefinitions and facts in Riemannian geometry which are used in thisarticle.
Acknowledgements.
C.E.K. is partly supported by NSF grant DMS-0456583. M.S. is supported in part by the Academy of Finland. G.U.would like to acknowledge partial support of NSF and a Walker FamilyEndowed Professorship. We would like to express our deepest thanks toJohannes Sj¨ostrand who made substantial contributions to this paper.
IMITING CARLEMAN WEIGHTS 9
His unpublished notes on characterizing limiting Carleman weights inthe Euclidean case are the basis for sections 2 and 3. In particular heproved that the level sets of limiting Carleman weights in the Euclideancase are either hyperspheres or hyperplanes (see section 3). We wouldalso like to thank David Jerison for helpful discussions on limiting Car-leman weights, and Robin Graham for useful suggestions on conformalgeometry. 2.
Limiting Carleman weights
We refer the reader to the appendix for a short overview on Rie-mannian geometry. We use h· , ·i and | · | to denote the Riemannianinner product and norm both on the tangent and the cotangent spaceand D to denote the Levi-Civita connection. Throughout this papersemiclassical conventions are used; we refer to [6] for an expositionof this theory. The principal symbol of the conjugated semiclassicalLaplace-Beltrami operator (1.3) is given by p ϕ = | ξ | − | dϕ | + 2 i h ξ, dϕ i . (2.1)There are two reasons to use limiting Carleman weights in the con-struction of complex geometrical optics solutions u = e − h ( ϕ + iψ ) ( a + r )of the Schr¨odinger equation ( − ∆ g + q ) u = 0. Given a function ϕ ,the construction amounts to looking for solutions of the conjugatedequation P ,ϕ v + h qv = 0of the form v = e − ih ψ ( a + r ) and then applying the usual WKB method.This includes solving the eikonal equation p ϕ ( x, dψ ) = 0and a transport equation on a . Note that P ,ϕ is not a self-adjointoperator and that the symbol p ϕ is complex valued. The existence ofa solution ψ to the eikonal equation implies { p ϕ , p ϕ } ( x, dψ ) = 0 . Hence using limiting Carleman weights is a way to ensure that theformer (necessary) equality is fulfilled. The other reason lies in thefact that one wants the conjugated operator P ,ϕ to be locally solvablein the semiclassical sense, in order to find the remainder term r andgo from an approximate solution to an exact solution. This means the principal symbol p ϕ of the conjugated operator needs to satisfyH¨ormander’s local solvability condition { p ϕ , p ϕ } ≤ p ϕ = 0 . Since applications of the complex geometrical optics construction toinverse problems require to construct solutions with both exponentialweights e ϕ/h and e − ϕ/h (in order to cancel possible exponential be-haviour in the product of two solutions — see section 6) and since p − ϕ = p ϕ , it seems natural to impose the bracket condition (1.4).We now proceed to the analysis of limiting Carleman weights onopen manifolds. We first observe that the notion of limiting Carlemanweight relates to a conformal class of Riemannian manifolds. Lemma 2.1.
Let ( M, g ) be an open Riemannian manifold, and ϕ alimiting Carleman weight. If c is a smooth positive function, then ϕ isa limiting Carleman weight in ( M, cg ) . In particular, if f : ( ˜ M , ˜ g ) → ( M, g ) is a conformal transformation, then f ∗ ϕ is a limiting Carlemanweight in ( ˜ M , ˜ g ) .Proof. The claim follows from the fact that the principal symbol of theconjugated Laplace-Beltrami operator e ϕ/h ∆ cg e − ϕ/h is ˜ p ϕ = c − p ϕ andthat 12 i { ˜ p ϕ , ˜ p ϕ } = c − i { p ϕ , p ϕ } + c − Im (cid:0) p ϕ { p ϕ , c − } (cid:1) . Both term in the right-hand side vanish when p ϕ = 0. (cid:3) Remark 2.2.
This lemma gives a way to construct limiting Carlemanweights. If we already know a limiting Carleman weight ϕ , then anyfunction of the form ϕ ◦ f , where f is a conformal transformation on( M, g ), is a limiting Carleman weight.In particular, from the linear Carleman weight h x, ξ i , using the in-version x → x/ | x | which is a conformal transformation on R n \ { } (endowed with the Euclidean metric e ), we obtain another limitingCarleman weight h x, ξ i / | x | on M = R n \ { } .Now we know that the existence of limiting Carleman weights onlydepends on a conformal class of geometries. Let a and b denote re-spectively the real and imaginary parts of the principal symbol of theconjugated operator (1.3), so p ϕ = a + ib. IMITING CARLEMAN WEIGHTS 11
They are given by the following expressions a = | ξ | − | dϕ | = | ξ ♯ | − |∇ ϕ | , (2.2) b = 2 h dϕ, ξ i = 2 h∇ ϕ, ξ ♯ i . Note that { p ϕ , p ϕ } / i = { a, b } so that the condition (1.4) also reads { a, b } = 0 when a = b = 0 . (2.3)We start with the computation of this Poisson bracket in terms ofthe Hessian D ϕ . Lemma 2.3.
The Poisson bracket may be expressed as { a, b } ( x, ξ ) = 4 D ϕ ( ξ ♯ , ξ ♯ ) + 4 D ϕ ( ∇ ϕ, ∇ ϕ ) . Proof.
Consider a = | ξ | and a = | dϕ | , so that a = a − a , thePoisson bracket is given by { a, b } = { a , b } + { b, a } = H a b + H b a . Since a is a function only depending on x , and b is linear in ξ , thesecond bracket is easily calculated and by (A.1) we obtain H b a = 2 L ∇ ϕ (cid:0) |∇ ϕ | (cid:1) = 4 D ϕ ( ∇ ϕ, ∇ ϕ ) . It remains to compute the first Poisson bracket. The Hamiltonian flowgenerated by a is the cogeodesic flow, t → ( x ( t ) , ξ ( t )) , x ( t ) a geodesic , ξ ( t ) = ˙ x ♭ ( t ) , therefore using (A.2) one has H a b = 2 ∂∂t b (cid:0) x ( t ) , ξ ( t ) (cid:1)(cid:12)(cid:12) t =0 = 4 ∂∂t dϕ ( x ( t )) (cid:0) ˙ x ( t ) (cid:1)(cid:12)(cid:12) t =0 = 4 ∂ ∂t ϕ ( x ( t )) (cid:12)(cid:12) t =0 = 4 D ϕ ( ˙ x (0) , ˙ x (0)) . This finishes the proof of the equality since ˙ x (0) = ξ ♯ . (cid:3) The bracket condition (2.3) now reads(2.4) D ϕ ( X, X ) + D ϕ ( ∇ ϕ, ∇ ϕ ) = 0for all X ∈ T ( M ) such that | X | = |∇ ϕ | , h X, ∇ ϕ i = 0 . The situation in dimension 2 is particularly simple.
Lemma 2.4.
In the case of a -dimensional Riemannian manifold ( M, g ) , limiting Carleman weights are exactly the harmonic functionswith non-vanishing differential. The musical notation is recalled in the appendix.
Proof.
This comes from the fact that when M has dimension 2, wehave D ϕ ( X, X ) + D ϕ ( ∇ ϕ, ∇ ϕ ) = |∇ ϕ | Tr D ϕ = |∇ ϕ | ∆ ϕ if | X | = |∇ ϕ | and h∇ ϕ, X i = 0. (cid:3) We will therefore continue our investigation in dimension n ≥
3. Theexpression of the Poisson bracket { a, b } suggests that it is convenientto work with Carleman weights which are also distance functions, inthe sense that |∇ ϕ | = 1, since in that case D ϕ ( ∇ ϕ, ∇ ϕ ) = 0. Onecan always reduce to this case by using the conformal metric˜ g = |∇ ϕ | g (2.5)since the notion of limiting Carleman weights only depends on a con-formal class of metrics. Lemma 2.5.
Among distance functions on an open Riemannian man-ifold ( M, g ) , limiting Carleman weights can be characterized by the fol-lowing equivalent properties. (1) The Hessian of ϕ vanishes identically. (2) The gradient of ϕ is a Killing field. (3) The gradient of ϕ is a parallel field. (4) If x ∈ M and v is in the domain of exp x then ϕ (exp x v ) = ϕ ( x ) + h∇ ϕ ( x ) , v i . Proof.
If we assume |∇ ϕ | = 1, then h D Z ∇ ϕ, ∇ ϕ i = 0 for any vectorfield Z and the bracket condition (2.4) is equivalent to D ϕ ( X, X ) = h D X ∇ ϕ, X i = 0 when h X, ∇ ϕ i = 0 . By bilinearity and by the fact that D ϕ ( ∇ ϕ, Z ) = h D Z ∇ ϕ, ∇ ϕ i = 0one can actually drop the orthogonality condition, and after polariza-tion (2.4) is furthermore equivalent to D ϕ = 0. The equivalence of (1)and (2) comes from the first equality in (A.1). The equivalence of (1)and (3) follows from the second equality in (A.1). Finally, properties(1) and (4) are equivalent because of the identities ∂∂t ϕ (cid:0) exp x ( tv ) (cid:1)(cid:12)(cid:12) t =0 = dϕ ( v ) = h∇ ϕ, v i ,∂ ∂t ϕ (cid:0) exp x ( tv ) (cid:1)(cid:12)(cid:12) t =0 = D ϕ ( v, v )and of Taylor’s expansion. (cid:3) Remark 2.6.
According to (4), in Riemannian manifolds, functionswith null Hessian are the analogue of linear Carleman weights in theEuclidean setting.
IMITING CARLEMAN WEIGHTS 13
Remark 2.7.
According to Lemmas 2.3 and 2.5, when ϕ is both alimiting Carleman weight and a distance function, the bracket { a, b } in(2.3) vanishes everywhere instead of just on the characteristic variety a = b = 0. Proof of Theorem 1.
Suppose that the manifold (
M, g ) has a limitingCarleman weight ϕ , then ϕ is both a distance function and a limitingCarleman weight in ( M, ˜ g ) where ˜ g is the metric (2.5) conformal to g .According to Lemma 2.5, this means that ∇ ˜ g ϕ is a unit parallel field.Conversely, if ( M, g ) is a simply connected Riemannian manifoldsuch that (
M, cg ) has a unit parallel field X , then according to LemmaA.3, X is both a gradient field X = ∇ cg ϕ and a Killing field. Thanks toLemmas 2.5 and 2.1, this implies that ϕ is a limiting Carleman weightin ( M, cg ) and in (
M, g ). (cid:3) Remark 2.8.
It is now easy to justify the local coordinate expressionin the introduction for metrics which admit limiting Carleman weights.If (
M, g ) admits a limiting weight ϕ , the proof of Theorem 1 shows that ∇ ˜ g ϕ is a unit parallel field for the conformal metric ˜ g = c − g where c = |∇ ϕ | − . Lemma A.4 implies that near any point of M there existlocal coordinates such that˜ g ( x , x ′ ) = (cid:18) g ( x ′ ) (cid:19) and ∇ ˜ g ϕ = ∂/∂x = ∇ ˜ g ( x ) . One obtains, after a translation of coordinates if necessary, that ϕ ( x ) = x and g ( x , x ′ ) = c ( x ) (cid:18) g ( x ′ ) (cid:19) . Conversely, if g is of this form and ˜ g = c − g , then ∂/∂x = ∇ ˜ g ( x )is a unit parallel field by Lemma A.4. Thus ϕ ( x ) = x is a limitingCarleman weight.The purpose of the next lemma is to give several properties of limitingCarleman weights, in particular the fact that their level sets are totallyumbilical hypersurfaces . Lemma 2.9.
A function ϕ with non-vanishing differential is a limitingCarleman weight if and only if |∇ ϕ | − ∇ ϕ is a conformal Killing field.In particular, if ϕ is a limiting Carleman weight then (1) the Hessian of ϕ is determined by the knowledge of ∇ ϕ and D ∇ ϕ ∇ ϕ , that is D ϕ = λg + |∇ ϕ | − (cid:0) dϕ ⊗ ( D ∇ ϕ ∇ ϕ ) ♭ + ( D ∇ ϕ ∇ ϕ ) ♭ ⊗ dϕ (cid:1) cf. Definition A.6. where λ = −|∇ ϕ | − D ϕ ( ∇ ϕ, ∇ ϕ ) , (2) the trace of the Hessian is given by ∆ ϕ = Tr D ϕ = ( n − λ, (3) the level sets of ϕ in ( M, g ) are totally umbilical submanifoldswith normal |∇ ϕ | − ∇ ϕ , with principal curvatures equal to µ = −|∇ ϕ | − D ϕ ( ∇ ϕ, ∇ ϕ ) , (4) the eigenvalues of the Hessian are λ , κ and − κ with κ = | D ∇ ϕ ∇ ϕ ||∇ ϕ | . Proof.
Let ˜ g denote the metric (2.5) conformal to g . Suppose that ϕ isa limiting Carleman weight in ( M, g ) then according to Lemma 2.5 ∇ ˜ g ϕ = |∇ g ϕ | − ∇ g ϕ is a Killing field in ( M, ˜ g ). This implies that it is a conformal Killingfield in ( M, g ). Conversely, if ∇ ˜ g ϕ is a conformal Killing field in ( M, g ),it is both a conformal Killing field and a unit gradient in ( M, ˜ g ). Evalu-ating the equality L ∇ ˜ g ϕ ˜ g = γ ˜ g at ( ∇ ˜ g ϕ, ∇ ˜ g ϕ ) gives γ = 0 since by (A.1) L ∇ ˜ g ϕ ˜ g ( ∇ ˜ g ϕ, ∇ ˜ g ϕ ) = 2 h ˜ D ∇ ˜ g ϕ ∇ ˜ g ϕ, ∇ ˜ g ϕ i = L ∇ ˜ g ϕ |∇ ˜ g ϕ | = 0 . This implies that ∇ ˜ g ϕ is a Killing field in ( M, ˜ g ) hence that ϕ is alimiting Carleman weight in ( M, ˜ g ) thanks to Lemma 2.5, and in ( M, g )thanks to Lemma 2.1.Formulas (A.3) and (A.1) give L |∇ ϕ | − ∇ ϕ g = 2 |∇ ϕ | − (cid:16) D ϕ − dϕ ⊗ ( D ∇ ϕ ∇ ϕ ) ♭ |∇ ϕ | − ( D ∇ ϕ ∇ ϕ ) ♭ |∇ ϕ | ⊗ dϕ (cid:17) . If ϕ is a limiting Carleman weight, then |∇ ϕ | − ∇ ϕ is a conformalKilling field, and the former expression equals to γg . It is easy tocompute γ by evaluating the former expression at ( X, X ) such that h∇ ϕ, X i = 0 and | X | = 1, and by using (2.4) γ = 2 |∇ ϕ | − D ϕ ( X, X ) = − |∇ ϕ | − D ϕ ( ∇ ϕ, ∇ ϕ ) . The combination of L |∇ ϕ | − ∇ ϕ g = γg and those two relations gives (1).Besides, one also has γ = 2 n div (cid:0) |∇ ϕ | − ∇ ϕ (cid:1) = 2 n (cid:0) |∇ ϕ | − ∆ ϕ − |∇ ϕ | − D ϕ ( ∇ ϕ, ∇ ϕ ) (cid:1) and the two expressions of γ yield the trace of the Hessian as in (2). IMITING CARLEMAN WEIGHTS 15
The level sets of ϕ are submanifolds of M , with unit normal givenby ν = |∇ ϕ | − ∇ ϕ . The tangent vectors of ϕ − ( t ) are orthogonal to ∇ ϕ and the second fundamental form is given by ℓ ( X, Y ) = h D X ν, Y i = |∇ ϕ | − D ϕ ( X, Y ) = µ h X, Y i for X and Y satisfying h X, ∇ ϕ i = h Y, ∇ ϕ i = 0. Thus the principalcurvatures of ϕ − ( t ) are all equal to µ .A consequence of the expression of the Hessian in (1) is D Z ∇ ϕ = λZ + |∇ ϕ | − (cid:0) h D ∇ ϕ ∇ ϕ, Z i∇ ϕ + h∇ ϕ, Z i D ∇ ϕ ∇ ϕ (cid:1) . (2.6)Since D ϕ = λg on ker dϕ ∩ ker( D ∇ ϕ ∇ ϕ ) ♭ , λ is an eigenvalue of theHessian of multiplicity at least n −
2. Because of the trace, the tworemaining eigenvalues are opposite. Considering the orthogonal vectors u = ∇ ϕ |∇ ϕ | + D ∇ ϕ ∇ ϕ | D ∇ ϕ ∇ ϕ | v = ∇ ϕ |∇ ϕ | − D ∇ ϕ ∇ ϕ | D ∇ ϕ ∇ ϕ | and using (2.6), one has : D u ∇ ϕ = κu, D v ∇ ϕ = − κv. One of the two vectors u, v is nonzero, therefore κ and − κ are the tworemaining eigenvalues. Note that when ∇ ϕ and D ∇ ϕ ∇ ϕ are dependentthen κ = | λ | . In this case the eigenvalues are λ (with multiplicity n − − λ . (cid:3) Remark 2.10.
It is an accepted fact in geometry that generic mani-folds in dimension n ≥ ϕ = log | x | plays an interestingrole (cf. [13], [7] and [14]). The former computation allows us to givea partial answer to the question: if ρ > |∇ ρ | = 1), when is ϕ = log ρ a limiting Carleman weight?Indeed, one has D ϕ = D ρρ − dρ ⊗ dρρ , D ∇ ϕ ∇ ϕ = − ∇ ρρ , and λ = 1 ρ , thus ϕ = log ρ is a limiting Carleman weight if and only if the metric g has the form g = ρD ρ + dρ ⊗ dρ. If ϕ = log ρ is a limiting Carleman weight, then by Remark 2.8 thereare coordinates near any point of M so that log ρ = x , |∇ ϕ | − = e x ,and g ( x ) = e x (cid:18) g ( x ′ ) (cid:19) = e x ˜ g ( x ′ ) . Conversely, any metric of this form admits a logarithmic Carlemanweight ϕ ( x ) = x = log ρ with ρ = e x . Therefore if log ρ is a lim-iting Carleman weight, one computes the Christoffel symbols in thecoordinates ( x , x ′ ) to beΓ = 1 and Γ ij = − ˜ g ij ( x ′ ) if i = 1 or j = 1 . Thus if we denote D b = D ∂ b , the curvature tensor for b = 1 and c = 1satisfies R bc = h D D b ∂ c − D b D ∂ c , ∂ i = h D (Γ dbc ∂ d ) − D b (Γ d c ∂ d ) , ∂ i = ( ∂ Γ bc − ∂ b Γ c + Γ dbc Γ d − Γ d c Γ bd ) g = (Γ bc − Γ d c Γ bd ) g . A direct computation using the special form of g shows that the lastexpression vanishes. Then the sectional curvature K ( X, Y ) = R ( X, Y, Y, X ) | X | | Y | − h X, Y i vanishes whenever X = ∂ . This implies that any manifold with non-vanishing sectional curvature at a point p cannot admit a limiting Car-leman weight of the form log ρ near p . In particular, the sphere orhyperbolic space do not admit limiting Carleman weights of this form.3. The Euclidean case
So far, we know at least three examples of limiting Carleman weightsdefined on open subsets of the Euclidean space: the linear weight, thelogarithmic weight, and the hyperbolic weight h x, ξ i , log | x | , h x, ξ i| x | . Note that the first is defined on R n while the two others are on R n \{ } . The purpose of this section is to determine all possible limitingCarleman weights for open subsets of the Euclidean space.The following result is well known. IMITING CARLEMAN WEIGHTS 17
Lemma 3.1.
The only connected totally umbilical hypersurfaces in theEuclidean space of dimension n ≥ are parts of either hyperplanes orhyperspheres.Proof. Let Σ be a connected totally umbilical hypersurface in an opensubset Ω ⊂ R n , let ν denote its unit exterior normal, and let λ be thecommon value of the principal curvatures. First, let us prove that λ isconstant along Σ. We have D X ν = λX for all vector fields X tangent to Σ. Therefore we deduce D X D Y ν − D Y D X ν = ( L X λ ) Y − ( L Y λ ) X + λ [ X, Y ] | {z } = D [ X,Y ] ν . Since the Euclidean space is flat, we have D X D Y ν − D Y D X ν − D [ X,Y ] ν = R ( X, Y ) ν = 0therefore we deduce ( L X λ ) Y − ( L Y λ ) X = 0 for all vectors tangent toΣ, which means that λ is constant along Σ.We now consider the vector field V = P nj =1 x j ∂ x j and we have D X ( ν − λV ) = λX − λD X V = 0for all vector fields X tangent to Σ. This means that ν − λx is constantalong the hypersurface. If λ = 0, the normal is constant along thehypersurface and Σ is part of a hyperplane, and if λ = 0 then α = λ − ( λx − ν ) is constant along the hypersurface, and Σ is a part of thehypersphere | x − α | = 1 / | λ | . (cid:3) This provides the additional information that the level sets of a lim-iting Carleman weight are parts of either hyperspheres or hyperplanes.Let Ω be a bounded open connected subset of R n , n ≥
3, let ϕ be alimiting Carleman weight on Ω, and consider J = ϕ (Ω) which is anopen interval. The curvature µ ( t ) = −|∇ ϕ | − h ϕ ′′ ∇ ϕ, ∇ ϕ i (cid:12)(cid:12) ϕ − ( t ) of the level sets, is a smooth function of t ∈ J .We will begin by proving that ϕ is locally of one of the forms inTheorem 2, thus we will suppose that Ω is a small open neighborhoodof a point x and J is a small interval. First suppose that µ is identically0, i.e. that all level sets of ϕ are hyperplanes. Possibly after doing arotation, we may suppose that ∇ ϕ ( x ) = ( ∂ x ϕ ( x ) , , . . . , = 0 . (3.1) If Ω is small enough, this implies that we can take ( t, x ′ ) as coordinateson Ω, and that the level sets of ϕ are of the form x = h ω ′ ( t ) , x ′ i + s ( t )where as usual the prime notation stands for x ′ = ( x , . . . , x n ). We willbegin with the relation ϕ (cid:0) h ω ′ ( t ) , x ′ i + s ( t ) , x ′ (cid:1) = t. (3.2)Differentiation with respect to x ′ gives ω ′ ∂ x ϕ + ∂ x ′ ϕ = 0 on ϕ − ( t )(3.3)which expresses the fact that ∇ ϕ is normal to the hyperplane ϕ − ( t ).Differentiating with respect to t the relation (3.2) we get the followingequations (cid:0) h ˙ ω ′ , x ′ i + ˙ s (cid:1) ∂ x ϕ = 1 , (3.4) and (cid:0) h ˙ ω ′ , x ′ i + ˙ s (cid:1) ∂ x ϕ + (cid:0) h ¨ ω ′ , x ′ i + ¨ s (cid:1) ∂ x ϕ = 0 . (3.5)It remains to compute ∂ x ϕ in order to obtain ordinary differentialequations for ω ′ and s .Differentiating (3.3) with respect to x ′ gives ω j ∂ x ϕ + 2 ω j ∂ x x j ϕ + ∂ x j ϕ = 0 , ≤ j ≤ n. Remember that we have supposed µ = 0, this implies that ϕ is har-monic because of (2) in Lemma 2.9. Summing up the relations aboveand using the fact that ∆ ϕ = 0, we get( ω ′ − ∂ x ϕ + 2 h ω ′ , ∂ x x ′ ϕ i = 0 . (3.6)Differentiating (3.3) with respect to t we also get (cid:0) ω j ∂ x ϕ + ∂ x x j ϕ (cid:1)(cid:0) h ˙ ω ′ , x ′ i + ˙ s (cid:1) + ˙ ω j ∂ x ϕ = 0 , ≤ j ≤ n. Multiplying the former relations by ω j and summing up yields (cid:0) ω ′ ∂ x ϕ + h ω ′ , ∂ x x ′ ϕ i (cid:1)(cid:0) h ˙ ω ′ , x ′ i + ˙ s (cid:1) + h ˙ ω ′ , ω ′ i ∂ x ϕ = 0 . (3.7)The combination of (3.6) and (3.7) gives (cid:0) h ˙ ω ′ , x ′ i + ˙ s (cid:1) ∂ x ϕ = − h ˙ ω ′ , ω ′ i ω ′ ∂ x ϕ. (3.8)Finally using (3.4), (3.5) and (3.8), we have − h ˙ ω ′ , ω ′ i ω ′ (cid:0) h ˙ ω ′ , x ′ i + ˙ s (cid:1) + (cid:0) h ¨ ω ′ , x ′ i + ¨ s (cid:1) = 0 . for all t ∈ J and all x ′ in a neighborhood of x ′ . IMITING CARLEMAN WEIGHTS 19
We end up with the following system of equations ¨ ω ′ − h ˙ ω ′ , ω ′ i ω ′ ˙ ω ′ = 0¨ s − h ˙ ω ′ , ω ′ i ω ′ ˙ s = 0 on J. (3.9)Solving the first ordinary differential equation yields˙ ω j = a j (1 + ω ′ )for some constant a j . After a rotation in the x ′ variable, we maysuppose that a j = 0 for j ≥
3. Because of the form (3.1) of thegradient of ϕ at x , we have ω ( t ) = tan( at + b ) , ω j = 0 , ≤ j ≤ n. Injecting the former solution into the equation in s leads to the follow-ing differential equation¨ s − a tan( at + b ) ˙ s = 0which can be integrated: if a = 0 then s ( t ) = ct + d and if a = 0 then s ( t ) = c tan( at + b ) + d. Finally, this gives two possible types of limiting Carleman weights1 . a h x − ˜ x , ξ i + b . a arg h x − ˜ x , ω + iω i + b with ω , ω unit orthogonal vectors. Remark 3.2.
The use of complex variables simplified computationsin [7]. Note that the second weight can be written as a arg z + b, with z = h x − ˜ x , ω i + i h x − ˜ x , ω i and a, b real numbers.Now we assume that µ does not vanish identically on J , and con-sider a subinterval I ⊂ µ − ( R ∗ + ) (the study on µ − ( R ∗− ) can be doneby considering − ϕ ). When t ∈ I , the level sets ϕ − ( t ) are thereforehyperspheres of radii r ( t ) = 1 /µ ( t ), whose centers we denote by α ( t ).Both functions r and α are smooth on I , and we have ϕ (cid:0) α ( t ) + r ( t ) ω (cid:1) = t, ∀ ( t, ω ) ∈ I × Γ(3.10)for some open subset Γ of the unit hypersphere. The normal to thehypersphere ϕ − ( t ) is ω = |∇ ϕ | − ∇ ϕ (cid:0) α ( t ) + r ( t ) ω (cid:1) . (3.11) Differentiating the identity (3.10) with respect to t , we obtain h∇ ϕ (cid:0) α ( t ) + r ( t ) ω (cid:1) , ˙ α + ˙ rω i = 1which together with (3.11) gives h∇ ϕ, ω i (cid:0) h ˙ α, ω i + ˙ r (cid:1) = 1 . (3.12)Then taking ω = y/ | y | in the former equality, multiplying it by | y | and differentiating with respect to y gives (cid:0) h ˙ α, ω i + ˙ r (cid:1)(cid:0) ∇ ϕ + rϕ ′′ ω − r h ϕ ′′ ω, ω i ω | {z } = −∇ ϕ (cid:1) + h∇ ϕ, ω i (cid:0) ˙ α + ˙ rω (cid:1) = 2 ω and thus ϕ ′′ ω = − |∇ ϕ | r ( ˙ α + ˙ rω ) . (3.13)We go back to (3.12) and differentiate it with respect to t h ϕ ′′ ω, ˙ α + ˙ rω i (cid:0) h ˙ α, ω i + ˙ r (cid:1) + h∇ ϕ, ¨ α + ¨ rω i = 0 . Using (3.13) and (3.11), this gives − r |∇ ϕ | (cid:0) h ˙ α, ω i + ˙ r (cid:1)| {z } =1 | ˙ α + ˙ rω | + h ¨ α, ω i + ¨ r = 0which leads to the following system of equations ¨ α − rr ˙ α = 0¨ r − r (cid:0) | ˙ α | + ˙ r (cid:1) = 0 on I. (3.14)The first equation implies that the centers of the hyperspheres ϕ − ( t )are moving along a line with fixed direction k ∈ R n α ( t ) = α + k Z t r ( s ) ds (3.15)(the indefinite integral denotes a primitive on I ). Thus we have | ˙ α | = | k | r and the second equation reads r ¨ r − r | k | − ˙ r = 0 . It is convenient to rewrite the former equation in terms of the curvature µ = 1 /r : µ ¨ µ + | k | − ˙ µ = 0which is equivalent to det (cid:18) ˙ µ µ ¨ µ ˙ µ (cid:19) = | k | . (3.16) IMITING CARLEMAN WEIGHTS 21
Before proceeding to the resolution of this differential equation, letus first notice that it actually holds on J . Indeed, if we differentiate itwith respect to t , then we obtain an equation on µ det (cid:18) ¨ µ µµ (3) ˙ µ (cid:19) = 0(3.17)which holds on any subinterval I of µ − ( R ∗ ), but also evidently on theinterior of µ − (0). This implies that the equation (3.17) holds on J bycontinuity, hence that the equation (3.16) holds on J by integration.To solve the equation (3.16) there are two cases to consider. Case k = 0 . This is the case where the hyperspheres ϕ − ( t ) are concen-tric. The vectors ( µ, ˙ µ ) and its derivative are linearly dependent, thus(3.16) is equivalent to the first order differential equation ˙ µ = cµ , with c constant. To be more precise, either µ vanishes identically or one canpick t ∈ J such that µ ( t ) = 0, and consider the maximal interval I on which µ doesn’t vanish. By (3.16) the derivative of ˙ µ/µ on I is zero,therefore µ solves the equation ˙ µ = cµ on I , with c constant. Thenthe curvature is an exponential function of t which never vanishes, thus I = J . The function ϕ is easily determined to be a logarithmic weight: ϕ ( x ) = a log | x − ˜ x | + b. Remark 3.3.
Note that the logarithmic weight can be written as ϕ = a Re(log z ) + b with z = y + i | y ′ | , y = x − x = ( y , y ′ ). Case k = 0 . The equation (3.17) shows that the vectors ( µ, ˙ µ ) and itssecond derivative are linearly dependent, this implies that µ solves asecond order differential equation of the form ¨ µ = cµ with c a constant.To see this, consider the function σ = ¨ µµ which is a priori only defined outside the set of zeros of µ . Note thatif µ ( τ ) = 0 then by (3.16) one has ˙ µ ( τ ) = 0 and therefore the zeros of µ are isolated. Moreover, by (3.17) one has ¨ µ ( τ ) = 0. We can extend σ as a continuous function on all of J , because as t tends to τ we have σ ( t ) = ¨ µ ( t ) − ¨ µ ( τ ) t − τ t − τµ ( t ) − µ ( τ ) −→ µ (3) ( τ )˙ µ ( τ ) . The Taylor expansion at higher orders actually shows that σ is of class C . Differentiating the function σ on J \ µ − (0) gives˙ σ = µ (3) µ − ¨ µ ˙ µµ = 0and ˙ σ = 0 on J by continuity.Hence the function σ is constant and this implies that µ satisfies theequation ¨ µ − cµ = 0 with c constant. Depending on whether c is zero,negative or positive, the curvature is one of the following functions ±| k | ( t + b ) , | k | a sin( at + b ) , ± | k | a sinh( at + b ) . (3.18)If t = ϕ ( x ) and µ ( t ) = 0, we may choose J so small that µ doesn’tvanish on J . If µ ( t ) = 0, then either µ vanishes identically near 0,and that case was covered first, or t is an isolated zero of µ by (3.16)with k = 0, and we may choose J so small that µ doesn’t vanish on J \ { t } . In conclusion, one may assume that the curvature vanishes atmost once on J . The corresponding expressions for the centers of thespheres α are α − k | k | t + b , α − k | k | a tan( at + b ) ,α − k | k | a tanh( at + b ) . If µ ( t ) = 0 then these expressions depend a priori on the connectedcomponent I ± of J \ { t } on which µ is positive or negative. The normof the vector k however is determined by the equation (3.16) whichholds on J and its direction k/ | k | by the normal to the hyperplane ϕ − ( t ). This may be seen in the following way: consider the family ofhyperspheres (cid:12)(cid:12)(cid:12) x − α ± + k ± | k | a tanh( at + b ) (cid:12)(cid:12)(cid:12) = a | k | sinh ( at + b )when t ∈ I ± . After expansion, we obtain(3.19) 2 h x − α ± , k ± i = | k | tanh( at + b ) a (cid:16) − a | k | − | x − α ± | (cid:17) and letting t tend to t = − b/a , we get the equation of the hyperplane ϕ − ( t ) h x − α ± , k ± i = 0 . This implies that either k + = k − or k + = − k − . Besides, by (3.11) and(3.12), we have h ˙ α, ω i + ˙ r = |∇ ϕ | − on I + , thus h k + , ∇ ϕ i − ˙ µ |∇ ϕ | = µ IMITING CARLEMAN WEIGHTS 23 and by letting x tend to x , this gives h∇ ϕ ( x ) , k + i = ˙ µ ( t ) |∇ ϕ ( x ) | . Considering I − , we obtain the same identity with k + replaced by k − .This additional information removes the uncertainty on the sign andwe have k + = k − . Moreover the components of α + and α − along k arealso equal. The computations are similar in the two other cases.To determine explicitly the function ϕ , let us deal with the casewhere the curvature is | k | sin( at + b ) /a : to fix the ideas, suppose that aJ + b ⊂ ( − θ, − θ + 2 π ) with θ ∈ R and that µ vanishes at t = ϕ ( x )(the non-vanishing case is simpler). Then we have J \ { t } = I − ∪ I + and x = α ± − a | k | tan( aϕ ( x ) + b ) k + a | k | sin( aϕ ( x ) + b ) ω for all x ∈ ϕ − ( I ± ). Thus, if we take ξ = − ak/ | k | , we have e iθ ( x − α ± + iξ ) = 2 | ξ | sin ( aϕ + b ) (cid:16) cos( aϕ + b ) − h ω, k i| k | (cid:17) e i ( aϕ + b + θ ) = 2 | a | − | ξ | (cid:0) h ˙ α, ω i + ˙ r (cid:1) e i ( aϕ + b + θ − π ) and since aϕ + b + θ − π ∈ ( − π, π ), and h ˙ α, ω i + ˙ r > aϕ ( x ) + b = arg (cid:0) e iθ ( x − α ± + iξ ) (cid:1) + π − θ on ϕ − ( I ± ). With the former expression of ϕ we have | a ∇ ϕ ( x ) | = 2 | ξ || x − α ± | − | ξ | on h x − α ± , ξ i = 0this shows that | x − α + | = | x − α − | in a neighborhood of x in thehyperplane ϕ − ( t ), hence that α + = α − . The former expression of ϕ therefore holds on Ω.The two other cases are similar. This finally gives three possibletypes of limiting Carleman weights1 . a h x − ˜ x , ξ i| x − ˜ x | + b . a arg (cid:0) e iθ ( x − ˜ x + iξ ) (cid:1) + b . a arctanh 2 h x − ˜ x , ξ i| x − ˜ x | + | ξ | + b = a log | x − ˜ x + ξ | | x − ˜ x − ξ | + b. Remark 3.4.
As in Remark 3.3, these functions take a simple formwith respect to some complex variable. Take y = x − ˜ x , and ( y , y ′ ) such that y = y ξ/ | ξ | + y ′ where y ′ is orthogonal to ξ . Denote z = y + i | y ′ | , the Carleman weights take the form1 . a Re 1 z + b, . a Im log e iθ ( z + ic ) z − ic + b, . a Re log z + cz − c + b, with a, b, c real numbers.This proves Theorem 2. Note that this result allows to determinelimiting Carleman weights locally on the sphere or on the hyperbolicspace by conformal transformation. Remark 3.5.
While the linear weight is smooth on R n , the weightslog | x | and h x, ξ i / | x | are only well defined and smooth on R n \ { } ,and the function log | x + ξ | / | x − ξ | on R n \ { ξ, − ξ } . The weightarg h x, ω + iω i is defined on R n \ {h x, ω i = 0 , h x, ω i ≤ } and thefunction arg (cid:0) e iθ ( x + iξ ) (cid:1) on R n \ n | x + cotan θ ξ | = | ξ | sin θ , h x, ξ i ⋚ o if θ ∈ (0 , π ) (with the ≥ inequality) or θ ∈ ( π, π ) (with the ≤ inequal-ity) and on R n \ (cid:8) h x, ξ i = 0 , | x | ⋚ | ξ | (cid:9) if θ = 0 (with the ≤ inequality) or θ = π (with the ≥ inequality).Let us end this section with some comments about the global aspectof Theorem 2 when Ω is an open connected set. Let ϕ be a limitingCarleman weight on (Ω , e ). The function ϕ is real analytic on Ω: indeedif x ∈ Ω, the function ϕ is one of the limiting Carleman weightscalculated above in the neighourhood of x , hence real analytic. Thisensures that if ϕ is locally equal to one of the functions a h x − ˜ x , ξ i + b, a log | x − ˜ x | + b,a h x − ˜ x , ξ i| x | + b, a log | x − ˜ x + ξ | | x − ˜ x − ξ | + b then ϕ is equal to this function on the whole set Ω. Indeed if ϕ is equalto a linear weight near some point x ∈ Ω then ϕ is equal to this weighton the whole set Ω by analytic continuation. If ϕ is equal to the function a log | x − ˜ x | + b near x ∈ Ω then Ω cannot contain the singularity˜ x of this function. Otherwise, ϕ would be equal to this function on IMITING CARLEMAN WEIGHTS 25 Ω \ { ˜ x } by analytic continuation and would not blow up at ˜ x . Thus ϕ is equal to a log | x − ˜ x | + b on Ω. The proof is similar if ϕ is one of thefunctions a h x − ˜ x , ξ i / | x − x | + b or a log( x − ˜ x + ξ ) / ( x − ˜ x − ξ ) + b .For the two argument forms, we need some additional assumptionson Ω. Suppose that ϕ is equal to a arg (cid:0) e iθ ( x − ˜ x + iξ ) (cid:1) + b near x ,and that the image of the set Ω by x ( x − ˜ x + iξ ) is containedin a simply connected set U ⊂ C ∗ , then by analytic continuation, ϕ isequal to a arg U ( x − ˜ x + iξ ) + b where arg U is the determination of the argument on U which coincideswith arg( e iθ z ) at z = ( x − ˜ x + iξ ) . The point is similar for thefunction a arg h x − ˜ x , ω + iω i . In particular, if Ω is contained in oneof the domains of existence computed in Remark 3.5 and ϕ is locallyof the corresponding argument form, then this is still true globally onΩ. 4. Carleman estimates
Let (
M, g ) be a compact Riemannian manifold with boundary. By dV we denote the volume form on ( M, g ), and by dS = ν y dV theinduced volume form on ∂M . The L norm of a function is then givenby k u k L ( M ) = (cid:16) Z M | u | dV (cid:17) and the corresponding scalar product by( u | v ) = Z M u v dV. Similarly on the boundary, the norm and scalar products are given by k f k L ( ∂M ) = (cid:16) Z ∂M | f | dS (cid:17) ( f | h ) ∂M = Z ∂M f h dS. We write for short k∇ u k L ( M ) = (cid:13)(cid:13) |∇ u | (cid:13)(cid:13) L ( M ) = (cid:16) Z M |∇ u | dV (cid:17) and we denote by H ( M ) the semiclassical Sobolev space associatedto the norm k u k H ( M ) = (cid:0) k u k L ( M ) + k h ∇ u k L ( M ) (cid:1) . We assume that (
M, g ) is embedded in a compact manifold (
N, g ) with-out boundary, and that ϕ is a limiting Carleman weight on ( U, g ) where U is open in N and M ⋐ U . The goal of this section is to prove thefollowing Carleman estimate. Theorem 4.1.
Let ( U, g ) be an open Riemannian manifold and ( M, g ) a compact Riemannian submanifold with boundary such that M ⋐ U .Suppose that ϕ is a limiting Carleman weight on ( U, g ) . Let X bea smooth vector field on M and q a smooth function on M . Thereexist two constants C > and < h ≤ such that for all functions u ∈ C ∞ ( M ◦ ) and all < h ≤ h , one has the inequality k e ϕh u k H ( M ) ≤ Ch k e ϕh (∆ + X + q ) u k L ( M ) . (4.1)To lighten the notations we will forget the subscript L ( M ) wheneverit is not needed. Proof.
We first observe that the result is invariant under conformalchange of metrics since we have c n +24 (cid:0) ∆ g + X + q (cid:1) u = (∆ c − g + cX + q c ) (cid:0) c n − u (cid:1) with q c = cq − n − Xc + c n +24 ∆ g (cid:0) c − n − (cid:1) . Therefore if needed, we canassume the limiting Carleman weight to be a distance function by re-placing g by the conformal metric (2.5).Our next observation is that the estimate may be perturbed by zeroorder terms since this gives rise to an error of the form O ( h ) k e ϕ/h u k ,which may be absorbed in the left-hand side if h is assumed smallenough. Therefore we can neglect the potential q , and assume that q = 0 from the start. Let us first assume that we also have X = 0.Then the Carleman estimate (4.1) is equivalent to the following a priori estimate k v k H ( M ) ≤ C h − k P ,ϕ v k . (4.2)One goes from one inequality to another by taking v = e ϕ/h u . Theconjugated operator is given by P ,ϕ = − h ∆ − |∇ ϕ | + 2 h∇ ϕ, h ∇i + h ∆ ϕ. Then we have in particular k h ∇ v k = ( P ,ϕ v | v ) + (cid:13)(cid:13) |∇ ϕ | v (cid:13)(cid:13) − (cid:0) h∇ ϕ, h ∇ v i (cid:12)(cid:12) v (cid:1) − h (∆ ϕ v | v )therefore using Cauchy-Schwarz inequality one sees that k h ∇ v k ≤ k P ,ϕ v k + C k v k . (4.3)This means that the gradient of v may be controlled and that it sufficesto prove the a priori estimate k v k ≤ C h − k P ,ϕ v k IMITING CARLEMAN WEIGHTS 27 to obtain (4.2).We decompose P ,ϕ into its self-adjoint and skew-adjoint parts P ,ϕ = A + iB, A = − h ∆ − |∇ ϕ | , B = 2 h∇ ϕ, h ∇i + h ∆ ϕ and we have by integration by parts k P ,ϕ v k = k Av k + k Bv k + i ([ A, B ] v | v ) . (4.4)A direct application of the commutator method will not be enoughto get an a priori estimate assuming the bracket condition (2.3), oneneeds to use convexification. This classical argument consists in takinga modified weight f ◦ ϕ where f is a convex function chosen so thatthe bracket in (2.3) becomes positive. We decompose the operator P ,f ◦ ϕ = ˜ A + i ˜ B into its self-adjoint and skew-adjoint parts, and denoteby ˜ a and ˜ b the corresponding principal symbols. We now suppose, aswe may according to our first observation, that ϕ is both a limitingCarleman weight and a distance function. We have ∇ ( f ◦ ϕ ) = ( f ′ ◦ ϕ ) ∇ ϕD ( f ◦ ϕ ) = ( f ′′ ◦ ϕ ) dϕ ⊗ dϕ + ( f ′ ◦ ϕ ) D ϕ | {z } =0 therefore using Lemma 2.3 (cid:8) ˜ a, ˜ b } ( x, ξ ) = 4( f ′′ ◦ ϕ ) ( f ′ ◦ ϕ ) |∇ ϕ | + 4( f ′′ ◦ ϕ ) h∇ ϕ, ξ ♯ i = 4( f ′′ ◦ ϕ ) ( f ′ ◦ ϕ ) + ( f ′′ ◦ ϕ )( f ′ ◦ ϕ ) − | {z } = β ˜ b . At the operator level, this gives i [ ˜ A, ˜ B ] = 4 h ( f ′′ ◦ ϕ ) ( f ′ ◦ ϕ ) + h ˜ Bβ ˜ B + h R where R is a first order semiclassical differential operator. For thefunction f , we choose the following convex polynomial f ( s ) = s + h ε s , f ′ ( s ) = 1 + hε s, f ′′ ( s ) = hε . We choose h/ε ≤ ε < ε small enough so that f ′ > on ϕ ( M )and denote ϕ ε = f ◦ ϕ . Note that the coefficients of R , as well as β ,are uniformly bounded with respect to h and ε .We finally obtain i (cid:0) [ ˜ A, ˜ B ] v (cid:12)(cid:12) v (cid:1) ≥ h ε k v k − C h k ˜ Bv k − C h k v k H k v k and using (4.3) i (cid:0) [ ˜ A, ˜ B ] v (cid:12)(cid:12) v (cid:1) ≥ h ε (1 − C ε ) k v k − C h k ˜ Bv k − C k P ,ϕ ε v k . Going back to (4.4), this gives(1 + C ) k P ,ϕ ε v k ≥ k ˜ Av k + (1 − C h ) k ˜ Bv k (4.5) + h ε (1 − C ε ) k v k . If we don’t assume X = 0, the conjugated operator P ,ϕ ε has to beperturbed by an additional term of the form h X ϕ ε = h e ϕ ε /h Xe − ϕ ε /h = h X − hf ′ ◦ ϕXϕ. By (4.3) and the estimate k h X ϕ ε v k ≤ C h k v k H , the inequality (4.5)may easily be perturbed into2(1 + C ) k P ,ϕ ε v + h X ϕ ε v k ≥ h ε (1 − C ε ) k v k if h is small enough. Taking ε small enough, we obtain C k P ,ϕ ε v + h X ϕ ε v k ≥ h ε k v k (4.6)which implies with the choice of f = s + hs / ε that C ε k e ϕ ε e ϕh (∆ + X ) u k ≥ h k e ϕ ε e ϕh u k therefore we obtain the desired estimate since 1 ≤ e ϕ ε ≤ C ′ ε . (cid:3) Remark 4.2.
The use of G˚arding’s inequality could give a strongerCarleman estimate, as in [26]. The present proof makes it possible toinclude boundary terms, which is useful in the study of inverse problemswith partial data (see [13], [7] and [14]).In order to prove suitable solvability results, we need to shift theindices of the Sobolev spaces in the Carleman estimate by using pseu-dodifferential calculus. Recall that we assume that (
M, g ) is embeddedin a compact manifold (
N, g ) without boundary, and that ϕ is a limitingCarleman weight near ( ¯ U , g ) where U is open in N and M ⋐ U . TheLaplace-Beltrami operator − ∆ on N , with domain C ∞ ( N ) ⊂ L ( N ),is essentially self-adjoint with spectrum in [0 , ∞ ). By the spectral the-orem we may define for s ∈ R the semiclassical Bessel potentials J s = (1 − h ∆) s/ . IMITING CARLEMAN WEIGHTS 29
One has J s J t = J s + t , and J s commutes with any function of − ∆.Define for s ∈ R the semiclassical Sobolev spaces via k u k H s scl ( N ) = k J s u k L ( N ) , so H s scl ( N ) is the completion of C ∞ ( N ) in this norm. It is easy to seethat the dual of H s scl ( N ) may be isometrically identified with H − s scl ( N ).It is a basic fact that J s is a semiclassical pseudodifferential operatorof order s in N (see [34] and [6]). This implies pseudolocal estimates: if ψ, χ ∈ C ∞ ( N ) with χ = 1 near supp ψ , and if s, α, β ∈ R and K ∈ N ,then(4.7) k (1 − χ ) J s ψu k H α scl ( N ) ≤ C K h K k u k H β scl ( N ) . We will also use commutator estimates in the form(4.8) k [ A, J s ] u k L ( N ) ≤ Ch k u k H s scl ( N ) whenever A is a first order semiclassical differential operator in N . Lemma 4.3.
Under the above assumptions on
M, N, U and under theassumptions of Theorem 4.1, given s ∈ R there are two constants C s > and < h s ≤ such that for all functions u ∈ C ∞ ( M ◦ ) and all < h ≤ h s one has the inequality k e ϕh u k H s +1scl ( N ) ≤ C s h k e ϕh (∆ g + X + q ) u k H s ( N ) . (4.9) Proof.
We consider the conjugated operator P ϕ ε = e ϕεh h (∆ g + X + q ) e − ϕεh where ϕ ε is the weight defined in the proof of Theorem 4.1. Let χ ∈ C ∞ ( U ) with χ = 1 near M . Then the estimates (4.6), (4.7) imply h k u k H s +1scl ≤ h k χJ s u k H + h k (1 − χ ) J s u k H ≤ C √ ε k P ϕ ε ( χJ s u ) k L + C h k u k H s +1scl . By the estimate k [ P ϕ ε , χ ] J s u k L ≤ C h k u k H s +1scl , and by absorbing theerror terms k u k H s +1scl in the left hand-side if h is small enough, we obtain(4.10) h k u k H s +1scl ≤ C √ ε k J s P ϕ ε u k L + C √ ε k χ [ P ϕ ε , J s ] u k L . Note that in estimating the last term one may extend ϕ smoothlyoutside U if desired. Since P ϕ ε = − h ∆ − |∇ ϕ ε | + 2 h∇ ϕ ε , h ∇i + h ∆ ϕ ε + h X ϕ ε + h q and since [ − h ∆ , J s ] = 0, the commutator estimates (4.8) imply thatfor h ≪ ε ≪ ≤ e ϕ ε ≤ C ε . (cid:3) Proposition 4.4. If A is a smooth 1-form and q is a smooth functionon M , there exists a constant < h ≤ such that for any func-tion f ∈ L ( M ) there exists a solution u ∈ H ( M ) to the equation e ϕ/h L g,A,q e − ϕ/h u = f satisfying k u k H ( M ) ≤ Ch k f k L ( M ) . Proof.
We consider the conjugated operator P ∗ ϕ = e − ϕh h L g, ¯ A, ¯ q e ϕh . Let f ∈ L ( M ), we consider the subspace E = P ∗ ϕ ( C ∞ ( N )) of H − ( N )and the linear form defined on E by L ( P ∗ ϕ v ) = ( f | v ) L ( M ) , v ∈ C ∞ ( M ◦ ) . By Lemma 4.9 applied to P ∗ ϕ , this form is well defined and | L ( P ∗ ϕ v ) | . k f k L ( M ) h − k P ∗ ϕ v k H − ( N ) . By the Hahn-Banach theorem, there is an extension ˆ L of L which is abounded functional on H − ( N ) with norm less than h − k f k L . Sincethe dual of H − ( N ) is H ( N ), there exists a function ˜ u ∈ H ( N )such that ˆ L ( v ) = (˜ u | v ) and h k ˜ u k H ( N ) . k f k L ( M ) . Then u = ˜ u | M isthe desired solution, since for all v ∈ C ∞ ( M ◦ )( P ϕ u | v ) = ( u | P ∗ ϕ v ) = ˆ L ( P ∗ ϕ v ) = L ( P ∗ ϕ v ) = ( f | v ) . This completes the proof. (cid:3) Complex geometrical optics
Let ϕ be a limiting Carleman weight in an admissible manifold( M, g ). We will construct solutions to L g,q u = 0 in M of the form(5.1) u = e − h ( ϕ + iψ ) ( a + r ) . Here the real valued phase ψ and complex amplitude a are obtainedfrom a WKB construction, and the function r will be a correction termwhich is small when h is small.We write ρ = ϕ + iψ for the complex phase. It will be convenient toextend the notations h · , · i and | · | to complex tangent vectors by h ζ , η i = h Re ζ , Re η i − h Im ζ , Im η i + i ( h Re ζ , Im η i + h Im ζ , Re η i ) , | ζ | = h ζ , ζ i . IMITING CARLEMAN WEIGHTS 31
We make similar extensions for the inner product of cotangent vectors.With this notation, the conjugated operator P ρ = e ρ/h h L g,q e − ρ/h hasthe expression P ρ = −|∇ ρ | + h (2( ∇ ρ ) + ∆ ρ ) + h L g,q . Then (5.1) will be a solution of L g,q u = 0 provided that P ρ ( a + r ) = 0.Following the WKB method, this results in the three equations |∇ ρ | = 0 , (5.2) (2( ∇ ρ ) + ∆ ρ ) a = 0 , (5.3) P ρ r = − h L g,q a. (5.4)These equations will be solved in special coordinates in the admissiblemanifold ( M, g ). We know that (
M, g ) is conformally embedded in R × ( M , g ) for some compact simple ( n − M , g ).Assume, after replacing M with a slightly larger simple manifold ifnecessary, that for some simple ( D, g ) ⋐ (int M , g ) one has(5.5) ( M, g ) ⋐ ( R × int D, g ) ⋐ ( R × int M , g ) . Here R × M is covered by a global coordinate chart in which g hasthe form(5.6) g ( x ) = c ( x ) (cid:18) g ( x ′ ) (cid:19) , where c > g is simple. The limiting Carleman weight will be ϕ ( x ) = x .5.1. The eikonal equation.
Since ϕ was given, the eikonal equation(5.2) for the complex phase becomes a pair of equations for ψ , |∇ ψ | = |∇ ϕ | , h∇ ϕ, ∇ ψ i = 0 . One has ϕ ( x ) = x and the metric is of the form (5.6), so ∇ ϕ = c ∂∂x and |∇ ϕ | = c . The eikonal equation now reads |∇ ψ | = 1 c , ∂ x ψ = 0 . Under the given assumptions on (
M, g ), there is an explicit constructionfor ψ . Let ω ∈ D be a point such that ( x , ω ) / ∈ M for all x . Denotepoints of M by x = ( x , r, θ ) where ( r, θ ) are polar normal coordinatesin ( D, g ) with center ω . That is, x ′ = exp Dω ( rθ ) where r > θ ∈ S n − . In these coordinates (which depend on the choice of ω ) the metric has the form g ( x , r, θ ) = c ( x , r, θ ) m ( r, θ ) , where m is a smooth positive definite matrix.To solve the eikonal equation it is enough to take ψ ( x ) = ψ ω ( x ) = r .With this choice one has ρ = x + ir and ∇ ρ = c ∂ where ∂ = 12 (cid:16) ∂∂x + i ∂∂r (cid:17) . The transport equation.
We now consider (5.3). In the coor-dinates ( x , r, θ ) one has∆ ρ = | g | − / ∂ x (cid:16) | g | / c (cid:17) + | g | − / ∂ r (cid:16) | g | / c i (cid:17) = 1 c ∂ log | g | c and the transport equation reads4 ∂a + (cid:16) ∂ log | g | c (cid:17) a = 0 . We choose a as the function a = | g | − / c / a ( x , r ) b ( θ )where ∂a = 0 and b ( θ ) is smooth.5.3. Complex geometrical optics solutions.
Finally, the equation(5.4) may be written as e ϕ/h h L g,q e − ϕ/h ( e − iψ/h r ) = − e − iψ/h h L g,q a. This can be solved by using Lemma 4.4. We obtain r satisfying k r k H ( M ) . h. We record the properties of the solution just obtained.
Proposition 5.1.
Assume that ( M, g ) satisfies (5.5) , (5.6) and let q ∈ C ∞ ( M ) . Let ω ∈ D be such that ( x , ω ) / ∈ M for all x . If ( r, θ ) are polar normal coordinates in ( D, g ) with center ω , then, whenever ∂a ( x , r ) = 0 and b ( θ ) is smooth, the equation L g,q u = 0 in M has a solution of the form u = e − h ( x + ir ) (cid:0) | g | − / c / a ( x , r ) b ( θ ) + r (cid:1) where k r k H ( M ) . h . IMITING CARLEMAN WEIGHTS 33
Next we consider the case where a magnetic field is present. Theconstruction of complex geometric solutions is similar to the case with-out magnetic field, except that we have an additional factor in theamplitude.
Proposition 5.2.
Assume that ( M, g ) satisfies (5.5) , (5.6) , and let A be a smooth -form in M and q ∈ C ∞ ( M ) . Let ω ∈ D be such that ( x , ω ) / ∈ M for all x . If ( r, θ ) are polar normal coordinates in ( D, g ) with center ω , then, whenever ∂a = 0 and b is smooth, the equation L g,A,q u = 0 in M has a solution of the form u = e − h ( x + ir ) (cid:0) | g | − / c / e i Φ a ( x , r ) b ( θ ) + r (cid:1) where Φ satisfies (5.7) ∂ Φ + 12 ( A + iA r ) = 0 , and k r k H ( M ) . h .Proof. If P ρ = e ρ/h h L g,A,q e − ρ/h , one computes P ρ = −|∇ ρ | + h (cid:0) ∇ ρ + ∆ ρ + 2 i h dρ, A i (cid:1) + h L g,A,q . If ρ = x + ir then |∇ ρ | = 0, and the transport equation for theamplitude a will be4 ∂a + ∂ (cid:16) log | g | c (cid:17) a + 2 i ( A + iA r ) a = 0 . A solution is given by a = | g | − / c / e i Φ a ( x , r ) b ( θ ), where Φ is asolution of (5.7) and ∂a = 0. The equation for r becomes P ρ r = − h L g,A,q a, and Lemma 4.4 finishes the proof. (cid:3) Uniqueness results
Let (
M, g ) be an admissible manifold. In this section we will provethe global uniqueness results, Theorems 3 to 5. As often in inverseboundary problems, the starting point is an integral identity whichrelates the boundary measurements to solutions inside the manifold.
Lemma 6.1.
Let A , A be smooth -forms in M and q , q ∈ C ∞ ( M ) .If Λ g,A ,q = Λ g,A ,q , then (6.1) Z M (cid:2) i h A − A , udv − vdu i + (cid:0) | A | − | A | + q − q (cid:1) uv (cid:3) dV = 0 , for any u, v ∈ H ( M ) satisfying L g,A ,q u = 0 and L g, − A ,q v = 0 in M .Proof. One has Λ ∗ g,A,q = Λ g, ¯ A, ¯ q and(Λ g,A,q f | h ) ∂M = ( d A u | d ¯ A v ) + ( qu | v ) , whenever L g,A,q u = 0 in M and u | ∂M = f , v | ∂M = h . These facts implythe identity (cid:0) (Λ g,A ,q − Λ g,A ,q ) f | h (cid:1) ∂M = Z M (cid:2) i h A − A , ud ¯ v − ¯ vdu i + ( | A | − | A | + q − q ) u ¯ v (cid:3) dV where u, v are H ( M ) solutions of L g,A ,q u = 0 and L g, ¯ A , ¯ q v = 0 in M , which satisfy u | ∂M = f and v | ∂M = h . The result follows. (cid:3) Recovering a potential.
Suppose that we have two potentials q , q ∈ C ∞ ( M ) for which the corresponding DN maps are equal. Ac-cording to Lemma 6.1 one has Z M ( q − q ) u u dV = 0(6.2)for any u j ∈ H ( M ) which satisfy L g,q j u j = 0 in M . We use Proposition5.1 and choose solutions of the form u = e − h ( x + ir ) (cid:0) | g | − / c / e iλ ( x + ir ) b ( θ ) + r (cid:1) ,u = e h ( x + ir ) (cid:0) | g | − / c / + r (cid:1) , where λ is a real number and k r j k H ( M ) . h . Inserting these solutionsin (6.2) and letting h → Z R Z Z M x e iλ ( x + ir ) ( q − q ) c ( x , r, θ ) b ( θ ) dr dθ dx = 0 , with M x = { ( r, θ ) ; ( x , r, θ ) ∈ M } . We can extend q − q smoothlyby zero since q = q up to infinite order on ∂M by Theorem 8.4,and we may then assume that the integral is over R × D . Taking the x -integral inside and varying b gives Z e − λr (cid:16) Z ∞−∞ e iλx ( q − q ) c ( x , r, θ ) dx (cid:17) dr = 0 , for all θ. We denote the expression in parentheses by f ( r, θ ), and obtain that Z γ f ( γ ( r )) exp h − Z r λ ds i dr = 0 IMITING CARLEMAN WEIGHTS 35 for all D -geodesics γ issued from the point ω . Varying ω and using theinjectivity of the attenuated geodesic ray transform given in Theorem7.1, with constant attenuation − λ , we obtain Z ∞−∞ e iλx ( q − q ) c ( x , r, θ ) dx = 0 , for all r, θ, which holds for small enough λ . Since ( q − q ) c is compactly supportedin x , its Fourier transform is analytic, and we obtain that q = q . Thisproves Theorem 3.6.2. Recovering a magnetic field.
Next we show that also a mag-netic field can be recovered from the DN map. We will need the follow-ing standard reduction of the problem to a simply connected domain.
Proposition 6.2.
Let M , M be compact manifolds with boundary with M ⋐ M , and let ( g, A j , q j ) be smooth coefficients in M such that isnot a Dirichlet eigenvalue of L g,A j ,q j in M ( j = 1 , . Suppose that A = A , q = q in M \ M . If the DN maps Λ g,A j ,q j in M coincide, then the integral identity (6.1) is valid for any H ( M ) solutions of L g,A ,q u = 0 and L g, − A ,q v = 0 in M .Proof. Since the DN maps coincide in M , one has by Lemma 6.1 Z M (cid:2) i h A − A , u dv − v du i + ( | A | − | A | + q − q ) u v dV (cid:3) = 0 , for any H ( M ) solutions of L g,A ,q u = 0 and L g, − A ,q v = 0 in M .If u and v are as above, then the restrictions to M solve the corre-sponding equations in M , and we obtain (6.1) since the coefficientscoincide outside M . (cid:3) Proof of Theorem 4.
By boundary determination, Theorem 8.4, andafter a gauge transformation, we may extend A j and q j smoothly sothat A = A and q = q outside M . We are now in the setting ofProposition 6.2. Therefore, replacing M by a larger manifold inside R × M if necessary, we may assume that M is convex and A j and q j are compactly supported in M , and also that the identity in Lemma6.1 holds whenever u, v ∈ H ( M ) are solutions of L g,A ,q u = 0 and L g, − A ,q v = 0 in M .Use Proposition 5.2 to choose solutions of the form u = e − ρ/h (cid:0) | g | − / c / e i Φ a ( x , r ) b ( θ ) + r (cid:1) ,v = e ρ/h (cid:0) | g | − / c / e i Φ + r (cid:1) , where ρ = x + ir , ∂a = 0, and Φ = Φ + Φ is a solution of theequation(6.3) ∂ Φ + 12 ( ˜ A + i ˜ A r ) = 0 in M .
Here ˜ A = A − A , and ˜ A and ˜ A r are the components in the x and r coordinates. Inserting these solutions in (6.1), multiplying both sidesby h , and letting h → h → Z M h ˜ A, dρ i uv dV = 0 . Writing the integral in local coordinates gives Z ( ˜ A + i ˜ A r ) e i Φ a ( x , r ) b ( θ ) dx dr dθ = 0 . Varying b leads to Z Ω θ ( ˜ A + i ˜ A r ) e i Φ a ( x , r ) d ¯ ρ ∧ dρ = 0 , for all θ, where Ω θ = { ( x , r ) ∈ R ; ( x , r, θ ) ∈ M } is identified with a domainin the complex plane, with complex variable ρ . Integrating by partsand using (6.3) gives(6.4) Z ∂ Ω θ e i Φ a dρ = 0 . The arguments in [7, Section 5], see also [14, Section 7], then implythat e i Φ | ∂ Ω θ = F | ∂ Ω θ where F ∈ C (Ω θ ) is a nonvanishing holomorphicfunction, and F = e G where G ∈ C (Ω θ ) is holomorphic and G − i Φ isconstant on ∂ Ω θ . We choose a = Ge − G e iλ ( x + ir ) where λ is a real number, and then (6.3), (6.4), and integration byparts imply Z Ω θ ( ˜ A + i ˜ A r ) e iλ ( x + ir ) d ¯ ρ ∧ dρ = 0 , for all θ. We define f ( x ′ ) = Z e iλx ˜ A ( x , x ′ ) dx ,α ( x ′ ) = n X j =2 (cid:16) Z e iλx ˜ A j ( x , x ′ ) dx (cid:17) dx j . IMITING CARLEMAN WEIGHTS 37
The identity above may be written as Z e − λr (cid:2) f ( γ ( r )) + iα ( ˙ γ ( r )) (cid:3) dr = 0 , for all θ. The r -integrals are integrals over geodesics γ in D . By varying thepoint ω in Proposition 5.2 on ∂D and using Theorem 7.1, we see for λ small that f = − λp and α = − idp where p ∈ C ∞ ( D ) and p | ∂D = 0.The definition of α and analyticity of the Fourier transform imply that ∂ k ˜ A j − ∂ j ˜ A k = 0 , j, k = 2 , . . . , n. Also Z e iλx ( ∂ j ˜ A − ∂ ˜ A j )( x , x ′ ) dx = ∂ j f + iλα j = 0 , showing that d ˜ A = 0 in M and the magnetic fields coincide.Since M is convex and ˜ A | ∂M = 0, one has ˜ A = dp where p ∈ C ∞ ( M )can be chosen so that p | ∂M = 0. By a gauge transformation, we mayassume that A = A , and this 1-form will be denoted by A . Theintegral identity (6.1) becomes(6.5) Z M ( q − q ) uv dV = 0 , for any solutions of L g,A,q u = 0 and L g, − A,q v = 0. Use Proposition5.2 and choose solutions u = e − ρ/h (cid:0) | g | − / c / e i Φ e iλ ( x + ir ) b ( θ ) + r (cid:1) ,v = e ρ/h (cid:0) | g | − / c / e − i Φ + r (cid:1) , where ρ = x + ir , and Φ satisfies ∂ Φ + 12 ( A + iA r ) = 0 in M .
Letting h → Z e iλ ( x + ir ) ( q − q ) c ( x , r, θ ) b ( θ ) dx dr dθ = 0 . Proceeding as in the proof of Theorem 3 shows that q = q . (cid:3) Recovering a conformal factor.
The results on the Schr¨odingerinverse problem can be used to recover a conformal factor from the DNmap. Recall that we use the notation Λ g = Λ g, when the potential q is zero. Proof of Theorem 5.
It is enough to show that if (
M, g ) is admissibleand c is smooth and positive, and if Λ cg = Λ g , then c = 1. We have c | ∂M = 1 and ∂ ν c | ∂M = 0 by Proposition 8.1, and then the assumptionand Proposition 8.2 implyΛ cg, = Λ g, = Λ cg,q , where q = − ∆ g ( c n − ) /c n +24 . We conclude from Theorem 3 that q = 0,so ∆ g ( c n − ) = 0 in M . Since c n − = 1 on ∂M , uniqueness of solutionsfor the Dirichlet problem shows that c ≡ (cid:3) Attenuated ray transform
For the uniqueness results in inverse problems, we have used thatcertain geodesic ray transforms are injective. If (
M, g ) is a compactmanifold with smooth boundary, geodesics can be parametrized bypoints on the unit sphere bundle SM = { ( x, ξ ) ∈ T M ; | ξ | = 1 } . For( x, ξ ) ∈ SM let γ x,ξ ( t ) be the geodesic with γ (0) = x and ˙ γ (0) = ξ . Weassume that ( M, g ) is nontrapping, which means that the time τ ( x, ξ )when γ x,ξ exits M is always finite.Given a smooth real function a on M , the attenuated geodesic raytransform of a function f is given by I a f ( x, ξ ) = Z τ ( x,ξ )0 f ( γ x,ξ ( t )) exp h Z t a ( γ x,ξ ( s )) ds i dt for ( x, ξ ) ∈ ∂ + SM . Here we use the sets of inward and outward pointingunit vectors ∂ ± S ( M ) = { ( x, ξ ) ∈ SM ; x ∈ ∂M, ±h ξ, ν ( x ) i < } , and ν is the outer unit normal vector to ∂M .We will also need to integrate 1-forms over geodesics. Let f be asmooth function and α = α i dx i a smooth 1-form on M , and consider(7.1) F ( x, ξ ) = f ( x ) + α i ( x ) ξ i for ( x, ξ ) ∈ SM . The attenuated geodesic ray transform of F is I a F ( x, ξ ) = Z τ ( x,ξ )0 F ( γ x,ξ ( t ) , ˙ γ x,ξ ( t )) exp h Z t a ( γ x,ξ ( s )) ds i dt. This transform always has a kernel: if p is a smooth function on M with p | ∂M = 0, then a direct computation shows that I a ( ap + dp ( ξ )) = 0 . The main result in this section states that for simple manifolds andsmall attenuation, this is the only obstruction to injectivity.
IMITING CARLEMAN WEIGHTS 39
Theorem 7.1.
Let ( M, g ) be a compact simple manifold with smoothboundary. There exists ε > such that the following assertion holdsfor any smooth real function a on M satisfying | a | ≤ ε : If F is givenby (7.1) and if I a F ( x, ξ ) = 0 for all ( x, ξ ) ∈ ∂ + SM , then F = ap + dp ( ξ ) for some smooth function p on M with p | ∂M = 0 . Note that if α = 0, this shows that any function f whose attenuatedray transform vanishes must be identically zero. If f = 0 and a = 0everywhere, then any 1-form whose attenuated ray transform vanishesmust be identically zero. Injectivity of the geodesic ray transform forfunctions and 1-forms on simple manifolds in the case a = 0 is wellknown [1], [20], [27]. The injectivity for functions and small a is provedin [27], [28] under conditions which involve a modified Jacobi equationor the size and curvature of M . We give a proof which works on simplemanifolds.We remark that the notation in this section is somewhat differentfrom the other sections. For instance, we will denote by ∇ the covariantderivative and more generally the horizontal derivative. The notationwill be explained in more detail below.7.1. Preliminaries.
The proof will be based on energy estimates anda Pestov identity, which is the standard approach to such problems.First we need to recall the definition of horizontal (or semibasic) ten-sor fields on
T M . These are ( p, q ) tensor fields on
T M which havecoordinate representations u = ( u j ··· j q i ··· i p ) = u j ··· j q i ··· i p ∂∂ξ j ⊗ · · · ⊗ ∂∂ξ j q ⊗ dx i ⊗ · · · ⊗ dx i p with respect to coordinates ( x, ξ ) on T M associated to charts x of M .The components transform in the same way as tensors on M underchanges of charts. Tensor fields on M may be considered as ξ -constanthorizontal tensor fields on T M . See [27] for an invariant definition andother details on horizontal tensor fields.For our purposes, it is sufficient to know that smooth functions on
T M are horizontal tensors of degree (0 , h ∇ u ) j ··· j q i ··· i p i = h ∇ i u j ··· j q i ··· i p := ˜ ∇ i u j ··· j q i ··· i p − Γ lik ξ k ∂∂ξ l u j ··· j q i ··· i p , ( v ∇ u ) j ··· j q i ··· i p i = v ∇ i u j ··· j q i ··· i p := ∂∂ξ i u j ··· j q i ··· i p , are invariantly defined operators which map horizontal ( p, q ) tensors tohorizontal ( p + 1 , q ) tensors. Here˜ ∇ i u j ··· j q i ··· i p = ∂∂x i u j ··· j q i ··· i p + q X r =1 u j ··· s ··· j q i ··· i p Γ j r is − p X r =1 u j ··· j q i ··· s ··· i p Γ sii r . Thus ˜ ∇ acts in the same way as the usual covariant derivative.Below, we will work with smooth functions and tensors on SM . Let p : T M \ { } → SM be the map ( x, ξ ) ( x, ξ/ | ξ | ). The horizontaland vertical derivatives on SM are defined by ∇ u = h ∇ ( u ◦ p ) | SM , ∂u = v ∇ ( u ◦ p ) | SM . Also ∇ i u = h ∇ i ( u ◦ p ) | SM and ∂ i u = v ∇ i ( u ◦ p ) | SM . If u is horizontaltensor field on SM , then ∇ u and ∂u are also horizontal tensor fields.For a smooth function on SM , these derivatives are given by ∇ i u = ∂∂x i ( u ( x, ξ/ | ξ | )) − Γ lik ξ k ∂ l u,∂ i u = ∂∂ξ i ( u ( x, ξ/ | ξ | )) . We mention the following formulas ∇ g = 0 , ∇ ξ = 0 , ∂ j ξ i = δ ij − ξ i ξ j , [ ∇ i , ∂ j ] = 0 , [ ∂ i , ∂ j ] = ξ i ∂ j − ξ j ∂ i , [ ∇ i , ∇ j ] u = − R ijkl ξ k ∂ l u, where R is the curvature tensor and u is a scalar function. We write ∂ i u = g ij ∂ j u, h ∂u, ∂v i = ∂ i u∂ i v, | ∂u | = ∂ i u∂ i u. Let H be the geodesic vector field on SM that generates geodesicflow. In local coordinates Hu ( x, ξ ) = ξ i ∂∂x i ( u ( x, ξ/ | ξ | )) − Γ lik ( x ) ξ i ξ k ∂ l u, ξ ∈ S x , where u is a smooth function on SM . We may apply the operator H to horizontal tensor fields on SM by defining Hu = ξ i ∇ i u .7.2. Ray transform of functions.
We now consider the boundaryvalue problem for transport equation: Hu + au = − f, u | ∂ − S ( M ) = 0 , IMITING CARLEMAN WEIGHTS 41 where the absorption a and f are smooth functions on ( M, g ). Thesolution is given by u fa ( x, ξ ) = Z τ ( x,ξ )0 f ( γ x,ξ ( t )) exp h Z t a ( γ x,ξ ( s )) ds i dt, The trace I a f = u fa | ∂ + S ( M ) is the attenuated geodesic ray transform of the function f . It is naturalto define τ | ∂ − SM = 0, and then u fa indeed vanishes on ∂ − SM .We will prove that f is uniquely determined by I a f under the follow-ing assumption. Let γ = γ x,ξ , ( x, ξ ) ∈ ∂ + SM , be an arbitrary geodesic,and consider the quadratic form(7.2) E aγ ( X ) = Z τ (cid:0) | DX | − h R γ X, X i − a | X | (cid:1) ( t ) dt where τ = τ ( x, ξ ), X ( t ) is a vector field on γ belonging to the space H ( γ ) = { X ∈ H ([0 , τ ]; T ( γ )) ; X (0) = X ( τ ) = 0 } ,D is the covariant derivative along γ , and R γ X = R ( X, ˙ γ ) ˙ γ . We assumefor all geodesics γ the positive definiteness of this quadratic form, ( E aγ ( X ) ≥ X ∈ H ( γ ) ,E aγ ( X ) = 0 iff X = 0 . (7.3)If (7.3) holds we say that any geodesic has no conjugate points withrespect to (7.2). If a = 0 we obtain the usual index form E γ = E γ .Then clearly there are no conjugate points in the usual sense if thereare none with respect to (7.2). Proposition 7.2.
Let ( M, g ) be compact and ∂M strictly convex. As-sume that any geodesic has no conjugate points with respect to (7.2) .Then any smooth function f on the manifold ( M, g ) is uniquely deter-mined by its attenuated geodesic X-ray transform.Proof. Let I a f = 0. We will assume that u = u fa is smooth on SM (otherwise one can work in a slightly smaller manifold than M , andpass to the limit using the smoothness properties of τ as in [27]). Thefunction u satisfies ∂Hu + a∂u = 0and therefore(7.4) | ∂Hu | = a | ∂u | . Using the formulas for ∇ and ∂ , it is not difficult to check the followingidentity (valid for any u ∈ C ∞ ( SM )):(7.5) | ∂Hu | = | H∂u | + δV + θW − R ( ∂u, ξ, ξ, ∂u ) , where δ and θ are the vertical and horizontal divergences, δX = ∇ i X i , θX = ∂ i X i , and V and W are defined by V i = h ∂u, ∇ u i ξ i − ( Hu ) ∂ i u, W i = ( Hu ) ∇ i u. From (7.4) and (7.5) we obtain | H∂u | − R ( ∂u, ξ, ξ, ∂u ) − a | ∂u | + δV + θW = 0 . Now integrate this equality over the manifold SM . Before this we recallthe integration formulas (see [27]): Z SM v d ( SM ) = Z M dM Z S x v dS x , Z S x θX dS x = ( n − Z S x h X, ξ i dS x , ∇ Z S x u dS x = Z S x ∇ u dS x , where v is a scalar, X is a horizontal vector field and u is horizontaltensor field, Z M δX dM = Z ∂M h X, ν i d ( ∂M ) , where X is a vector field on ( M, g ). In these formulas the volume formsof corresponding manifolds are naturally defined using the metric g .After integration we have(7.6) Z SM (cid:0) | H∂u | − R ( ∂u, ξ, ξ, ∂u ) − a | ∂u | (cid:1) d ( SM )+ ( n − Z SM | Hu | d ( SM ) = 0 . We used the fact that h V, ν i vanishes on ∂ ( SM ) since u | ∂ ( SM ) = 0.We next show that (7.3) implies E a ( Y ) = Z SM (cid:0) | HY | − R ( Y, ξ, ξ, Y ) − a | Y | (cid:1) d ( SM ) ≥ , for any horizontal vector field Y ∈ C ∞ ( SM ; T M ) with Y | ∂ ( SM ) = 0,and that the equality holds iff Y = 0. Santal´o’s formula (see [27]) IMITING CARLEMAN WEIGHTS 43 states Z SM v d ( SM ) = − Z ∂ + S ( M ) Z τ ( x,ξ )0 v ( γ x,ξ ( t ) , ˙ γ x,ξ ( t )) h ξ, ν i dt d ( ∂ ( SM )) , for v ∈ C ∞ ( SM ). Let X ( x, ξ, t ) = Y ( γ x,ξ ( t ) , ˙ γ x,ξ ( t )) , which implies HY ( γ x,ξ ( t ) , ˙ γ x,ξ ( t )) = DX ( x, ξ, t ). Then we have E a ( Y ) = − Z ∂ + S ( M ) E aγ x,ξ ( X )( x, ξ ) h ξ, ν ( x ) i d ( ∂ ( SM )) ≥ . Equality holds iff Y = 0. We have from (7.6) that Hu = 0, whichimplies u = 0 and f = 0. (cid:3) Ray transform of -forms. Let f be a smooth function and α = α i ( x ) dx i a smooth 1-form in M , and consider the boundary valueproblem Hu + au = − F, u | ∂ − SM = 0 , where a is a smooth function on M and F is as in (7.1). The solution u = u Fa is given by u Fa ( x, ξ ) = Z τ ( x,ξ )0 F ( γ x,ξ ( t ) , ˙ γ x,ξ ( t )) exp h Z t a ( γ x,ξ ( s )) ds i dt, and the trace I a F = u Fa | ∂ + SM is the attenuated geodesic X-ray transform of F . Proposition 7.3.
Let ( M, g ) be compact and ∂M strictly convex, andsuppose that any geodesic has no conjugate points with respect to (7.2) .If I a F = 0 , then F = ap + dp ( ξ ) for some smooth function p on M which vanishes on ∂M .Proof. We follow the proof of Proposition 7.2. Again assume that u issmooth in SM . Then u satisfies ∂Hu + a∂u = − ∂F, and | ∂Hu | = a | ∂u | + 2 a h ∂u, ∂F i + | ∂F | . The identity (7.5) then implies | H∂u | − R ( ∂u, ξ, ξ, ∂u ) − a | ∂u | + δV + θW − a h ∂u, ∂F i − | ∂F | = 0 . Integrating this over SM , it follows that(7.7) E a ( ∂u ) + Z SM (cid:2) ( n − | Hu | − a h ∂u, ∂F i − | ∂F | (cid:3) d ( SM ) = 0 . Using the integration formula Z S x ∂ i v dS x = ( n − Z S x vξ i dS x and the identities | Hu | = ( au ) + 2 auF + F ,∂ i ∂ i F = − ( n − α i ξ i , we easily obtain from (7.7) that E a ( ∂u ) + ( n − Z SM ( au + f ) d ( SM ) = 0 . The assumption (7.3) and Santal´o’s formula imply E a ( ∂u ) ≥
0, sofrom the last equality we obtain E a ( ∂u ) = 0 and au + f = 0. Now E a ( ∂u ) = 0 implies ∂u = 0, so u = u ( x ) and h du, ξ i + au = − f − h α, ξ i , x ∈ M, ξ ∈ S x . The claim follows. (cid:3)
It remains to show that simple manifolds satisfy the condition ofProposition 7.3.
Proof of Theorem 7.1.
Let (
M, g ) be a compact simple manifold withsmooth boundary. We need to show that there is ε > E γ x,ξ ( X ) ≥ ε Z τ ( x,ξ )0 | X | dt, for all ( x, ξ ) ∈ ∂ + SM and for all X ∈ H ( γ x,ξ ). If (7.8) holds and | a | < √ ε then any geodesic on ( M, g ) has no conjugate points withrespect to (7.2), and Proposition 7.3 implies the desired result.Let first ( x, ξ ) ∈ ∂ + SM , and consider the unbounded operator on L ( γ x,ξ ) with domain H ∩ H ( γ x,ξ ), given by L γ x,ξ : X
7→ − D X − R ( X, ˙ γ x,ξ ) ˙ γ x,ξ . This operator is self-adjoint and has discrete spectrum, which lies in(0 , ∞ ) since the corresponding quadratic form E γ x,ξ is positive definite.Therefore E γ x,ξ ( X ) ≥ λ ( x, ξ ) Z τ ( x,ξ )0 | X | dt, X ∈ H ( γ x,ξ ) , where λ ( x, ξ ) > L depend smoothly on ( x, ξ ) and τ is smooth and positive in ∂ + SM , soit is not hard to see that (7.8) holds in a neighborhood of any fixedpoint in ∂ + SM . IMITING CARLEMAN WEIGHTS 45
For tangential directions we use the Poincar´e inequality Z L | u ( t ) | dt ≤ L π Z L | ˙ u ( t ) | dt, u ∈ H ([0 , L ]) , where the constant L /π is optimal [23]. If ( x, ξ ) ∈ S ( ∂M ) and δ > τ there is a neighborhood U of ( x, ξ ) in ∂ + SM such that τ ( y, η ) ≤ δ in that neighborhood. Choosing δ small enough,the Poincar´e inequality implies E γ y,η ( X ) = Z τ ( y,η )0 ( | DX | − h R γ y,η X, X i ) dt ≥ Z τ ( y,η )0 (cid:16) π τ ( y, η ) | X | − h R γ y,η X, X i (cid:17) dt ≥ π δ Z τ ( y,η )0 | X | dt whenever ( y, η ) ∈ U and X ∈ H ( γ y,η ). This shows (7.8) near anypoint of S ( ∂M ). It follows that for some ε >
0, (7.8) holds on thecompact set ∂ + SM . (cid:3) Boundary determination
To deal with the inverse problems we are interested in, we need touse the fact that the DN map determines the Taylor expansions atthe boundary of the different quantities involved. In the case of theLaplace-Beltrami operator, the relevant result is proved in [18] and isas follows.
Proposition 8.1.
Let ( M, g ) and ( M, g ) be compact manifolds withsmooth boundary, with dimension n ≥ . If Λ g = Λ g , then the Taylorseries of g and g in boundary normal coordinates are equal at eachpoint on the boundary. In this section we extend the previous result to the case where electricand magnetic potentials are present. First we need to consider thegauge invariance of the DN map.
Proposition 8.2.
Let ( M, g ) be a compact manifold with boundary,and let A be a smooth 1-form and q a smooth function on M . If c and ψ are smooth functions such that c > , c | ∂M = 1 , ∂ ν c | ∂M = 0 , ψ | ∂M = 0 , then we have Λ g,A,q = Λ c − g,A + dψ,c ( q − q c ) where q c = c n − ∆ g (cid:0) c − n − (cid:1) . Proof.
It follows from a direct computation that c n +24 L g,A,q ( c − n − u ) = L c − g,A,c ( q − q c ) u,e − iψ L g,A,q ( e iψ u ) = L g,A + dψ,q u. Let f ∈ C ∞ ( ∂M ), and let u be the solution of L g,A,q u = 0 in M whichsatisfies u | ∂M = f . If(˜ g, ˜ A, ˜ q ) = ( c − g, A + dψ, c ( q − q c ))and if ˜ u = c n − e − iψ u , we have L ˜ g, ˜ A, ˜ q ˜ u = 0 in M and ˜ u | ∂M = f . ThenΛ ˜ g, ˜ A, ˜ q f = d ˜ A ˜ u ( ν ˜ g ) | ∂M = d A + dψ ( e − iψ u )( ν g ) | ∂M = Λ g,A,q f, by using the boundary values of c and ψ and the fact that ν ˜ g = ν g . (cid:3) Remark 8.3.
The conformal gauge invariance is related to the factthat the conformal Laplace-Beltrami operator ˜∆ g on ( M, g ), definedby ˜∆ g = ∆ g − n − n −
1) Scal g , transforms under a conformal change of metrics by˜∆ cg u = c − n +24 ˜∆ g (cid:0) c n − u (cid:1) . We use the notation f ≃ f to denote that f and f have the sameTaylor series. Our main boundary determination result is as follows. Theorem 8.4.
Let ( M, g ) and ( M, g ) be compact manifolds withboundary, of dimension n ≥ , and let A , A be two smooth -formsand q , q two smooth functions in M . If Λ g ,A ,q = Λ g ,A ,q and if p ∈ ∂M , then there exist smooth positive functions c j with c j | ∂M = 1 and ∂ ν c j | ∂M = 0 , and smooth functions ψ j with ψ j | ∂M = 0 , such that the gauge trans-formed coefficients (˜ g j , ˜ A j , ˜ q j ) = (cid:0) c − j g j , A j + dψ j , c j ( q j − d − j ∆ g j d j ) (cid:1) with d j = c − n − j , satisfy in boundary normal coordinates at p ˜ g ≃ ˜ g , ˜ A ≃ ˜ A , ˜ q ≃ ˜ q . Furthermore, if g ≃ g in boundary normal coordinates on all of ∂M ,then ˜ A ≃ ˜ A and q ≃ q on ∂M . IMITING CARLEMAN WEIGHTS 47
Boundary normal coordinates ( x ′ , x n ) at a boundary point p are suchthat x ′ parametrizes ∂M near p and x n is the distance to the boundaryalong unit speed geodesics normal to ∂M . See [18] for more details.We prove Theorem 8.4 by showing that Λ g,A,q is a pseudodifferentialoperator whose symbol determines the boundary values of the coeffi-cients. This method was used in [33] for the conductivity equation,in [18] for the Laplace-Beltrami operator, and in [22] for the magneticSchr¨odinger operator with Euclidean metric. We follow the proof in[18].The first issue to consider is the gauge invariance in the operatorΛ g,A,q . This will be dealt with by normalizing the coefficients ( g, A, q )in a way which is suitable for boundary determination. Fix a point p ∈ ∂M and boundary normal coordinates ( x ′ , x n ) near p . In thesecoordinates ∂M corresponds to { x n = 0 } and g = g αβ dx α ⊗ dx β + dx n ⊗ dx n . We use the convention that Greek indices run from 1 to n − n . Lemma 8.5.
Let ( g, A, q ) be smooth coefficients in M and p ∈ ∂M .There exist a positive smooth function c with c | ∂M = 1 and ∂ ν c | ∂M = 0 ,and a smooth function ψ with ψ | ∂M = 0 , such that in boundary normalcoordinates near p the quantities ˜ g = c − g and ˜ A = A + dψ satisfy ∂ jn ˜ A n ( x ′ ,
0) = 0 , j ≥ , (8.1) ∂ jn (˜ g αβ ∂ n ˜ g αβ )( x ′ ,
0) = 0 , j ≥ . (8.2) Proof.
One can find a smooth function ψ near p with ψ ( x ′ ,
0) = 0 and ∂ j +1 n ψ ( x ′ ,
0) = − ∂ jn A n ( x ′ ,
0) for j ≥
0. Extending this in a suitableway, one obtains ψ ∈ C ∞ ( M ) with ψ | ∂M = 0 such that ˜ A = A + dψ will satisfy (8.1).Further, we construct a smooth function c near p which satisfies c ( x ′ ,
0) = 1, ∂ n c ( x ′ ,
0) = 0, and ∂ jn (log det( cg αβ ))( x ′ ,
0) = 0 , j ≥ . In fact, one may take c = e µ where µ ( x ′ ,
0) = ∂ n µ ( x ′ ,
0) = 0 and ∂ jn µ ( x ′ ,
0) = − n − ∂ jn (log det( g αβ ))( x ′ ,
0) for j ≥
2. There is an ex-tension of c to a positive function c ∈ C ∞ ( M ) with c | ∂M = 1 and ∂ ν c | ∂M = 0. Since ∂ n (log det(˜ g αβ )) = ˜ g αβ ∂ n ˜ g αβ , one also has the second condition (8.2). (cid:3) Replacing coefficients ( g, A, q ) by the gauge equivalent coefficients( c − g, A + dψ, c ( q − d − ∆ g d )) does not affect the DN map. Therefore,below we will assume that (8.1) and (8.2) are valid. Note that boundarynormal coordinates for g are also boundary normal coordinates for anyconformal multiple of g , if the conformal factor is 1 on ∂M .The next aim is to write Λ g,A,q as a pseudodifferential operator andto compute the symbol in a small neighborhood of p . Here we use theusual (not semiclassical) pseudodifferential calculus and left quantiza-tion, so that a symbol r ( x, ξ ) in T ∗ R n corresponds to the operator Rf ( x ) = (2 π ) − n Z R n Z R n e i ( x − y ) · ξ r ( x, ξ ) f ( y ) dy dξ. We denote by p ∼ P p j the asymptotic sum of symbols, see [10] forthese basic facts. Lemma 8.6. Λ g,A,q is a pseudodifferential operator of order on ∂M .Its full symbol (in left quantization) in boundary normal coordinatesnear p is − b ∼ − P j ≤ b j , where b j are given in (8.5) – (8.8) .Proof. In the x coordinates, one has L g,A,q = − ∆ g + 2 g jk A j D k + G, where G = | g | − / D j ( | g | / g jk A k ) + g jk A j A k + q and | g | = det( g jk ) = det( g αβ ) . From (8.1) we know that ∂ K A n ( x ′ ,
0) = 0 for all multi-indices K ∈ N n .One would like to have A n = 0 also inside M . To achieve this, weintroduce as in [22] the conjugated operator M = e − ih L g,A,q e ih , where h ( x ) = − R x n A n ( x ′ , s ) ds . Note that ∂ K h ( x ′ ,
0) = 0. Writing˜ A j = A j + ∂ j h , we obtain ˜ A n = 0 and M = − ∆ g + 2 g αβ ˜ A α D β + ˜ G, where ˜ G = | g | − / D α ( | g | / g αβ ˜ A β ) + g αβ ˜ A α ˜ A β + q . We then have M = D n + iE ( x ) D n + Q ( x, D x ′ ) + Q ( x, D x ′ ) + 2 g αβ ˜ A α D β + ˜ G, with E , Q , and Q given by E ( x ) = 12 g αβ ∂ n g αβ ,Q ( x, D x ′ ) = g αβ D α D β ,Q ( x, D x ′ ) = − i ( 12 g αβ ∂ α (log | g | ) + ∂ α g αβ ) D β . IMITING CARLEMAN WEIGHTS 49
As in [18], one would like to have a factorization(8.3) M = ( D n + iE ( x ) − iB ( x, D x ′ ))( D n + iB ( x, D x ′ ))modulo a smoothing operator , where B is a pseudodifferential operator of order 1 with symbol b ( x, ξ ′ ).Using left quantization for symbols, (8.3) implies on the level of symbolsthat(8.4) ∂ n b − E ( x ) b + X K ∂ Kξ ′ bD Kx ′ bK ! = q + q +2 g αβ ˜ A α ξ β + ˜ G mod S −∞ . Let b ∼ P j ≤ b j where b j ( x, ξ ′ ) is homogeneous of order j in ξ ′ . Insert-ing this in (8.4) and collecting terms with the same order of homogene-ity, one obtains b j as follows: b = −√ q , (8.5) b = 12 b ( − ∂ n b + Eb − ∇ ξ ′ b · D x ′ b + q + 2 g αβ ˜ A α ξ β ) , (8.6) b − = 12 b ( − ∂ n b + Eb − X ≤ j,k ≤ j + k = | K | ∂ Kξ ′ b j D Kx ′ b k K ! + ˜ G ) , (8.7) b m − = 12 b ( − ∂ n b m + Eb m − X m ≤ j,k ≤ j + k −| K | = m ∂ Kξ ′ b j D Kx ′ b k K ! ) ( m ≤ − . (8.8)With these choices, b ∈ S and (8.3) is valid. By the argument in [18,Proposition 1.2], one hasΛ g,A,q f ( x ′ ) = − B ( x ′ , , D x ′ ) f ( x ′ ) + Rf ( x ′ )where R is a smoothing operator. (cid:3) From the symbol of the DN map, one can recover the following in-formation on the coefficients.
Lemma 8.7.
The knowledge of Λ g,A,q determines on { x n = 0 } thequantities (8.9) g αβ , ∂ n g αβ , ∂ K A α , ∂ K l αβ . Here K ∈ N n is any multi-index and l αβ = 14 ∂ n k αβ + qg αβ , with k αβ = ∂ n g αβ − ( g γδ ∂ n g γδ ) g αβ . Proof.
By Lemma 8.6, Λ g,A,q determines b | x n =0 and each b j | x n =0 . From b | x n =0 one recovers g αβ | x n =0 . If T ( g αβ ) denotes any linear combinationof tangential derivatives of g αβ , one has for x n = 0 b = 12 b ( − ∂ n b + Eb + 2 g αβ ˜ A α ξ β ) + T ( g αβ )= − ∂ n q q + 12 E − √ q g αβ ˜ A α ξ β + T ( g αβ )= − k αβ ω α ω β − g αβ ˜ A α ω β + T ( g αβ ) , where ω = ξ ′ / | ξ ′ | g . Evaluating at ± ω and varying ω one recovers ˜ A α and k αβ , and consequently also ∂ n g αβ , for x n = 0.Moving to b − , one has for x n = 0 b − = 12 b ( − ∂ n b + Eb + ˜ G ) + T ( g αβ , ∂ n g αβ , ˜ A α )= 12 b ( l αβ ω α ω β + g αβ ( ∂ n ˜ A α ) ω β ) + T ( g αβ , ∂ n g αβ , ˜ A α )where T denotes tangential derivatives of the given quantities. Thus,one recovers l αβ and ∂ n ˜ A α on { x n = 0 } . By induction, we prove thatfor j ≥ b − j = − (cid:16) − b (cid:17) j (( ∂ j − n l αβ ) ω α ω β + g αβ ( ∂ jn ˜ A α ) ω β )+ T ( g αβ , ∂ n g αβ , l αβ , . . . , ∂ j − n l αβ , ˜ A α , . . . , ∂ j − n ˜ A α ) . Indeed, this is true for j = 1, and assuming this for j one gets b − j − = − b ∂ n b − j + T ( g αβ , ∂ n g αβ , l αβ , . . . , ∂ j − n l αβ , ˜ A α , . . . , ∂ jn ˜ A α )= − (cid:16) − b (cid:17) j +1 (( ∂ jn l αβ ) ω α ω β + g αβ ( ∂ j +1 n ˜ A α ) ω β ) + T ( · ) . Thus one recovers ∂ jn l αβ and ∂ jn ˜ A α on { x n = 0 } for all j ≥
0. Theresult follows since ∂ K ˜ A α = ∂ K A α when x n = 0. (cid:3) We may now prove the main result on boundary determination.
Proof of Theorem 8.4.
Let Λ g,A,q = Λ ˜ g, ˜ A, ˜ q . Replacing both sets of co-efficients by gauge equivalent ones as discussed after Lemma 8.5, wemay assume that (8.1) and (8.2) are valid. Then Lemma 8.7 impliesthat the quantities (8.9) with and without tildes coincide on { x n = 0 } .We prove by induction that for j ≥
0, one has on x n = 0(8.10) ∂ jn q = ∂ jn ˜ q, ∂ j +2 n g αβ = ∂ j +2 n ˜ g αβ , g αβ ∂ j +3 n g αβ = ˜ g αβ ∂ j +3 n ˜ g αβ . IMITING CARLEMAN WEIGHTS 51
We first note that(8.11) g αβ ∂ n g αβ = ˜ g αβ ∂ n ˜ g αβ on x n = 0 . This follows from (8.2) for g and ˜ g , since ∂ jn g αβ = ∂ jn ˜ g αβ on x n = 0 for j = 0 ,
1. Note also that (8.2) implies ∂ jn k αβ = ∂ j +1 n g αβ − ( g γδ ∂ n g γδ ) ∂ jn g αβ on x n = 0 , and therefore ∂ jn l αβ = 14 ( ∂ j +2 n g αβ − ( g γδ ∂ n g γδ ) ∂ j +1 n g αβ ) + ∂ jn ( qg αβ ) on x n = 0 . To prove (8.10) for j = 0, we use that l αβ = ˜ l αβ on x n = 0. Thisimplies, upon multiplying by g αβ and summing, that q = ˜ q on x n = 0.Here we used (8.11). Then l αβ = ˜ l αβ also implies ∂ n g αβ = ∂ n ˜ g αβ on x n = 0. The equality g αβ ∂ n g αβ = ˜ g αβ ∂ n ˜ g αβ follows by using (8.2).Assume now that (8.10) holds for j ≤ k . Moving to k + 1, theequality ∂ k +1 n l αβ = ∂ k +1 n ˜ l αβ on x n = 0 implies upon multiplying by g αβ ,summing, and using the induction hypothesis, that g αβ ∂ k +1 n ( qg αβ ) = ˜ g αβ ∂ k +1 n (˜ q ˜ g αβ ) on x n = 0 . The induction hypothesis again gives ∂ k +1 n q = ∂ k +1 n ˜ q on x n = 0, and ∂ k +3 n g αβ = ∂ k +3 n ˜ g αβ then follows by going back to the equality ∂ k +1 n l αβ = ∂ k +1 n ˜ l αβ on x n = 0. The last statement in (8.10) for j = k + 1 is aconsequence of (8.2). This ends the induction.The outcome of the above argument is that g αβ ≃ ˜ g αβ , A α ≃ ˜ A α ,and q ≃ ˜ q at p . This shows the first statement in Theorem 8.4. If g ≃ ˜ g at each p , then it is easy to obtain q ≃ ˜ q at each p from l αβ ≃ ˜ l αβ .Also, the function ψ constructed locally in Lemma 8.5 can be obtainedglobally on ∂M by a suitable partition of unity. Therefore A ≃ ˜ A onall of ∂M . (cid:3) Appendix A. Riemannian geometry
In this appendix we include basic definitions and facts which are usedthroughout the text. For more details see [12]. We are using Einstein’ssummation convention: repeated indices in lower and upper positionare summed. In the following (
M, g ) is a Riemannian manifold. Whenno confusion may occur, we will use the following standard notationsfor the inner product and the norm: h X, Y i = g ( X, Y ) , | X | = p g ( X, X ) . A.1.
Connection and Hessian.
The Riemannian metric g on M in-duces a natural isomorphism between the tangent and cotangent bun-dles given by T ( M ) → T ∗ ( M )( x, X ) ( x, X ♭ )where X ♭ ( Y ) = h X, Y i , and whose inverse is T ∗ ( M ) → T ( M )( x, ξ ) ( x, ξ ♯ )where ξ ♯ is defined by ξ ( X ) = h ξ ♯ , X i . In local coordinates, if themetric is given by g = g jk dx j ⊗ dx k , this reads X ♭ = g jk X j dx k , ξ ♯ = g jk ξ j ∂ x k . In particular, the gradient field is defined by ∇ ϕ = dϕ ♯ . The mu-sical isomorphisms allow to lift the metric to the cotangent bundle.The cotangent bundle is hence naturally endowed with the Riemann-ian metric g − given in local coordinates by g − = g jk dξ j ⊗ dξ k . It is natural to use h· , ·i and | · | to denote the inner product and thenorm both on the tangent and cotangent bundles.We denote by D the Levi-Civita connection on ( M, g ). A connectionis a bilinear map on the vector space of vector fields which satisfies thefollowing conditions:( i ) D fX Y = f D X Y, and D X ( f Y ) = ( L X f ) Y + f D X Y if f is asmooth function on M ,( ii ) D X Y − D Y X = [ X, Y ].Here L X is the Lie derivative. On a Riemannian manifold, there isprecisely one connection, called the Levi-Civita connection, which isconsistent with the metric, i.e. which satisfies( iii ) L X h Y, Z i = h D X Y, Z i + h Y, D X Z i .This connection is determined in local coordinates by D ∂ xj ∂ x k = Γ ljk ∂ x l where the Christoffel symbols Γ ljk are given byΓ ljk = 12 g lm (cid:0) ∂ x j g km + ∂ x k g jm − ∂ x m g jk (cid:1) . IMITING CARLEMAN WEIGHTS 53
Note that Γ ljk = Γ lkj . If X is a vector field on M , then the endomor-phism D X has a unique extension as an endomorphism on the space oftensors satisfying the following requirements:( i ) D X is type preserving,( ii ) D X ( c ( S )) = c ( D X S ) for any tensor S and any contraction c ,( iii ) D X ( S ⊗ T ) = D X T ⊗ S + T ⊗ D X S for any tensors S, T .In particular, if f is a function we have D X ϕ = dϕ ( X ) = L X ϕ , and for1-forms the connection is given in local coordinates by D ∂ xj dx k = − Γ kjl dx l . The total derivative DS of S is the tensor DS ( X, · ) = D X S ( · ).The Hessian of a smooth function ϕ is the symmetric (2 , D ϕ = Ddϕ . The expression of the Hessian in local coordinates is D ϕ = (cid:16) ∂ x j x k ϕ − Γ ljk ∂ x l ϕ (cid:17) dx j ⊗ dx k . The following identities will be useful: D ϕ ( X, Y ) = 12 L ∇ ϕ g ( X, Y ) = h D X ∇ ϕ, Y i , (A.1) D ϕ ( X, X ) = d dt ϕ ( γ ( t )) (cid:12)(cid:12)(cid:12) t =0 . (A.2)Here γ is the geodesic with ˙ γ (0) = X .A.2. Parallel and Killing fields.
First we recall the following iden-tities for the Lie derivative: if f is a function and X a vector fieldthen L fX g = f L X g + df ⊗ X ♭ + X ♭ ⊗ df, (A.3)and if S is a (2 , L X S )( Y, Z ) = L X ( S ( Y, Z )) − S ([ X, Y ] , Z ) − S ( Y, [ X, Z ]) . (A.4) Definition A.1.
A vector field X in ( M, g ) is a Killing field if L X g = 0 . Note that (A.4) implies( L X g )( Y, Z ) = h D Y X, Z i + h Y, D Z X i . Definition A.2.
A vector field X is said to be parallel if its covariantderivative vanishes identically, that is DX = 0 . The following characterization is used in the proof of Theorem 1.
Lemma A.3.
Let ( M, g ) be a simply connected Riemannian manifold.A vector field on ( M, g ) is parallel if and only if it is both a gradientfield and a Killing field.Proof. In a simply connected manifold, a vector field X is a gradientfield if and only if the one form ω = X ♭ is closed. We have dω ( Y, Z ) = L Y (cid:0) ω ( Z ) (cid:1) − L Z (cid:0) ω ( Y ) (cid:1) − ω ([ Y, Z ])= L Y h X, Z i − L Z h X, Y i − h X, [ Y, Z ] i = h D Y X, Z i + h X, D Y Z i − h D Z X, Y i − h
X, D Z Y i− h X, D Y Z i + h X, D Z Y i . Thus, X is a gradient field if and only if for all vector fields Y, Z h D Y X, Z i − h D Z X, Y i = 0 . On the other hand, X is a Killing field if and only if L X g ( Y, Z ) = h D Y X, Z i + h Y, D Z X i = 0for all vector fields Y, Z . The result ensues. (cid:3)
The next result states that the existence of a unit parallel vectorfield implies a local product structure on the manifold.
Lemma A.4.
Let X be a unit parallel vector field in a manifold ( M, g ) .Near any point of M , there are local coordinates x such that X = ∂/∂x and the metric has the form g ( x , x ′ ) = (cid:18) g ( x ′ ) (cid:19) . Conversely, if such coordinates exist then X = ∂/∂x is unit parallel.Proof. Let X be unit parallel and let Γ be the distribution orthogonalto X . If V, W are vector fields orthogonal to X then h [ V, W ] , X i = h D V W − D W V, X i = V h W, X i − W h V, X i = 0 . Thus Γ is involutive, and by the Frobenius theorem there is a hypersur-face S (through any point of M ) which is normal to X . Let x ′ q ( x ′ )be local coordinates on S , and let ( x , x ′ ) exp q ( x ′ ) ( x X ( q ( x ′ )) becorresponding semigeodesic coordinates. In fact integral curves of X are geodesics (if ˙ γ ( t ) = X ( γ ( t )) then D ˙ γ ˙ γ = 0), so X = ∂ . Then g ( x , x ′ ) = (cid:18) g ( x , x ′ ) (cid:19) . If j, k ≥ ∂ g jk = h D ∂ ∂ j , ∂ k i + h ∂ j , D ∂ ∂ k i = 0 since ∂ is parallel.Therefore g = g ( x ′ ).The converse follows since D ∂ j ∂ = 0 by a direct computation. (cid:3) IMITING CARLEMAN WEIGHTS 55
Finally, we need the definition of conformal Killing fields.
Definition A.5.
A vector field X in ( M, g ) is called a conformalKilling field if L X g = λg. Note that if L X g = λg , by taking traces one has λ = n div X . Thenotion of conformal Killing field is of course invariant under conformalchange of the metric.A.3. Submanifolds.
Let (
M, g ) be a Riemannian manifold and let S be an embedded hypersurface in M . Fix a unit vector field N which isnormal to S . The second fundamental form of S is defined by ℓ ( X, Y ) = h D X N, Y i , where X and Y are vector fields tangent to S , and D is the Levi-Civitaconnection in ( M, g ). The eigenvalues of the symmetric bilinear form ℓ are the principal curvatures of S ; their sign depends on the choice ofnormal. Definition A.6.
A point of a hypersurface is called umbilical if all theprincipal curvatures are equal at that point. A hypersurface is calledtotally umbilical if every point is umbilical.
Definition A.7.
A hypersurface is called strictly convex if the secondfundamental form is positive definite.
A.4.
Curvature tensors.
Next we consider curvature tensors on aRiemannian manifold (
M, g ). The Riemann curvature endomorphismis a (3 , M , defined by R ( X, Y ) Z = D X D Y Z − D Y D X Z − D [ X,Y ] Z, whenever X, Y, Z are vector fields on M . By lowering indices, oneobtains the Riemann curvature tensor which is the (4 , R ( X, Y, Z, W ) = h R ( X, Y ) Z, W i . In local coordinates, with coordinate vector fields ∂ a = ∂/∂x a and with D a = D ∂ a , the components of the curvature tensor are given by R abcd = h ( D a D b − D b D a ) ∂ c , ∂ d i . By taking traces of the Riemann curvature tensor, we obtain the Riccitensor which is a symmetric (2 , R bc = g ad R abcd . The scalar curvature is the functionScal = g bc R bc . A manifold (
M, g ) is said to be flat if the Riemann curvature tensorvanishes identically. Euclidean space is flat, and any flat manifold islocally isometric to a subset of Euclidean space.Finally, we wish to define the conformal curvature tensors. First con-sider the rho-tensor, which is a symmetric 2-tensor given in componentsby P ab = 1 n − (cid:16) R ab − Scal2( n − g ab (cid:17) . The Weyl tensor of (
M, g ) is the 4-tensor W abcd = R abcd + P ac g bd + P bd g ac − P bc g ad − P ad g bc , and the Cotton tensor is the 3-tensor C abc = D a P bc − D b P ac . If the metric g is replaced by a conformal metric cg , then the Weyltensor transforms as W cg = cW g . If n = 3 then W ≡
0, but one has C cg = C g .A manifold ( M, g ) is called conformally flat if some conformal man-ifold (
M, cg ) is flat. Any 2-dimensional manifold is locally conformallyflat. A 3-dimensional manifold is locally conformally flat if and onlyif its Cotton tensor vanishes identically, and a manifold of dimension n ≥ References [1] Yu. E. Anikonov,
Some methods for the study of multidimensional inverseproblems for differential equations , Nauka Sibirsk. Otdel, Novosibirsk (1978).[2] K. Astala, M. Lassas, L. P¨aiv¨arinta,
Calder´on’s inverse problem for anisotropicconductivity in the plane , Comm. Partial Differential Equations, (2005),207–224.[3] K. Astala, L. P¨aiv¨arinta, Calder´on’s inverse conductivity problem in the plane ,Ann. of Math., (2006), 265–299.[4] D. C. Barber, B. H. Brown,
Progress in electrical impedance tomography , in In-verse problems in partial differential equations, edited by D. Colton, R. Ewing,and W. Rundell, SIAM, Philadelphia (1990), 151–164.[5] R. M. Brown, G. Uhlmann,
Uniqueness in the inverse conductivity problemfor nonsmooth conductivities in two dimensions , Comm. Partial DifferentialEquations, (1997), 1009–1027.[6] M. Dimassi, J. Sj¨ostrand, Spectral asymptotics in the semi-classical limit , Cam-bridge University Press, 1999.[7] D. Dos Santos Ferreira, C. E. Kenig, J. Sj¨ostrand, G. Uhlmann,
Determining amagnetic Schr¨odinger operator from partial Cauchy data , Comm. Math. Phys., (2007), 467–488.[8] L. Eisenhart,
Riemannian geometry , 2nd printing, Princeton University Press,1949.
IMITING CARLEMAN WEIGHTS 57 [9] C. Guillarmou, A. Sa Barreto,
Inverse problems for Einstein manifolds ,preprint (2007), arXiv:0710.1136.[10] L. H¨ormander,
The Analysis of Linear Partial Differential Operators III ,Springer-Verlag, 1985.[11] H. Isozaki,
Inverse spectral problems on hyperbolic manifolds and their applica-tions to inverse boundary value problems in Euclidean space , Amer. J. Math., (2004), 1261–1313.[12] J. Jost,
Riemannian geometry and geometric analysis , Springer-Verlag, 2002.[13] C. E. Kenig, J. Sj¨ostrand, G. Uhlmann,
The Calder´on problem with partialdata , Ann. of Math., (2007), 567–591.[14] K. Knudsen, M. Salo,
Determining non-smooth first order terms from partialboundary measurements , Inverse Problems and Imaging, (2007), 349–369.[15] R. Kohn, M. Vogelius, Identification of an unknown conductivity by means ofmeasurements at the boundary , in Inverse Problems, edited by D. McLaughlin,SIAM-AMS Proc. No. 14, Amer. Math. Soc., Providence (1984), 113–123.[16] M. Lassas, M. Taylor, G. Uhlmann,
The Dirichlet-to-Neumann map for com-plete Riemannian manifolds with boundary , Comm. Anal. Geom., (2003),207–221.[17] M. Lassas, G. Uhlmann, On determining a Riemannian manifold from theDirichlet-to-Neumann map , Ann. Sci. ´Ecole Norm. Sup., (2001), 771–787.[18] J. Lee, G. Uhlmann, Determining anisotropic real-analytic conductivities byboundary measurements , Comm. Pure Appl. Math., (1989), 1097–1112.[19] W. Lionheart, Conformal uniqueness results in anisotropic electrical impedanceimaging , Inverse Problems, (1997), 125-134.[20] R. G. Mukhometov, The reconstruction problem of a two-dimensional Rie-mannian metric, and integral geometry (Russian), Dokl. Akad. Nauk SSSR, (1977), 32-35.[21] A. Nachman,
Global uniqueness for a two-dimensional inverse boundary valueproblem , Ann. of Math., (1996), 71–96.[22] G. Nakamura, Z. Sun, G. Uhlmann,
Global identifiability for an inverse problemfor the Schr¨odinger equation in a magnetic field , Math. Ann., (1995), 377–388.[23] L. E. Payne, H. F. Weinberger,
An optimal Poincar´e inequality for convexdomains , Arch. Rat. Mech. Anal., (1960), 286–292.[24] P. Petersen, Riemannian geometry , Springer-Verlag, 1998.[25] M. Salo,
Inverse boundary value problems for the magnetic Schr¨odinger equa-tion , J. Phys. Conf. Series, (2007), 012020.[26] M. Salo, L. Tzou, Carleman estimates and inverse problems for Dirac opera-tors , preprint (2007), arXiv:0709.2282.[27] V. Sharafutdinov,
Integral geometry of tensor fields , in
Inverse and Ill-PosedProblems Series , VSP, Utrecht, 1994.[28] V. Sharafutdinov,
On emission tomography of inhomogeneous media , SIAM J.Appl. Math., (1995), 707–718.[29] Z. Sun, G. Uhlmann, Generic uniqueness for an inverse boundary value prob-lem , Duke Math. J., (1991), 131–155.[30] Z. Sun, G. Uhlmann, Anisotropic inverse problems in two dimensions , InverseProblems, (2003), 1001–1010. [31] J. Sylvester, An anisotropic inverse boundary value problem , Comm. PureAppl. Math., (1990), 201–232.[32] J. Sylvester, G. Uhlmann, A global uniqueness theorem for an inverse boundaryvalue problem , Ann. of Math., (1987), 153–169.[33] J. Sylvester, G. Uhlmann,