The Poisson embedding approach to the Calder\'on problem

We introduce a new approach to the anisotropic Calder\'on problem, based on a map called Poisson embedding that identifies the points of a Riemannian manifold with distributions on its boundary. We give a new uniqueness result for a large class of Calder\'on type inverse problems for quasilinear equations in the real analytic case. The approach also leads to a new proof of the result by Lassas and Uhlmann (2001) solving the Calder\'on problem on real analytic Riemannian manifolds. The proof uses the Poisson embedding to determine the harmonic functions in the manifold up to a harmonic morphism. The method also involves various Runge approximation results for linear elliptic equations.


Introduction
The anisotropic Calderón problem consists in determining a conductivity matrix of a medium, up to a change of coordinates fixing the boundary, from electrical voltage and current measurements on the boundary. In dimensions n ≥ 3 this problem may be written geometrically as the determination of a Riemannian metric on a compact manifold with boundary from Dirichlet and Neumann data of harmonic functions. More precisely, if (M, g) is a compact oriented Riemannian manifold with smooth boundary, we consider the Dirichlet problem for the Laplace-Beltrami operator ∆ g , u| ∂M = f and denote u = u f . We define the Dirichlet-to-Neumann map (DN map) where ∂ ν is the normal derivative on ∂M . One has the coordinate invariance Λ g = Λ φ * g for any diffeomorphism φ : M → M fixing the boundary. It is a long-standing conjecture [LU89] that if two Riemannian manifolds (M 1 , g 1 ) and (M 2 , g 2 ) with mutual boundary have the same DN maps, then there is a boundary fixing isometry between the manifolds. In this paper we give a candidate for this isometry: The points x 1 ∈ M 1 and x 2 ∈ M 2 are to be identified if and only if for every f ∈ C ∞ (∂M ) the harmonic extensions u 1 f and u 2 f of f to (M 1 , g 1 ) and (M 2 , g 2 ) satisfy u 1 f (x 1 ) = u 2 f (x 2 ). (1) For general Riemannian manifolds such an identification does not exist. However, if such an identification exists, we show that it induces a mapping M 1 → M 2 , and that this mapping is the boundary fixing isometry required for the solution of the Calderón problem. The Calderón problem thus reduces to showing that the equality of DN maps implies that the above identification exists. In this case we say that the manifolds can be identified by their harmonic functions.
We introduce a tool for studying the existence of this identification. This will be an embedding P of a Riemannian manifold into the linear dual of the space of smooth functions on its boundary. If (M, g) is a compact Riemannian manifold with C ∞ boundary and x ∈ M , then the value of P at x is a linear functional given by the formula where u f ∈ C ∞ (M ) solves the Dirichlet problem We call the mapping P the Poisson embedding, and it will be the main object of study of this paper. If P 1 (M 1 ) ⊂ P 2 (M 2 ), then is a well-defined map M 1 → M 2 , and we prove that it is a boundary fixing isometry if n ≥ 3. If n = 2, it is a conformal mapping. The condition P 1 (M 1 ) ⊂ P 2 (M 2 ) is equivalent to existence of the identification (1).
There are different points of view to the Poisson embedding: 1. The Poisson formula for solutions of the Dirichlet problem gives that where G(x, y) is the Green function for ∆ g in M and ∂ νy G(x, · ) is the Poisson kernel. Thus P identifies the point x in M with the Poisson kernel ∂ νy G(x, · ) on ∂M (hence the name Poisson embedding). 2. One also has the formula where ω x is the harmonic measure for ∆ g at x. Thus P identifies points in M with measures on ∂M ; points of ∂M are identified with the corresponding Dirac measures, and points in M int are identified with C ∞ functions since dω x = ∂ νy G(x, · ) dS in this case. 3. For x ∈ M int one has G(x, · ) = 0 on ∂M , and thus the knowledge of ∂ νy G(x, · )| ∂M determines the Green function G(x, · ) in M by elliptic unique continuation. Thus, instead of identifying points of M with the corresponding Green functions in M as in [LU02], we use the normal derivatives of the Green functions on ∂M . This change of point of view allows one to work on the boundary, which is natural since the measurements are given on ∂M .
The problem of finding the isometry, or conformal mapping if n = 2, from the knowledge of the DN map is known as the geometric Calderón problem. This problem has been solved in [LU02] in the following cases: Theorem. Let (M, g) be a compact connected C ∞ Riemannian manifold with C ∞ boundary.
(a) If n = 2, the DN map Λ g determines the conformal class of (M, g). (b) If n ≥ 3 and if M , ∂M and g are real-analytic, then the DN map Λ g determines (M, g).
As a first application of our techniques we give new proofs of these results. The proofs consist of three steps.
(1) The first step is to determine the harmonic functions near the boundary, using a standard boundary determination result [LU89] and real-analyticity.
(2) The second step uses a unique continuation argument for harmonic functions where the manifold and harmonic functions are continued simultaneously. The use of harmonic coordinates (see e.g. [DK81]) and the Runge approximation property are key ingredients in this step.
(3) Finally, we show that we can read the metric (conformal metric if n = 2) from the knowledge of harmonic functions.
The works [LU89,LU02,LTU03] study the Calderón problem on realanalytic manifolds, and an analogous result for Einstein manifolds (which are real-analytic in the interior) is proved in [GS09]. See [LLS16] for a recent result for a conformal Calderón problem. It remains a major open problem to remove the real-analyticity condition in dimensions ≥ 3; for recent progress in the case of conformally transversally anisotropic manifolds see [DKSU09,DKLS16] and also [DKLLS17,GST18] for the linearized problem. Other interesting approaches for related problems may be found in [Be09] and [Ce17], and counterexamples for disjoint data are given in [DKN17].
The inverse problems may not be uniquely solvable even when the metric is a priori known to be real analytic. Indeed, the above theorem proven in [LU02] does not hold for non-compact manifolds in the two-dimensional case. It is shown in [LTU03] that there are a compact and a non-compact complete, two-dimensional manifold for which the boundary measurements are the same. This counterexample was obtained using a blow-up map. Analogous non-uniqueness results have been studied in the invisibility cloaking, where an arbitrary object is hidden from measurements by coating it with a material that corresponds to a degenerate Riemannian metric [GLU03,GKLU07,GKLU09a,GKLU09b,DLU17].
Our main tool for studying the Poisson embedding and constructing the metric from harmonic functions is the Runge approximation property. This property allows one to approximate local solutions to an elliptic equation by global solutions. In particular this implies that harmonic functions separate points and have prescribed Taylor expansions modulo natural constraints. There are many results of this type in the literature, see e.g. [La56,Ma56,Br62,RS17]. We require specific approximation results for uniformly elliptic operators, and for completeness they will be given in Appendix A together with proofs.
1.1. An inverse problem for quasilinear equations. As a new result we prove a determination result for Calderón type inverse problems for quasilinear equations on manifolds (M, g) with boundary ∂M , including models that are both anisotropic and nonlinear. We consider the equation where f ∈ C ∞ c (W ), W ⊂ M is a fixed open set, and Q is a uniformly elliptic quasilinear operator of the form Qu(x) = A ab (x, u(x), du(x))∇ a ∇ b u(x) + B(x, u(x), du(x)). (4) Here and below we use Einstein summation rule, which means that repeated indices are always summed over 1, . . . , n. The source-to-solution mapping S : C ∞ c (W ) → C ∞ (W ) for this problem is defined as where u f solves (3). We assume that (M, g) and the matrix valued function A and the function B are real analytic. In this case we show that the source-to-solution mapping, even for small data, determines the manifold and the coefficients A and B up to a diffeomorphism and a built in "gauge symmetry" of the problem: Theorem 1.1. Let (M 1 , g 1 ) and (M 2 , g 2 ) be compact connected real analytic Riemannian manifolds with mutual boundary and assume that Q j , j = 1, 2, are quasilinear uniformly elliptic operators of the form (4) satisfying (22)- (27). Assume that the coefficients A j , B j are real analytic in all their arguments. Let W j ⊂ M int j , j = 1, 2, be open sets so that there is a diffeomorphism φ : W 1 → W 2 , and assume that the local source-to-solution maps S j for Q j on W j agree, φ * S 2 f = S 1 φ * f, for small f ∈ C ∞ c (W 2 ). Then there is a real analytic diffeomorphism J : M int 1 → M int 2 such that A 1 = J * A 2 := A and B 1 − J * B 2 = A ab (Γ(g) k ab − Γ(J * g) k ab )σ k .
(5) The mapping J satisfies In Theorem 1.1, the maps A, B 1 and J * B 2 have (x, c, σ) ∈ M 1 ×R×T * x M 1 as their argument, and Γ(g) k ab and Γ(J * g) k ab refers to the Christoffel symbols of the metrics g and J * g respectively.
The reason why B can not be determined independently of A and Γ, as presented in (5), is due to the fact that the covariant Hessian in the definition of Q contains first order terms.
The proof proceeds by linearizing the problem and using a slightly modified Poisson embedding for the linearized equation. Using the linearization we can first use the Poisson embedding approach to construct the manifold, but not yet the coefficients A and B. The source-to-solution mapping determines the coefficients in the measurement set W . Since the manifold is now known, the proof is completed by determining the coefficients on the whole manifold by analytic continuation from the set W . The linearization method goes back to [Is93] and has been used in various inverse problems for nonlinear equations, including anisotropic problems [SU97,HS02]. We refer to [Su05,SZ12] for further references.
1.2. Further aspects of Poisson embedding. The governing principle of this paper is that instead of trying to find the metric in the anisotropic Calderón problem directly, one can focus on finding the harmonic functions. This principle is implemented by Poisson embedding and by the fact, which we prove, that the metric can be determined from the knowledge of harmonic functions. We will now discuss in more detail some aspects the Poisson embedding approach.
If J is an isometry (M 1 , g 1 ) → (M 2 , g 2 ), then J can be locally represented in various coordinate charts. Useful coordinate charts include boundary normal coordinates and harmonic coordinates [DK81,Ta06]. In the study of the Calderón problem, boundary normal coordinates have been used to locally identify real analytic manifolds near boundary points by showing that in boundary normal coordinates the metrics g 1 and g 2 agree. See e.g. [LU89,LU02,LTU03,GS09,LLS16]. However, local representations do not yield a global candidate for the isometry J required for solving the Calderón problem. In contrast, the Poisson embeddings P j are globally defined objects, and they yield a candidate P −1 2 • P 1 for the isometry (in fact we prove that if P −1 2 •P 1 is well defined, then it gives the required isometry). The representation formula P −1 2 • P 1 also gives uniqueness of the boundary fixing isometry directly if it exists.
In [LTU03] the authors introduce an embedding of real analytic Riemannian manifolds by using Green functions to study the Calderón problem. Their approach involves an analytic continuation argument based on the implicit function theorem applied to the embedding. In contrast, the analogous step for Poisson embedding can be done simply by using harmonic coordinates. This feature also emphasizes the role of choosing suitable coordinates in the study of the anisotropic Calderón problem. Moreover, the recovery of the metric using Poisson embedding is an elementary linear algebra argument that yields a representation of the metric in terms of harmonic functions (see Proposition 4.1). In [LTU03] an asymptotic expansion of Green functions near the diagonal is used.
The basic principle of Poisson embedding is to control the points on a manifold by the values of solutions to Dirichlet problems on the manifold. This principle generalizes to nonlinear equations, to linear systems or to nonlocal operators where Green functions might not be easily accessible. The Poisson embedding also generalizes directly to less regular, say piecewise smooth or C k regular, settings.
Finally, we mention that ideas related to the Poisson embedding have been used in other fields as well. In [GW75] an embedding by finitely many harmonic functions is used to embed an open Riemannian manifold into a higher dimensional Euclidean space. Their embedding is similar to the Poisson embedding, in the sense that the Poisson embedding parametrizes the manifold by the data of all, instead of a finite number, of harmonic functions on the manifold. Another related result is that when the mapping P −1 2 • P 1 : M 1 → M 2 exists, it is necessarily a harmonic morphism. The study of harmonic morphisms, which are mappings that preserve harmonic functions, has applications in the study of minimal surfaces and in mathematical physics [BW03]. In Section 4 we give a new proof of the characterization of harmonic morphisms as homotheties [BW03, Corollary 3.5.2] based on harmonic coordinates.
1.3. Outline of the paper. In Section 2 we introduce the Poisson embedding and its basic properties, in a way that does not directly involve the Calderón problem. In Section 3 we determine the harmonic functions from the knowledge of the DN map on real analytic manifolds by using the Poisson embedding. From the knowledge of harmonic functions, we then determine the metric in Section 4, which gives a new proof of the main result of [LU89] in dimension n ≥ 3. Section 5 gives a new proof of the two-dimensional result of [LU89]. In Section 6 we use the Poisson embedding approach and linearization to prove Theorem 1.1, yielding uniqueness in the inverse problem for quasilinear equations. Appendix A is independent of the rest of the paper and contains Runge approximation results. In Appendix B we include the proofs of some basic auxiliary results.

Poisson embedding
We begin by introducing the Poisson embedding of a Riemannian manifold (M, g) with boundary. Throughout this section, we assume that M is connected and M and g are C ∞ . In particular, we do not require real analyticity in the definition of the Poisson embedding. We will solve Dirichlet problems with boundary values supported on a nonempty open set Γ of the boundary ∂M . The domain of the Poisson embedding will be Definition 2.1 (Poisson embedding). Let (M, g) be a compact Riemannian manifold with boundary, and let Γ be a nonempty open subset of ∂M . The Poisson embedding of the manifold M is defined to be the mapping where u f (x) solves the Dirichlet problem In the definition D ′ (Γ) is the space of distributions on Γ, i.e. D ′ (Γ) is the dual of C ∞ c (Γ). We call P the Poisson embedding due to the connection with the representation formula for the solution u f (x) in terms of the Poisson kernel ∂ νy G(x, y), where x ∈ M and y ∈ ∂M , as Here G(x, y) is the Dirichlet Green's function of (M, g) and dS g ∂M is the induced metric on the boundary. Thus P (x) can be identified with the We have the following basic properties of P .
Proposition 2.1. Let (M, g) be a compact manifold with boundary. For any x ∈ M Γ , one has P (x) ∈ H −s (Γ) whenever s + 1/2 > n/2. The mapping P is continuous M Γ → H −s−1 (Γ) and k times Fréchet differentiable considered as a mapping M Γ → H −s−1−k (Γ). In particular, P : The Fréchet derivative of P at x is a linear mapping given by where u f solves ∆ g u f = 0 in M and u f | ∂M = f ∈ C ∞ c (Γ). In the proposition · refers to the canonical pairing of vectors and covectors on M Γ , that is, We have placed the proof of the proposition, which just applies standard estimates for solutions of elliptic equations, in Appendix B. However, let us formally calculate where the formula for the derivative of P comes from. Let x ∈ M Γ and V ∈ T x M Γ . By definition V is given by a path γ : Let us next show that the mapping P is indeed an embedding, which means a C ∞ injective mapping with injective Fréchet derivative. The main tool that we encounter here for the first time is Runge approximation, which allows one to approximate locally defined harmonic functions by global harmonic functions. It is known since [La56,Ma56] that approximation results of this type follow by duality from the unique continuation principle. We have devoted Appendix A to various Runge approximation results. We will mostly use the following consequence, whose proof may also be found in Appendix A.
Proposition 2.2. Let (M, g) be a compact Riemannian manifold with smooth boundary, and let Γ be a nonempty open subset of ∂M .
Proposition 2.3 (P is an embedding). Let (M, g) be a compact manifold with boundary. The mapping P : M Γ → D ′ (Γ) is a C ∞ embedding in the sense that it is injective with injective Fréchet derivative on T M Γ .
Proof. Let x 1 , x 2 ∈ M Γ and assume that P (x 1 ) = P (x 2 ). That is, for all boundary value functions f ∈ C ∞ c (Γ), we have We need to show that x 1 = x 2 . We argue by contradiction and suppose that This is a contradiction and we must have x 1 = x 2 . Thus P is injective.
To show that the differential of P is injective, let x ∈ M Γ and V ∈ T x M Γ , and assume that DP x V = 0. By the formula (6) for the differential DP x , we have for all boundary value functions f ∈ C ∞ c (Γ). To conclude that V = 0, we use Proposition 2.2 again and choose f ∈ C ∞ c (Γ) such that ∇u f (x) = V . Here ∇u is the Riemannian gradient. Thus the condition (7) yields showing that V = 0. Thus the differential of P is injective on T M Γ .
2.1. Composition of Poisson embeddings. Let (M 1 , g 1 ) and (M 2 , g 2 ) be compact manifolds with mutual boundary ∂M . One of the aims of this paper is to give a candidate for the isometry that one hopes to construct in the anisotropic Calderón problem. This candidate is J := P −1 2 • P 1 . We will see that, whenever this mapping is well defined from M 1 to M 2 , it is exactly the mapping that one seeks in the Calderón problem. In dimensions n ≥ 3, it is an isometry. In dimension 2, it is a conformal mapping. It also fixes the boundary if we consider full data problem, or the part Γ ⊂ ∂M , if we consider partial data problem with measurements made on Γ.
We now begin to study the composition P −1 2 • P 1 . We include considerations related to partial data problem on the part Γ of the boundary. For this purpose we denote throughout this section Since we already know that the Poisson embeddings P j , j = 1, 2, are injective, we know that the mapping is well defined and bijective if the image sets of P j : (M Γ j , g j ) → D ′ (Γ) agree: Thus solving the Calderón problem would reduce to verifying this condition from the knowledge that the DN maps of (M 1 , g 1 ) and (M 2 , g 2 ) agree. The next lemma considers the smoothness properties of the mapping J assuming it is defined on some open set of M 1 .
Lemma 2.4. Let (M j , g j ), j = 1, 2, be compact manifolds with mutual boundary ∂M . Assume that Proof. We first note that, writing u j This follows from the computation P 1 (x) = P 2 (P −1 2 • P 1 (x)) = P 2 • J(x). Now, to show that J(B) ⊂ M Γ 2 , we argue by contradiction and assume that there is some , which is impossible by Proposition 2.2. Thus J(B) ⊂ M Γ 2 . We next prove that J : B → M 2 is continuous (this perhaps surprisingly uses compactness of M 2 ). Let x ∈ B. If J would not be continuous at x, there would be ε > 0 and a sequence (x l ) ⊂ M 1 with x l → x such that J(x l ) / ∈ B(J(x), ε). By the compactness of M 2 , passing to a subsequence (still denoted by (x l )), we may assume that J(x l ) converges to y ∈ M 2 . We have d(y, J(x)) ≥ ε. Now, using Proposition 2.2 in M 2 and the fact that J(x) ∈ M Γ 2 , we can find f ∈ C ∞ c (Γ) such that the harmonic function u 2 f ∈ C ∞ (M 2 ) satisfies u 2 f (J(x)) = u 2 f (y). The formula (8) shows that ). Since harmonic functions are continuous, taking the limit l → ∞ yields 0 = u 2 f (y) − u 2 f (J(x)), which is a contradiction. Thus J is continuous.
We will next show that J : B → M 2 is C ∞ . This follows an idea from [Ta06] related to smoothness of Riemannian isometries. Fix x ∈ B, and choose harmonic coordinates U = (u 2 f 1 , . . . , u 2 fn ) with f j ∈ C ∞ c (Γ), j = 1, . . . , n, near J(x). This can be done by Proposition 2.2 upon choosing {du 2 f 1 (J(x)), . . . , du 2 fn (J(x))} linearly independent. Write V = (u 1 f 1 , . . . , u 2 fn ). By the formula (8) we have V = U • J in B. Now U is bijective in some neighborhood Ω ⊂ M 2 of J(x), and since J is continuous there is a neighborhood W of x with J(W ) ⊂ Ω. Thus we have It remains to show that the differential of J is invertible on B. Since J : B → M 2 is injective, the claim will then follow from the inverse function theorem. Let f ∈ C ∞ c (Γ), x ∈ B and X ∈ T x M 1 . By (8), we have u 1 f = u 2 f • J. Together with (6) this gives: Thus for any f ∈ C ∞ c (Γ) we have the equation This equation will be used again later on and we name it "the equation of injectivity". (This equation can be interpreted as the infinite dimensional counterpart of the chain rule for the composition P 2 • (P −1 2 • P 1 ).) Assume that (DJ(x))X = 0 and choose by Proposition 2.2 a harmonic function u 1 f so that the Riemannian gradient satisfies ∇u 1 f (x) = X. Then |X| 2 = du 2 f · (DJ(x))X = 0. Thus X = 0. It follows that DJ(x) is injective at x and since the manifolds are of the same dimension, it is invertible. This proves the claim.

Determination of harmonic functions
We have so far acquired the basic properties of the Poisson embedding. We now move on to give a new proof of the fact [LU02] that for real analytic Riemannian manifolds with dim(M ) ≥ 3, the knowledge of the DN-map determines the Riemannian manifold up to isometry. Throughout this section we will assume that the manifolds M j and metrics g j , j = 1, 2, are real analytic, and n = dim(M j ) ≥ 3. We continue to denote M Γ j = M int j ∪ Γ. We first show that near any boundary point there exist coordinates in which the coordinate representations of harmonic functions, corresponding to a common boundary value f , agree. This follows from boundary determination [LU89] and unique continuation.
Lemma 3.1 (Determination near the boundary). Let (M 1 , g 1 ) and (M 2 , g 2 ) be compact real analytic manifolds with mutual boundary whose DN maps agree on an open set Γ ⊂ ∂M . Assume also that Γ is real analytic. Then, for any p ∈ Γ there are boundary normal coordinates ψ j , j = 1, 2, defined on neighborhoods U j ⊂ M Γ j of p, such that ψ 1 and ψ 2 agree on Γ and such that for any boundary function Here the functions u j f • ψ −1 j are the coordinate representations of the harmonic functions u j f on (M j , g j ) with boundary value f . Proof. Let p ∈ Γ ⊂ ∂M and let ψ j , j = 1, 2, be boundary normal coordinates near p on manifolds (M j , g j ), respectively, so that ψ 1 | Γ = ψ 2 | Γ . Then by the boundary determination result in [LU89], we have that in these coordinates, the jets of the Riemannian metrics g j agree. Since g j and Γ are real analytic, it follows that ψ j are real analytic coordinate charts, and thus the coordinate representations ψ −1 * j g j of g j agree near x = ψ 1 (p) = ψ 2 (p) ∈ {x n = 0}.
with the same Cauchy data (since the DN-maps agree) Proof. By Lemma 3.1, in its notation, we have that for any p ∈ Γ there exist gives an open set B ⊂ M Γ 1 such that Γ ⊂ B and P 1 (B) ⊂ P 2 (M Γ 2 ). By Lemma 2.4 it is enough to set F = P −1 2 • P 1 in B. We have now shown that knowledge of the DN map on Γ determines harmonic functions near Γ. We proceed in the real analytic case to determine the harmonic functions globally. From this knowledge we will then determine Riemannian manifolds up to isometry in Section 4.
In the following result a (global) harmonic morphism means a mapping that preserves solutions to the Dirichlet problem. Precisely, a C ∞ mapping Here u 1 f and u 2 f are the solutions to the Dirichlet problems for ∆ g 1 and ∆ g 2 as usual. For more details on harmonic morphisms, we refer to [BW03]. (In [BW03] a harmonic morphism is a mapping that also preserves local harmonic functions instead of just global harmonic functions, but our results will show that there is no difference at least when dim(M 1 ) = dim(M 2 ).) Theorem 3.3. Let (M 1 , g 1 ) and (M 2 , g 2 ) be compact real analytic manifolds with mutual boundary. Let Γ be an open subset of ∂M . Assume that there is a neighborhood B ⊂ M Γ 1 of a boundary point p ∈ Γ and a mapping F : Then we have P 1 (M Γ 1 ) = P 2 (M Γ 2 ) and . By our assumption B is nonempty. We will show that B is closed and thus B = M Γ 1 . Then P 1 (M Γ 1 ) ⊂ P 2 (M Γ 2 ), and from Lemma 2.4 it will follow that We argue by contradiction and assume that B is not closed. Then there is a sequence We actually have that This cannot be true by Proposition 2.2, so indeed x 2 must be in M Γ 2 . Let U = (u 2 f 1 , u 2 f 2 , . . . , u 2 fn ) be harmonic coordinates on a neighborhood Ω 2 ⊂ M Γ 2 of x 2 , as in Lemma 2.4. Here u 2 f l , l = 1, . . . , n, are global harmonic functions on M 2 with boundary values is well defined by Lemma 2.4. Now comes the punch line: we set . . , u 1 fn ). We will prove shortly that V is a coordinate system in some neighborhood Ω 1 ⊂ M Γ 1 of x 1 . The map U −1 • V will then give us a local identification of neighborhoods Ω 1 ⊂ M Γ 1 of x 1 and Ω 2 ⊂ M Γ 2 of x 2 , such that Ω 1 intersects the complement of the closure of B in M Γ 1 (if nonempty). This will allow us to extend B and reach a contradiction.
We now take a small deviation from the main line of the proof, and show that V is also a harmonic coordinate system around x 1 . To see this, first observe that V (x 1 ) = U (x 2 ) since Next we note that we have at p k the equation of injectivity (10) This holds for all X k ∈ T p k M 1 . Since near x 1 on B, we have J = U −1 • V , see (9), we can substitute this to the equation of injectivity yielding is a continuous (in fact smooth) matrix field even though we still do not know if it is invertible. Taking the limit as k → ∞ in (13) gives Choosing f so that ∇u 1 f (x 1 ) = X (by Proposition 2.2) shows that X = 0 and consequently that D( Let us continue on the main line of the proof of the proposition. Let What we will show is that Note that we require (15) to hold in Ω 1 , not only in B ∩ Ω 1 . In particular we will have Since Ω 1 in this case extends B to a neighborhood of the point x 1 ∈ ∂B \ B, this will give a contradiction and prove the theorem since M Γ 1 is connected. So far we have not used the assumption of real analyticity, but we use it now to prove (15). To prove it, let f ∈ C ∞ c (Γ). Now u j f ∈ C ω (M int j ) and V and U are local C ω diffeomorphisms in Ω 1 ∩M int 1 and Ω 2 ∩M int 2 , respectively. Thus we have Since on B ∩ Ω 1 , we have by (11) and (14) that . By the real analyticity, stated in (16), this holds on the whole V (Ω 1 ). This means that the coordinate representations of u 1 f and u 2 f in the (harmonic) coordinates V and U agree.
Since we have proven the latter, and since the boundary function f was arbitrary, we have proven (15). Consequently we have . Thus B extends to a neighborhood of the point x 1 ∈ ∂B \ B, which gives a contradiction. We have now proved that B is closed. Since M Γ 1 is connected, Inverting the role of (M 1 , g 1 ) and (M 2 , g 2 ), and replacing F and J in the statement of the theorem by F −1 and J −1 , shows that P 1 (M Γ 1 ) = P 2 (M Γ 2 ) and that J is surjective onto M Γ 2 . Consequently J : M Γ 1 → M Γ 2 is diffeomorphism by Lemma 2.4. Since we already showed that J is a global harmonic morphism at the beginning of the proof in (11), the claim follows.
Combining Lemma 3.2 and Theorem 3.3, we have proved the following statement.
Theorem 3.4. Let (M 1 , g 1 ) and (M 2 , g 2 ) be compact real analytic Riemannian manifolds, n ≥ 3, with mutual boundary whose DN maps agree on an open set Γ ⊂ ∂M . Assume also that Γ is real analytic. Then there is a diffeomorphic (global) harmonic morphism J : M Γ 1 → M Γ 2 such that J is real analytic in M Γ 1 and J| Γ = Id. Proof. We only need to show that J is real analytic in M Γ 1 . In the interior this follows from the representation (9) in terms of harmonic coordinates, which are real analytic in the interior. Near points of Γ this follows from the statement of Lemma 3.1, which implies that near points of Γ one has J = ψ −1 2 • ψ 1 where ψ j are real analytic boundary normal coordinates.

Recovery of the Riemannian metric from a harmonic morphism
In the previous section we showed that the Poisson embedding can be used to determine the manifold up to a global harmonic morphism from the knowledge of the DN map in the real analytic case, n ≥ 3. To give a new proof for the Calderón problem in the real analytic case, we need to show that a global harmonic morphism in this case is an isometry. Throughout this section, we assume that (M j , g j ), j = 1, 2, are compact connected and C ∞ smooth, n ≥ 3.
It is known that a mapping between Riemannian manifolds having the same dimension n ≥ 3 that pulls back local harmonic functions to local harmonic functions is in fact a homothety [Fu78], see also [BW03, Cor. 3.5.2]. Our definition of harmonic morphisms assumes that the mapping pulls back global harmonic functions to global harmonic functions. Our condition is seemingly slightly different, but it follows from the next result that, for manifolds having the same dimension, these conditions are equivalent.
We give a proof that a global harmonic morphism is in fact a homothety when n ≥ 3 by using harmonic coordinates. This seems to give a new proof for the result in the local case as well.
Proposition 4.1. Let (M 1 , g 1 ) and (M 2 , g 2 ) be C ∞ Riemannian manifolds having the same dimension n ≥ 3 and having a mutual boundary ∂M . Let Γ ⊂ ∂M be a nonempty open set, and let J : (M Γ 1 , g 1 ) → (M Γ 2 , g 2 ) be a locally diffeomorphic C ∞ global harmonic morphism. Then J is a homothety.
Proof. Let x ∈ M Γ 1 and let U = (u 2 f 1 , . . . , u 2 fn ) be an n-tuple of global harmonic functions that define harmonic coordinates on . This can be done by Proposition 2.2 upon choosing Since J is a global harmonic morphism, the coordinate system V is harmonic.
In the coordinates V and U the coordinate representations of harmonic In any g j -harmonic coordinates, j = 1, 2, the Laplace-Beltrami operator is particularly simple: ∆ g j u = −g ab j ∂ a ∂ b u, j = 1, 2. Here we abuse notation and denote by g 1 and g 2 the coordinate representations (V −1 ) * g 1 and (U −1 ) * g 2 , respectively. Define z := V (x) = U (J(x)) so z ∈ R n , and Ω : For any symmetric matrix (H ij ) such that g ij 2 (z)H ij = 0 we may find by Runge approximation a global g 2 -harmonic function u = u f with f ∈ C ∞ c (Γ) whose Hessian at z is (H ij ) in the g 2 -harmonic coordinates U . This is proved in Proposition A.5 (note that Hess g (w) corresponds to (∂ jk w) in harmonic coordinates). Thus we have that since the coordinate representations of the harmonic functions v = v f and u = u f , and thus their Hessians, agree in the coordinates V and U .
Since H can be any symmetric matrix with g ij 2 (z)H ij = 0, the above means that g −1 1 and g −1 2 have the same orthocomplement at z with respect to the Hilbert-Schmidt inner product in the space of symmetric matrices. Thus g −1 1 (z) = λ(z) −1 g −1 2 (z) for some nonzero real number λ(z). Due to the positive definiteness of g j , j = 1, 2, we have that λ(z) > 0. The argument above can be repeated for all z ∈ Ω, and we have where λ is a positive smooth function (λ is smooth since g 1 and g 2 are). We next show that the function λ is constant in Ω. For this we use the fact that the coordinates, where (18) holds, are harmonic. This is equivalent to saying that the contracted Christoffel symbols g ab j Γ c ab (g j ) vanish. By lowering the index, this means that . By taking the contracted Christoffel symbol of the equation 18, we have Here we have used the following simple computation for the conformal scaling of the contracted Christoffel symbols: Since n ≥ 3, the equation 19 shows that λ is constant in Ω. Recalling that g 1 (z) and g 2 (z) were the coordinate representations of g 1 and g 2 in coordinates V and U , with V = J * U , we have g 1 = λJ * g 2 on Ω 1 . Since this identity holds near an arbitrary point x ∈ M Γ 1 and since M Γ 1 is connected, we have proved the claim.
Remark 4.2. We remark that in the setting of the proof above we can by equation 17 actually express (a multiple of ) g −1 j at z in terms of Hessians of solutions u j f at z for some f ∈ C ∞ c (Γ), j = 1, 2. Let H k j , k = 1, . . . , m, m = n(n+1) 2 − 1, be a basis for the orthocomplement {g j (z) −1 } ⊥ in the space of symmetric matrices equipped with the Hilbert-Schmidt inner product. By the Runge approximation of Proposition A.5, we may find f k so that H j k = Hess g 1 (u 1 f k (z)) = Hess g 2 (u 2 f k (z)). Then we have Here * is the Hodge star operator in the space of symmetric matrices defined with respect to the Hilbert-Schmidt inner product and ∧ is the wedge product in that space. The equation (20) holds because it follows from the definition of the Hodge star that the right hand side is orthogonal to each H k j . Thus the right hand side has the same orthocomplement as g −1 j (x 0 ) has. Another remark is that since a homothety maps harmonic functions to harmonic functions, we have that a mapping between same dimensional Riemannian manifolds is global harmonic morphism if and only if it is a local harmonic morphism as defined in [BW03, Definition 4.1.1].
Next we show that if the DN maps agree, the homothety constant λ is 1.
Proposition 4.3. Assume the conditions of Proposition 4.1, and assume in addition that the DN maps of (M 1 , g 1 ) and (M 2 , g 2 ) agree on an open subset Γ of the boundary. Also assume that J| Γ = Id. Then Proof. By Proposition 4.1, we know that J is homothety and that Fix a point p ∈ Γ, and use boundary determination (see the proof of Lemma 3.1) to deduce that , the first equation and the fact that J| Γ = Id imply that Thus λ = 1.
Combining the results so far, we have achieved a new proof of the uniqueness in the Calderón problem in the real analytic case when n ≥ 3 [LU02]: Theorem. Let (M 1 , g 1 ) and (M 2 , g 2 ) be compact real analytic Riemannian manifolds, n ≥ 3, with mutual boundary whose DN maps agree on an open set Γ ⊂ ∂M . Assume also that Γ is real analytic. Then there is a real analytic diffeomorphism J : M Γ 1 → M Γ 2 such that g 1 = J * g 2 and J| Γ = Id.

Uniqueness in the 2D Calderón problem
In this section we use the Poisson embedding technique to give a new proof of uniqueness in the Calderón problem in dimension 2. This result is also due to [LU02]. In this section we assume (M 1 , g 1 ) and (M 2 , g 2 ) are compact, connected C ∞ Riemannian manifolds with mutual boundary ∂M . Note that it is not required that the manifolds are real analytic.
Theorem 5.1 (Uniqueness in the Calderón problem in 2D). Let (M 1 , g 1 ) and (M 2 , g 2 ) be two-dimensional compact connected C ∞ Riemannian manifolds with mutual boundary ∂M . Assume that the DN maps agree on an open subset Γ ⊂ ∂M . Then there is a conformal diffeomorphism J : Here λ is a smooth positive function in M Γ 1 , λ| Γ = 1, and J| Γ = Id. There is no assumption on real analyticity in this result. The proof relies on the fact that on two-dimensional manifolds there exist isothermal coordinates near any point, i.e. coordinates (u 1 , u 2 ) such that du 1 = * du 2 , see [Ta96, Section 5.10]. In these coordinates the metric looks like g jk = cδ jk for some positive function c, see Lemma B.1. Isothermal coordinates are also harmonic coordinates in dimension 2. We will use both of these facts.
We first prove local determination of harmonic functions near a boundary point, and then extend local determination to global determination. These are analogues of Lemma 3.2 and Theorem 3.3. After this, a two-dimensional version of Proposition 4.1 determines the metric up to a conformal mapping.
For the determination of harmonic functions near a boundary point, we note that in isothermal coordinates a g-harmonic function actually satisfies the Laplace equation in a subset R 2 . We show that the boundary determination result [LU89] of the metric in boundary normal coordinates implies determination of the metric on the boundary also in isothermal coordinates. Determination of harmonic functions near the boundary then follows from unique continuation for harmonic functions on R 2 .
The determination of harmonic functions near the boundary in isothermal coordinates involves some technicalities. These are consequences of the fact that the boundary determination result of [LU89], that we rely on, is given in boundary normal coordinates instead of isothermal coordinates. We address the technicalities in the next lemma, whose proof is given in the Appendix B.2.
Lemma 5.2. Let (M 1 , g 1 ) and (M 2 , g 2 ) be two-dimensional Riemannian manifolds with boundary whose DN maps agree on an open subset Γ ⊂ ∂M . Then for any p ∈ Γ there are isothermal coordinates U j , j = 1, 2, defined on neighborhoods Ω j ⊂ M Γ j of p such that the following statements hold: Proof. Let f ∈ C ∞ c (Γ) and let u j f , j = 1, 2, be the corresponding harmonic functions on (M j , g j ). Let U j be the isothermal coordinates of the previous lemma. Then the Cauchy data of u j f agree onΓ ⊂ R 2 in coordinates U j . In isothermal coordinates, which are always also harmonic coordinates in dimension 2, the Laplace-Beltrami equation for u j f reads Thus we see that u j f • U −1 j satisfy the same Euclidean Laplace equation with the same Cauchy data locally on a smooth mutual part of the boundary of the domain. By elliptic unique continuation, see e.g. [Is06, Theorem 3.3.1], and by setting F = U −1 2 • U 1 we obtain the claim with B replaced by U −1 1 (U 1 (Ω 1 ) ∩ U 2 (Ω 2 )). We can then enlarge B as in Lemma 3.2 to conclude the proof.
We record the following: Proposition 5.4. Let (M 1 , g 1 ) and (M 2 , g 2 ) be two-dimensional C ∞ Riemannian manifolds with mutual boundary. Let J : M Γ 1 → M Γ 2 be a locally diffeomorphic C ∞ global harmonic morphism. Then J is conformal, for some positive function λ ∈ C ∞ (M Γ 1 ). The proof is identical to that of Proposition 4.1 except that we cannot deduce that λ(x) is constant by the argument using equation (19). We omit the proof.
We will now prove global determination of harmonic functions.
Theorem 5.5. Let (M 1 , g 1 ) and (M 2 , g 2 ) be compact 2-dimensional C ∞ smooth Riemannian manifolds with mutual boundary. Let Γ be an open subset of ∂M . Assume that there is a neighborhood B ⊂ M Γ 1 of a boundary point p ∈ Γ and a mapping F : B → F (B) ⊂ M Γ 2 diffeomorphic onto its image such that P 1 = P 2 • F.
Proof. We proceed as in the proof of Theorem 3.3 to which we refer the reader for more details. Let us recall the notation and some facts from there. We redefine ∅ = B ⊂ M Γ 1 to be the largest open connected set such that P 1 (B) ⊂ P 2 (M Γ 2 ). The task is to show that B is closed. We argue by contradiction and assume that it is not. Then the points x 1 ∈ ∂B \ B and x 2 ∈ M Γ 2 are limits of sequences (p k ) ⊂ B and (J(p k )) ⊂ J(B).
We construct isothermal coordinates on neighborhoods Ω 1 of x 1 ∈ M Γ 1 and Ω 2 of x 2 ∈ M Γ 2 as follows. Let us first choose by Runge approximation a boundary function f ∈ C ∞ c (Γ) such that du 2 f (x 2 ) = 0. Let us denote by u 1 a harmonic conjugate of u 2 f near x 2 . This is a function solving near x 2 the equation f , where * g 2 is the Hodge star of g 2 . A local solution u 1 exists since the right hand side is closed, because u 2 f is harmonic. We may assume u 1 (x 2 ) = 0. Let us denote U = (u 1 , u 2 f ). Then U is an isothermal coordinate system on a neighborhood Ω 2 of x 2 , see Lemma B.1.
Likewise, let v 1 be a harmonic conjugate of u 1 f near x 1 with v 1 (x 1 ) = 0. Thus v 1 solves dv 1 = − * g 1 du 1 f . We set V = (v 1 , u 1 f ). These are isothermal coordinates on a neighborhood Ω 1 of x 1 . The fact that the Jacobian of V is invertible at x 1 follows from We next show that on B ∩ Ω 1 we have v 1 = u 1 • J. By using u 1 By Proposition 5.4 applied with M Γ 1 replaced by B and M Γ 2 replaced by J(B) (the proof of Proposition 5.4 is really a pointwise argument and applies in this case), we have that J is a conformal mapping on B, J * g 2 = λg 1 . Thus we have * g 1 J * du 2 f = J * ( * (g 2 /(λ•J −1 )) du 2 f ). Since Hodge star is conformally invariant when operating on 1-forms in dimension 2, the above is (The point here is that U −1 • V is defined on the whole Ω 1 and gives us a good candidate for a local extension of J onto the whole Ω 1 .) Let f ∈ C ∞ c (Γ). To conclude the proof, we will show that on V (Ω 1 ), we have , the above holds on the open set V (Ω 1 ∩ B). Since the coordinates in question are isothermal and harmonic (where Γ a (g j ) = 0) we have that Since these functions agree on the open set V (Ω 1 ∩B) they agree everywhere on V (Ω 1 ) by unique continuation. This shows that we may indeed extend J to B ∪ Ω 1 , which gives a contradiction and concludes the proof.
Proof of Theorem 5.1. By Lemma 5.3, we have that there is B ⊂ M Γ 1 and a diffeomorphic harmonic morphism F : B → F (B) ⊂ M Γ 2 . By Theorem 5.5 the mapping F extends to a global harmonic morphism J : M Γ 1 → M Γ 2 . Proposition 5.4 shows that J is a conformal mapping. That the implied conformal factor is 1 on Γ follows from calculations in the proof of Proposition 4.3.

On determining the coefficients of quasilinear elliptic operators from source-to-solution mapping
In this section we apply the Poisson embedding technique for a Calderón type inverse problem for second order quasilinear elliptic operators on Riemannian manifolds. Let (M, g) be a Riemannian manifold with boundary. The quasilinear operators we study are assumed to have the coordinate representation Here we assume that a 2-tensor field and B(x, c, σ) is a function. The covariant derivative ∇ is determined by g as usual. We consider A and B as mappings and Here π is the canonical projection T 2 0 (M ) → M for 2-contravariant tensors. The notation M × R ⊗ T * M refers to the subset We can also think of M × R ⊗ T * M as tensor product of the trivial line bundle M × R and the vector bundle T * M .
We assume that the coefficients A and B are C ∞ smooth and in the main theorem of this section, Theorem 6.2, we assume that they are real analytic. We will assume throughout that 0 is a solution, i.e. Q(u) = 0, which is equivalent with the condition The linearization of Q at u = 0 is the operator where in the last term we have identified T * x M with R n . We will also assume that, for some fixed α with 0 < α < 1, It follows from these assumptions that there are unique small solutions of Q(u) = f , u| ∂M = 0 when f is small. The proof of the next standard result is given in Appendix B.
Proposition 6.1. Let (M, g) be a compact manifold with smooth boundary, let A, B be C ∞ maps satisfying (22)-(25), and let Assume that L is the linearization of Q at u = 0 given in (26), and assume that (27) holds.
Operators of the above form appear e.g. in the study of minimal surfaces or prescribed scalar curvature questions (Yamabe problem), see [GT98,Ta96] for more information.
We describe the inverse problem we are about to study. Let Q, ε, δ be as in Proposition 6.1. We assume that we know the source-to-solution mapping of Q on an open subset W of M . The source-to-solution mapping is defined as where u is the unique solution to Note that indeed u ∈ C ∞ (M ), using Schauder estimates for linear elliptic equations and the fact that u ∈ C 2,α (M ). The aim of the inverse problem is to determine the coefficients A and B of Q up to a diffeomorphism and possible other symmetries of the problem. When the coefficients of Q are real analytic, our main theorem shows that in this case there is only one additional symmetry, which we describe next. We note that there is a simple transformation between coefficient B and the Christoffel symbols Γ k ab contracted by A ab that leaves the source-tosolution mapping intact: Assume that u solves Qu = f on M. (31) Then u also solvesQ u = f, and whereB is defined as Here∇ andΓ k ab denote the Levi-Civita connection and Christoffel symbols of some other Riemannian metricg on M . Therefore the source-to-solution mapping defined with respect toQ coincides with the source-to-solution map S of Q. Note that even though Christoffel symbols does not constitute a tensor field, the difference of two Christoffel symbolsΓ k ab − Γ k ab is a tensor field. It follows that we can not make the symmetry vanish by choosing a suitable coordinate system. This symmetry will be called the gauge symmetry of the inverse problem.
The gauge symmetry is an obstruction for finding B, A and Γ k ab independently of each other in the general case. However, in some cases when we have extra information about the coefficients A and B, the gauge symmetry vanishes. We give examples of conditions when this happens in Corollary 6.3.
We remark that if the coefficients are not real analytic, other symmetries in the inverse problem can appear. An easy example is the standard Laplace-Beltrami operator in dimension 2 where one can scale the metric by a positive function that is constant 1 on the measurement set W without affecting the source-to-solution mapping. Another similar example is given by the conformal Laplacian in dimensions n ≥ 3 [LLS16].
Our main theorem of this section is the following determination result.
Theorem 6.2. Let (M 1 , g 1 ) and (M 2 , g 2 ) be compact connected real analytic Riemannian manifolds with mutual boundary and assume that Q j , j = 1, 2, are quasilinear operators of the form (21) having coefficients A j , B j satisfying (22)-(27). Moreover, assume that A j and B j are real analytic in all their arguments. Let W j , j = 1, 2, be open subsets of M j , and assume that there is a diffeomorphism φ : W 1 → W 2 so that the source-to-solutions maps S j for Q j agree in the sense that We describe our strategy for proving the theorem. By the arguments in the preceding sections, it would be natural to define a mapping analogous to the Poisson embedding for the quasilinear elliptic operator Q and then use tools analogous to those we built around the Poisson embedding. However, as far as we know, Runge approximation for quasilinear operators is not known. This prevents us of using this natural approach for the moment.
Instead we do the following. We linearize the source-to-solution mapping (at the mutual solution 0) that yields a linear Calderón type inverse problem for a linear second order elliptic operator whose source-to-solution map is known. For this linearized problem we use the Poisson embedding technique modified slightly to deal with the source-to-solution map instead of the DN map. In this way we will find the manifold up to a real analytic diffeomorphism. This is the first step. The modified Poisson embedding is given in Definition 6.1.
The second step is the following. We will see that knowing the sourceto-solution map on the open set W determines the coefficients A and B on W , up to the gauge symmetry described in (32). In this step we read the coefficients A and B in W from the solutions, which is similar to the argument in Proposition 4.1.
Since we have determined the manifold up to a real analytic diffeomorphism, we can view the coefficients A and B on a single fixed manifold and use standard real analytic unique continuation there. This determines the coefficients of the quasilinear operator on the whole manifold up to a diffeomorphism and the gauge symmetry.
As already mentioned, with some suitable extra information about the coefficients B and A the gauge symmetry vanishes and we can determine A and B independently. Corollary 6.3. Assume the conditions and notation in Theorem 6.2, and assume also one of the following: (1) A 1 (x, c, σ) (or A 2 (x, c, σ)) is s-homogeneous in the σ-variable and B 1 (x, c, σ) and B 2 (x, c, σ) are s ′ -homogeneous with s ′ = s + 1 for all x ∈ W 1 and c ∈ R; or ) is an isometry.
Then we have A 1 = J * A 2 and B 1 = J * B 2 and also A ab 1 Γ k ab (g 1 ) = (J * A 2 ) ab Γ k ab (J * g 2 ). Note that the corollary does not claim that we can find the Riemannian metrics g 1 and g 2 up to J. An example satisfying the first condition is the nonlinear Schrödinger operator u ∈ C ∞ → −∆ g u + q|u| 2 u, q ∈ C ∞ .
An example satisfying the second condition is the case, where one knows the Riemannian metric and the manifold (φ = Id) on a measurement set W . We have now seen that linearizing the source-to-solution mapping S of a quasilinear equation leads to a Calderón type inverse problem for a linear equation. Next we show that in the real analytic case the source-to-solution mapping of the linearized problem determines the manifold up to a real analytic diffeomorphism. Precisely we will prove: Theorem 6.5. Let (M 1 , g 1 ) and (M 2 , g 2 ) be compact real analytic manifolds with mutual boundary. Let L 1 and L 2 be second order uniformly elliptic partial differential operators on M 1 and M 2 , respectively, of the general form where A ab j and B a j are the components of a 2-tensor field A and vector field B j , and C is a function. Assume that L j are injective on C ∞ (M j )∩H 1 0 (M j ), and assume that A, B and C are real analytic up to boundary.
Let W j ⊂ M j be open sets, and let φ : W 1 → W 2 be a diffeomorphism so the source-to-solutions maps S L j of L j satisfy In the theorem u 1 f •φ and u 2 f are the solutions to L 1 u 1 In the definition D ′ (W ) means [C ∞ c (W )] * as usual. The reason why we consider R to be defined only in the interior of M is because we assume that the boundary values of the solutions u f vanish on the boundary. Thus we have no control on the points on the boundary by using solutions u f . Even though R is not the Poisson embedding of the previous section, we use the same name for R, and we note that R is related to the linear elliptic operator L.
The basic properties of the Poisson embedding R are as follows. The proof of the following proposition is in Appendix B.1. Proposition 6.6. Let (M, g) be smooth compact manifold with boundary. For any x ∈ M int , one has R(x) ∈ H −s (W ) whenever s + 2 > n/2. The mapping R is continuous M int → H −s−1 (W ) and k times Fréchet differentiable considered as a mapping M int → H −s−1−k (W ). In particular, R : M int → D ′ (W ) is C ∞ smooth in the Fréchet sense. The mapping R can be extended continuously to a mapping M → H −s (W ) by defining R| ∂M = 0.
The Fréchet derivative of R at x ∈ M int is a linear mapping given by , and · refers to the canonical pairing of vectors and covectors on M .
In the statement, we are not claiming that the continuation of R onto M is injective on ∂M .
To prove that R is an embedding, we use the following analogue of Proposition 2.2 which follows from a suitable Runge approximation result. Its proof is in Appendix A.
Proposition 6.8 (R is an embedding). Let M , L, R be as in Definition 6.1. The mapping R : M int → D ′ (W ) is a C ∞ embedding (and a C k embedding M int → H −s−1−k (W )) in the sense that it is injective with injective Fréchet differential on T M int .
Proof. The injectivity of R follows from Proposition 6.7(a). Let x ∈ M int and V ∈ T x M . Assume that 0 = (DR x V )f = du f (x) · V for all f ∈ C ∞ c (W ). By Proposition 6.7(b) one can find f ∈ C ∞ c (W ) so that du f (x) · V = 0, unless V = 0. This shows injectivity of the differential.
We construct next local coordinate systems from solutions u f to Lu f = f , u f | ∂M = 0. We call these coordinates solution coordinates. These coordinates are constructed by Runge approximating local solutions to Lu = 0.
If the coefficients of L are real analytic, then (u f 1 , . . . , u fn ) is real analytic on Ω.
Proof. Let x 0 ∈ M int . If x 0 ∈ W , we redefine W as a smaller open set such that x 0 / ∈ W . By Proposition 6.7(b) we may find f 1 , . . . , f n ∈ C ∞ c (W ) such that the Jacobian matrix of U = (u f 1 , . . . , u fn ) is the identity matrix at x 0 . Thus U is a coordinate system in some neighborhood Ω of x 0 , and by shrinking Ω if necessary we may assume that f j = 0 in Ω.
If the coefficients of L are real analytic, the coordinate system U is real analytic on Ω by elliptic regularity, since f j | Ω = 0.
The next lemma is an analogue of Lemma 2.4, where we considered the composition of Poisson embeddings. If W 1 and W 2 are open subsets of M 1 and M 2 and φ : W 1 → W 2 is a diffeomorphism, we define a mapping with x ∈ M int 2 and f ∈ C ∞ c (W 1 ). From Proposition 6.6 and Proposition 6.8, it follows that φ * R 2 is a C k embedding M int 2 → H −s−1−k (W ), k = 0, 1, . . .. Lemma 6.10. Let (M 1 , g 1 ) and (M 2 , g 2 ) be compact manifolds with mutual boundary ∂M . Let W j ⊂ M j , j = 1, 2, be open subsets and let φ be a diffeomorphism W 1 → W 2 . Assume that for some open set We prove the continuity of J differently than we did in the corresponding situation in Lemma (2.4). It follows from Proposition 6.6 that we may continue φ * R 2 by zero to a continuous mapping M 2 → H −s−1 (W 1 ). Let E ⊂ M int 2 be closed in M 2 . It follows that φ * R 2 (E) is closed in H −s−1 (W 1 ) by continuity and by compactness of M 2 . Since φ * R 2 is injective on M int 2 , we have that I := (φ * R 2 ) −1 is defined as a mapping φ * R 2 (M int 2 ) → M int 2 and we have that I −1 (E) equals φ * R 2 (E), which is closed. Thus I is continuous and consequently J is continuous B → J(B) by Proposition 6.6. Let x 0 ∈ M int 1 and let U = (u 2 f 1 , . . . , u 2 fn ) be solution coordinates of Lemma 6.9 on a neighborhood Ω 2 of J(x 0 ), where f k ∈ C ∞ c (W 2 ), k = 1, . . . , n. In case x 0 ∈ W 1 , we may assume that and It follows from Proposition 6.7 that DJ(x 0 ) is invertible. Since J is also injective by Proposition 6.8, it follows that J : We prove next the main result of this subsection.
Proof of Theorem 6.5. Assume that the source-to-solutions maps agree: By shrinking W j if necessary, we may assume that W j are small geodesic balls. Redefine B to be the largest connected open set of M int To see this, let f ∈ C ∞ c (W 1 ) and x ∈ W int 1 . We have by using definitions and (35) that , which by Proposition 6.7 leads to a contradiction to the fact that x 1 ∈ M int 1 . We conclude that B = ∅. We show that B is closed in M int 1 and thus B = M int 1 . ) and thus, by the definition (34), By (36), we directly also have that To show that B is closed, we argue by contradiction and let p k → x 1 ∈ ∂B \ B, with p k ∈ B. By passing to a subsequence, we have be solution coordinates as in Lemma 6.9 on a neighborhood Ω 2 ⊂ M int 2 of the point x 2 . Here f j ∈ C ∞ c (W 2 ), j = 1, . . . , n, and the limit x 2 is found by first passing to a subsequence.
We set We have J = U −1 • V on B near x 1 by (37). We have that V is a solution coordinate system on a neighborhood Ω 1 ⊂ M int 1 of x 1 , and the "equation of injectivity" , holds for X ∈ T x 1 M 1 . These facts can be proved similarly following the argument in Theorem 3.3. We redefine the sets Ω 1 and Ω 2 , if necessary, so that we have V (Ω 1 ) = U (Ω 2 ).
We next show that This will imply , which is a contradiction and proves the theorem.
Let f ∈ C ∞ c (W 1 ). We claim that 1 there is nothing to prove since then x 1 ∈ W int 1 ⊂ B and it is a contradiction to x 1 ∈ ∂B \ B. Thus we may assume x 1 ∈ M int 1 \ supp(f ). If x 2 ∈ W int 2 , then by (38) and by the injectivity of J and continuity of J −1 we have that Thus we may also assume that . It follows that there are small geodesic balls Ω ′ 1 ⊂ Ω 1 and Ω ′ 2 ⊂ Ω 2 of x 1 and x 2 , respectively, such that u 1 f and u 2 f •φ −1 are C ω on Ω ′ 1 and Ω ′ 2 , and that V (Ω ′ 1 ) ⊂ U (Ω ′ 2 ). Consequently, the functions are real analytic on V (Ω ′ 1 ). The functions in (40) agree on V (B ∩ Ω 1 ) by (37). Consequently, they agree on V (Ω ′ 1 ) by real analyticity. The set V (Ω ′ 1 ) ∩ V (Ω 1 \ B) is open and non-empty. Note that the functions in (40) are real analytic on V (Ω 1 \ B). This is because supp(f ) ⊂ W int 1 ⊂ B and because y ∈ supp(f • φ −1 ) ∩ Ω 2 implies that By real analyticity the functions in (40) agree on V (Ω 1 \B) and consequently on V (Ω 1 ). We have (39) since it is equivalent to Thus B extends to a neighborhood of the point x 1 ∈ ∂B \ B, which gives a contradiction. Thus B is closed. Since M int 1 is connected, we conclude that B = M int 1 . By Lemma 6.10, J is C ∞ diffeomorphism M int 1 → J(M int 1 ). Inverting the role of M 1 and M 2 , we have J(M int 1 ) = M int 2 . Since by Lemma 6.9 we may locally represent J as U −1 • V near any point, where U and V are C ω solution coordinates, we have that J and J −1 are real analytic. If u 2 and u 2 f | ∂M = 0, then (37) shows that u 1 f •φ = J * u 2 f . 6.2. Local determination of the coefficients. We determine the coefficients of a quasilinear elliptic operator on open sets where the source-tosolution mapping of the operator is known. Precisely, we prove the following: Proposition 6.11. Let (M 1 , g 1 ) and (M 2 , g 2 ) be compact connected Riemannian manifolds with mutual boundary and assume that Q j , j = 1, 2, are quasilinear operators of the form (21) having coefficients A j , B j satisfying (22)-(27).
Let W j , j = 1, 2, be open subsets of M j , and assume that there is a diffeomorphism φ : W 1 → W 2 so that the source-to-solutions maps S j for Q j agree in the sense that be an open set. Then there is δ > 0 such that for all x ∈ W and (c, σ) ∈ T x W , with |c| + |σ| g 1 ≤ δ, we have and To prove the proposition, we begin with the following observation.
Lemma 6.12. Assume the conditions and notation in Proposition 6.11. Let x ∈ W int 1 and let U 2 be coordinates on a neighborhood Ω 2 ⊂⊂ W 2 of φ(x) ∈ W int 2 . Let U 1 = φ * U 2 be coordinates on a neighborhood φ −1 (Ω 2 ) ⊂⊂ W 1 of the point x.
Proof. Let x ∈ W int 1 and let U 1 and U 2 be coordinate systems as in the statement of the lemma. Let ε j , δ j > 0, j = 1, 2, be as in the definition (28)-(30) of the source-to-solution mapping of Q j . Set ε = min(ε 1 , ε 2 ). Let Such a δ ′ can be found since B 2 (x, 0, 0) = 0 by assumption (25), since Q 2 : C 2,α (W 2 ) → C α (W 2 ) is continuous and since composing a compactly supported function with C ∞ diffeomorphism is continuous on Hölder spaces.
Let v ∈ C ∞ c (Ω) with v C 2,α (Ω) ≤ δ ′ . Then the problem Q 2 u 2 has a unique solution u 2 F with u 2 F C 2,α (M 2 ) ≤ δ 2 by Proposition 6.1. Let us extend v • U 2 by zero onto M 2 . By the condition B 2 (x, 0, 0) = 0 and we have Since v • U 2 C 2,α (M 2 ) ≤ δ 2 , by uniqueness we have that By the definition of δ ′ we have that Q 1 u 1 F = φ * F with u 1 F | ∂M = 0 has a unique solution with u 1 F C 2,α (M 1 ) ≤ δ 1 . Since the source-to-solutions mappings agree, we have We also have on W 2 that Using the coordinate invariance of Q j , we have by (43)-(45) that The lemma tells us that we can use any test function v with small enough C 2,α norm to solve for the coefficients ofQ j from the equationQ 1 v =Q 2 v in the coordinates U 1 and U 2 . The local determination result of Proposition 6.11 is a consequence of this observation. Its proof is similar to that of Proposition 4.1 where we used harmonic functions to solve for (a multiple of) the metric in the Calderón problem.
Here A is the Hilbert-Schmidt norm of matrices and δ is independent of y ∈ Ω. Let y ∈ Ω and let (c, σ) ∈ R × T * y Ω with |c| + |σ| g 2 ≤ δ. Let A be a matrix with A ≤ δ and let v = v (y,c,σ,A) be the function defined in (47). Since v C 2,α (Ω) < δ ′ by (48), the equation 46 implies that for x ∈ Ω. Here Γ k j,ab are the Christoffel symbols of g j , and A j and B j are understood as their coordinate representations in coordinates U j .
Let us first choose as the matrix A a one with A ≤ δ and that satisfies have A ⊥ A 1 (y, v(y), dv(y)) − A 2 (y, v(y), dv(y)).
Here ⊥ is defined with respect to the Hilbert-Schmidt inner product of matrices. We have v C 2,α (Ω) ≤ 2δ and thus the equation 49 holds. It follows that (50) We next choose the matrix A as A ab = ρ(A ab 1 (y, c, σ) − A ab 2 (y, c, σ)), where ρ > 0 is a number small enough so that A ≤ δ. It follows from (49) and by using (50) that ρ A 1 (y, c, σ) − A 2 (y, c, σ) 2 = 0, where · is the Hilbert-Schmidt norm of matrices. Thus A ab 1 (y, c, σ) = A ab 2 (y, c, σ), which combined with (50), yields The equations (51) and (52) hold for all (y, c, σ) ∈ Ω × R ⊗ T * Ω with |c| + |σ| g 2 ≤ δ. Recall that (51) and (52) are equations for the coordinate representations of A 1 , B 1 and A 2 , B 2 in coordinates U 1 and U 2 respectively, where U 1 = φ * U 2 . Thus we have, by redefining δ if necessary, that for (x, c, σ) ∈ (φ −1 Ω) × R ⊗ T * (φ −1 Ω) with |c| + |σ| g 1 ≤ δ. Let W be an open set compactly contained in W int 1 . Since we may carry out the argument above on a neighborhood of any point x 0 ∈ W int 1 , we have the claim by compactness for some δ > 0. 6.3. Global determination of the coefficients. We prove the main theorem of this section.
Proof of Theorem 6.2. By Theorem 6.5 we know that there is a real analytic diffeomorphism J : It follows that J * A 2 and J * B 2 , which are given by are real analytic for all x ∈ M int 1 and (c, σ) ∈ R × T * x M int 1 . By Proposition 6.11 we have that there is a non-empty open set W ⊂ W int 1 and δ > 0 such that for all (c, σ) ∈ R × T * x W with |c| + |σ| g 1 ≤ δ, we have A 1 (x, c, σ) = J * A 2 (x, c, σ) and (54) is s-homogeneous in the σ-variable and that B 1 and B 2 are s ′ -homogeneous with s ′ = s + 1. It follows from (53) that J * B 2 is s ′homogeneous in the σ-variable. By Theorem 6.2 we know that J * A 2 = A 1 is s + 1 homogeneous in the σ-variable. Thus Γ(x, c, σ) := A ab (x, c, σ)(Γ(g 1 ) k ab (x) − Γ(J * g 2 ) k ab (x))σ k , where A = A 1 = J * A 2 , is (s + 1)-homogeneous in the σ-variable. Since is s ′ -homogeneous, s ′ = s + 1, in the σ-variable, we must have B 1 − J * B 2 = 0 and Γ(x, c, σ) = 0 by Theorem 6.2.

Appendix A. Runge approximation results
Here we prove the Runge approximation results that are used repeatedly in this paper. In this section, which is independent of the other sections, we will assume that (M, g) is a compact connected oriented Riemannian manifold with boundary, and dim(M ) ≥ 2. We will mostly use the following easy consequence of Runge approximation.
The main Runge approximation result is the following.
Proposition A.2. Let s ≥ 1, let Γ be a nonempty open subset of ∂M , and let U ⊂⊂ M int be a domain with C ∞ boundary such that M \U is connected. Let also L be a second order uniformly elliptic differential operator on M , and let P be the Poisson operator for L. Then the set The proof is a standard Runge approximation argument, which boils down to the solvability of the adjoint equation in negative order Sobolev spaces and the unique continuation property. It will be convenient to embed (M, g) in a closed manifold (N, g) and to extend the operator L in M as a second order uniformly elliptic operator A in N . In this way we avoid having to consider boundary values of solutions in negative order Sobolev spaces. The required solvability result will follow from the next lemma, where A * is the formal L 2 -adjoint. Proof. Let r ∈ R, and consider the map By [Hö85,Theorem 19.2.1] this map is a Fredholm operator with finite dimensional kernel N * := {v ∈ C ∞ (N ) ; A * v = 0}, and the range of T r is given by In particular, the kernel and cokernel are independent of r. This proves the first statement.
Assume now that F | U ≡ 0, let V ⊂⊂ U and let m ≥ 0. Let χ, χ 1 ∈ C ∞ c (U ) satisfy χ = 1 near V and χ 1 = 1 near supp(χ). Let also Q ∈ Ψ −2 (N ) be a parametrix for A * , so that using that F | U ≡ 0. Using the pseudolocal property of Q and the fact that R is smoothing, we get the estimate We will also need the following simple lemma.
Lemma A.4. Let (N, g) be a closed manifold, let A be a second order uniformly elliptic differential operator on N , let N be the kernel of A, and let {ψ 1 , . . . , ψ m } be an L 2 (N )-orthonormal basis of N . Let also V ⊂ N be any nonempty open set. There exist η 1 , . . . , Proof. Consider the map . Now if S a = 0, then in particular S a · a = a 1 ψ 1 + . . . + a m ψ m 2 L 2 (V ) = 0 and thus a 1 ψ 1 | V + . . . + a m ψ m | V = 0. It follows that a 1 ψ 1 + . . . + a m ψ m = 0 in N by elliptic unique continuation, showing that a = 0 and that S is invertible.
Since the matrix S is invertible, T is surjective. Finally, since C ∞ c (V ) is dense in L 2 (V ) we have that T (C ∞ c (V )) is a dense subspace of R m . Since R m is finite dimensional, it follows that T (C ∞ c (V )) = R m . We can thus choose η j ∈ C ∞ c (V ) with T η j = e j for 1 ≤ j ≤ m. Proof of Proposition A.2. Let F ∈ (H s (U )) * satisfy F (Pf | U ) = 0 for all f ∈ C ∞ c (Γ). By the Hahn-Banach theorem, it is enough to show that F (u) = 0 for all u ∈ S.
Let (M, g) ⊂⊂ (M 1 , g) ⊂⊂ (N, g), where M 1 is a compact manifold with boundary and N is a closed connected manifold. Let A be an elliptic second order operator in N such that Aw| M = L(w| M ) for w ∈ C ∞ (N ). Also, defineF ∈ H −s (N ) bŷ We defineF ∈ H −s (N ) with supp(F ) ⊂ U ∪ V bỹ Lemma A.4 ensures thatF is L 2 -orthogonal to N .

Now, by Lemma A.3 there is a unique solution
It will be convenient to choose smooth approximationsF j =F j + η j , wherê In particular, one has A * v = 0 near ∂M , so v is C ∞ near ∂M by elliptic regularity. We will next solve In the last expression, E(Pf ) is supported in M int 1 and A * v j | M 1 vanishes outside a neighborhood of U . Thus this expression can be understood as an integral over M . Integrating by parts, we get Here ∂ A * ν is the normal derivative associated with A * , and we used that ϕ j | ∂M = 0. Now, if W is a small neighborhood of ∂M , the higher regularity estimate in Lemma A.3 implies that v j → v in H m (W ) for any m ≥ 0. Thus one has ∂ A * ν ϕ j → ∂ A * ν ϕ in H m (∂M ) for any m ≥ 0, and consequently 0 = (∂ A * ν ϕ, f ) ∂M . Since the previous result holds for all f ∈ C ∞ c (Γ), we see that ϕ solves Finally, let u ∈ S, and let E be a bounded extension operator from H s (U ) to H s (N ) so that supp(Ew) ⊂ M int for any w ∈ H s (U ). Then, since Using that ψ j ∈ C ∞ c (U ) and Lu = 0 in U , we may integrate by parts in U and obtain that F (u) = 0 for all u ∈ S. This concludes the proof.
As a consequence of the Runge approximation property we can find global harmonic functions with prescribed 2-jet at a given point, up to the natural restrictions given by the equation ∆ g u = 0. The existence of a local harmonic function with this property is found e.g. in [BW03, Lemma A.1.1]), but we give the details for completeness. Proof. It is enough to do the proof in the case where p ∈ M int . For if this has been done, and if p ∈ Γ, we may extend the manifold M near p to a slightly larger manifold M 1 so that p ∈ M int 1 and ∂M \ Γ ⊂⊂ ∂M 1 . Since p is an interior point of M 1 , we can find a harmonic function u 1 in M 1 having the correct second order Taylor expansion at p and satisfying u 1 | ∂M \Γ = 0. Choosing f = u 1 | ∂M implies that the solution of −∆ g u = 0 in M with u| ∂M = f satisfies u = u 1 | M and has the required behaviour at p.
Thus, assume that p ∈ M int . The proof will be given in four steps.
Step 1. First we find a local g(p)-harmonic function with prescribed 2jet at p. Let x = (x 1 , . . . , x n ) be normal coordinates in a small geodesic ball U centered at p. In these coordinates, let ξ 0 = (ξ 0 ) j dx j | p and H 0 = H jk dx j ⊗ dx k | p . Define the function Clearly u 0 ∈ C ∞ (U ), u 0 (p) = a 0 and du 0 (p) = ξ 0 . The Hessian, computed in normal coordinates, is given by Since H 0 is trace free it follows that ∆ g(p) u 0 = n j=1 H jj = 0 in U , i.e. u 0 is harmonic in U with respect to the metric g(p) with coefficients frozen at p.
Step 2. Next we find a local g-harmonic function near p with 2-jet (0, 0, H 0 ). This is done by perturbing the functions u 0 from Step 1 in small balls, see e.g. [Ta96, Proposition 5.10.4]. Let u ∈ C ∞ (B(p, ε)) solve ∆ g u = 0 in B(p, ε), u| ∂B(p,ε) = u 0 that U ⊂⊂ M int and M \ U is connected. Consider the harmonic function u 0 in U with u 0 | B 1 = 1 and u 0 | B 2 = 0. By Proposition A.2, there exist By Sobolev embedding we also have u f j | U − u 0 L ∞ (U ) → 0. Choosing f = f j for large enough j implies that u f (x) = u f (y). Now assume that x ∈ M int and y ∈ ∂M . We choose Γ ′ ⊂⊂ Γ \ {y}, and use the argument above to find f ∈ C ∞ c (Γ ′ ) with u f (x) = 0. Then one also has u f (x) = u f (y) = 0. Next, the case where x ∈ Γ and y ∈ M int reduces to the previous case by interchanging x and y. Finally, if x ∈ Γ and y ∈ ∂M with x = y, choose some f ∈ C ∞ c (Γ) with f (x) = f (y) to obtain that u f (x) = u f (y) as required.
(b) The result follows from Proposition A.5.
Finally, let us give a Runge approximation result for a linear source problem used for studying the inverse problem for nonlinear equations. Then the set Proof. Let F ∈ (H s (U )) * satisfy F (Kf | U ) = 0 for all f ∈ C ∞ c (W ). By the Hahn-Banach theorem, it is enough to show that F (u) = 0 for all u ∈ S. As in the proof of Proposition A.2, we take (M, g) ⊂⊂ (M 1 , g) ⊂⊂ (N, g) and extend L to an elliptic second order operator A on N . Given F ∈ (H s (U )) * , we defineF ∈ H −s (N ), supp(F ) ⊂ U , bŷ Continuing as in the proof of Proposition A.2, we may findF ∈ H −s (N ) so thatF =F + η with supp(η) ⊂ V , where V ⊂ N \ M 1 , andF is L 2 -orthogonal to the kernel of A. Moreover we may findF j ∈ C ∞ c (M int ∪ V ) withF j →F in H −s (N ), and using the same procedure as in the proof of Proposition A.2 we can representF andF j as Similarly, in M we have In the last expression, E(Kf ) is supported in M int 1 and A * v j | M 1 vanishes outside a neighborhood of U ⊂ M int . Thus this expression can be understood as an integral over M . Integrating by parts, we get Here we used that Kf and ϕ j vanish on ∂M and thus the boundary terms vanish. The higher regularity estimate in Lemma A.3 implies that ϕ j → ϕ in H m (W ) for any m ≥ 0, and consequently Since the previous result holds for all f ∈ C ∞ c (W ), we see that ϕ solves Also, ϕ is smooth in the connected set M \ U . The unique continuation principle implies that ϕ| M \U = 0. Thus ϕ ∈ H −s+2 U (N ). As in the proof of Proposition A.2, we have ψ j ∈ C ∞ c (U ), ψ j → ϕ in H −s+2 (N ). If E denotes a bounded extension operator from H s (U ) to H s (N ) with supp(Ew) ⊂ M int for any w ∈ H s (U ), we have If u ∈ S, we can integrate by parts to see that F (u) = 0.
As a consequence, we obtain an analogue of Proposition A.1 where boundary sources are replaced by interior sources.
Proof. The proof is analogous to that of Proposition A.1, except that we need a version of Proposition A.5 for 0-and 1-jets where −∆ g is replaced by L (this can be found in [BJS79, Theorem I.5.4.1]), and we need to use the Runge approximation result in Proposition A.6.
Proof of Proposition 6.6. The proof is completely analogous to the proof of Proposition 2.1 above. Let x 0 ∈ M int 1 and s + 2 > n/2. We have by elliptic regularity [Ta96, Proposition 5.11.2] and by Sobolev embedding that Thus R maps M int to H −s (W ).
That R is k times Fréchet differentiable R : M int → H −s−1−k (W ), and C ∞ smooth R : M int → D ′ (W ), follows as in the proof of Proposition 2.1 above by using elliptic regularity [Ta96, Proposition 5.11.2].
B.2. Auxiliary results used in 2D Calderón problem. We prove the existence of the special isothermal coordinates used in Lemma 5.3 to prove the determination result near the boundary for harmonic functions in dimension 2.
We set Γ ′ = ψ 1 (Γ ∩ Ω 1 ) ∩ ψ 2 (Γ ∩ Ω 2 ). We next transform to isothermal coordinates. For this, let u j , j = 1, 2, denote the globally harmonic functions on (M j , g j ) solving ∆ g j u j = 0 on M j u j = h on ∂M Since the gradient of h is nonvanishing on Γ, so is the gradient of u j on Γ. Since u j are global harmonic functions, the fact that the DN maps agree on Γ implies that ∂ ν 1 u 1 | Γ = ∂ ν 2 u 2 | Γ .
We form local harmonic conjugates toũ j by solving * j dũ j = dṽ j , j = 1, 2, forṽ j locally. Here * j is the Hodge star of (the coordinate representation of) the metric g j . A local solutionṽ j exists by Poincaré lemma since * jũ j is closed by the harmonicity ofũ j (the proof of the Poincaré lemma shows thatṽ j is smooth up to the boundary). We may assumeṽ j (0, 0) = 0. Since now * 2 j is −Id on 1-forms and d 2 = 0, we have thatṽ j is harmonic. Since near the origin (0, 0) we have 0 = |dũ j | 2 g j dV g j = dũ j ∧ dṽ j , we have that (ũ j ,ṽ j ) is a coordinate system on a neighborhood Ω ⊂ R 2 ∩ {t ≥ 0} of the origin (0, 0) = ψ 1 (p) = ψ 2 (p). Redefine Γ ′ as Γ ′ ∩ Ω.
We have that the Jacobian matrices of T j agree on Γ ′ . This is due to (58) and because dṽ 1 = * 1 dũ 1 = * 2 dũ 2 = dṽ 2 (60) holds since Hodge star operators * j agree in the coordinates ψ j by (56). Finally we are ready to pass to isothermal coordinates The coordinates U j = (u j , v j ) are isothermal coordinates, since * g j du j = dv j due to (59) and the coordinate invariance of d and * . Even though we might have that the set Γ ′ is not mapped to {t = 0} by T j , we have the origin on {x 2 = 0}, we have that the Cauchy data of u j f agree in these isothermal coordinates. This is (2), which concludes the proof.
Finally, for completeness we give the standard characterization of isothermal coordinates in terms of the coordinate expression of the metric.
Since this holds for all 1-forms α, we have that du 1 = − * du 2 .
B.3. Wellposedness and linearization of quasilinear equations. This subsection contains the proofs of Propositions 6.1 and 6.4.
Proof of Proposition 6.1. Let X = C α (M ), Y = C 2,α (M ) ∩ H 1 0 (M ) and Z = C α (M ) be Banach spaces, where the norm and the topology of Y is given by the C 2,α (M ) norm. Define the map F : X × Y → Z, F (f, u) = Q(u) − f.
Then F (0, 0) = 0 by (25). To check that F is continuously differentiable, we use that the coefficients of Q are C 2,α and note that The expression for d 2 ds 2 Q(u + sv) in terms of the coefficients A and B and derivatives of v is long. It suffices to say that it is an expression that contains up to second order derivatives of A ab and B evaluated at (x, u+sv, du+sdv), and up second order derivatives of u and v. Thus there is a constant C, uniform over (u, v, s) in a bounded set of Y × Y × R, such that d 2 ds 2 Q(u + sv) where L u : Y → Z is the linearization of Q at u ∈ Y , given by The linearization L u depends continuously on u ∈ C 2,α (M ). It follows that We write u t = tv t , so that for t > 0 one has A ab (x, tv t , tdv t )∇ a ∇ b v t + 1 t B(x, tv t , tdv t ) = f, v t | ∂M = 0.
Let also v 0 be the unique solution of We wish to show that v t → v 0 in C 1 (M ) as t → 0.
If this is true, then also which proves the theorem. We have In the second to last equality we used that B(x, 0, 0) = 0. We wish to show that F t L ∞ (M ) ≤ Ct.
This is obtained from the expression for F t given above, the estimate This concludes the proof.