The Linearized Calderón Problem on Complex Manifolds

In this note we show that on any compact subdomain of a K¨ahler manifold that admits sufficiently many global holomorphic functions, the products of harmonic functions form a complete set. This gives a positive answer to the linearized anisotropic Calder´on problem on a class of complex manifolds that includes compact subdomains of Stein manifolds and sufficiently small subdomains of K¨ahler manifolds. Some of these manifolds do not admit limiting Carleman weights, and thus cannot be treated by standard methods for the Calder´on problem in higher dimensions. The argument is based on constructing Morse holomorphic functions with approximately prescribed critical points. This extends earlier results from the case of Riemann surfaces to higher dimensional complex manifolds.


Introduction
If (M, g) is a compact connected oriented Riemannian manifold with C ∞ boundary ∂M and if q ∈ C ∞ (M ), we consider the Dirichlet problem for the Schrödinger equation, Here we assume that f vanishes to infinite order at ∂M since we know by boundary determination [8] that Λ g,q determines q| ∂M to infinite order, and thus the linearization can be reduced to the case of smooth potentials vanishing to any order at ∂M . This assumption removes the possible boundary contributions in the stationary phase argument in the proof of Theorem 1.4.
The methods of [15] and [10] also yield a positive answer to Conjecture 1.2 if dim(M ) = 2 or if dim(M ) ≥ 3, (M, g) is conformally transversally anisotropic, and the geodesic X-ray transform on the transversal manifold is injective (see [10] for more details). However, even the linearized Calderón problem remains open in general when dim(M ) ≥ 3. We refer to [11] for a recent result related to recovering transversal singularities, and to [9,30] for results on the linearized partial data problem in Euclidean space.
In this work, our objective is to extend the powerful Riemann surface methods of [15] to some class of higher dimensional complex manifolds. It turns out that the linearized problem can be solved on complex manifolds that have sufficiently many global holomorphic functions. Let us first give some related definitions on complex manifolds with boundary. The above definitions often appear in connection with Stein manifolds. Let X be complex manifold without boundary, let dim C (X) = n, and let O(X) be the set of holomorphic functions on X. One says that X is a Stein manifold if . . , f n ∈ O(X) which form a complex coordinate system near p; and (iii) X is holomorphically convex, meaning that for any compact K ⊂ X the holomorphically convex hull is also compact.
Examples of Stein manifolds include open Riemann surfaces, domains of holomorphy in C n , and complex submanifolds of C N that are closed in the relative topology. Conversely, any Stein manifold of complex dimension n admits a proper holomorphic embedding in C 2n+1 (hence nontrivial Stein manifolds are noncompact). See e.g. [12] for these basic facts on Stein manifolds.
The next result gives a positive answer to the linearized Calderón problem on complex manifolds that satisfy (a) and (b) above.

Theorem 1.4
Let M be a compact complex manifold with C ∞ boundary. Assume that M is holomorphically separable and has local charts given by global holomorphic functions, and let g be any Kähler metric on M . If f ∈ C ∞ (M ) vanishes to infinite order at ∂M and satisfies Here are examples of manifolds covered by the above theorem: (1) If X is a Stein manifold and g is a Kähler metric on X (for instance g could be the metric induced by the embedding of X in some C N ), then any compact C ∞ subdomain M of X satisfies the conditions in the theorem.
(2) More generally, the condition (iii) in the definition of Stein manifolds is not required. For example, any complex submanifold of C N satisfies (i) and (ii) since the functions f and f j can be constructed from the coordinate functions z 1 , . . . , z N in C N . Hence any compact C ∞ subdomain M of some complex submanifold of C N , equipped with a Kähler metric g, satisfies the conditions in the theorem.
(3) Let (X, g) be a Kähler manifold, and let U be a complex coordinate neighborhood in X. If M is a compact C ∞ subdomain of U , then (M, g) satisfies the required conditions: if f 1 , . . . , f n are complex coordinates in U , the conditions (a) and (b) hold just by using the functions f 1 , . . . , f n .
Most of the earlier progress in Conjectures 1.1 and 1.2 is based on the notion of limiting Carleman weights (LCWs), introduced in the Euclidean case in the fundamental work of Kenig, Sjöstrand and Uhlmann [23]. The notion of LCWs placed the method of complex geometrical optics solutions, which has been used intensively since [31], in an abstract general context. LCWs have been particularly useful for partial data problems [15,18,19,23], see also the surveys [16,20]. The study of LCWs in the geometric case was initiated in [8] and applied further e.g. in [7,21,22]. In [8] it was also shown that the existence of an LCW (at least with nonvanishing gradient) is locally equivalent to a certain conformal symmetry that a generic manifold in dimensions ≥ 3 will not satisfy [1,27]. This conformal symmetry has been studied further in [2][3][4] in terms of the structure of the Cotton or Weyl tensors of the manifold.
One of the results in [3] states that CP 2 with the Fubini-Study metric g does not admit an LCW near any point. Since (CP 2 , g) is a Kähler manifold, compact C ∞ subdomains in CP 2 provide examples of manifolds where Theorem 1.4 applies but where methods based on LCWs fail.
The proof of Theorem 1.4 extends the arguments in [5] for domains in C and [15] for Riemann surfaces to the case where dim C (M ) ≥ 2. Here is an outline of the argument: (1) Since g is a Kähler metric, we have the factorization Thus holomorphic and antiholomorphic functions are harmonic.
(2) We show that there is a dense subset S of M such that for any p ∈ S, there exists a Morse holomorphic function Φ in M having a critical point at p. This is based on the fact that M has local charts given by global holomorphic functions. This allows us to construct a global holomorphic function Φ 0 in M having a prescribed nondegenerate critical point, and further to find Morse functions arbitrarily close to Φ 0 using a transversality argument.
(3) The next step is to show that if Φ is a Morse holomorphic function with critical point at p, there is a holomorphic amplitude a such that a(p) = 1 and a vanishes at all other critical points of Φ. This follows from the fact that M is holomorphically separable.
(4) Finally, for h > 0 we define Here u 1 is holomorphic and u 2 is antiholomorphic, hence both are harmonic, and one has Since Im(Φ) is a Morse function having critical point at p ∈ S, and since a(p) = 1 but a vanishes at the other critical points, letting h → 0 (after multiplying by a suitable power of h) and applying the stationary phase argument implies that f (p) = 0. Since this holds for all p in the dense set S, we obtain f ≡ 0.
Our argument for treating the linearized Calderón problem relies on the fact that we can produce solutions of Δ g u = 0 out of holomorphic or antiholomorphic functions. Dealing with the full nonlinear Calderón problem would require solutions to the Schrödinger equation (−Δ g + q)u = 0, which are typically obtained via Carleman estimates when LCWs are present. However, the manifolds that we are considering do not necessarily admit LCWs, and thus at the moment the methods in this paper are restricted to the linearized problem.
This paper is organized as follows. Section 1 is the introduction. Section 2 includes some preliminaries related to complex manifolds, and Section 3 gives the proof of Theorem 1.4.

Preliminaries
In this section we recall standard facts concerning complex and Kähler manifolds. We refer to [17,28] for more details.

Complex Manifolds
If M is a complex manifold, let z = (z 1 , . . . , z n ) be a holomorphic chart U α → C n , and write z j = x j + iy j with x j and y j real. There is a canonical almost complex structure J on M , defined for holomorphic charts by Conversely, if M is a differentiable manifold equipped with an almost complex structure J (so it is necessarily even dimensional and orientable), then by the Newlander-Nirenberg theorem M has the structure of a complex manifold if J satisfies an additional integrability condition. Let M be a complex manifold. From now on we denote by T p M the complexified tangent space, and by T * p M the complexified cotangent space. We also denote by J the C-linear extension of the almost complex structure. Locally in holomorphic coordinates, and the dual basis is Since J 2 = −Id, the eigenvalues of J acting on T p M are ±i. We define These are the holomorphic and antiholomorphic tangent spaces. Locally Since J also acts on T * p M , we define T 1,0 * p M and T 0,1 * p M in the same way and have that locally We move on to differential forms. Because of the splitting Locally, any form u ∈ A p,q (M ) can be written as If π p,q : A k (M ) → A p,q (M ) is the natural projection (here k = p + q), we can define the ∂ and ∂ operators as follows: Since d = ∂ + ∂ and d 2 = 0 on forms of type (p, q), we have (∂ + ∂) 2 = 0 on such forms and consequently

Hermitian Metrics
For the inverse problem, we want to have a Laplace-Beltrami operator on M . For this one needs a Riemannian metric. In the case of complex manifolds, it is natural to assume a compatibility condition.

Definition 2.4
Let M be a complex manifold with almost complex structure J. We say that a Riemannian metric g on M is compatible with the almost complex structure if g(Jv, Jw) = g(v, w) for all v, w.
If z ∈ M , let h z ( · , · ) = ( · , · ) z = ( · , · ) be the sesquilinear extension of g z to the complexified tangent space. Then Conversely, if ( · , · ) z is a family on symmetric sesquilinear forms on the complex tangent spaces, satisfying the above three conditions and varying smoothly with z, then the restriction to the real tangent bundle is a compatible Riemannian metric.
We call h a Hermitian metric on M , and (M, h) a Hermitian manifold. It naturally induces a metric on the complex cotangent spaces. One obtains inner products on the exterior powers and also on the space L 2 A k (M ) of complex k-forms with L 2 coefficients, is orthogonal with respect to this inner product. We extend the Hodge star operator * as a complex linear operator on complex forms. Since then the orthogonality implies that * maps A p,q (M ) to A n−q,n−p (M ). The L 2 inner product induced by the Hermitian metric allows us to define the adjoints of ∂ and ∂ as operators In terms of the Hodge star operator they may be expressed as

Kähler Manifolds
If (M, h) is a Hermitian manifold and g is the corresponding Riemannian metric, we would like to factor the Laplace-Beltrami operator Δ = Δ g = d * d acting on functions in terms of the ∂ and ∂ operators. The situation is particularly simple on Kähler manifolds. The manifold is Kähler iff dω = 0 (and then ω is called a Kähler form). Clearly, the metric g can be recovered from the Kähler form and vice versa. One knows that on a Kähler manifold, near any point there is a smooth function f such that

Definition 2.5 A Hermitian manifold (M, h) is called
Thus, the metric on a Kähler manifold locally only depends on one function.
The important fact for our purposes is the following (this is a special case of the Kähler identities when acting on 0-forms).

Lemma 2.6 If (M, h) is Kähler, then the Laplace-Beltrami operator on functions satisfies
Here ∂ * and ∂ * are the formal adjoints of ∂ and ∂ in the L 2 inner product as discussed above. Thus The following immediate consequence is the crucial point for solving the linearized Calderón problem.

Remark 2.8
In the solution of the inverse problem on Riemann surfaces, one needs to construct holomorphic functions in M with prescribed zeros or Taylor series in a finite set of points. This was done in [15] by using a version of the Riemann-Roch theorem. There are well known extensions of the Riemann-Roch theorem to higher dimensional complex manifolds, such as the Riemann-Roch-Hirzebruch theorem. In this work, instead of using Riemann-Roch type results, we will construct the required holomorphic functions directly from the assumption that the manifold has local charts given by global holomorphic functions.

Linearized Inverse Problem
In this section we will prove Theorem 1.4. This will be done by constructing complex geometrical optics solutions to the Laplace equation, obtained from holomorphic functions given in the next proposition.  for all u j ∈ C ∞ (M ) with Δ g u j = 0 in M . Since any harmonic function in H 1 (M ) can be approximated by C ∞ harmonic functions by smoothing out its boundary data, we may assume that the above identity holds for harmonic functions in C k (M ) where k ≥ 2.
We now use Proposition 3.1: for any point p in the dense subset of S and for h > 0, we choose holomorphic functions with critical points {p, p 1 , . . . , p N }, a(p) = 1, and a(p 1 ) = · · · = a(p N ) = 0. Define u 2 =v 2 . Since g is a Kähler metric on M , u 1 and u 2 are harmonic functions in M . We obtain that M f e 2iIm(Φ)/h |a| 2 dV g = 0 for all h > 0. Since p is a critical point of Im(Φ), since a(p) = 1 but a vanishes at all the other critical points of p, and since f vanishes to infinite order on ∂M , the stationary phase argument implies that f (p) = 0. Since this is true for all p in a dense subset of M , it follows that f ≡ 0.
The next result is the first step in constructing holomorphic functions with prescribed critical points. We will later need to perturb these functions so that their real and imaginary parts become Morse. Before the proof, we give an elementary lemma related to critical points. Recall that if u is a C ∞ real valued function in a real manifold M , then we can define the Hessian as the 2-tensor D 2 u where D is the Levi-Civita connection. If for a given point p we have du(p) = 0, then the Hessian D 2 u(p) does not depend on the metric and if x are local coordinates near p, one has Note that the holomorphic Hessian of f at p is simply given by the T 1,0 * Proof The first claim is obvious. To prove the second claim, assume that df (p) = 0 and extend D 2 u(p) and D 2 v(p) as complex bilinear forms on CT p M . Let z be complex local coordinates Since f l (p) = 0 one has dΦ(p) = 0, so p is a critical point of both Re(Φ) and Im(Φ).
The holomorphic Hessian of Φ in the (f 1 , . . . , f n ) coordinates is given by We may identify the 2-tensor D 2 hol Φ(p) with the corresponding operator mapping vectors to 1-forms. Thus D 2 hol Φ(p)(a j ∂ f j ) = 2a l df l , showing that D 2 hol Φ(p) is nondegenerate on T 1,0 p M . Lemma 3.3 implies that p is a nondegenerate critical point both for Re(Φ) and Im(Φ).
The next step is to show that we may approximate the functions in Lemma 3.2 by holomorphic functions whose real and imaginary parts are Morse. For Riemann surfaces such a result was given in [15], based on a transversality argument from [32]. We will follow the same approach.
Fix some C ∞ Riemannian metric g on M and define where ∇ is the Levi-Civita connection and L ∞ (M ) = L ∞ (M, dV g ). Equip H with the norm which makes H a Banach space. We claim that the set is dense in H. If this holds, then for any point p 0 ∈ M int we may use Lemma 3.2 to find Φ = ϕ + iψ ∈ O(M ) having a nondegenerate critical point at p 0 . The density of (3.2) implies that there are Φ (l) = ϕ (l) + iψ (l) in C k (M ) ∩ O(M int ) such that ϕ (l) and ψ (l) are Morse and ϕ (l) → ϕ, ψ (l) → ψ in C k (M ). It follows that Hess g (ϕ (l) ) → Hess g (ϕ) on M . In particular, if x are Riemannian normal coordinates at p 0 , both matrices (∂ x j ∂ x k ϕ (l) ) and (∂ x j ∂ x k ϕ) are nondegenerate at p 0 for l large, and by the inverse function theorem the vector fields (∂ x j ϕ (l) ) and (∂ x j ϕ) are invertible maps in some neighborhood of p which is independent of l. It follows that there exist p (l) ∈ M int with dϕ (l) (p (l) ) = 0 and p (l) → p 0 . This proves that the set is dense in M . The result then follows. To prove that (3.2) is dense in H, we repeat the argument in [15] and consider the map ((u, v), p) = (p, du(p)).
We also write m u,v : M → T * M, m u,v (p) = (p, du(p)). We first observe that p is a critical point of u iff m u,v (p) ∈ T * 0 M , where T * 0 M is the zero section of T * M . If p is a critical point, X ∈ T p M and if γ(t) is a smooth curve in M withγ(0) = X, we compute = (X, D 2 u(X)).
Thus we have proved that for (u, v) ∈ H, This means that u is Morse iff m u,v is transverse to the zero section T * 0 M in T * M (see [32] for this terminology).
We next show that the set is residual (i.e., a countable intersection of open dense sets) in H. In fact, this follows from [32, Transversality Theorem 2], see also [15], provided that we can show that m is transverse to T * 0 M . Let (u, v) ∈ H, p ∈ M be such that p is a critical point of u (i.e., m(u, v, p) ∈ T * 0 M ), let (û,v) ∈ H and X ∈ T p M , and let γ be a smooth curve in M withγ(0) = X. We compute (D u,v,p Thus m is transverse to T * 0 M provided that {dû(p) ; (û,v) ∈ H} spans T * p M for any p ∈ M . This last fact follows since M has local charts given by global holomorphic functions. To see this, let p ∈ M and choose f 1 , . . . , f n ∈ O(M ) which form a complex chart near p. Write f j = u j + iv j and assume that for some a j , b j ∈ R one has n j=1 a j du j (p) + b j dv j (p) = 0.
Since 2u j = f j +f j and 2iv j = f j −f j , we obtain 0 = n j=1 c j df j +c j df j where c j = a j − ib j . Since {df j , df j } span the complex cotangent space at p, we get c j = 0 for j = 1, . . . , n. Thus a j = b j = 0, and {du j (p), dv j (p)} j=1,...,n span the real cotangent space at p. The fact that Re(−if ) = Im(f ) implies that real parts of functions in O(M ) span T * p M as well.
We have now proved that the set (3.4) is residual. The observation (3.3) and Lemma 3.3 show that the set (3.2) is dense in H as required.
Finally, we invoke the assumption that M is holomorphically separable to construct the amplitude in Theorem 3.1.
Proof of Proposition 3.1 Let S be as in Lemma 3.4, let p ∈ S, and let Φ ∈ C k (M ) ∩ O(M int ) be the function given in Lemma 3.4 so that dΦ(p) = 0 and Im(Φ) is Morse. Let {p, p 1 , . . . , p N } be the critical points of Im(Φ) in M . Since M is holomorphically separable, for each j we may find a j ∈ O(M ) such that a j (p) = 1 and a j (p j ) = 0. It is enough to choose a = a 1 · · · a N .