Stability for a magnetic Schr\"odinger operator on a Riemann surface with boundary

We consider a magnetic Schr\"odinger operator $(\nabla^X)^*\nabla^X+q$ on a compact Riemann surface with boundary and prove a $\log\log$-type stability estimate in terms of Cauchy data for the electric potential and magnetic field under the assumption that they satisfy appropriate a priori bounds. We also give a similar stability result for the holonomy of the connection 1-form $X$.


1.
Introduction. Let (M, g) be a compact Riemann surface with boundary ∂M . We will consider a connection ∇ X on a complex line bundle over M , with connection 1-form X. By the associated connection Laplacian we mean the differential operator (1) ∆ X := (∇ X ) * ∇ X = (d + iX) * (d + iX) = − (d +iX ∧ )(d + iX), where d denotes the exterior derivative, i = √ −1 and is the Hodge star with respect to the metric g. In particular, when X is real valued, ∆ X is often called the magnetic Laplacian associated with the magnetic field dX. We will restrict our attention to the case of real-valued X in this work.
By adding a complex-valued potential q we get the magnetic Schrödinger operator associated with the couple (X, q) (2) L = L X,q := ∆ X + q.
We denote by H s (M ) the Sobolev space containing functions on M with s derivatives in L 2 (M ). The Cauchy data space C L of L is defined by (3) C L := {(u, ∇ X ν u)| ∂M ∈ H 1/2 (∂M ) × H −1/2 (∂M ); u ∈ H 1 (M ), Lu = 0}, where ν denotes the outward pointing unit normal vector field to ∂M and ∇ X ν u := (∇ X u)(ν) is the normal derivative associated with X.
In this work we assume that we are working with two different magnetic Schrödinger operators L j := L Xj ,qj and their corresponding Cauchy data spaces C j := C Lj , j = 1, 2. Assuming certain a priori bounds for the norms of X j and q j , we illustrate that if the Cauchy data spaces are sufficiently similar, then so are the q j :s and X j :s respectively. The main results of this paper are: Theorem 1.1. Suppose that q 1 , q 2 are complex-valued functions and X 1 , X 2 are real-valued 1-forms such that for some p > 2, (4) q j W 1,p (M ) ≤ K, X j W 2,p (T * M ) ≤ K, j = 1, 2.
Denote by C j := C Lj the Cauchy data spaces as defined in 3 for the corresponding magnetic Schrödinger operators L j := L Xj ,qj , as defined in 1-2. Then if the distance d(C 1 , C 2 ) is small enough, there is an α > 0 such that where C = C(K, M, α).
By interpolation it is then also quite immediate to deduce: Corollary 1. If in addition to the assumptions in Theorem 1.1 we have for some k ≥ 3 q j H k (M ) ≤ K, X j H k (T * M ) ≤ K, j = 1, 2. Then there is an α > 0 such that where C = C(K, M, β), 0 ≤ s < k.
These results further quantify the uniqueness results by Guillarmou and Tzou from 2011, [7]. They showed that the Cauchy data of the magnetic Schrödinger operator L X,q uniquely determines the potential q and uniquely determines the connection X up to so-called gauge isomorphism. This result was extended, by the same authors together with Albin and Uhlmann [1], to identification of coefficients (up to gauge) in elliptic systems in 2013. We have borrowed plenty of notations and conventions from these works.
The main idea in [7] is to rewrite L j u = 0 to∂-systems with matrix-valued potentials, and then apply the idea by Bukhgeim [2]. In the identifiability case, one is able to do this due to a certain orthogonality condition on the boundary which allows one to judiciously choose conjugation factors so that they agree on the boundary. In this work the orthogonality condition will only be an approximate one and we quantify the boundary conditions of the conjugation factors by this approximate orthogonality condition (see Section 5). This process unfortunately causes the extra logarithm in the end result.
Another additional feature which we must consider is the uniformity of the estimates in the stationary phase expansion. As such we must construct slightly different phase functions than in [7] to be used in these solutions and refine several estimates from mentioned works. This work is done in Section 2.
To see that the Cauchy data space cannot determine the couple (X, q) completely, consider introducing a real-valued function f (say in H 2 (M )) whose restriction to the boundary ∂M is zero. Then the Cauchy data spaces associated with the magnetic Schrödinger operators L X,q and L X+df,q can be seen to coincide.
In [7] it is shown that the Cauchy data space determines the relative cohomology class of X. This is done through a number of steps. First by showing that C 1 = C 2 implies that d(X 1 −X 2 ) = 0 and q 1 = q 2 . If M is simply connected this would imply that X 1 and X 2 differ by an exact 1-form. Furthermore, by boundary determination, ι * ∂M (X 1 −X 2 ) = 0, where ι * ∂M denotes pullback by inclusion to the boundary. In the case of a more general Riemann surface, not necessarily simply connected, there are further obstructions so that X 1 and X 2 may no longer only differ by an exact 1-form. Namely, if E is a complex line bundle over M and there is a so-called unitary bundle isomorphism F : E → E that preserves the Hermitian inner product and satisfies ∇ X1 = F * ∇ X2 F with F = Id on E| ∂M it can again be seen that C 1 = C 2 . Having such a unitary bundle isomorphism corresponds to multiplication by a function F on M with the properties |F | = 1 and F | ∂M = 1. This is again equivalent with ι * ∂M (X 1 − X 2 ) = 0 on ∂M , d(X 1 − X 2 ) = 0 and that all periods of X 1 − X 2 is an integer multiple of 2π. I.e. for every closed loop γ in M it holds that In this case one can deduce that which is a basis for the first relative cohomology H 1 (M, ∂M ), dual to {γ j } N j=1 which are some non-homotopically equivalent loops on M , f is a function whose restriction to the boundary ∂M is zero and δ jk is the usual Kronecker delta.
In our framework we quantify the above statement by showing: is small enough, then for every closed loop γ in M it holds that where C = C(K, M, p, α).
The above result is a consequence of Theorem 1.1 and the Picard-Lindelöf Theorem. Some precaution is needed when defining relevant functions on M since in our case X 1 − X 2 is not necessarily closed, compare with [7]. This introduces extra complications and a suitable expression for X 1 − X 2 is derived to quantify its distance from an exact gauge (see Section 6).
We also refer to [8] for more background on the Calderón inverse problem for Schrödinger operators in dimension 2. In particular, [5] proves uniqueness for the usual Schrödinger operator ∆ g + q on a Riemann surface and [6] handles the corresponding partial data case. For similar results in Euclidean domains, see [15,12,11]. Related stability estimates can also be found in [16,17]. A constructive method for the reconstructing an isotropic conductivity on a Riemann surface was first obtained in [9].

2.
A primer on Riemann surfaces. We aim here to give only a brief introduction to the geometry of compact Riemann surfaces and refer to the books [3,4,13] for more comprehensive treatments.
For a closed Riemann surface (M, g) we denote by Λ k (M ), k = 0, 1, 2 the bundles of complex-valued k-forms on M . In particular, Λ 0 (M ) contains functions on M and Λ 1 (M ) = CT * M is the complexified cotangent bundle on M , containing 1-forms.
A Hermitian inner product can be defined on the vector space Λ k (M ) according to (6) λ We will often assume that our functions and forms belong to certain Sobolev spaces. Recall that for non-negative integers k and 1 ≤ p ≤ ∞, a function f is said to belong to the Sobolev space W k,p (M ) if it is k times weakly differentiable and all partial derivatives up to order k belong to L p (M ). For k = 2 we use the common notation H k (M ) := W k,2 (M ), or H s (M ) when considering non-integers s = k, and for k = 0 we have L p (M ) = W 0,p (M ). For a more thorough discussion on Sobolev spaces we refer to e.g. Section 1.3 of [18], that also covers the spaces W k,p (Λ l (M )) that we also consider in the cases l = 1, 2.
In coordinates this is It is clear that d = ∂ +∂ holds for both functions and 1-forms. The adjoints of ∂ and∂ are simply given by ∂ * = i ∂ and∂ * = −i ∂ respectively. We can now define the Laplacian of a function f on M by where δ = d * is the codifferential, i.e. the adjoint of d with respect to the inner product, and the is the induced Hodge star that maps 2-forms to 0-forms.

2.2.
Existence of suitable phase functions on M . We will now construct functions that are holomorphic and Morse, with uniformly bounded from below Hessian outside a neighborhood of their stationary points. These functions will be used as phase functions in latter arguments that allow us to estimate the degeneracy near stationary points. This must be taken into account when deriving correct remainder estimates in e.g. stationary phase expansions. Start out with any Φ ∈ H(M ) which is Morse, meaning that ifp ∈ M is a stationary point, ∂Φ(p) = 0, then the Hessian ∂ 2 p Φ(p) = 0. For the construction of such functions see [5]. Suppose that {p 1 , . . . , p n } is the set of stationary points of Φ and for k = 1, . . . , n denote by p j,k , j = 1, . . . , m k the set of points for which the values Φ(p j,k ) coincide with the critical values Φ(p k ). Then we have the following lemma: Lemma 2.1. Let Φ be a holomorphic Morse function on a compact Riemann surface M with boundary, having critical points {p k } n k=1 ⊂ M , and let P cv = {p j,k ∈ M ; Φ(p j,k ) = Φ(p k ), j = 1, . . . , m k , k = 1, . . . , n} be the set of points where the values of Φ coincides with a critical value. For any fixed δ > 0 sufficiently small we let N δ be a neighborhood of P cv defined by: where (U, φ) are charts such that φ : U → C and Then for anyp ∈ M \N δ there exists a holomorphic Morse function Φp with a critical point atp such that at all critical points p of Φp there is a c > 0 independent of the choice ofp ∈ M \ N δ such that |∂ 2 p Φp(p)| ≥ cδ 4 . Proof. The sought after function will be defined by Then so clearly ∂Φp(p) = 0, and we will show that Φp is Morse. Observe that, in a coordinate system (U, φ) containing p = φ(z) Then, sincep is bounded away from critical values of Φ, we have the following two mutually exclusive cases 1. ∂Φ(p) = 0 in which case p = p k for some k ∈ {1, . . . , n} or, 2. Φ(p) = Φ(p).
In case 1, we have with z k = φ(p k ), and since Φ is Morse there is a holomorphic chart in which ∂ 2 z Φ(φ −1 (z k )) = a k = 0. Also sincep / ∈ N δ , Φ(p k ) − Φ(p) = 0 and furthermore (for small enough δ > 0 and p close enough to p k ) we have in local coordinates, (Observe that |Φ(p k ) − Φ(p)| → 0 only ifp → p j,k for some p j,k ∈ P cv , and thus Since we only have finitely many critical points we can thus choose an a > 0, such that in a coordinate where p = φ(z) Clearly the right hand side approaches 0 only when p → p k for some k ∈ {1, . . . , n} and we have by a similar argument as in case 1 that where c > 0 is the Lipschitz constant for Φ•φ −1 . So the lower bound |ẑ−z k | ≥ δ will yield a lower bound on |z − z k | δ 2 , hence near p k we have (in local coordinates) Thus for p near p k we can again choose b > 0 such that |∂ 2 z Φp(φ −1 (z))| > bδ 4 , for allp ∈ M \ N δ , for p ∈ M : Φ(p) = Φ(p), and this holds uniformly inp ∈ M \ N δ . Remark 1. When the Riemann surface M is of a particularly simple type, e.g. if it is a equipped with a global holomorphic coordinate z (such as when M is just a domain in C) we do not need the construction in the above lemma. Since in the latter case it holds that for everyẑ = φ(p), Φ(z) = (z −ẑ) 2 is a Morse holomorphic function with a critical point atẑ.
We are going to use the method of stationary phase with the above constructed phase function in order to later derive estimates. For convenience we replace δ by √ δ in the above lemma, i.e. we consider the phase function Φ = Φp on M \ N √ δ , where |∂ 2 p Φp| ≥ cδ 2 . Recall that C ∞ 0 as usual denotes smooth and compactly supported functions/form. The dependence on the parameters h and δ will be important. Lemma 2.2 (Stationary phase). Suppose ψ is a smooth function and K ⊂ C is a set containing only one critical pointẑ of ψ and that |∂ 2 z ψ(ẑ)| ≥ δ 2 > 0. Then for every u ∈ C ∞ 0 (K), 1.
2.3. The inverses of ∂ and∂. Section 2 in [7] contain lemmas regarding the construction and boundedness of right inverses of the Cauchy-Riemann operators. These results ensures that the constructed solutions to the Dirac-systems that are considered in the paper are well-behaved. As we will use a very similar approach in Section 3, we recite some of these essential lemmas below. Lemma 2.3 (Right inverse to∂, [7]). There exists an operator are supported in complex charts U j , bi-holomorphic to a bounded open set Ω ⊂ C with complex coordinate z, and such that χ = χ j is equal to 1 on M , then as operators So by (ii), the inverse can be expressed by the usual Cauchy operator, which is calledT here, plus a smoothing term, in local coordinates. A similar result for∂ * is given by are supported in complex charts U j , bi-holomorphic to a bounded open set Ω ⊂ C with complex coordinate z, and such that χ = χ j is equal to 1 on M , then as operators Here we again observe that (ii) says that the inverse can be expressed as a Cauchy-type operator plus a smoothing term, in local coordinates. For the proof we again refer to [7]. The main use of Lemma 2.3 and 2.4 in [7] is to prove Lemma 2.5 and 2.7. We could also make use of Lemma 2.3 and 2.4 in order to prove estimates for the solutions of the∂-systems we will consider in later sections. In order to prove Theorem 1.1 we will require more refined and explicit estimates than what is needed in order to prove identifiability of the pair (X, q), so this is another reason for restating also the above lemmas.
Let us now assume that M is strictly contained in some larger surface N . Suppose p, q ∈ [1, ∞] and define the continuous extension operator from M to N by ) denotes the subspace of compactly supported type (0, 1)-forms in W k,p (Λ 0,1 (N )), k = 1, 2, with a range made of type (0, 1)-forms with support in N δ = {n ∈ N ; d(n,M ) ≤ δ} for some δ > 0. We also denote by Lemma 2.5 (Lemma 2.2 from [7]). Let ψ be a real-valued smooth Morse function on N and let∂ −1 As we will be required to use the special phase functions constructed in Lemma 2.1 we state below a more explicit version of the above lemma in order to see the δ-dependence. Lemma 2.6 (Refinement of Lemma 2.2 from [7]). Let Φp be a holomorphic Morse function as in Lemma 2.1 and let ψ = Im Φp. Define∂ −1 ψ := R∂ −1 e −2iψ/h E wherē ∂ −1 is the right inverse of∂ : W 1,p (M ) → L p (Λ 0,1 (M )). Let p > 2, then there are constants ε > 0, C > 0 independent of h, δ, andp such that for all ω ∈ W 1,p (Λ 0,1 (M )), Proof. The last inequality comes from interpolating the estimates 8 and 9 so we will prove 8 and 9. We recall the Sobolev embedding W 1,p (M ) ⊂ C α (M ) for α ≤ 1−2/p if p > 2, and we shall denote by T the Cauchy-Riemann inverse of ∂z in C: If Ω, Ω ⊂ C are bounded open sets, then the operator 1l Ω T maps L p (Ω) to L p (Ω ). Clearly, since E and R are continuous operators, it suffices to prove the estimates for compactly supported forms ω ∈ W 1,p (T * 0,1 M ) on M . Thus by partition of unity, it suffices to assume that ω is compactly supported in a chart biholomorphic to a bounded domain Ω ∈ C, and since the estimates will be localized, we can assume with no loss of generality that ψ has only one critical point, say z 0 ∈ Ω (in the chart). The expression of∂ −1 ψ (f dz) in complex local coordinates in the chart Ω satisfies where K is an operator with smooth kernel and χ ∈ C ∞ 0 (C).
Let us first prove 8. Let ϕ ∈ C ∞ 0 (C) be a function which is equal to 1 for |z − z 0 | > 2 0 and to 0 in |z − z 0 | ≤ 0 , where 0 > 0 is a parameter that will be chosen later (it will depend on h). Using Minkowski inequality, one can write when q < 2 On the support of ϕ, we observe that since ϕ = 0 near z 0 , we can use and the boundedness of T on L q to deduce that for any q < 2 The first term is clearly bounded by The second term can be bounded by || f∂φ δ 4 ,∂ϕ is only supported in a neighbourhood of radius 2 0 . Therefore we obtain The third term can be estimated by Combining these four estimates with 11 we obtain ).
Combining this and 10 and optimizing by taking 0 = h 1/3 , we deduce that We now move on to the smoothing part given by K(e −2iψ/h f ). Take χ to be a compactly supported function in Ω such that it is equal to 1 on the support of f , we see that . By applying stationary phase and Part 1 of Lemma 2.2, we easily see that For the first term, we write f := f − χf (z 0 ) and we integrate by parts to get, for some smoothing operator K By the fact that K and K are smoothing, we see that for all k ∈ N Using the Hessian estimate of the phase in Lemma 2.1, the Sobolev embedding W 1,p ⊂ C α for α = 1−2/p and f (z 0 ) = 0, we can estimate both terms by C for any q ∈ [1, ∞] and p > 2. Combining 13 and 12 we see that 8 is established.
Let us now turn our attention to the case when ∞ > q ≥ 2, one can use the boundedness of T on L q and thus (14) ||χT and the boundedness of T on L q to deduce that for any q ≤ p, 11 holds again with all the terms satisfying the same estimates as before so that since now q ≥ 2. Now combine the above estimate with 14 and take 0 = h 1 2 we get The smoothing operator K is controlled by 13 for all q ∈ [1, ∞] and therefore we obtain 9.
The final lemma of this section follows Lemma 2.5, and is proved in exactly the same way, but for a corresponding∂ * −1 ψ . Here the restriction and extension operators must be interpreted in a different way, namely that R restricts sections of Λ 0,1 (N ) to M and E is a continuous extension from W k,p (M ) to W k,p c (N ), k = 0, 1 where the functions in its range are supported in some N δ . Lemma 2.7 (Refinement of Lemma 2.3 in [7]). Let Φp be a holomorphic Morse function as in Lemma 2.1 and let ψ = Im Φp. Define∂ * −1 ψ q > 1, p > 2 there exists constants ε > 0, C > 0 independent of h, δ, andp such that for all v ∈ W 1,p (M ), By interpolation, there is thus an ε > 0 and C > 0 such that for all v ∈ W 1,p (M ) 3. Inverse boundary problems for systems. Our approach mimic the idea of Bukhgeim [2] that makes it possible to study a first order differential operator represented by a matrix instead of 2.
The idea is to consider the bundle Σ(M ) := Λ 0 (M ) ⊕ Λ 0,1 (M ) and the∂-system The operator D is often called a Dirac operator and is formally self-adjoint. The potential V will be built up by functions Q ± on the diagonal and 1-forms A, A on the antidiagonal. The action of V on Σ(M ) must be interpreted in the correct way, e.g. in our case it will be of the form In [7] it is assumed that V is a diagonal endomorphism of Σ(M ). This condition was relaxed in [1] and we will mainly follow the methodology of this work in this section.
We will make use of the following boundary integral identity, which is easily proved by integration by parts. In the above, ι * ∂M denotes pullback by inclusion and Our goal will be to reduce to the case when V is in fact diagonal. This is the situation first studied by Bukhgeim in the planar case in [2], laying the foundation to the manifold version in Proposition 2.5 in [1]. In our case, due to the modified phase function Φ that we must resort to, our version of that proposition reads: Proposition 1. Let Φp be a holomorphic Morse function as in Lemma 2.1.

If
for some p > 2, then there exist solutions to (D + V )F h = 0 on M of the form for any anti-holomorphic one form b and so that for some ε > 0

There exists solutions to
for any holomorphic function a and so that for some ε > 0, 15 still hold.
Proof. We prove item 1 only as item 2 is analogous. The proof closely resembles the one of Proposition 3.1 in [7] but makes use of the refined Lemma 2.6 (or similar) instead of Lemma 2.2 in [7] (and it's variants respectively). We make use of the fact that the mentioned lemmas contain very explicit expressions for the remainder terms r h ∈ Λ 0 (M ), s h ∈ Λ 0,1 (M ). It can be seen, following the same computation as in [7], that the remainder terms must solve the system In the caseQ ∈ L ∞ (M ),F ∈ W 1,p (M ) we can choose a = 0 and it follows that r h must satisfy ψQ . The idea is now to solve through a Neumann series. From Lemma 2.6 it follows that S h L 2 →L 2 ≤ C δ 4 (h 1/2−ε ), 0 < ε < 1/2, c.f. Lemma 2.4 in [1]. We remark that also this result (or the corresponding Lemma 3.1 in [7]) must be modified slightly. However, as we will later require that h ε δ −7/2 to be small for some (preferebly as large as possible) 0 < ε < 1/2. This will ensure that the bound: 3.1. The distance between Cauchy data for systems. We would like to compare the Cauchy data for different potentials V 1 , V 2 in a meaningful and quantitative manner. One standard method is to use a pseudo-distance inspired by the so-called Hausdorff distance.
Recall the definition of the Cauchy data spaces C Lj , as in 3. Assuming that for f j ∈ H k−1/2 (∂M ), for some k ≥ 1. Then ∇ Xj ν u j = g j ∈ H k−3/2 (∂M ) and we may consider a norm on C Lj defined by, and define .
Correspondingly for the system formulation, we think of the Cauchy data C V is made up of boundary values ι * ∂M (u, ω) T for H k -solutions U = (u, ω) T to (D + V )U = 0. We may consider C V being a subset of H s (∂M ) ⊕ H s−1 (∂M ), whose norm we can use to introduce a distance when considering two potentials. Suppose we have two traces, (f j , λ j ) T ∈ C Vj , j = 1, 2, then we can consider the quantity The H s (Σ(∂M ))-norm is defined, for s ≥ 1/2, by Then we can define a distance between Cauchy data as Proposition 2. If (D + V j )U j = 0 are the system formulations of the problems L j u j = 0, then Proof. Pick any (f j , g j ) ∈ C Lj , then f j = u j | ∂M , g j = ∇ Xj ν u j | ∂M where u j ∈ H k (M ) solves L j u j = 0. Then the corresponding solutions U j = (u j , ω j ) to (D + V j )U j = 0 has Cauchy data (u j | ∂M , ι * ∂M ω j ) = (u j | ∂M , ι * ∂M π 0,1 ∇ Xj u j (ν)). So clearly, g 1 ), (f 2 , g 2 )).
We will from now on use the notation d(C 1 , C 2 ) = d(C L1 , C L2 ) and not bother much with estimates that could be expressed with d (C V1 , C V2 ) instead. However, we will still need the definition of d when we later will solve a diagonalized version of the problems (D + V j )U j = 0 and in that case Proposition 2 will become weaker.
Furthermore, we will from now on only consider the case when k = 1 (or s = 1/2), in which case

System reduction and estimates.
Suppose now that we are given two magnetic Schrödinger operators L j := L Xj ,qj , j = 1, 2. Introduce A j := π 0,1 X j , B j := π 1,0 X j , where π 0,1 and π 1,0 are the projections discussed in Section 2. We first observe that we can rewrite L j from the form 2 into If α j are primitive functions in the sense that∂α j = A j (the existence of such α j is guaranteed by Lemma 2.3), then we can furthermore rewrite 19 (using integrating factor) to (20) L j = 2e −iᾱj∂ * e iᾱj e −iαj∂ e iαj + Q j .
In order to abbreviate, let us denote by F j = e iαj , then we can once again rewrite 20 as Introducing ω = (∂ + iA j )u we see that we can rewrite the second order partial differential equations L j u = 0 as the first order∂-system Using notations from Section 3 we can abbreviate 22 by (D + V j )U = 0. By a similar argument leading to the form 21 of L j we can also observe that we can split the system 22 further into (c.f. [1]) or equivalently, since the leftmost matrix is invertible Denoting the potential matrix in 23 byṼ j , the system can be abbreviated as (D + V j )Ũ = 0. Now we are in the case discussed above, with diagonal potential matrices, that was treated by Bukhgeim. Suppose now that we have a solution to (D +Ṽ 1 )Ũ 1 h = 0, then by Proposition 1 we can assume that the solution has the form It follows that a solution to (D + V 1 )U 1 h = 0 can be found on the form where a is an arbitrary holomorphic function.
Similarly, a solution to (D + V * 2 )U 2 h = (D + V 2 )U 2 h = 0 can be found on the form .
Suppose now that U h is a solution to (D+V 2 )U h = 0, and U 1 h , U 2 h are the solutions described above. Then by Lemma 3.1 (26) . So we have derived the boundary integral identity h ∂M , from which we will derive an estimate in order to compare Cauchy data for the potential matrices V j and their diagonalized counterpartsṼ j , in the sense of 22-23. First we show the following auxiliary estimate. Lemma 4.1. Assume that there is a constant K > 0 such that for some p > 2, and consider the systems corresponding to the problems L j u = 0 as described in 22. Then the boundary integral identity 27 implies the following inequality for small h > 0, δ > 0: where C = C(K, M, p), c > 0, A j = π 0,1 X j , d(C 1 , C 2 ) is the distance between Cauchy data for the problems L j u = 0 and a, b, F j are the quantities defined above appearing in the solutions U j h , j = 1, 2 of the systems.
implies in particular that holds. Then by Sobolev embedding and elliptic regularity (for∂, see e.g. Theorem 4.6.9 in [14]), it follows that Proof of Lemma 4.1. Expanding the left hand side in 27, we find where A = A 2 − A 1 , Q = Q 2 − Q 1 , a can be any holomorphic function and b any antiholomorphic 1-form. The next step will be to use the estimates from Proposition 1. Under the assumption in 28 and the following remark, we get (by applying Cauchy-Schwarz inequality) the estimates By examining the boundary term in 27 we find where we temporarily have abbreviated the solutions according to the earlier convention, U j h = (u j , ω j ) T . By Cauchy-Schwarz inequality, by the boundedness of the trace operator. Since this holds for any solution U h to (D + V 2 )U h = 0 it follows by taking infimum over the corresponding Cauchy data space that . The proof is finished by rearranging the terms in 27 and applying the triangle inequality.
The next step is to estimate the Sobolev norms of the solutions appearing in the right hand side of 36. This requires a quite detailed discussion but will yield results that we will use more than once. 4.1. H 1 -estimates of solutions to (D + V )U = 0. We will need to estimate solutions of systems (D + V )U = 0 in H 1 -norm sense. For the general (nondiagonal) potentials V , that we must consider, we saw that solutions were given by 24 and 25. Our goal will be to establish:  For the solutions in 24 and 25, this means that we need to estimate , where we are free to choose the holomorphic function a and antiholomorphic 1-form b. We start by observing that there is a constant c > 0 so that |e ±Φ/h | ≤ e c/h , since ϕ = Re Φ is harmonic. By the a priori assumptions on the A j :s in 29 and the discussion leading to 33, we will also have no trouble bounding the first order partial derivatives of the F j by some constant depending on the a priori bounding constant K.
We need to be a bit careful with the remainders r j h , s j h . By studying the system 16 and invoking elliptic regularity we can however see that we are in no danger, assuming sufficient regularity onQ,F . Rewriting the system we see that, + r h )). Now it is more clear that the right hand side should belong to H 1 (M ). To get an estimate, observe that by writing out the operators as in Section 2,∂ −1 ψ = R∂ −1 e −2iψ/h E,∂ * −1 ψ = R ∂ * −1 e 2iψ/h E we have (on M ) that for small enough h > 0 (by Lemma 2.3(i), 2.4(i) and Proposition 1), Furthermore, since ∂∂ −1 ,∂∂ −1 are bounded operators (related to the so-called Beurling transform) on L p (M ). In local coordinates the kernels are of Calderón-Zygmund type, modulo smoothing terms by Lemma 2.3 and 2.4, c.f. [19].
Remark 3. Continuing the argument in the above proof it is in fact also more or less almost immediate that ifQ,F ∈ W k,p (M ), k ≥ 1, p > 2, then

Conclusion of the system reduction step. Adding up 26-37 we have managed to show that
Choosing for some 1 − ε < α < 1 we get the estimate (for some large enough C = C(K, M, p) > 0) Looking at the left hand side-integral we also observe that The first equality follows from the fact that F −1 j∂ F j = −F j∂ F −1 j = A j by construction, while the third equality is just Green's integral identity. Finally, the last equality follows since b is antiholomorphic. Thus we have shown Lemma 4.2. If for some p > 2, an a priori assumption of type max{ q j W 1,p (M ) , X j W 2,p (M,Λ 0,1 (M )) } ≤ K hold, then for some small ε > 0, where C j is the Cauchy data associated with the operators L j = L Xj ,qj , as defined in 3, and a ∈ H(M ), b ∈ Λ 0,1 (M ).
Later we will choose δ = (log | log d(C 1 , C 2 )|) −ε for some ε < 1/2 + ε so that 38 can be replaced by with 0 < β < 1/2 + ε − ε γ where γ > 0 can be choosen arbitrarily small. We will from now on assume that this choice of δ has been made and use 39 in place of 38 when referring to Lemma 4.2.
5. Reduction to diagonal system. One of the key observations in [1] is that if (This is also a consequence of Lemma 4.2 of course.) The following lemma is then used extensively: The proof can be found in [7] where the above result is contained in Lemma 4.1. One must show that the harmonic extension of f is actually holomorphic, and this can be done by considering the so-called Hodge-Morrey decomposition of (0, 1)-forms.
Using the above lemma, the authors are thus able to conclude that (F −1 2 F 1 )| ∂M is indeed the restriction of a holomorphic function. Hence they may reduce the case of a general matrix potential V j to the diagonalized caseṼ j .
Clearly, we are not in the same situation here so we need to motivate a similar reduction step in a slightly different way. 5.1. Auxiliary estimate. Let us introduce the subspace U = {ω = ι * ∂Mb ;∂b = 0, b ∈ Λ 1 (M )}, of X = H 1/2 (Λ 1 (∂M )) (or H s (Λ 1 (∂M )), for any s ≥ 0) and argue abstractly from the viewpoint of Hilbert space theory. Consider the following subspace of the dual space X * , ker U := {x * ∈ X * ; x * (x) = 0 for all x ∈ U }. Take a Schauder basis {x * n } ∞ n=1 such that ker U = span {x * n } ∞ n=1 and a dual basis Then in particular x * (x n ) = 0 for all n ∈ N. Thus x * / ∈ ker U , but at the same time we must have x * (x) = 0 for all x ∈ U and thus x * ∈ ker U . An obvious contradiction unless x * = 0. To see that it is indeed a direct sum, pick x ∈ U ∩ span {x n } ∞ n=1 , then x = a n x n ∈ U , but then 0 = x * n (x) = a n for all n ∈ N. Now take another Schauder basis {y n } ∞ n=1 such that span {y n } ∞ n=1 = U and a dual basis {y * n } ∞ n=1 ⊂ X * , then we can in a very similar way show that X * = ker U ⊕ span {y * n } ∞ n=1 . Introducing the projection π : X * → ker U we have by the above splitting of X * = H −1/2 (Λ 1 (∂M )) that any linear functional on X may be written This equality follows since (1 − π)x * (x n ) = 0, n ∈ N since (1 − π)x * ∈ span {y * n }. In particular, if we consider the linear functional we can interpret Lemma 4.2 as 1 )| ∂M equals the norm in the left hand side by the argument above. Furthermore, by the a priori assumptions in 29-34, we have for some δ > 0, and interpolation where 0 < α < 1, 0 < γ(α, β) < (1 − α)β and r > 3/2. Now we have arrived at a stage where we have and by Lemma 5.1 we know that Clearly it also holds that∂F 1 = iA 1F1 ,∂F 2 = iA 2 F 2 and by our calculations above and a priori assumptions on the A j :s So by Sobolev embedding we can conclude the following lemma: Lemma 5.2. LetF 1 , F 2 be as defined above, then there is a 0 < γ ≤ 1/2 such that where C = C(K, M, p) and d(C 1 , C 2 ) measures the distance between the Cauchy data spaces as defined above.

An inequalty between Cauchy data.
From 24-25 we see that solutions to systems (D + V j )U j = 0 is related to the corresponding system (D +Ṽ j )Ũ j = 0 with diagonalized potentialsṼ j (in the sense described above) viã The next lemma relates the distances for Cauchy data for the diagonalized potential V j and the Cauchy data for the corresponding partial differential equations with the pairs (X j , q j ). Lemma 5.3. Suppose that we have reduced the boundary value problems for L j u j = 0 to boundary value problems with diagonal potential matrices (D +Ṽ j )Ũ j = 0 as above. Then the distance of Cauchy data corresponding to the diagonalized problems is bounded in terms of the distance between Cauchy data for the initial problem according to The quantities d (CṼ 1 , CṼ 2 ) and d(C 1 , C 2 ) are those defined in Section 3.1 and 0 < γ ≤ 1/2.
Proof. We begin by writing Starting with the first term where the last inequality follows from Lemma 5.2. Similarly, the second term satisfies Adding 40 and 41 we get This implies that 2 is bounded from below and ι * ∂M ω 2 H −1/2 (∂M ) ≤ C u 2 | ∂M H 1/2 (∂M ) . As we have a bijective correspondence between solutions to the L j , V j andṼ j problems, it follows that which is what we wanted to prove.
The important conclusion of Lemma 5.3 is that small differences in Cauchy data for the non-diagonalized V j -case implies small differences in Cauchy data for the diagonal counterparts. This will allow us to deduce stability by considering the latter case.

5.3.
Estimates for the potential q and magnetic field dX. We are now ready to complete the proof of Theorem 1.1 (and 1). Recall our a priori assumptions 4: If C j are the Cauchy data spaces as defined in 3 for the corresponding magnetic Schrödinger operators L j := L Xj ,qj , as defined in 1-2. Then if the distance d(C 1 , C 2 ) is small enough, there is an α ∈ (0, 1/2) such that where C = C(K, M, α).
We will split up the proof into three parts, using our reduction to a diagonal system as described in the above sections.
Proof. We will consider two cases, first when the solutionsŨ j h are of the forms where a 1 , a 2 are holomorphic functions. Then we have In particular if a 1 = a 2 = a we claim that we can estimate, for h small, Furthermore, if ψ has a non-degenerate stationary point at p 0 ∈ M , we claim that where C depends on the a priori bounds on the potentials. Similarly, if we instead consider solutions where b 1 , b 2 are antiholomorphic 1-forms. Then In particular if b 1 = b 2 = b we claim that we can, similarly as for 42, estimate, for h small, Again, if ψ has a non-degenerate stationary point at where C depends on the a priori bounds on the potentials. From the discussion in Section 4 and Lemma 5.3, we also have the estimate Combining either 42-43 or 44-45 with 46, we get  To justify 44 for the terms that are not immediately obvious we use arguments similar to those in [7], e.g.
and use Lemma 2.6 and Proposition 1. Similarly one shows 42.
Next we will prove an L 2 -estimate where we take care of those exceptional points that are not included in Lemma 5.4.
Lemma 5.5. There is an ε > 0 such that It follows from this and the priori assumptions 4 that there is a C = C(K, M, p, γ, ε) such that Choosing δ = | log H| −2ε/15 , we get together with the fact that |F 1 | + |F 2 | is uniformly bounded from below we can then also to conclude that In the last steps of the proof we indicate how to also get higher Sobolev regularity estimates.
Similarly we want to give an estimate for the potentials q j , starting from Q L 2 (M ) = Q 2 /(2|F 2 | 2 ) − Q 1 /(2|F 1 | 2 ) L 2 (M ) ≤ C| log H| −2ε/15 , where Q j := − dX j + q j , j = 1, 2. Some simple algebra and using that the |F j | are bounded away from zero one can see that Q 1 − Q 2 L 2 (M ) = q 1 − q 2 − d(X 1 − X 2 ) L 2 (M ) ≤ C| log H| −2ε/15 . from which it simply follows, similarly as above, that For s = 0 we get Theorem 1.1, and η > 1 would allow for larger s > 0. 6. Stability for the holonomy. We finally focus our attention towards the holonomy of the 1-form X := X 1 − X 2 . We can for every closed loop γ based at any m ∈ M , consider parallel transport for the connection ∇ X = d + iX on the bundle M × C. This defines an isomorphism P X γ : C → C, so we may view P X γ as a non-zero complex number and define the holonomy group of ∇ X at m by Hol(∇ X , m) := {P X γ ∈ C \ {0}; γ is a closed loop based at m}.
For real X, P X γ is in fact unitary and it can be observed that P X γ = e i γ X . When the curvature dX = 0 there is a natural group morphism ρ X m : π 1 (M, m) → Hol(∇ X , m), where π 1 (M, m) is the first fundamental group, consisting of equivalence classes of closed curves up to homotopy. The morphism ρ X is called the holonomy representation into GL(C) and it is trivial if and only if P X γ = 1 for all closed loops based at m, independent of m. We will show Theorem 1.2, or more precisely: where ω = ω(x) = C(log | log x|) −ε → 0 as x → 0, C = C(K, M p, ε). Thus the holonomy representation is in this sense close to being trivial.
Let now γ be any closed curve on M , it will be made up of a finite number of non-self-intersecting loops. So without loss of generality we can assume that γ is made up of a single simple loops. We will show that for any such loop, and hence it will follow that inf k∈Z γ which is what we want to show.
To show 49, suppose we remove a small subarc γ q,p between two points q, p ∈ γ, so that we are left with another arc γ p,q consisting of the remaining points on γ between p and q. Suppose then that we take a simply connected tubular neighborhood N (γ p,q ) around γ p,q . In this neighborhood, is well-defined (since the 1-form dΘ Θ is closed) and by the fundamental theorem of calculus, dg(q) = dΘ Θ (q).
Then Θ(q) = Θ(p)e g(q) . Taking the limit as q → p (strictly enlarging the neighborhood N (γ p,q )) we thus find 1 = exp lim q→p q p dΘ Θ .
We can then conclude that 49 must indeed be true and the theorem follows.