# Calderón problem for Yang–Mills connections

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##### Citations

42 citations

### Cites methods from "Calderón problem for Yang–Mills con..."

...The case of systems was considered in [13] and Yang–Mills potentials with arbitrary geometry in [9]....

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20 citations

### Cites methods from "Calderón problem for Yang–Mills con..."

...The case of systems was considered in [Esk01] and Yang-Mills potentials with arbitrary geometry in [Cek17b]....

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7 citations

^{1}, Gunther Uhlmann

^{2}, Gunther Uhlmann

^{3}, Hanming Zhou

^{4}•Institutions (4)

6 citations

4 citations

##### References

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### "Calderón problem for Yang–Mills con..." refers background or methods in this paper

...t us denote the complement of the zero set T= MnN(g); obviously Mn([M i) ˆ T and T open. Let x 0 2Mbe a point in the open neighbourhood of V where B= h(A) and ybe any point in T. Consider any path : [0;1] !Mwith (0) = x 0 18 Mihajlo Cekic and 0(1) = y. We will construct a path from x 0 to y, lying in T, by slightly perturbing the path , such that and 0are arbitrarily close. Let dbe the usual complete...

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...pology as the weak topology given by the seminorms. Furthermore, this also induces a Frechet metric to the space C 1([0;1];Rm) = m i=1 C ([0;1];R) for all m2N. Moreover, we may consider the space C1([0;1];M) for any compact Riemannian manifold (M;g) by isometrically embedding M into a Euclidean space RN for some N, as a closed subspace of C1([0;1];RN). Now we prove the following lemma for the continui...

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...ang-Mills equations, which are considered as a generalisation of Maxwell’s equations in electromagnetism and which provide a framework to write the latter equations in a coordinate-free way (see e.g. [1] or [6] for a geometric overview and denitions). More concretely, let us consider a Hermitian bundle Eover a compact oriented Riemannian manifold (M;g) equipped with a unitary connection A; we will d...

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...ary facts that we use in the paper. We need the metric space of smooth curves in the proof of our main theorem { here are some properties: Remark 6.1. We are using the standard metric on the space C1([0;1];R) induced by the seminorms kfk k= sup t2[0;1] dkf dtk . Then a choice of the metric on this space is: d(f;g) = X1 k=0 2 k kf gk k 1 + kf gk k and it is a standard fact that this space is a Frechet ...

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...ed subset. Assume also that for any two points x;y2 nEand any smooth path in between xand y, there exist smooth paths i from xto y, lying in nE, for i= 1;2:::, that converge to in the metric space C1([0;1];Rn). Let f: nE!C be a smooth function, such that @ fextend continuously to for all multi-indices . Then there exists a unique smooth extension f~: !C with f~j nE = f. Proof. This is a local claim, so...

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683 citations

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### "Calderón problem for Yang–Mills con..." refers background or methods or result in this paper

... and deduce that its full symbol determines the full Taylor series of the connection, metric and a potential at the boundary. This was rst proved in the case of a Riemannian metric by Lee and Uhlmann [22] and later considered in the m= 1 case with a connection in [11]. In this paper, we generalise this approach to the case of systems and prove the analogous result. 1.1. Motivation. Let us explain some...

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...ple of remarks are in place. Remark 3.5 (Boundary determination for surfaces). There are a few reasons to exclude the case dimM= 2 in Theorem 3.4. To start with, after the proof of Proposition 1.3 in [22], the authors (considering the case E= M C, A= 0 and Q= 0) remark that all the symbols of Bsatisfy b j = 0 for j0 (easily checked for b 0 by direct computation and for the rest by induction); in othe...

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...mmation will always be assumed to go over 1;:::;n 1. We use the notation D x j= @i@ x = i @xj and jgj= det(g ij) = det(g ). We start by proving an analogue of Lemma 8.6 in [11] and Proposition 1.1 in [22]. Lemma 3.2. Let us assume A satises condition (3.12). There exists a C mvalued pseudodierential operator B(x;D x0) of order one on @M, depending smoothly on xn2[0;T] for some T>0, such that the ...

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...uch that (3.2) holds. We have established the existence of the factorisation (3.2) and now it is time to use it to prove facts about the DN map. The following claim is analogous to Proposition 1.2 in [22] { the main dierence is that now we are using matrix valued pseudodierential operators, so we need to make sure that appropriate generalisations hold. Proposition 3.3. The DN map g;A;Q is a C m-valu...

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^{1}, Carlos E. Kenig

^{2}, Mikko Salo

^{3}, Gunther Uhlmann

^{4}•Institutions (4)

275 citations

### "Calderón problem for Yang–Mills con..." refers methods or result in this paper

...es of the connection, metric and a potential at the boundary. This was rst proved in the case of a Riemannian metric by Lee and Uhlmann [22] and later considered in the m= 1 case with a connection in [11]. In this paper, we generalise this approach to the case of systems and prove the analogous result. 1.1. Motivation. Let us explain some motivation for considering this problem. Partly, the idea came ...

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...) + Q c). We will call a triple fg;A;Qgthat satises conditions (3.11) and (3.12) normalised. Moreover, we will use the notation f 1 ’f 2 to denote that f 1 and f 2 have the same Taylor series (as in [11]). Theorem 3.4. Assume M satises dimM = n 3 and the triple fg;A;Qgis normalised. Let W ˆ@M open, with a local coordinate system fx1;:::;xn 1gand let fb j jj1gdenote the full symbol of B(see Lemma 3...

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...e metric appropriately, we may determine the full Taylor series of a connection, metric and matrix potential from the DN map on a vector bundle with m>1. The case of m= 1 was already considered in [11] (Section 8) and we generalise the result proved there. The approach is based on constructing a factorisation of the operator d A d A+ Qmodulo smoothing, from which we deduce that g;A;Q is a pseudodi...

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...g Greek indices and , the summation will always be assumed to go over 1;:::;n 1. We use the notation D x j= @i@ x = i @xj and jgj= det(g ij) = det(g ). We start by proving an analogue of Lemma 8.6 in [11] and Proposition 1.1 in [22]. Lemma 3.2. Let us assume A satises condition (3.12). There exists a C mvalued pseudodierential operator B(x;D x0) of order one on @M, depending smoothly on xn2[0;T] for...

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