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Journal ArticleDOI

Calderón problem for Yang–Mills connections

14 May 2020-Vol. 10, Iss: 2, pp 463-513

Abstract: We consider the problem of identifying a unitary Yang-Mills connection $ abla$ on a Hermitian vector bundle from the Dirichlet-to-Neumann (DN) map of the connection Laplacian $ abla^* abla$ over compact Riemannian manifolds with boundary. We establish uniqueness of the connection up to a gauge equivalence in the case of trivial line bundles in the smooth category and for the higher rank case in the analytic category, by using geometric analysis methods and essentially only one measurement. Moreover, by using a Runge-type approximation argument along curves to recover holonomy, we are able to uniquely determine both the bundle structure and the connection, but at the cost of having more measurements. Also, we prove that the DN map is an elliptic pseudodifferential operator of order one on the restriction of the vector bundle to the boundary, whose full symbol determines the complete Taylor series of an arbitrary connection, metric and an associated potential at the boundary.
Topics: Differential geometry (51%)

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CALDER
´
ON PROBLEM FOR YANG-MILLS CONNECTIONS
MIHAJLO CEKI
´
C
Abstract. We consider the problem of identifying a unitary Yang-Mills connection
on a Hermitian vector bundle from the Dirichlet-to-Neumann (DN) map of the
connection Laplacian
over compact Riemannian manifolds with boundary.
We establish uniqueness of the connection up to a gauge equivalence in the case
of trivial line bundles in the smooth category and for the higher rank case in the
analytic category, by using geometric analysis methods and essentially only one
measurement.
Moreover, by using a Runge-type approximation argument along curves to re-
cover holonomy, we are able to uniquely determine both the bundle structure and
the connection, but at the cost of having more measurements. Also, we prove that
the DN map is an elliptic pseudodifferential operator of order one on the restriction
of the vector bundle to the boundary, whose full symbol determines the complete
Taylor series of an arbitrary connection, metric and an associated potential at the
boundary.
1. Introduction
In this paper, we consider the Calder´on inverse problem for a special type of con-
nections, called the Yang-Mills connections. Given a Hermitian vector bundle E of
rank m over a compact Riemannian manifold (M, g) with non-empty boundary and a
unitary connection A () on E, one may consider the connection Laplacian denoted
by d
A
d
A
(
), where d
A
(
) denotes the formal adjoint of d
A
() with respect
to the Hermitian and Riemannian structures. Sometimes this operator is called the
magnetic Laplacian because it is used to represent the magnetic Schr¨odinger equation,
where A corresponds to the magnetic potential.
Given this, we may define the associated Dirichlet-to-Neumann (DN map in short)
Λ
A
: C
(M; E|
M
) C
(M; E|
M
)
1
by solving the Dirichlet problem:
d
A
d
A
(u) = 0, u|
M
= f (1.1)
and setting Λ
A
(f) = d
A
(u)(ν), where ν is the outwards pointing normal at the bound-
ary. The problem can then be posed as asking whether the map A 7→ Λ
A
is injective
modulo the natural obstruction, or in other words whether Λ
A
= Λ
B
implies the
existence of a gauge automorphism F : E E with F
(A) = B and F |
M
= Id.
This problem was considered in [6, 1113]; for a survey of the Calder´on problem
for metrics, see [31]. In this paper, we take two approaches to uniqueness: one is
via geometric analysis and the other by constructing special gauges along curves
via the Runge approximation property of elliptic equations. As far as we know,
this paper is the first one that considers the connection problem and does not rely
1
By C
(M; E) we denote the space of smooth sections of E over M.
1
arXiv:1704.01362v2 [math.AP] 12 Jun 2018

2 Mihajlo Ceki
´
c
on the Complex Geometric Optics solutions (see any of [6, 1113]), but on unique
continuation principles and geometric analysis of the zero set of a solution to an
elliptic equation.
The Yang-Mills connections generalise flat connections and are important in physics
and geometry. They satisfy the following equation:
D
A
F
A
= 0
where D
A
= d
End
A
is the induced connection on the endomorphism bundle EndE and
F
A
is the curvature of A (see the preliminaries for more details).
Firstly, we prove that the DN map Λ
A
is an elliptic pseudodifferential operator of
order 1 on the restriction of the vector bundle to the boundary and deduce that its
full symbol determines the full Taylor series of the connection, metric and a potential
at the boundary. This was first 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 from the analogy between Einstein metrics in Riemannian geometry
and Yang-Mills connections on Hermitian vector bundles. Also, Guillarmou and a
Barreto in [14] prove the recovery of two Einstein manifolds from the DN map for
metrics. The method of their proof relies on a reconstruction near the boundary,
where in special harmonic coordinates Einstein equations become quasi-linear ellip-
tic (the metric is thus also analytic in such coordinates). Hence, by combining the
boundary determination result and a unique continuation result for elliptic systems
they prove one can identify the two metrics in a neighbourhood of the boundary.
Moreover, by exploiting this analytic structure they observe that the method of Las-
sas and Uhlmann [20] who prove the analytic Calder´on problem for metrics, may be
used to extend this local isometry to the whole of the manifold.
2
In our case, the conventionally analogous concept to harmonic coordinates to con-
sider would be the Coulomb gauge [30] which transforms the connection to a form
where d
(A) = 0, so that the Yang-Mills equations become an elliptic system with
principal diagonal part. However, this gauge does not tie well with the DN map, so
in Lemma 4.1 we construct an analogue of the harmonic gauge for connections. In
this gauge, we may use a similar unique continuation property (UCP in short) result
to yield the equivalence of connections close to the boundary. However, for going
further into the interior we designed new methods.
1.2. Uniqueness via geometric analysis. We believe this approach to be entirely
new. Here is one of the main theorems of the paper.
Theorem 1.1 (Global result). Assume dim M 2, let E = M × C be a Hermitian
line bundle with standard metric and 6= Γ M an open, non-empty subset of the
boundary. Let A and B be two unitary Yang-Mills connections on E. If Λ
A
(f)|
Γ
=
2
This works by embedding the two manifolds in a suitable Sobolev space using Green’s functions
of the metric Laplacians and showing the appropriate composition is an isometry.

Calder
´
on problem for Yang-Mills connections 3
Λ
B
(f)|
Γ
for all f C
0
(Γ; E|
Γ
), then there exists a gauge automorphism (unitary) h
with h|
Γ
= Id such that h
(A) = B on the whole of M.
We now explain this geometric analysis type method in more detail. Our gauge F
from Lemma 4.1 (m × m matrix function on M) satisfies the equation d
A
d
A
F = 0
and so we cannot guarantee that it is non-singular globally. We show that the zero
set of the determinant of F is suitably small in the smooth case when m = 1 and in
the analytic case for arbitrary m it is covered by countably many submanifolds of
codimension one, or in the language of geometric analysis it is (n 1)-C
-rectifiable.
Since (the complement of) this singular set can be topologically non-trivial (see Figure
1), we end up with barriers consisting of singular points of F that prevent us to use
the UCP and go inside the manifold. This is addressed by looking at the sufficiently
nice points of the barriers and locally near these points, using a degenerate form of
UCP (in the smooth case) or a suitable form of analytic continuation (in the analytic
case) to extend an appropriate gauge equivalence between the two given connections
beyond the barriers; we name this procedure as “drilling”. Since we show there is
a dense set of such nice points, we may perform the drilling to extend our gauges
globally.
Here is what we prove in the analytic case, for arbitrary m:
Theorem 1.2. Let (M, g) be an analytic Riemannian manifold with dim M 2 and
let Γ be as in Theorem 1.1.
3
If E = M × C
m
is a Hermitian vector bundle with
the standard structure and if A and B are two unitary Yang-Mills connections on E,
then Λ
A
(f)|
Γ
= Λ
B
(f)
Γ
for all f C
0
(Γ; E|
Γ
) if and only if there exists a gauge
automorphism H of E, with H|
Γ
= Id, such that H
(A) = B.
We briefly remark that the proof the above theorem also relies on using the Coulomb
gauge locally, since in this gauge the connection is analytic, so det F satisfies the
SUCP (see Lemma 5.1); we additionally apply this gauge in the drilling procedure.
The main difficulty in proving uniqueness via the geometric analysis method for the
smooth, higher rank (m > 1) case is that the strong unique continuation property
(SUCP) for the determinant det F of a solution to d
A
d
A
F = 0 might not hold
see Remark 5.2 for more details. Indeed, in the subsequent work [5] we treat this
question in more detail and prove a positive answer for n = 2 and also provide some
counterexamples.
1.3. Uniqueness via Runge approximation. Next, we outline our second ap-
proach to uniqueness by using Runge-type approximation for elliptic equations to
recover holonomy, which we use to prove the stronger statement of uniqueness for
arbitrary bundles. In general, Runge approximation is known to be applicable to
inverse problems (see e.g. [2, 19, 27]).
Theorem 1.3. Let (M, g) be a smooth compact Riemannian manifold with boundary
of dimension dim M 2, Γ M a non-empty open set, E and E
0
Hermitian vector
bundles over M such that we have the identification E|
Γ
= E
0
|
Γ
. Let A and B be two
smooth unitary Yang-Mills connections on E and E
0
respectively, such that Λ
A
(f)|
Γ
=
3
The metric g is only assumed to be analytic in the interior of M and smooth up to the boundary.

4 Mihajlo Ceki
´
c
Λ
B
(f)|
Γ
for all f C
0
(Γ; E). Then there exists a unitary bundle isomorphism
H : E
0
E with H|
Γ
= Id, such that H
A = B.
This theorem clearly generalises Theorems 1.1 and 1.2. Note in particular that
we are able to uniquely determine the topology of the bundles and the Hermitian
structures.
Let us point out the main differences between the two approaches. Although Theo-
rem 1.3 is stronger than the first two theorems, the advantage of the former approach
is in its method of proof. More precisely, the geometric analysis technique is minimal
with respect to the necessary data essentially, we only need one arbitrary measure-
ment to uniquely identify the connections see Remark 4.6 for more details. Also,
this method is entirely new, so it gives hope that it can be generalised to different
settings, such as the metric Calder´on problem.
On the other hand, in the Runge-type density approach we need many measure-
ments that concentrate in a suitable sense on closed loops (see Lemma 6.1). In the
context of the Calder´on problem for connections, this method is also new and gives
hope to be generalised to other settings.
1.4. Organisation of the paper. The paper is organised as follows: in the next
section, we recall some formulas from differential geometry and a make a few obser-
vations about choosing appropriate gauges. In the third section we prove that Λ
A
is a pseudodifferential operator of order 1 for systems and prove that its full symbol
determines the full jet of A at the boundary. Furthermore, in section four we consider
the smooth case and prove the global result for m = 1, Theorem 1.1. In the same
section, we construct the new gauge and deduce the UCP result we need. In section
five we consider the m > 1 case for analytic metrics, Theorem 1.2, by adapting the
proof of the line bundle case and exploiting real-analyticity. Finally, in section five
we apply the Runge-type approximation property and prove Theorem 1.3.
This paper also has two appendices: in Appendix A we recall some well-posedness
condition for the heat equation and prove a few elementary statements about extend-
ing functions smoothly over small sets. Next, in Appendix B we lay out the technical
results needed to prove the Runge type approximation result we need this requires
some well-posedness for Dirichlet problem in negative Sobolev spaces and a duality
argument; the aim is to prove that for L
A
= d
A
d
A
, we can build m smooth solutions
that span the bundle over a given curve.
Acknowledgements. The author would like to express his gratitude for the support
of his supervisor, Gabriel Paternain. Furthermore, he would also like to acknowledge
the help of Mikko Salo, Rafe Mazzeo, Colin Guillarmou and Gunther Uhlmann, es-
pecially in clarifying the unique continuation results used in the paper. The author
thanks Guillaume Bal and Francois Monard for telling him about Runge type ap-
proximation results which inspired the last section of the paper.
He is grateful to the Trinity College for financial support, where the most of this
research took place and to Max-Planck Institute for Mathematics in Bonn, where a
part of this research took place.

Calder
´
on problem for Yang-Mills connections 5
2. Preliminaries
2.1. Yang-Mills connections. As mentioned previously, Yang-Mills (YM) connec-
tions are very important in physics and geometry. They satisfy the so called Yang-
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 [10] for a geometric overview and definitions). The
Yang-Mills connections are critical points of the functional:
F
Y M
(A) =
Z
M
|F
A
|
2
g
Here F
A
= dA + A A is the curvature 2-form with values in the endomorphism
bundle of E determined by the map d
2
A
s = F
A
s on sections s C
(M; E) and ω
g
is the volume form. It can then be shown by considering variations of this functional,
that the equivalent conditions for A being its critical point are (the Euler-Lagrange
equations):
(D
A
)
F
A
= 0 and D
A
F
A
= 0 (2.1)
where D
A
= d
End
A
is the induced connection on the endomorphism bundle, given
locally by D
A
S = dS + [A, S] or equivalently by D
A
S = [d
A
, S], where [·, ·] denotes
the commutator. The second equation in (2.1) is actually redundant, since it is the
Bianchi identity.
Yang-Mills connections clearly generalise flat connections, for which the curvature
vanishes, i.e. F
A
= 0.
They have been a point of unification between pure mathematics and theoretical
physics, but moreover have brought a few areas of pure mathematics together, such
as e.g. PDE theory and vector bundles over complex projective spaces, or algebraic
geometry.
Example 2.1 (Yang-Mills connections over Riemann surfaces). We give an idea of
the size of the set of YM connections in the simplest non-trivial example of Riemann
surfaces. First recall that connections on bundles modulo gauges are classified by
their holonomy representation on the so called loop group modulo conjugation (see
Kobayashi and Nomizu [17]). In the setting of flat connections, this correspondence
simplifies significantly for a Riemann surface Σ:
ρ : π
1
(Σ) U(m)
/conj. {unitary flat bundles of rank m}
since homotopic loops have the same holonomy. The direct map (going left to right)
here is the one taking a representation ρ and defining an associated flat bundle via
e
Σ ×
ρ
C
m
, where
e
Σ is the universal cover of Σ and ×
ρ
means we identified the two by
the diagonal action. Somewhat surprisingly, we may still obtain a correspondence in
the case of YM connections, where π
1
(Σ) is replaced by a certain central extension
bπ
1
(Σ) (see [1] for more details). This has an analogous geometric interpretation: the
difference to the flat case is that we now identify homotopic only if they enclose the
same area. In particular, for the sphere S
2
this simplifies, so that we have bπ
1
(S
2
) = S
1
.

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References
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Book
01 Jan 1972
Abstract: 7 Scalar and Vector Ultra-Distributions.- 1. Scalar-Valued Functions of Class Mk.- 1.1 The Sequences {Mk}.- 1.2 The Space $${D_{{M_k}}}\left( H \right)$$.- 1.3 The Spaces $${D_{{M_k}}}\left( H \right)$$ and $${\varepsilon _{{M_k}}}\left( H \right)$$.- 2. Scalar-Valued Ultra-Distributions of Class Mk Generalizations.- 2.1 The Space $$D{'_{{M_k}}}\left( \Omega \right)$$.- 2.2 Non-Symmetric Spaces of Class Mk.- 2.3 Scalar Ultra-Distributions of Beurling-Type.- 3. Spaces of Analytic Functions and of Analytic Functionals.- 3.1 The Spaces H(H) and H'(H).- 3.2 The Spaces H(?) and H(?).- 4. Vector-Valued Functions of Class Mk.- 4.1 The Space $${D_{{M_k}}}\left( {\phi F} \right)$$.- 4.2 The Spaces $${D_{{M_k}}}\left( {H,F} \right)$$ and $${E_{{M_k}}}\left( {\phi F} \right)$$.- 4.3 The Spaces $${D_{ \pm ,{M_k}}}\left( {\phi F} \right)$$.- 4.4 Remarks on the Topological Properties of the Spaces $${D_{{M_k}}}\left( {\phi F} \right),{E_{{M_k}}}\left( {\phi F} \right),{D_{ \pm ,{M_k}}}\left( {\phi F} \right)$$.- 5. Vector-Valued Ultra-Distributions of Class Mk Generalizations.- 5.1 Recapitulation on Vector-Valued Distributions.- 5.2 The Space $$D{'_{{M_k}}}\left( {\phi F} \right)$$.- 5.3 The Space $$D{'_{ \pm ,{M_k}}}\left( {\phi F} \right)$$.- 5.4 Vector-Valued Ultra-Distributions of Beurling-Type.- 5.5 The Particular Case: F = Banach Space.- 6. Comments.- 8 Elliptic Boundary Value Problems in Spaces of Distributions and Ultra-Distributions.- 1. Regularity of Solutions of Elliptic Boundary Value Problems in Spaces of Analytic Functions and of Class Mk Statement of the Problems and Results.- 1.1 Recapitulation on Elliptic Boundary Value Problems.- 1.2 Statement of the Mk-Regularity Results.- 1.3 Reduction of the Problem to the Case of the Half-Ball.- 2. The Theorem on "Elliptic Iterates": Proof.- 2.1 Some Lemmas.- 2.2 The Preliminary Estimate.- 2.3 Bounds for the Tangential Derivatives.- 2.4 Bounds for the Normal Derivatives.- 2.5 Proof of Theorem 1.3.- 2.6 Complements and Remarks.- 3. Application of Transposition Existence of Solutions in the Space D'(?) of Distributions.- 3.1 Generalities.- 3.2 Choice of the Form L the Space ?(?) and its Dual.- 3.3 Final Choice of the Form L the Space Y.- 3.4 Density Theorem.- 3.5 Trace Theorem and Green's Formula in Y.- 3.6 The Existence of Solutions in the Space Y.- 3.7 Continuity of Traces on Surfaces Neighbouring ?.- 4. Existence of Solutions in the Space $$D{'_{{M_k}}}\left( \Omega \right)$$ of Ultra-Distributions.- 4.1 Generalities.- 4.2 The Space $${\Xi _{{M_k}}}\left( \Omega \right)$$ and its Dual.- 4.3 The Space $${Y_{{M_k}}}$$ and the Existence of Solutions in $${Y_{{M_k}}}$$.- 4.4 Application to the Regularity in the Interior of Ultra-Distribution Solutions of the Equation Au = f.- 5. Comments.- 6. Problems.- 9 Evolution Equations in Spaces of Distributions and Ultra-Distributions.- 1. Regularity Results. Equations of the First Order in t.- 1.1 Orientation and Notation.- 1.2 Regularity in the Spaces D+.- 1.3 Regularity in the Spaces $${D_{ + ,{M_k}}}$$.- 1.4 Regularity in Beurling Spaces.- 1.5 First Applications.- 2. Equations of the Second Order in t.- 2.1 Statement of the Main Results.- 2.2 Proof of Theorem 2.1.- 2.3 Proof of Theorem 2.2.- 3. Singular Equations of the Second Order in t.- 3.1 Statement of the Main Results.- 3.2 Proof of Theorem 3.1.- 4. Schroedinger-Type Equations.- 4.1 Statement of the Main Results.- 4.2 Proof of Theorem 4.1.- 4.3 Proof of Theorem 4.2.- 5. Stability Results in Mk-Classes.- 5.1 Parabolic Regularization.- 5.2 Approximation by Systems of Cauchy-Kowaleska Type (I).- 5.3 Approximation by Systems of Cauchy-Kowaleska Type (II).- 6. Transposition.- 6.1 Orientation.- 6.2 The Parabolic Case.- 6.3 The Second Order in t Case and the Schroedinger Case.- 7. Semi-Groups.- 7.1 Orientation.- 7.2 The Space of Vectors of Class Mk.- 7.3 The Semi-Group G in the Spaces D(A? Mk). Applications.- 7.4 The Transposed Settings. Applications.- 7.5 Another Mk-Regularity Result.- 8. Mk -Classes and Laplace Transformation.- 8.1 Orientation-Hypotheses.- 8.2 Mk -Regularity Result.- 8.3 Transposition.- 9. General Operator Equations.- 9.1 General Results.- 9.2 Application. Periodic Problems.- 9.3 Transposition.- 10. The Case of a Finite Interval ]0, T[.- 10.1 Orientation. General Problems.- 10.2 Space Described by v(0) as v Describes X.- 10.3 The Space $${\Xi _{{M_k}}}$$.- 10.4 Choice of L.- 10.5 The Space Y and Trace Theorems.- 10.6 Non-Homogeneous Problems.- 11. Distribution and Ultra-Distribution Semi-Groups.- 11.1 Distribution Semi-Groups.- 11.2 Ultra-Distribution Semi-Groups.- 12. A General Local Existence Result.- 12.1 Statement of the Result.- 12.2 Examples.- 13. Comments.- 14. Problems.- 10 Parabolic Boundary Value Problems in Spaces of Ultra-Distributions.- 1. Regularity in the Interior of Solutions of Parabolic Equations.- 1.1 The Hypoellipticity of Parabolic Equations.- 1.2 The Regularity in the Interior in Gevrey Spaces.- 2. The Regularity at the Boundary of Solutions of Parabolic Boundary Value Problems.- 2.1 The Regularity in the Space $$D\left( {\bar Q} \right)$$.- 2.2 The Regularity in Gevrey Spaces.- 3. Application of Transposition: The Finite Cylinder Case.- 3.1 The Existence of Solutions in the Space D'(Q): Generalities, the Spaces X and Y.- 3.2 Space Described by ?v as v Describes X.- 3.3 Trace and Existence Theorems in the Space Y.- 3.4 The Existence of Solutions in the Spaces D's,r(Q) of Gevrey Ultra-Distributions, with r > 1, s ? 2m.- 4. Application of Transposition: The Infinite Cylinder Case.- 4.1 The Existence of Solutions in the Space D' (R D'(?)): The Space X_.- 4.2 The Existence of Solutions in the Space D'+ (R D'(?)): The Space Y+ and the Trace and Existence Theorems.- 4.3 The Existence of Solutions in the Spaces D'+,s(R D'r(?)), with r > 1, s ? 2m.- 4.4 Remarks on the Existence of Solutions and the Trace Theorems in other Spaces of Ultra-Distributions.- 5. Comments.- 6. Problems.- 11 Evolution Equations of the Second Order in t and of Schroedinger Type.- 1. Equations of the Second Order in t Regularity of the Solutions of Boundary Value Problems.- 1.1 The Regularity in the Space $$D\left( {\bar Q} \right)$$.- 1.2 The Regularity in Gevrey Spaces.- 2. Equations of the Second Order in t Application of Transposition and Existence of Solutions in Spaces of Distributions.- 2.1 Generalities.- 2.2 The Space $${D_{ - ,\gamma }}\left( {\left[ {0,T} \right] {D_\gamma }\left( {\bar \Omega } \right)} \right)$$ and its Dual.- 2.3 The Spaces X and Y.- 2.4 Study of the Operator ?.- 2.5 Trace and Existence Theorems in the Space Y.- 2.6 Complements on the Trace Theorems.- 2.7 The Infinite Cylinder Case.- 3. Equations of the Second Order in t Application of Transposition and Existence of Solutions in Spaces of Ultra-Distributions.- 3.1 The Difficulties in the Finite Cylinder Case.- 3.2 The Infinite Cylinder Case for m > 1.- 4. Schroedinger Equations Complements for Parabolic Equations.- 4.1 Regularity Results for the Schroedinger Equation.- 4.2 The Non-Homogeneous Boundary Value Problems for the Schroedinger Equation.- 4.3 Remarks on Parabolic Equations.- 5. Comments.- 6. Problems.- Appendix. Calculus of Variations in Gevrey-Type Spaces.

5,794 citations


Journal ArticleDOI
Michael Atiyah1, Raoul Bott2Institutions (2)
Abstract: The Yang-Mills functional over a Riemann surface is studied from the point of view of Morse theory. The main result is that this is a ‘perfect9 functional provided due account is taken of its gauge symmetry. This enables topological conclusions to be drawn about the critical sets and leads eventually to information about the moduli space of algebraic bundles over the Riemann surface. This in turn depends on the interplay between the holomorphic and unitary structures, which is analysed in detail.

2,181 citations


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

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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...

    [...]

  • ...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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Journal ArticleDOI
Karen Uhlenbeck1Institutions (1)
Abstract: We show by means of the implicit function theorem that Coulomb gauges exist for fields over a ball inR n when the integralL n/2 field norm is sufficiently small. We then are able to prove a weak compactness theorem for fields on compact manifolds withL p integral norms bounded,p>n/2.

683 citations


Journal ArticleDOI
John M. Lee1, Gunther Uhlmann1Institutions (1)
Abstract: If an electrical potential is applied to the surface of a solid body, the current flux across the surface depends on the conductivity in the interior of the body. We want to consider the inverse problem of determining the conductivity by these boundary measurements

410 citations


"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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Journal ArticleDOI
Abstract: In this article we consider the anisotropic Calderon problem and related inverse problems. The approach is based on limiting Carleman weights, introduced in Kenig et al. (Ann. Math. 165:567–591, 2007) in the Euclidean case. We characterize those Riemannian manifolds which admit limiting Carleman weights, and give a complex geometrical optics construction for a class of such manifolds. This is used to prove uniqueness results for anisotropic inverse problems, via the attenuated geodesic ray transform. Earlier results in dimension n≥3 were restricted to real-analytic metrics.

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 ...

    [...]

  • ...) + 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...

    [...]