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Gauge theory for embedded surfaces
 I, Topology
, 1993
"... (i) Topology of embedded surfaces. Let X be a smooth, simplyconnected 4manifold, and ξ a 2dimensional homology class in X. One of the features of topology in dimension 4 is the fact that, although one may always represent ξ as the fundamental class of some smoothly ..."
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Cited by 68 (6 self)
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(i) Topology of embedded surfaces. Let X be a smooth, simplyconnected 4manifold, and ξ a 2dimensional homology class in X. One of the features of topology in dimension 4 is the fact that, although one may always represent ξ as the fundamental class of some smoothly
Minimization of Conformally Invariant Energies in Homotopy Classes
, 1996
"... this paper is to analyse the behavior of a minimizing sequence u k 2 ff for a conformally invariant functional, e.g. the nenergy E(u) = ..."
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Cited by 5 (0 self)
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this paper is to analyse the behavior of a minimizing sequence u k 2 ff for a conformally invariant functional, e.g. the nenergy E(u) =
The Atiyah–Jones conjecture for rational surfaces
, 2008
"... We show that if the Atiyah–Jones conjecture holds for a surface X, then it also holds for the blowup of X at a point. Since the conjecture is known to hold for P 2 and for ruled surfaces, it follows that the conjecture is true for all rational surfaces. If P → X is a principal SU(2) bundle over a R ..."
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Cited by 3 (3 self)
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We show that if the Atiyah–Jones conjecture holds for a surface X, then it also holds for the blowup of X at a point. Since the conjecture is known to hold for P 2 and for ruled surfaces, it follows that the conjecture is true for all rational surfaces. If P → X is a principal SU(2) bundle over a Riemannian fourmanifold X, with c2(P) = k> 0, and A is a connection on P, the YangMills functional Y M(A) = FA  2 is minimal precisely when the curvature FA is antiself dual, i.e. FA = −∗FA, in which case A is called an instanton of charge k on X. Let MIk(X) denote the moduli space of framed instantons on X with charge k and let Ck(X) denote the space of all framed gauge equivalence classes of connections on X with charge k. In 1978, Atiyah and Jones [AJ] conjectured that the inclusion MIk(X) → Ck(X) induces an isomorphism in homology and homotopy through a range that grows with k. The original statement of the conjecture was for the case when X is a sphere, but the
Geometry of the Moduli Space of SelfDual Connections over the FourSphere
, 1992
"... A Riemannian metric on a compact fourmanifold induces a natural L 2 metric on the corresponding moduli space of (anti) selfdual connections on a principal G bundle P . When the bundle structure group G is SU(2) and \Gammac 2 (P ) = k, Groisser, Parker and others have found explicit formulas ..."
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Cited by 1 (0 self)
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A Riemannian metric on a compact fourmanifold induces a natural L 2 metric on the corresponding moduli space of (anti) selfdual connections on a principal G bundle P . When the bundle structure group G is SU(2) and \Gammac 2 (P ) = k, Groisser, Parker and others have found explicit formulas for the components of the L 2 metric on the moduli space M k when k = 1 and the fourmanifold is the sphere S 4 or the complex projective space C P 2 . The moduli space M 1 (S 4 ) is diffeomorphic to the open fiveball, while M 1 (C P 2 ) is diffeomorphic to the open cone over C P 2 : these moduli spaces have finite volume and diameter with respect to the L 2 metric. Donaldson, Groisser, and Parker have conjectured that the moduli space M k has finite volume and diameter with respect to the L 2 metric for any integer k. We consider the case where the fourmanifold is the spher...
Atiyah–Jones conjecture for blownup surfaces
, 2004
"... We show that if the Atiyah–Jones conjecture holds for a surface X, then it also holds for the blowup of X at a point. Since the conjecture is known to hold for P 2 and for ruled surfaces, it follows that the conjecture is true for all rational surfaces. Given a 4manifold X, let MIk(X) denote the m ..."
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We show that if the Atiyah–Jones conjecture holds for a surface X, then it also holds for the blowup of X at a point. Since the conjecture is known to hold for P 2 and for ruled surfaces, it follows that the conjecture is true for all rational surfaces. Given a 4manifold X, let MIk(X) denote the moduli space of rank 2 instantons on X with charge k and let Ck(X) denote the space of all gauge equivalence classes of connections on X with charge k. In 1978, Atiyah and Jones [AJ] conjectured that the inclusion MIk(X) → Ck(X) induces an isomorphism in homology and homotopy through a range that grows with k. The original statement of the conjecture was for the case when X is a sphere, but the question readily generalizes for other 4manifolds. The stable topology of these moduli spaces was understood in 1984, when Taubes [Ta] constructed instanton patching maps tk: MIk(X) → MIk+1(X) and showed that the stable limit lim MIk indeed has the ho
Compactness Theorems for Invariant Connections
, 2000
"... The PalaisSmale Condition C holds for the YangMills functional on principal bundles over compact manifolds of dimension ≤ 3. This was established by S. Sedlacek [17] and C. Taubes [18] Proposition 4.5 using the compactness theorem of K. Uhlenbeck [20]; see also [23]. It is well known that Conditio ..."
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The PalaisSmale Condition C holds for the YangMills functional on principal bundles over compact manifolds of dimension ≤ 3. This was established by S. Sedlacek [17] and C. Taubes [18] Proposition 4.5 using the compactness theorem of K. Uhlenbeck [20]; see also [23]. It is well known that Condition C fails for YangMills over compact manifolds of dimension ≥ 4. The example of SO(3)invariant SU(2)connections over S 4, see [2], [14], and [16], suggested that Condition C holds for YangMills over compact manifolds of any dimension when restricted to connections that are invariant under a group action on the manifold with orbits of codimension ≤ 3. Such a result, essentially Theorem 3 below, was established by T. Parker [14]. In this paper we generalize his result. Let X be a smooth compact Riemannian manifold of any dimension, let G be a compact Lie group, and let P be a smooth principal Gbundle over X. Let A denote the space of smooth connections on P and G the group of smooth bundle automorphisms of P. We denote the gauge equivalence class of A ∈ A by [A] ∈ A/G. We fix a Ginvariant positive definite inner product on the Lie algebra g. Then the YangMills functional is given by
The Nahm transform for calorons
, 705
"... Dedicated to Nigel Hitchin on the occasion of his sixtieth birthday. In this paper, we complete the proof of an equivalence given by Nye and Singer of the equivalence between calorons (instantons on S 1 × R 3) and solutions to Nahm’s equations over the circle, both satisfying appropriate boundary co ..."
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Dedicated to Nigel Hitchin on the occasion of his sixtieth birthday. In this paper, we complete the proof of an equivalence given by Nye and Singer of the equivalence between calorons (instantons on S 1 × R 3) and solutions to Nahm’s equations over the circle, both satisfying appropriate boundary conditions. Many of the key ingredients are provided by a third way of encoding the same data which involves twistors and complex geometry. 1
AND
, 1988
"... Let (S3, c) be the standard 3sphere, i.e., the 3sphere equipped with the standard metric. Let K be a C2 positive function on S3. The KazdanWarner problem [l] is the problem of finding suitable conditions on K such that K is the scalar curvature for a metric g on S3 conformally equivalent to c. Th ..."
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Let (S3, c) be the standard 3sphere, i.e., the 3sphere equipped with the standard metric. Let K be a C2 positive function on S3. The KazdanWarner problem [l] is the problem of finding suitable conditions on K such that K is the scalar curvature for a metric g on S3 conformally equivalent to c. The metric g then reads g=u4c and u is a positive function on S3 satisfying the partial differential equation 8 Au + 6u = K(x) us u> 0. (1) Let L = 8 Au + 624 be the conformal Laplacian. The same problem can be formulated for any compact Riemannian manifold (M”, g). Since this problem has been formulated, there have been some partial answers (see [37, 181). Obstructions have also been pointed out [l, 21. The main difficulty, arising when one tries to solve equations of type (4), consists of the failure of the PalaisSmale condition. We show, in this paper, how this difficulty may be overcome in the case Eq. (1). Our method consists of studying the critical points at infinity of the variational problem, in