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14
Metrics of Positive Scalar Curvature and Connections With Surgery
 Annals of Math. Studies
, 2001
"... this paper will be assumed to be smooth (C 1 ). For simplicity, we restrict attention to compact manifolds, although there are also plenty of interesting questions about complete metrics of positive scalar curvature on noncompact manifolds. At some points in the discussion, however, it will be ne ..."
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Cited by 45 (1 self)
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this paper will be assumed to be smooth (C 1 ). For simplicity, we restrict attention to compact manifolds, although there are also plenty of interesting questions about complete metrics of positive scalar curvature on noncompact manifolds. At some points in the discussion, however, it will be necessary to consider manifolds with boundary
Yamabe constants and the perturbed SeibergWitten equations
 Comm. Anal. Geom
, 1997
"... Among all conformal classes of Riemannian metrics on CP2, that of the FubiniStudy metric is shown to have the largest Yamabe constant. The proof, which involves perturbations of the SeibergWitten equations, also yields new results on the total scalar curvature of almostKähler 4manifolds. 1 ..."
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Cited by 42 (9 self)
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Among all conformal classes of Riemannian metrics on CP2, that of the FubiniStudy metric is shown to have the largest Yamabe constant. The proof, which involves perturbations of the SeibergWitten equations, also yields new results on the total scalar curvature of almostKähler 4manifolds. 1
Kodaira dimension and the Yamabe problem
 Comm. Anal. Geom
, 1999
"... The Yamabe invariant Y (M) of a smooth compact manifold is roughly the supremum of the scalar curvatures of unitvolume constantscalar curvature Riemannian metrics g on M. (To be absolutely precise, one only considers constantscalarcurvature metrics which are Yamabe minimizers, but this does not ..."
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Cited by 41 (4 self)
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The Yamabe invariant Y (M) of a smooth compact manifold is roughly the supremum of the scalar curvatures of unitvolume constantscalar curvature Riemannian metrics g on M. (To be absolutely precise, one only considers constantscalarcurvature metrics which are Yamabe minimizers, but this does not affect the sign of the answer.) If M is the underlying smooth 4manifold of a complex algebraic surface (M, J), it is shown that the sign of Y (M) is completely determined by the Kodaira dimension
Polarized 4manifolds, extremal Kähler metrics, and Seiberg–Witten
 653–662. GEOMETRY ON COMPLEX VECTOR BUNDLES 105
, 1995
"... Abstract. Using SeibergWitten theory, it is shown that any Kähler metric of constant negative scalar curvature on a compact 4manifold M minimizes the L 2norm of scalar curvature among Riemannian metrics compatible with a fixed decomposition H 2 (M) =H + ⊕ H −. This implies, for example, that any ..."
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Cited by 33 (12 self)
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Abstract. Using SeibergWitten theory, it is shown that any Kähler metric of constant negative scalar curvature on a compact 4manifold M minimizes the L 2norm of scalar curvature among Riemannian metrics compatible with a fixed decomposition H 2 (M) =H + ⊕ H −. This implies, for example, that any such metric on a minimal ruled surface must be locally symmetric. 1.
Indefinite KählerEinstein metrics on compact complex surfaces
 Commun. Math. Phys
, 1997
"... Indefinite Kähler solutions of the Einstein equations are studied, and it is almost completely determined which compact complex surfaces admit such metrics. 1 ..."
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Cited by 20 (0 self)
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Indefinite Kähler solutions of the Einstein equations are studied, and it is almost completely determined which compact complex surfaces admit such metrics. 1
ON THE SCALAR CURVATURE OF EINSTEIN MANIFOLDS
 MATHEMATICAL RESEARCH LETTERS 4, 843–854 (1997)
, 1997
"... We show that there are highdimensional smooth compact manifolds which admit pairs ofEinstein metrics for which the scalar curvatures have opposite signs. These are counterexamples to a conjecture considered by Besse [6, p. 19]. The proof hinges on showing that the Barlow surface has small deforma ..."
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Cited by 16 (3 self)
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We show that there are highdimensional smooth compact manifolds which admit pairs ofEinstein metrics for which the scalar curvatures have opposite signs. These are counterexamples to a conjecture considered by Besse [6, p. 19]. The proof hinges on showing that the Barlow surface has small deformations with ample canonical line bundle.
Blowing up Kähler manifolds with constant scalar curvature II
, 2005
"... In this paper we prove the existence of Kähler metrics of constant scalar curvature on the blow up at finitely many points of a compact manifold which already carries a Kähler constant scalar curvature metric. Necessary conditions of the number and locations of the blow up points are given. ..."
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Cited by 14 (3 self)
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In this paper we prove the existence of Kähler metrics of constant scalar curvature on the blow up at finitely many points of a compact manifold which already carries a Kähler constant scalar curvature metric. Necessary conditions of the number and locations of the blow up points are given.
Hermitian conformal classes and almost Kähler structures on four manifolds, Diff
 Geom. Appl
, 1999
"... structures on 4manifolds ..."
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Einstein Metrics on Complex Surfaces
"... Which smooth compact 4manifolds admit an Einstein metric with nonnegative Einstein constant? A complete answer is provided in the special case of 4manifolds that also happen to admit either a complex structure or a symplectic structure. A Riemannian manifold (M, g) is said to be Einstein if it ha ..."
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Cited by 6 (1 self)
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Which smooth compact 4manifolds admit an Einstein metric with nonnegative Einstein constant? A complete answer is provided in the special case of 4manifolds that also happen to admit either a complex structure or a symplectic structure. A Riemannian manifold (M, g) is said to be Einstein if it has constant Ricci curvature, in the sense that the function v ↦− → r(v, v) on the unit tangent bundle {v ∈ TM‖v‖g = 1} is constant, where r denotes the Ricci tensor of g. This is of course equivalent to demanding that g satisfy the Einstein equation r = λg for some real number λ. A fundamental open problem in global Riemannian geometry is to determine precisely which smooth compact nmanifolds admit Einstein metrics. For further background on this problem, see [4]. When n = 4, the problem is deeply intertwined with geometric and topological phenomena unique to this dimension; and our discussion here will therefore solely focus on this