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10
The curvature and the integrability of almostKähler manifolds: a survey
, 2003
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LOCAL MODELS AND INTEGRABILITY OF CERTAIN ALMOST Kähler 4manifolds
, 2001
"... We classify, up to a local isometry, all nonKähler almost Kähler 4manifolds for which the fundamental 2form is an eigenform of the Weyl tensor, and whose Ricci tensor is invariant with respect to the almost complex structure. Equivalently, such almost Kähler 4manifolds satisfy the third curvatu ..."
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Cited by 18 (4 self)
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We classify, up to a local isometry, all nonKähler almost Kähler 4manifolds for which the fundamental 2form is an eigenform of the Weyl tensor, and whose Ricci tensor is invariant with respect to the almost complex structure. Equivalently, such almost Kähler 4manifolds satisfy the third curvature condition of A. Gray. We use our local classification to show that, in the compact case, the third curvature condition of Gray is equivalent to the integrability of the corresponding almost complex structure. 2000 Mathematics Subject Classification.
Almost Kähler 4manifolds with Jinvariant Ricci tensor and . . .
 HOUSTON J. OF MATH
, 1999
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Einstein metrics and the number of smooth structures on a fourmanifold
, 2003
"... We prove that for every natural number k there are simply connected topological four–manifolds which have at least k distinct smooth structures supporting Einstein metrics, and also have infinitely many distinct smooth structures not supporting Einstein metrics. Moreover, all these smooth structur ..."
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Cited by 8 (2 self)
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We prove that for every natural number k there are simply connected topological four–manifolds which have at least k distinct smooth structures supporting Einstein metrics, and also have infinitely many distinct smooth structures not supporting Einstein metrics. Moreover, all these smooth structures become diffeomorphic to each other after connected sum with only one copy of the complex projective plane. We prove that manifolds with these properties cover a large geographical area.
Topological restrictions for circle actions and harmonic morphisms
, 2000
"... Let M m be a compact oriented smooth manifold which admits a smooth circle action with isolated fixed points which are isolated as singularities as well. Then all the Pontryagin numbers of M m are zero and its Euler number is nonnegative and even. In particular, M m has signature zero. We apply this ..."
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Cited by 5 (3 self)
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Let M m be a compact oriented smooth manifold which admits a smooth circle action with isolated fixed points which are isolated as singularities as well. Then all the Pontryagin numbers of M m are zero and its Euler number is nonnegative and even. In particular, M m has signature zero. We apply this to obtain nonexistence of harmonic morphisms with onedimensional fibres from various domains, and a classification of harmonic morphisms from certain 4manifolds.
The Weyl functional near the Yamabe invariant
, 2002
"... For a compact manifold M of dim M ≥ 4, we study two conformal invariants of a conformal class C on M. These are the Yamabe constant YC(M) and the L n 2norm WC(M) of the Weyl curvature. We prove that for any manifold M there exists a conformal class C such that the Yamabe constant YC(M) is arbitrari ..."
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Cited by 4 (1 self)
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For a compact manifold M of dim M ≥ 4, we study two conformal invariants of a conformal class C on M. These are the Yamabe constant YC(M) and the L n 2norm WC(M) of the Weyl curvature. We prove that for any manifold M there exists a conformal class C such that the Yamabe constant YC(M) is arbitrarily close to the Yamabe invariant Y (M), and, at the same time, the constant WC(M) is arbitrarily large. We study the image of the map YW: C ↦ → (YC(M),WC(M)) ∈ R2 near the line {(Y (M),w)  w ∈ R}. We also apply our results to certain classes of 4manifolds, in particular, minimal compact Kähler surfaces of Kodaira dimension 0, 1 or 2.
Einstein Metrics And The Yamabe Problem
, 1999
"... Which smooth compact nmanifolds admit Riemannian metrics of constant Ricci curvature? A direct variational approach sheds some interesting light on this problem, but by no means answers it. This article surveys some recent results concerning both Einstein metrics and the associated variational ..."
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Cited by 3 (0 self)
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Which smooth compact nmanifolds admit Riemannian metrics of constant Ricci curvature? A direct variational approach sheds some interesting light on this problem, but by no means answers it. This article surveys some recent results concerning both Einstein metrics and the associated variational problem, with the particular aim of highlighting the striking manner in which the 4dimensional case differs from the case of dimensions >= 5.
SeibergWitten invariants of nonsimple type and Einstein metrics Heberto del Rio Guerra ∗
, 2008
"... We construct examples of four dimensional manifolds with Spin cstructures, whose moduli spaces of solutions to the SeibergWitten equations, represent a nontrivial bordism class of positive dimension, i.e. the Spin cstructures are not induced by almost complex structures. As an application, we sh ..."
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We construct examples of four dimensional manifolds with Spin cstructures, whose moduli spaces of solutions to the SeibergWitten equations, represent a nontrivial bordism class of positive dimension, i.e. the Spin cstructures are not induced by almost complex structures. As an application, we show the existence of infinitely many nonhomeomorphic compact oriented 4manifolds with free fundamental group and predetermined Euler characteristic and signature that do not carry Einstein metrics (see [10]). 1