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12
Ricci Curvature, Minimal Volumes, and SeibergWitten Theory
, 2000
"... We derive new, sharp lower bounds for certain curvature functionals on the space of Riemannian metrics of a smooth compact 4manifold with a nontrivial SeibergWitten invariant. These allow one, for example, to exactly compute the infimum of the L 2norm of Ricci curvature for all complex surfaces ..."
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Cited by 37 (2 self)
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We derive new, sharp lower bounds for certain curvature functionals on the space of Riemannian metrics of a smooth compact 4manifold with a nontrivial SeibergWitten invariant. These allow one, for example, to exactly compute the infimum of the L 2norm of Ricci curvature for all complex surfaces of general type. We are also able to show that the standard metric on any complex hyperbolic 4manifold minimizes volume among all metrics satisfying a pointwise lower bound on sectional curvature plus suitable multiples of the scalar curvature. These estimates also imply new nonexistence results for Einstein metrics.
Einstein metrics on spheres
 Ann. of Math
, 2005
"... Any sphere S n admits a metric of constant sectional curvature. These canonical metrics are homogeneous and Einstein, that is the Ricci curvature is a constant multiple of the metric. The spheres S 4m+3, m> 1 are known to have another Sp(m + 1)homogeneous Einstein metric discovered by Jensen [Je ..."
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Cited by 32 (13 self)
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Any sphere S n admits a metric of constant sectional curvature. These canonical metrics are homogeneous and Einstein, that is the Ricci curvature is a constant multiple of the metric. The spheres S 4m+3, m> 1 are known to have another Sp(m + 1)homogeneous Einstein metric discovered by Jensen [Jen73]. In addition,
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.
Spin manifolds, Einstein metrics, and differential topology
 Math. Res. Lett
"... We show that there exist smooth, simply connected, fourdimensional spin manifolds which do not admit Einstein metrics, but nevertheless satisfy the strict HitchinThorpe inequality. Our construction makes use of the Bauer/Furuta cohomotopy refinement of the SeibergWitten invariant [4, 3], in conjun ..."
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Cited by 16 (8 self)
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We show that there exist smooth, simply connected, fourdimensional spin manifolds which do not admit Einstein metrics, but nevertheless satisfy the strict HitchinThorpe inequality. Our construction makes use of the Bauer/Furuta cohomotopy refinement of the SeibergWitten invariant [4, 3], in conjunction with curvature estimates previously proved by the second author [17]. These methods also easily allow one to construct examples of topological 4manifolds which admit an Einstein metric for one smooth structure, but which have infinitely many other smooth structures for which no Einstein metric can exist. 1
Almost Kähler 4manifolds with Jinvariant Ricci tensor and . . .
 HOUSTON J. OF MATH
, 1999
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Weyl curvature, Einstein metrics, and Seiberg–Witten theory
 Math. Research Letters
, 1998
"... We show that solutions of the SeibergWitten equations lead to nontrivial estimates for the L 2norm of the Weyl curvature of a smooth compact 4manifold. These estimates are then used to derive new obstructions to the existence of Einstein metrics on smooth compact 4manifolds with a nonzero Seibe ..."
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Cited by 10 (1 self)
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We show that solutions of the SeibergWitten equations lead to nontrivial estimates for the L 2norm of the Weyl curvature of a smooth compact 4manifold. These estimates are then used to derive new obstructions to the existence of Einstein metrics on smooth compact 4manifolds with a nonzero SeibergWitten invariant. These results considerably refine those previously obtained [21] by using scalarcurvature estimates alone. 1
Einstein metrics, fourmanifolds and differential topology, in
 Surveys in Differential Geometry, vol. VIII, Ed. S.T.Yau, International
, 2003
"... A Riemannian metric g on a smooth manifold M is said to be Einstein if it has constant Ricci curvature, or in other words if (1) r = λg, where r is the Ricci tensor of g and λ is some real constant [7]. We still do not know if there are any obstructions to the existence of Einstein metrics on highdi ..."
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Cited by 6 (1 self)
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A Riemannian metric g on a smooth manifold M is said to be Einstein if it has constant Ricci curvature, or in other words if (1) r = λg, where r is the Ricci tensor of g and λ is some real constant [7]. We still do not know if there are any obstructions to the existence of Einstein metrics on highdimensional manifolds, but it has now been known for three decades that not every 4manifold admits such metrics [20, 37]. Only recently, however, has it emerged that there are also obstructions to the existence of Einstein metrics which depend on the differentiable structure rather than just on the homotopy type of a 4manifold [27, 23, 28, 22]. This article will attempt to give a concise explanation of this state of affairs. Our current understanding of the problem rests largely on certain curvature estimates which are deduced from the SeibergWitten equations. The strongest of these [28] was originally proved indirectly, by invoking the solution to a generalized version of the Yamabe problem. One main purpose of the present article is to give (§4) a new and simpler proof of this estimate, using conformal rescaling only in order to introduce a generalized form of the SeibergWitten equations. But this article will also attempt to clarify the nature of the resulting obstructions, by systematically reformulating them in terms of a new diffeomorphism invariant, called α(M), which is introduced in §3. As this article will make abundantly clear, Blaine Lawson’s work on spin geometry and scalar curvature has had a deep and lasting impact on my own research. On a more personal level, Blaine has also been tremendous source of inspiration and encouragement throughout my many years at Stony Brook. I am lucky indeed to be able to call him a friend and colleague, and it is a very great pleasure for me to be able to contribute an article to this volume. 1. Differential Geometry on 4Manifolds The curvature and topology of 4manifolds are interrelated in a number of ways that have no adequate analogs in other dimensions. Many of these phenomena
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.