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42
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
On Einstein manifolds of positive sectional curvature, preprint
"... Let (M, g) be a compact oriented 4dimensional Einstein manifold. If M has positive intersection form and g has nonnegative sectional curvature, we show that, up to rescaling and isometry, (M, g) is CP2, with its standard FubiniStudy metric. 1 ..."
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Cited by 25 (0 self)
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Let (M, g) be a compact oriented 4dimensional Einstein manifold. If M has positive intersection form and g has nonnegative sectional curvature, we show that, up to rescaling and isometry, (M, g) is CP2, with its standard FubiniStudy metric. 1
The Yamabe invariant of simply connected manifolds
 J. Reine Angew. Math
"... Let M be any simply connected smooth compact manifold of dimension n ≥ 5. We prove that the Yamabe invariant of M is nonnegative. This is equivalent to say that the infimum, over the space of all Riemannian metrics on M, of the L n/2 norm of the scalar curvature is zero. 1 ..."
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Cited by 24 (4 self)
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Let M be any simply connected smooth compact manifold of dimension n ≥ 5. We prove that the Yamabe invariant of M is nonnegative. This is equivalent to say that the infimum, over the space of all Riemannian metrics on M, of the L n/2 norm of the scalar curvature is zero. 1
Singular Lefschetz pencils
 Geom. Topol
"... We consider structures analogous to symplectic Lefschetz pencils in the context of a closed 4manifold equipped with a \nearsymplectic" structure (i.e., a closed 2form which is symplectic outside a union of circles where it vanishes transversely). Our main result asserts that, up to blowups, ..."
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Cited by 17 (0 self)
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We consider structures analogous to symplectic Lefschetz pencils in the context of a closed 4manifold equipped with a \nearsymplectic" structure (i.e., a closed 2form which is symplectic outside a union of circles where it vanishes transversely). Our main result asserts that, up to blowups, every nearsymplectic 4manifold (X;!) can be decomposed into (a) two symplectic Lefschetz ¯brations over discs, and (b) a ¯bre bundle over S1 which relates the boundaries of the Lefschetz ¯brations to each other via a sequence of ¯brewise handle additions taking place in a neighbourhood of the zero set of the 2form. Conversely, from such a decomposition one can recover a near
The structure of pseudoholomorphic subvarieties for a degenerate almost complex structure and symplectic form on S1 × B3
 GEOMETRY & TOPOLOGY
, 1998
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ON YAMABE CONSTANTS OF RIEMANNIAN PRODUCTS
, 2006
"... Abstract. For a closed Riemannian manifold (M m, g) of constant positive scalar curvature and any other closed Riemannian manifold (N n, h), we show that the limit of the Yamabe constants of the Riemannian products (M × N, g + rh) as r goes to infinity is equal to the Yamabe constant of (M m × R n, ..."
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Cited by 14 (7 self)
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Abstract. For a closed Riemannian manifold (M m, g) of constant positive scalar curvature and any other closed Riemannian manifold (N n, h), we show that the limit of the Yamabe constants of the Riemannian products (M × N, g + rh) as r goes to infinity is equal to the Yamabe constant of (M m × R n, [g + g E]) and is strictly less than the Yamabe invariant of S m+n provided n ≥ 2. We then consider the minimum of the Yamabe functional restricted to functions of the second variable and we compute the limit in terms of the best constants of the GagliardoNirenberg inequalities. 1.
Yamabe invariants and Spin c structures
 Geom. Funct. Anal
, 1998
"... The Yamabe invariant of a smooth compact manifold is by definition the supremum of the scalar curvatures of unitvolume Yamabe metrics on the manifold. For an explicit infinite class of 4manifolds, we show that this invariant is positive but strictly less than that of the 4sphere. This is done by ..."
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Cited by 9 (1 self)
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The Yamabe invariant of a smooth compact manifold is by definition the supremum of the scalar curvatures of unitvolume Yamabe metrics on the manifold. For an explicit infinite class of 4manifolds, we show that this invariant is positive but strictly less than that of the 4sphere. This is done by using spin c Dirac operators to control the lowest eigenvalue of a perturbation of the Yamabe Laplacian. These results dovetail perfectly with those derived from the perturbed SeibergWitten equations [14], but the present method is much more elementary in spirit. 1
Hermitian conformal classes and almost Kähler structures on four manifolds, Diff
 Geom. Appl
, 1999
"... structures on 4manifolds ..."
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