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41
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.
Minimal entropy and collapsing with curvature bounded from below
 Invent. Math
"... Abstract. We show that if a closed manifold M admits an Fstructure (not necessarily polarized, possibly of rank zero) then its minimal entropy vanishes. In particular, this is the case if M admits a nontrivial S 1action. As a corollary we obtain that the simplicial volume of a manifold admitting ..."
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Cited by 37 (4 self)
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Abstract. We show that if a closed manifold M admits an Fstructure (not necessarily polarized, possibly of rank zero) then its minimal entropy vanishes. In particular, this is the case if M admits a nontrivial S 1action. As a corollary we obtain that the simplicial volume of a manifold admitting an Fstructure is zero. We also show that if M admits an Fstructure then it collapses with curvature bounded from below. This in turn implies that M collapses with bounded scalar curvature or, equivalently, its Yamabe invariant is nonnegative. We show that Fstructures of rank zero appear rather frequently: every compact complex elliptic surface admits one as well as any simply connected closed 5manifold. We use these results to study the minimal entropy problem. We show the following two theorems: suppose that M is a closed manifold obtained by taking connected sums of copies of S 4, CP 2, CP 2, S 2 × S 2 and the K3 surface. Then M has zero minimal entropy. Moreover, M admits a smooth Riemannian metric with zero topological entropy if and only if M is diffeomorphic to S 4, CP 2, S 2 × S 2, CP 2 #CP 2 or CP 2 #CP 2. Finally, suppose that M is a closed simply connected 5manifold. Then M has zero minimal entropy. Moreover, M admits a smooth Riemannian metric with zero topological entropy if and only if M is diffeomorphic to S 5, S 3 ×S 2, the nontrivial S 3bundle over S 2 or the Wumanifold SU(3)/SO(3). 1.
On the Topology and Area of HigherDimensional Black Holes
, 2001
"... Over the past decade there has been an increasing interest in the study of black holes, and related objects, in higher (and lower) dimensions, motivated to a large extent by developments in string theory. The aim of the present paper is to obtain higherdimensional analogues of some well known resul ..."
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Cited by 27 (4 self)
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Over the past decade there has been an increasing interest in the study of black holes, and related objects, in higher (and lower) dimensions, motivated to a large extent by developments in string theory. The aim of the present paper is to obtain higherdimensional analogues of some well known results for black holes in 3 + 1 dimensions. More precisely, we obtain extensions to higher dimensions of Hawking's black hole topology theorem for asymptotically flat (# 0) black hole spacetimes, and Gibbons' and Woolgar's genusdependent, lower entropy bound for topological black holes in asymptotically locally antide Sitter (#<0) spacetimes. In higher dimensions the genus is replaced by the socalled # constant, or Yamabe invariant, which is a fundamental topological invariant of smooth compact manifolds.
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
Constant mean curvature foliations of flat spacetimes
 Commun. Anal. Geom
"... Abstract. Let V be a maximal globally hyperbolic flat n+1–dimensional space–time with compact Cauchy surface of hyperbolic type. We prove that V is globally foliated by constant mean curvature hypersurfaces Mτ, with mean curvature τ taking all values in (−∞,0). For n ≥ 3, define the rescaled volume ..."
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Cited by 17 (7 self)
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Abstract. Let V be a maximal globally hyperbolic flat n+1–dimensional space–time with compact Cauchy surface of hyperbolic type. We prove that V is globally foliated by constant mean curvature hypersurfaces Mτ, with mean curvature τ taking all values in (−∞,0). For n ≥ 3, define the rescaled volume of Mτ by H = τ  n Vol(M, g), where g is the induced metric. Then H ≥ n n Vol(M, g0) where g0 is the hyperbolic metric on M with sectional curvature −1. Equality holds if and only if (M, g) is isometric to (M, g0). 1.
Entropy and collapsing of compact complex surfaces
 the Proceedings of the London Math. Soc
"... Abstract. We study the problem of existence of Fstructures (in the sense of Cheeger and Gromov, but not necessarily polarized) on compact complex surfaces. We give a complete classification of compact complex surfaces of Kähler type admitting Fstructures. In the nonKähler case we give a complete ..."
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Cited by 11 (2 self)
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Abstract. We study the problem of existence of Fstructures (in the sense of Cheeger and Gromov, but not necessarily polarized) on compact complex surfaces. We give a complete classification of compact complex surfaces of Kähler type admitting Fstructures. In the nonKähler case we give a complete classification modulo the gap in the classification of surfaces of class VII. In all these examples a surface admits an Fstructure if and only if it admits a Tstructure. We then use these results to study the minimal entropy problem for compact complex surfaces: we prove that, modulo the gap in the classification of surfaces of class VII, all compact complex surfaces of Kodaira dimension ≤ 1 have minimal entropy 0 and such a surface admits a smooth metric g with htop(g) = 0 if and only if it is CP 2, a ruled surface of genus 0 or 1, a Hopf surface, a complex torus, a Kodaira surface, a hyperelliptic surface or a Kodaira surface modulo a finite group. The key result we use to prove this, is a new topological obstruction to the existence of metrics with vanishing topological entropy. Finally we show that these results fit perfectly into Wall’s study of geometric structures on compact complex surfaces. For instance, we show that the minimal entropy problem can be solved for a minimal compact Kähler surface S of Kodaira dimension −∞, 0 or 1 if and only if S admits a geometric structure modelled on CP 2, S 2 × S 2, S 2 × E 2 or E 4. 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
The Yamabe invariant for nonsimply connected manifolds
 J. Differential Geom
, 2001
"... The Yamabe invariant is an invariant of a closed smooth manifold defined using conformal geometry and the scalar curvature. Recently, Petean showed that the Yamabe invariant is nonnegative for all closed simply connected manifolds of dimension # 5. We extend this to show that Yamabe invariant is ..."
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Cited by 8 (2 self)
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The Yamabe invariant is an invariant of a closed smooth manifold defined using conformal geometry and the scalar curvature. Recently, Petean showed that the Yamabe invariant is nonnegative for all closed simply connected manifolds of dimension # 5. We extend this to show that Yamabe invariant is nonnegative for all closed manifolds of dimension # 5 with fundamental group of odd order having all Sylow subgroups abelian. The main new geometric input is a way of studying the Yamabe invariant on Toda brackets. A similar method of proof shows that all closed manifolds of dimension # 5 with fundamental group of odd order having all Sylow subgroups elementary abelian, with nonspin universal cover, admit metrics of positive scalar curvature, once one restricts to the "complement" of manifolds whose homology classes are "toral." The exceptional toral homology classes only exist in dimensions not exceeding the "rank" of the fundamental group, so this proves important cases of the GromovLawsonRosenberg Conjecture once the dimension is su#ciently large. 1
3MANIFOLDS WITH YAMABE INVARIANT GREATER THAN THAT OF RP³
, 2006
"... We show that, for all nonnegative integers k, ℓ, m and n, the Yamabe invariant of ..."
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Cited by 8 (1 self)
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We show that, for all nonnegative integers k, ℓ, m and n, the Yamabe invariant of