Four-Manifolds, Curvature Bounds, and Convex Geometry

LeBrun, Claude

doi:10.1007/978-0-8176-4743-8_6

Cited by 9 publications

(24 citation statements)

References 31 publications

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“…We will see in the proof of Theorem 1 that both 2p(S 2 × S 2 ) and 2q(CP 2 #CP 2 ) admit infinitely many distinct reducible smooth structures that support non-zero monopole classes (see Definition 4), provided that p and q satisfy the hypothesis of Theorem 1. Proposition 3.3 in [16] tells us that a smooth 4-manifold having non-zero monopole classes cannot admit any positive scalar curvature metric.…”

Section: Corollary 3 Let P and Q Be Integers Satisfying One Of The Fmentioning

confidence: 99%

Reducible smooth structures on 4-manifolds with zero signature

Akhmedov

Ishida

Park

2013

Journal of Topology

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Section: Corollary 3 Let P and Q Be Integers Satisfying One Of The Fmentioning

confidence: 99%

Reducible smooth structures on 4-manifolds with zero signature

Akhmedov

Ishida

Park

2013

Journal of Topology

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“…First of all, let us recall Definition 6 ( [38,42,34,43]) Let X be a closed oriented smooth 4-manifold…”

Section: Curvature Bounds Arising From the Seiberg-witten Equationsmentioning

confidence: 99%

Families of 4-Manifolds with Nontrivial Stable Cohomotopy Seiberg–Witten Invariants, and Normalized Ricci Flow

Baykur

Ishida

2013

J Geom Anal

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In this article, we produce infinite families of 4-manifolds with positive first betti numbers and meeting certain conditions on their homotopy and smooth types so as to conclude the non-vanishing of the stable cohomotopy Seiberg-Witten invariants of their connected sums. Elementary building blocks used in [35] are shown to be included in our general construction scheme as well. We then use these families to construct the first examples of families of closed smooth 4-manifolds for which Gromov's simplicial volume is nontrivial, Perelman'sλ invariant is negative, and the relevant Gromov-Hitchin-Thorpe type inequality is satisfied, yet no non-singular solution to the normalized Ricci flow for any initial metric can be obtained. In [16], Fang, Zhang and Zhang conjectured that the existence of any non-singular solution to the normalized Ricci flow on smooth 4-manifolds with non-trivial Gromov's simplicial volume and negative Perelman'sλ invariant implies the Gromov-Hitchin-Thorpe type inequality. Our results in particular imply that the converse of this fails to be true for vast families of 4-manifolds.

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“…Definition 2 [Kronheimer 1999;Ishida and LeBrun 2003;LeBrun 2009]. Let X be a closed oriented smooth 4-manifold with b + (X ) ≥ 2.…”

Section: Obstruction To the Existence Of Einstein Metricsmentioning

confidence: 99%

“…Moreover, it is known that the convex hull Hull(C(X )) is symmetric, that is, Hull(C(X )) = − Hull(C(X )). See [LeBrun 2009] for more details. Since C(X ) is a finite set, we are able to write C(X ) = {a 1 , a 2 , .…”

Section: Obstruction To the Existence Of Einstein Metricsmentioning

confidence: 99%

“…It is now well-known [LeBrun 1995a;1995b;2001;2009] that the existence of monopole classes (see Definition 2 below) gives rise to obstructions to the existence of Einstein metrics on 4-manifolds. In particular, any Einstein 4-manifold X with a nonzero special monopole class (see Section 2 below) must satisfy the inequality (2) χ (X ) ≥ 3τ (X ).…”

Section: Introductionmentioning

confidence: 99%

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