Smooth Structures and Einstein Metrics on $CP^2#5,6,7\bar{CP^2}$

Abstract

We show that each of the topological 4-manifolds $CP^2#k\bar{CP^2}, for $k = 6, 7$admitsasmoothstructurewhichhasanEinsteinmetricofscalarcurvature$s > 0$,asmoothstructurewhichhasanEinsteinmetricwith$s < 0$andinfinitelymanynon-diffeomorphicsmoothstructureswhichdonotadmitEinsteinmetrics.Weshowthatthereareinfinitelymanymanifoldshomeomorphicnon-diffeomorphicto$CP^2#5\bar{CP^2}$ which do not admit an Einstein metric. We also exhibit new higher dimensional examples of manifolds carrying Einstein metrics of both positive and negative scalar curvature. The main ingredients are recent constructions of exotic symplectic or complex manifolds with small topological numbers.

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References in corpus (3)

• A simply connected surface of general type with p_g=0 and K^2=2
• Simply connected symplectic 4-manifolds with $b_2^{+}=1$ and $c_1^2=2$
• Constructing infinitely many smooth structures on small 4-manifolds

Cited by in corpus (1)

• The Yamabe invariant of a class of symplectic manifolds