Einstein metrics and the number of smooth structures on a four-manifold

Abstract

We prove that for every natural number k there are simply connected topological four-manifolds which have at leat k distinct smooth structures supporting Einstein metrics, and also have infinitely many distinct smooth structures not supporting Einstein metrics. Moreover, all these smooth structures become diffeomorphic after connected sum with only one copy of the complex projective plane. We prove that manifolds with these properties cover a large geographical area.

23 pages

References in corpus (4)

• Simply connected symplectic 4-manifolds with $b_2^{+}=1$ and $c_1^2=2$
• Ricci Curvature, Minimal Volumes, and Seiberg-Witten Theory
• Einstein metrics and smooth structures
• Monopole classes and Einstein metrics

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• Smooth Structures and Normalized Ricci Flows on Non-Simply Connected Four-Manifolds
• On Einstein metrics, normalized Ricci flow and smooth structures on $3\mathbb{CP}^2 # k \bar{\mathbb{CP}}^2$
• On finite symmetries of simply connected four-manifolds