Einstein Manifolds, Self-Dual Weyl Curvature, and Conformally Kaehler Geometry
Einstein Manifolds, Self-Dual Weyl Curvature, and Conformally Kaehler Geometry
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Claude LeBrun, Stony Brook University
There are certain compact 4-manifolds, such as real and complex hyperbolic 4-manifolds, 4-tori, and K3, where we completely understand the moduli space of Einstein metrics. But there are vast numbers of other 4-manifolds where we know that Einstein metrics exist, but cannot currently determine whether or not there听might also exist other Einstein metrics on them that are utterly different from the 听ones we currently know.
In this lecture, I will first present a characterization of the known Einstein metrics on del Pezzo surfaces which I proved听several years ago, I will 听then describe some recent results which considerably improve our 听understanding of Einstein metrics on听these 4-manifolds, and suggest new avenues of research.