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Einstein metrics, harmonic forms, and symplectic four-manifolds

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Abstract

If \(M\) is the underlying smooth oriented four-manifold of a Del Pezzo surface, we consider the set of Riemannian metrics \(h\) on \(M\) such that \(W^+(\omega , \omega )> 0\), where \(W^+\) is the self-dual Weyl curvature of \(h\), and \(\omega \) is a non-trivial self-dual harmonic two-form on \((M,h)\). While this open region in the space of Riemannian metrics contains all the known Einstein metrics on \(M\), we show that it contains no others. Consequently, it contributes exactly one connected component to the moduli space of Einstein metrics on \(M\).

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Acknowledgments

The author would like to thank Tedi Draghici for subsequently pointing out some of his own related work, and the anonymous referee for suggesting ways to streamline and clarify the exposition. This work was supported in part by NSF Grant DMS-1205953.

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  1. Department of Mathematics, State University of New York, Stony Brook, NY, 11794-3651, USA

    Claude LeBrun

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  1. Claude LeBrun
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Correspondence to Claude LeBrun.

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LeBrun, C. Einstein metrics, harmonic forms, and symplectic four-manifolds. Ann Glob Anal Geom 48, 75–85 (2015). https://doi.org/10.1007/s10455-015-9458-0

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