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The Einstein–Maxwell Equations and Conformally Kähler Geometry

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Abstract

Page’s Einstein metric on \({{\mathbb{CP}}_2\#\overline{\mathbb{CP}}_2}\) is conformally related to an extremal Kähler metric. Here we construct a family of conformally Kähler solutions of the Einstein–Maxwell equations that deforms the Page metric, while sweeping out the entire Kähler cone of \({{\mathbb{CP}}_2\#\overline{\mathbb{CP}}_2}\). The same method also yields analogous solutions on every Hirzebruch surface. This allows us to display infinitely many geometrically distinct families of solutions of the Einstein–Maxwell equations on the smooth 4-manifolds \({S^2 \times S^2}\) and \({{\mathbb{CP}}_2\#\overline{\mathbb{CP}}_2}\).

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Authors and Affiliations

  1. SUNY Stony Brook, Stony Brook, NY, USA

    Claude LeBrun

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  1. Claude LeBrun
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Correspondence to Claude LeBrun.

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Communicated by N. A. Nekrasov

C. LeBrun’s research was funded in part by NSF Grant DMS-1510094.

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LeBrun, C. The Einstein–Maxwell Equations and Conformally Kähler Geometry. Commun. Math. Phys. 344, 621–653 (2016). https://doi.org/10.1007/s00220-015-2568-5

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  • DOI: https://doi.org/10.1007/s00220-015-2568-5

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