The Einstein–Maxwell Equations and Conformally Kähler Geometry

Claude LeBrun

doi:10.1007/s00220-015-2568-5

The Einstein–Maxwell Equations and Conformally Kähler Geometry

Claude LeBrun

Mathematics

Research output: Contribution to journal › Article › peer-review

29 Scopus citations

Abstract

Page’s Einstein metric on CP₂# 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 CP₂# 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 S²× S² and CP₂# CP¯ ₂.

Original language	English
Pages (from-to)	621-653
Number of pages	33
Journal	Communications in Mathematical Physics
Volume	344
Issue number	2
DOIs	https://doi.org/10.1007/s00220-015-2568-5
State	Published - Jun 1 2016

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title = "The Einstein{\textendash}Maxwell Equations and Conformally K{\"a}hler Geometry",

abstract = "Page{\textquoteright}s Einstein metric on CP2\# CP¯ 2 is conformally related to an extremal K{\"a}hler metric. Here we construct a family of conformally K{\"a}hler solutions of the Einstein{\textendash}Maxwell equations that deforms the Page metric, while sweeping out the entire K{\"a}hler cone of CP2\# 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{\textendash}Maxwell equations on the smooth 4-manifolds S2{\texttimes} S2 and CP2\# CP¯ 2.",

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AB - Page’s Einstein metric on CP2# 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 CP2# 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 S2× S2 and CP2# CP¯ 2.

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JO - Communications in Mathematical Physics

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The Einstein–Maxwell Equations and Conformally Kähler Geometry

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