The Einstein-Maxwell Equations and Conformally Kaehler Geometry

TL;DR

This paper constructs a family of conformally Kähler solutions to the Einstein-Maxwell equations on certain 4-manifolds, generalizing known metrics and revealing infinitely many distinct solutions with rich geometric structures.

Contribution

It introduces a method to generate a continuous family of Einstein-Maxwell solutions on complex surfaces, expanding the known landscape of such solutions and connecting them to conformally Kähler geometry.

Findings

Constructed solutions deforming the Page metric.

Solutions sweep out the entire Kähler cone of CP_2 # (-CP_2).

Produced infinitely many geometrically distinct solutions on S^2 x S^2 and CP_2 # (-CP_2).

Abstract

Page's Einstein metric on CP_2 # (-CP_2) is conformally related to an extremal Kaehler metric. Here we construct a family of conformally K\"ahler solutions of the Einstein-Maxwell equations that deforms the Page metric, while sweeping out the entire Kaehler cone of CP_2 # (-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 x S^2 and CP_2 # (-CP_2).