# Strongly Hermitian Einstein–Maxwell solutions on ruled surfaces

## Abstract

This paper produces explicit strongly Hermitian Einstein–Maxwell solutions on the smooth compact 4-manifolds that are \(S^2\)-bundles over compact Riemann surfaces of any genus. This generalizes the existence results by LeBrun (J Geom Phys 91:163–171, 2015, The Einstein–Maxwell equations and conformally Kaehler geometry. arXiv:1504.06669, 2016). Moreover, by calculating the (normalized) Einstein–Hilbert functional of our examples, we generalize Theorem E of LeBrun (The Einstein–Maxwell equations and conformally Kaehler geometry. arXiv:1504.06669, 2016), which speaks to the abundance of Hermitian Einstein–Maxwell solutions on such manifolds. As a bonus, we exhibit certain pairs of strongly Hermitian Einstein–Maxwell solutions, first found in LeBrun (The Einstein–Maxwell equations and conformally Kaehler geometry. arXiv:1504.06669, 2016), on the first Hirzebruch surface in a form which clearly shows that they are conformal to a common Kähler metric. In particular, this yields a non-trivial example of non-uniqueness of positive constant scalar curvature metrics in a given conformal class.

### Keywords

Ruled surfaces Einstein–Maxwell metrics Conformally Kaehler metrics### Mathematics Subject Classification

53C55 (Primary) 14J26 53C25 53C80 83C15 83C22 (Secondary)## Notes

### Acknowledgments

We would like to warmly thank Claude LeBrun for his advice and encouragement as we were writing this paper. We would also like to thank Vestislav Apostolov and Gideon Maschler for their insightful comments. Further, we are grateful to the anonymous referee for making some very helpful comments that resulted in the addition of Sect. 5.

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