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 Kahler metric. In particular, this yields a non-trivial example of non-uniqueness of positive constant scalar curvature metrics in a given conformal class.