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.

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Notes

  1. 1.

    Here we just consider the linear operator \(y \mapsto ({\mathfrak {z}}+b)^2y''- 6({\mathfrak {z}}+b) y' + 12 y\).

  2. 2.

    In fact, there are no CSC Kähler metrics on \(S_n\) [5].

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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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Correspondence to Caner Koca.

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This work was partially supported by a grant from the Simons Foundation (208799 to Christina Tønnesen-Friedman).

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Koca, C., Tønnesen-Friedman, C.W. Strongly Hermitian Einstein–Maxwell solutions on ruled surfaces. Ann Glob Anal Geom 50, 29–46 (2016). https://doi.org/10.1007/s10455-016-9499-z

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Keywords

  • Ruled surfaces
  • Einstein–Maxwell metrics
  • Conformally Kaehler metrics

Mathematics Subject Classification

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