Strongly Hermitian Einstein-Maxwell Solutions on Ruled Surfaces

TL;DR

This paper constructs explicit strongly Hermitian Einstein-Maxwell solutions on certain 4-manifolds, extending previous existence results and demonstrating the abundance and non-uniqueness of such solutions in specific conformal classes.

Contribution

It generalizes previous existence results for Einstein-Maxwell solutions on ruled surfaces and calculates the Einstein-Hilbert functional for these solutions.

Findings

Explicit solutions on S^2-bundles over Riemann surfaces of any genus.

Generalization of the abundance of Hermitian Einstein-Maxwell solutions.

Examples of non-uniqueness of constant scalar curvature metrics in a conformal class.

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 C. LeBrun in arXiv:1411.3992 and arXiv:1504.06669. Moreover, by calculating the (normalized) Einstein-Hilbert functional of our examples we generalize Theorem E of arXiv:1504.06669, 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 arXiv:1504.06669, on the first Hirzebruch surface in a form which clearly shows that they are conformal to a common K\"ahler metric. In particular, this yields a non-trivial example of non-uniqueness of positive constant scalar curvature metrics in a given conformal class.