Selfdual Einstein metrics and conformal submersions

Abstract

Weyl derivatives, Weyl-Lie derivatives and conformal submersions are defined, then used to generalize the Jones-Tod correspondence between selfdual 4-manifolds with symmetry and Einstein-Weyl 3-manifolds with an abelian monopole. In this generalization, the conformal symmetry is replaced by a particular kind of conformal submersion with one dimensional fibres. Special cases are studied in which the conformal submersion is holomorphic, affine, or projective. All scalar-flat Kahler metrics with such a holomorphic conformal submersion, and all four dimensional hypercomplex structures with a compatible Einstein metric, are obtained from solutions of the resulting ``affine monopole equations''. The ``projective monopole equations'' encompass Hitchin's twistorial construction of selfdual Einstein metrics from three dimensional Einstein-Weyl spaces, and lead to an explicit formula for carrying out this construction directly. Examples include new selfdual Einstein metrics depending explicitly on an arbitrary holomorphic function of one variable or an arbitrary axially symmetric harmonic function. The former generically have no continuous symmetries.