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.

34 pages

References in corpus (1)

• Self-dual Einstein Hermitian four manifolds

Cited by in corpus (8)

• Integrable Background Geometries
• Einstein manifolds with torsion and nonmetricity
• Self-dual Einstein Hermitian four manifolds
• Selfdual 4-Manifolds, Projective Surfaces, and the Dunajski-West Construction
• Harmonic morphisms on heaven spaces
• Harmonic morphisms between Weyl spaces and twistorial maps II
• Hyper-Hermitian quaternionic Kaehler manifolds
• Twistorial harmonic morphisms with one-dimensional fibres on self-dual four-manifolds