Twistorial Harmonic Morphisms With One-Dimensional Fibres on Self-Dual Four-Manifolds

Abstract

We introduce a general notion of twistorial map and classify twistorial harmonic morphisms with one-dimensional fibres from self-dual four-manifolds.Such maps can be characterised as those which pull back Abelian monopoles to self-dual connections. In fact, the constructions involve solving a generalised monopole equation, and also the Beltrami fields equation of hydrodynamics, and lead to constructions of self-dual metrics.1991 Mathematics Subject Classification. Primary 58E20, Secondary 53C43.

Harmonic morphisms between Weyl spaces and twistorial maps

“…Similar descriptions, in higher dimensions, can be obtained if instead of (±)holomorphic functions, we use the more general notion of twistorial map [26]. A twistorial structure on a complex manifold M is given by a foliation F on a complex manifold P such that F ∩ ker dπ = {0}, where π : P → M is a proper complex analytic submersion; the leaf space of F is called the twistor space of the given twistorial structure (P, M, π, F ) and is usually denoted Z(M ) .…”

“…We start with a brief presentation of the examples of twistorial maps with which we shall work; more details can be found in [26] (and in [20], for the notions of almost twistorial structure and twistorial map in the smooth category). N 3 , c N , D N ) is anti-self-dual in the sense of [5]).…”

“…Thus, any twistorial structure on M determines a sheaf of twistorial functions (ii) Functions on a three-dimensional Einstein-Weyl space, which, locally, are the composition of a horizontally conformal submersion with geodesic fibres followed by a complex analytic function; here (M 3 , c, D) is the complexification of a three-dimensional Einstein-Weyl space, π : P → M is the bundle of two-dimensional degenerate spaces on (M 3 , c) and, for each p ∈ P , the subspace F p ⊆ T p P is the horizontal lift of p ⊆ T π(p) M with respect to D (see [26,Example 2.4] for details about this twistorial structure).…”

“…For example, any harmonic morphism from an Einstein manifold of dimension 4 with fibres of dimension 1 or 2 is twistorial ( [23,26,29]; cf. Corollaries 7.1 and 7.2 below).…”

“…In Section 5, we recall from [26] the basic examples of twistorial maps and study the conditions under which these are harmonic morphisms; the resulting harmonic morphisms are between Weyl spaces of dimensions m and n where (m, n) = (3, 2), (4, 2), (4,3). Thus, we obtain the following for an analytic submersion ϕ between Weyl spaces of dimensions m and n.…”