Twistors, nilpotent orbits and harmonic maps

Abstract

Twistor theory provides a useful tool which has applications in the theory of harmonic maps. A good example is the Calabi-Penrose twistor fibration ℂℙ3 → S 4. All harmonic spheres in S 4 can be obtained from projections of holomorphic horizontal curves in ℂℙ3 (a holomorphic curve is horizontal if it is tangent to the complex contact distribution Η ⊂ Tℂℙ3 which is perpendicular to the fibres of the twistor fibration with respect to the Fubini-Study metric). In order to construct holomorphic curves tangent to the distribution Η one can use the Bryant correspondence which maps ℂℙ3 birationally to ℙT*ℂℙ2 and maps Η to the canonical complex contact distribution on ℙT*ℂℙ2 (see [6] and [27]). The flag manifold F 12(ℂ3) ≃ ℙT*ℂℙ2 is the twistor space of ℂℙ2 and Burstall shows in [11] that in fact all twistor spaces of compact quaternion-Kähler symmetric spaces of the same dimension are birationally equivalent as complex contact manifolds.