Twistorial constructions of harmonic morphisms and Jacobi fields

Abstract

Twistor methods provide a powerful tool in the study of harmonic maps and harmonic morphisms. Indeed, their use has enabled us to produce a variety of examples of harmonic morphisms defined on 4-dimensional manifolds, and a complete classification in some cases. In the first part of this work, we generalize those constructions to obtain harmonic morphisms from higher-dimensional manifolds. The use of twistor methods in the study of Jacobi fields has proved quite fruitful, leading to a series of results. In the second part of this work we give a general treatment of the relations between Jacobi fields and variations in the twistor space, obtaining first-order analogues of twistorial constructions.