Infinitesimal Deformations of Harmonic Maps and Morphisms

Harmonic maps are mappings between Riemannian manifolds which extremize a natural energy functional. They have been studied for many years in differential geometry, and in particle physics as nonlinear sigma models. We shall report on recent progress in understanding their infinitesimal deformations, the so-called Jacobi fields. It is important to know whether the Jacobi fields along harmonic maps between given Riemannian manifolds are integrable, i.e., arise from genuine variations through harmonic maps. If they are, and the manifolds are real analytic, the space of harmonic maps between them is a smooth (in fact, real-analytic) manifold with tangent spaces given by the Jacobi fields; we also gain some information on the structure of the singular set of weakly harmonic maps. We shall outline what is known about integrability. Harmonic morphisms are less well known — these are mappings between Riemannian manifolds which preserve Laplace’s equation. They can be characterized as harmonic maps which enjoy an extra property called horizontal (weak) conformality or semiconformality. We shall give a brief introduction to the theory, and consider what is known about their infinitesimal deformations. This paper is dedicated to Dmitri Alekseevsky on the occasion of his 65th birthday. I hope that it reminds him of happy times at “Yorkshire Differential Geometry Days” which took place four times a year at the Universities of Hull, Leeds and York, with the support of the London Mathematical Society.