Harmonic Mappings

Abstract

When considering maps between Riemannian manifolds it is possible to associate a variety of invariantly defined ‘energy’ functionals that are of geometrical and physical interest. The core problem is that of finding maps which are ‘optimal’ in the sense of minimising the energy functional in some class; one of the techniques for finding minimisers (or more generally critical points) is to use a gradient descent flow to deform a given map to an extremal of the energy.

The first major study of Harmonic mappings between Riemannian mani– folds was made by Eells and Sampson [ES64]. They showed, under suitable metric and curvature assumptions on the target manifold, gradient lines do indeed lead to extremals.

We motivate the study of Harmonic maps by considering a simple problem related to geodesics. Following this we discuss the convergence result of Eells and Sampson. The techniques and ideas used for Harmonic maps provide some motivation for those use later for Ricci flow, and will appear again explicitly when we discuss the short-time existence for Ricci flow.