Introduction

Abstract

Harmonic maps are mappings between Riemannian or pseudo-Riemannian manifolds which extremise a certain natural energy integral generalising Dirichlet’s integral. Examples include geodesics, harmonic functions, complex analytic mappings between suitable (e.g. Miller) manifolds, the Gauss maps of constant mean curvature surfaces, and harmonic morphisms, these last being maps which preserve Laplace’s equation. The Euler-Lagrange equations for a harmonic map (the “harmonic equations” are a system of semi-linear equations, elliptic if the domain is Riemannian. If we impose sufficient symmetry or “equivariance”, the harmonic equations can frequently be reduced to an ordinary or partial differential equation which can be interpreted as a Hamiltonian system; such reductions to ordinary and partial differential equations have been exploited to find many harmonic maps and morphisms However, it is much more recently that the techniques of integrable systems used in soliton theory have been employed to find large families of harmonic maps, especially from a surface to a homogeneous space; in some cases these techniques have given all harmonic maps, for example from 2-tori to spheres or complex projective spaces. The method is to translate the harmonic equation into a Lax type differential equation for a map with values in a loop space, and to solve this either by finding commuting flows using r-matrices, or by using ideas of Kostant, Adler and Symes where the harmonic map appears as the projection of a complex geodesic. This book has a two-fold purpose: firstly, to explain the ideas and methods starting from an elementary level and secondly to bring the reader close to the current state of research in this area.