Harmonic maps via Adler—Kostant—Symes theory

Abstract

Over the past few years significant progress has been made in the understanding of various completely integrable nonlinear partial differential equations (soliton equations) and their relationship to classical problems in differential geometry. It has been shown in a series of recent papers [42, 32, 21, 25, 15, 8, 12] that constant mean and Gauss curvature surfaces, Willmore surfaces, minimal surfaces in spheres and projective spaces and generally harmonic maps from a Riemann surface M into various homogeneous spaces may be described as solutions to various soliton equations (see [7]). Moreover, these solutions are algebraic in the sense that they are obtained by integrating ordinary differential equations of Lax type which linearise on the Jacobian of an appropriate algebraic curve.