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Minimal Surfaces and the Toda Equations for the Classical Groups

The Toda equations, which were investigated by Toda in the 1960s (see 16]), provide a particularly interesting example of a completely integrable Hamiltonian system. There are two forms, the \open" and \aane" Toda equations. They were initially studied in a dynamical context but it has become increasingly apparent over the last thirty years that they arise naturally in many other contexts. A major step forward was made in the late 1970s by Adler 1], Kostant 12] and Symes 15] who gave a Lie algebra formulation of both forms of the equations for any simple Lie algebra g, the original equations studied by Toda being those associated to su(n + 1). In this paper we continue the investigation begun in 5] of the way in which solutions to these equations arise naturally in a geometrical context from special types of harmonic maps into certain homogeneous spaces associated to the Lie algebra. Speciically, it was shown in 5] that there is an essentially 1-1 correspondence between solutions of the aane Toda equations and certain maps, called-primitive maps, into the ag manifold G=T, where T is a maximal torus of the compact simple Lie group G with Lie algebra g. The geometry of-primitive maps was then studied in the cases g = su(n + 1), g = so(2n + 1) and g = g 2. In this note we describe the geometry of the-primitive maps for the remaining classical simple Lie algebras. We also consider the open case (see also 14]). Generally speaking, this is rather easier to deal with since the solutions correspond to holomorphic curves, but the aane case is in some ways more interesting since it has certain special features not enjoyed by the open case. We do not present detailed arguments in this brief note. A more complete account of the cases not discussed in 5] will appear elsewhere. The topics we discuss have also been investigated independently by Doliwa 10, 11]. For a discussion of the background to the ideas presented here see 6, 7], while a more general treatment of the Toda equations in the context of harmonic maps may be found in 9] .