Toda frames, harmonic maps and extended Dynkin diagrams

Emma Carberry and Katharine Turner

Abstract

We prove that all immersions of a genus one surface into $G/T$ possessing a Toda frame can be constructed by integrating a pair of commuting vector fields on a finite dimensional Lie algebra. Here $G$ is any simple real Lie group (not necessarily compact), $T$ is a Cartan subgroup and the $k$-symmetric space structure on $G/T$ is induced from the Coxeter automorphism. We provide necessary and sufficient conditions for the existence of a Toda frame for a harmonic map into $G/T$ and describe those $G/T$ to which the theory applies in terms of involutions of extended Dynkin diagrams.

Keywords: Harmonic map, Toda frame, Dynkin diagram.