## 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.

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 Friday, November 18, 2011