Some examples of `second order elliptic integrable systems associated to a 4-symmetric space'

Frédéric Hélein (1)

1 IMJ - Institut de Mathématiques de Jussieu

In this text we expound recent results by Idrisse Khemar on the construction of various geometric completely integrable systems generalizing the structure of Hamiltonian stationary Lagrangian surfaces (HSLS) discovered by F. Hélein and P. Romon. We first explain that, for surfaces in a 4-dimensional Euclidean space, the HSLS are a particular case of a more general theory of surfaces with harmonic left Gauss map and that this theory includes also the classical constant mean curvature surfaces in 3-space. Then we expound a generalization of this theory for surfaces in an 8-dimensional Euclidean space, using octonions. Lastly we discuss supersymmetric harmonic and primitive maps, a theory which also covers the HSLS cas.

Mots clés: Completely integrable systems; surfaces; quaternions; octonions

Domaines: Physique mathématique [math-ph]

Frédéric Hélein. Some examples of `second order elliptic integrable systems associated to a 4-symmetric space'. London Mathematical Society Durham symposium : Methods of Integrable Systems in Geometry, Aug 2006, Durham, United Kingdom. ⟨hal-00346771⟩