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

TL;DR

This paper explores advanced geometric integrable systems related to 4-symmetric spaces, generalizing Hamiltonian stationary Lagrangian surfaces to higher dimensions using octonions and supersymmetric maps.

Contribution

It introduces a unified framework for integrable systems associated with 4-symmetric spaces, extending classical surface theories to 8-dimensional spaces with octonions.

Findings

Hamiltonian stationary Lagrangian surfaces are special cases of harmonic Gauss map surfaces.

The theory is extended to 8-dimensional Euclidean space using octonions.

Supersymmetric harmonic and primitive maps encompass the classical cases.

Abstract

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. H\'elein 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.