Surfaces isotropes de $\mathbb{O}$ et syst\`{e}mes int\'{e}grables

Abstract

We define a notion of isotropic surfaces in $\mathbb{O}$, i.e. on which some canonical symplectic forms vanish. Using the cross-product in $\mathbb{O}$ we define a map $\rho\colon Gr\_2(\mathbb{O})\to S^6$ from the Grassmannian of $\mathbb{O}$ to $S^6$. This allows us to associate to each surface $\Sigma$ of $\mathbb{O}$ a function $\rho\_{\Sigma}\colon \Sigma\to S^6$. Then we show that the isotropic surfaces in $\mathbb{O}$ such that $\rho\_{\Sigma}$ is harmonic are solutions of a completely integrable system. Using loop groups we construct a Weierstrass type representation of these surfaces. By restriction to $ \mathbb{H}\subset\mathbb{O}$ we obtain as a particular case the Hamiltonian Stationary Lagrangian surfaces of $\mathbb{R}^4$, and by restriction to $\text{Im}(\mathbb{H})$ we obtain the CMC surfaces of $\mathbb{R}^3$.