Surfaces isotropes de O et systèmes intégrables

الأصل · EN

We define a notion of isotropic surfaces in O, i.e. on which some canonical symplectic forms vanish. Using the cross-product in O we define a map ρ Gr_2(O)→ S⁶ from the Grassmannian of O to S⁶. This allows us to associate to each surface Σ of O a function ρ_Σ Σ→ S⁶. Then we show that the isotropic surfaces in O such that ρ_Σ is harmonic are solutions of a completely integrable system. Using loop groups we construct a Weierstrass type representation of these surfaces. By restriction to H we obtain as a particular case the Hamiltonian Stationary Lagrangian surfaces of R⁴, and by restriction to Im(H) we obtain the CMC surfaces of R³.