Surfaces isotropes de \mathbbO et systèmes intégrables

2005/11/10 by Idrisse Khemar, Khemar, Idrisse
Mathematics · #Advanced Algebra and Geometry #Algebraic Geometry and Number Theory #Differential Geometry (math.DG) #FOS: Mathematics #Geometric Analysis and Curvature Flows #math.DG

paper · pdf · doi:10.48550/arxiv.math/0511258

openalex publication_date 2005/11/10 · arxiv created 2005/12/05 · arxiv updated 2009/12/01 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

We define a notion of isotropic surfaces in \mathbbO, i.e. on which some canonical symplectic forms vanish. Using the cross-product in \mathbbO we define a map ρ\colon Gr_2(\mathbbO)→ S⁶ from the Grassmannian of \mathbbO to S⁶. This allows us to associate to each surface Σ of \mathbbO a function ρ_Σ\colon Σ→ S⁶. Then we show that the isotropic surfaces in \mathbbO 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⊂\mathbbO 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.

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