Weierstrass representation of Lagrangian surfaces in four-dimensional space using spinors and quaternions

Frédéric Hélein (1) , Pascal Romon (2)

1 CMLA - Centre de Mathématiques et de Leurs Applications
2 LAMA - Laboratoire d'Analyse et de Mathématiques Appliquées

We derive a Weierstrass-type formula for conformal Lagrangian immersions in Euclidean 4-space, and show that the data satisfies an equation similar to Dirac equation with complex potential. Alternatively this representation has a simple formulation using quaternions. We apply it to the Hamiltonian stationary case and construct all possible tori, thus obtaining a first approach to a moduli space in terms of a simple algebraic-geometric problem on the plane. We also classify Hamiltonian stationary Klein bottles and show they self-intersect.

Mots clés: variational problem with constraint; Lagrangian surfaces; Weierstrass representation; Dirac equation; minimal surfaces; variational problem with constraint.

Domaines: Géométrie différentielle [math.DG]

Frédéric Hélein, Pascal Romon. Weierstrass representation of Lagrangian surfaces in four-dimensional space using spinors and quaternions. Commentarii Mathematici Helvetici, 2000, 75 (4), pp.668-680. ⟨10.1007/s000140050144⟩. ⟨hal-00858413⟩