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.
Publié le : 2000-07-05
Classification:
variational problem with constraint,
Lagrangian surfaces,
Weierstrass representation,
Dirac equation,
minimal surfaces,
variational problem with constraint.,
MSC : Primary 53C42, 53D12, Secondary 49Q10, 53A05,
[MATH.MATH-DG]Mathematics [math]/Differential Geometry [math.DG]
@article{hal-00858413,
author = {H\'elein, Fr\'ed\'eric and Romon, Pascal},
title = {Weierstrass representation of Lagrangian surfaces in four-dimensional space using spinors and quaternions},
journal = {HAL},
volume = {2000},
number = {0},
year = {2000},
language = {en},
url = {http://dml.mathdoc.fr/item/hal-00858413}
}
Hélein, Frédéric; Romon, Pascal. Weierstrass representation of Lagrangian surfaces in four-dimensional space using spinors and quaternions. HAL, Tome 2000 (2000) no. 0, . http://gdmltest.u-ga.fr/item/hal-00858413/