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

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. Mathematics Subject Classification (2000). Primary 53C42, 53D12, Secondary 49Q10, 53A05.