Abstract :

In a previous work Pascal Romon and I proposed a Weierstrass type formula for conformal parametrisations of Lagrangian surfaces in the 4-space, which relies on a kind of Dirac equation. Similarities with the Weierstrass formula found by B.G. Konopelchenko for surfaces in the 3-space were observed. Recently Franz Pedit pointed out to me that B.G. Konopelchenko did generalize his formula for representing surfaces in the 4-space. The purpose of this note is simply to stress the fact that our representation formula for Lagrangian surfaces can be recovered from Konopelchenko's one. As a byproduct we can deduce a simplification of Konopelchenko's representation by using quaternions.