A construction of special Lagrangian 3-folds via the generalized Weierstrass representation

Abstract

We show that certain holomorphic loop algebra-valued 1-forms over Riemann surfaces yield minimal Lagrangian immersions into the complex 2-dimensional projective space via the Weierstrass type representation, hence 3-dimensional special Lagrangian submanifolds of ℂ³. A particular family of 1-forms on ℂ corresponds to solutions of the third Painlevé equation which are smooth on (0, +∞).