Lagrangian submanifolds of the nearly Kähler 3 × 3 from minimal surfaces in 3

We study non-totally geodesic Lagrangian submanifolds of the nearly Kähler 3 × 3 for which the projection on the first component is nowhere of maximal rank. We show that this property can be expressed in terms of the so-called angle functions and that such Lagrangian submanifolds are closely related to minimal surfaces in 3. Indeed, starting from an arbitrary minimal surface, we can construct locally a large family of such Lagrangian immersions, including one exceptional example. We also show that locally all such Lagrangian submanifolds can be obtained in this way.