Special Lagrangian submanifolds in the complex sphere

Special Lagrangian submanifolds may be defined as those submanifolds which are both Lagrangian (an order 1 condition) and minimal (an order 2 condition). Alternatively, they are characterised as those submanifolds which are calibrated by a certain n-form (cf [HL]), so they have the remarkable property of being area minimizing. Their study have received many attention recently since connections with string theory have been discovered. More particularly, understanding fibrations of special Lagrangian (possibly with singularities) in Calabi-Yau manifolds of (complex) dimension 3 is crucial for mirror symmetry (cf [SYZ],[Jo]). Since the pioneering work of Harvey and Lawson [HL], where this notion were introduced, several authors ([Ha],[Jo],[CU2]) have discovered many families of special Lagrangian submanifolds in the complex Euclidean space, however very few examples (cf [Br]) of such submanifolds are known in other Calabi-Yau manifolds, where this notion is naturally extended. Maybe the main reason is that such manifolds are somewhat rare, in particular the existence of compact, Calabi-Yau manifolds is a hard result of S.-T. Yau [Y] involving non explicit solutions to some PDE. However