Minimal cones with isotropic links

We show that any closed oriented immersed isotropic minimal surface Σ with genus gΣ in S⁵ ⊂ ℂ³ is (1) Legendrian (and totally geodesic) if gΣ = 0; (2) either Legendrian or with exactly 2gΣ − 2 Legendrian points if gΣ ≥ 1. In general, any compact oriented immersed isotropic minimal submanifold L^{n − 1} ⊂ S^{2n − 1} ⊂ ℂⁿ must be Legendrian if its first Betti number is zero. Corresponding results for nonorientable links are also provided.