We present two generalizations for the celebrated works of Ferus-Karcher-M\"unzner \cite{ FKM81} and Wang \cite{ W94}. We first show that an isoparametric foliation on $\mathbb{ S}^{ 2n+1}$ constructed by Ferus-Karcher-M\"unzner naturally yields an isoparametric foliation on its submanifold $\mathbb{ S}^n \times \mathbb{ S}^n$ with one same focal variety. The second part concerns area-minimizing cones; all known regular area-minimizing hypercones are realized as real algebraic varieties: isoparametric cones (cf. \cite{ W94}). As a noteworthy application, we extend area-minimizing isoparametric hypercones in \cite{ W94} to codimension-two cases, and obtain infinitely many families (each containing infinitely many members) of area-minimizing subcones of Simons cones.
Categories: math.DG