On FKM isoparametric hypersurfaces in ${\mathbb{S}}^n\times {\mathbb{S}}^n$ and new area-minimizing cones

Abstract

We present two generalizations for the celebrated works of Ferus-Karcher-Münzner \cite{FKM81} and Wang \cite{W94}. We first show that an isoparametric foliation on ${\mathbb{S}}^{2n+1}$ constructed by Ferus-Karcher-Münzner 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.

Updated: 1. an possible connection between area-minimizing cones of low codimension and generic regularity; 2. an statement regarding relevant scholars' independent work on the classification of isoparametric hypersurfaces (Remark 1.3); 3. some comments on the potential applications and remaining questions of area-minimizing cones (Sect. 5)