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

Hongbin Cui

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arXiv · 2510.14650

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

Hongbin Cui

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Abstract

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.

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Hongbin Cui. 2025-10-16. On FKM isoparametric hypersurfaces in $\mathbb{S}^n \times \mathbb{S}^n$ and new area-minimizing cones. https://arxiv.org/abs/2510.14650

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