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

TL;DR

The paper constructs new codimension-two area-minimizing cones in $\mathbb{R}^{2n+2}$ by restricting the Ferus–Karcher–Münzner (FKM) isoparametric foliation to $\mathbb{S}^n\times\mathbb{S}^n$ inside $\mathbb{S}^{2n+1}(\sqrt{2})$ and exploring their isonormal geometry. By establishing that these restricted hypersurfaces are isoparametric and, in fact, isonormal, the authors compute their second fundamental forms and the normal radii, then apply Lawlor's curvature criterion via vanishing angles to certify area minimization of the resulting cones. They derive explicit dimension thresholds: for the irreducible case ($k=1$) one needs $n\ge 15$, while for the reducible case ($k\ge 2$) it suffices that $n\ge 11$, together with certain Clifford-algebra multiplicity constraints. Additionally, they extend the framework to minimal product cones over FKM isoparametric hypersurfaces, proving area-minimization when the ambient cone dimension satisfies $\dim C\ge 21$, thus broadening the landscape of non-holomorphic, non-homogeneous area-minimizing cones. These results connect isoparametric foliation theory with concrete geometric measure theory constructions, generalizing Urbano’s $n=2$ cases and providing new codimension-two examples.

Abstract

Via a new isoparametric foliation in $\mathbb{S}^n(1) \times \mathbb{S}^n(1)$, we find many new area-minimizing cones by applying Lawlor's curvature criterion, especially codimension-two area-minimizing cones in $\mathbb{R}^{2n+2}$ for $n\geq 63$. This new isoparametric foliation is a further restriction of the Ferus-Karcher-Münzner isoparametric foliation to $\mathbb{S}^n(1) \times \mathbb{S}^n(1) \subset \mathbb{S}^{2n+1}(\sqrt{2})$, which can also be defined on general $\mathbb{S}^n(a) \times \mathbb{S}^n(b)(a>0,b>0)$, and it extends the recent classification results of F. Urbano for $n=2$.