Area-minimizing Cones over Products of Grassmannian Manifolds

TL;DR

This paper extends the classification of area-minimizing cones over products of Grassmannian manifolds, including new cones in dimension 7 and higher, using Jacobian computations and geometric analysis.

Contribution

It proves that cones over minimal products of Grassmannians of dimension at least 8 are area-minimizing, and introduces new cones in the critical dimension 7.

Findings

Cones over products of Grassmannians of dimension ≥8 are area-minimizing.

New area-minimizing cones are identified in dimension 7.

Cones over minimal products of real Grassmannians are also area-minimizing.

Abstract

This paper is the continuation of the previous one \cite{Cui2021}, where we re-proved the area-minimization of cones over Grassmannians of $n$-planes $G(n,m;\mathbb{F})(\mathbb{F}=\mathbb{R},\mathbb{C},\mathbb{H})$, Cayley plane $\mathbb{O}P^2$ from the point view of Hermitian orthogonal projectors, and gave area-minimizing cones associated to oriented real Grassmannians $\widetilde{G}(n,m;\mathbb{R})$. In this paper, we make a further step on showing that the cones, of dimension no less than $\mathbf{8}$, over minimal products of $G(n,m;\mathbb{F})$ are area-minimizing. Moreover, those cones are very similar to the classical cones over products of spheres, and for the critical situation -- the cones of dimension $\mathbf{7}$ \cite{lawlor1991sufficient}, we gain more area-minimizing cones by carefully computing the Jacobian $inf_{v}det(I-tH^{v}_{ij})$. Certain minimizing cones among them had been found from the perspective of $R$-spaces\cite{Ohno2021area}, or isoparametric theory\cite{tang2020minimizing}, and others are completely new. We also prove that the cones over minimal product of $\widetilde{G}(n,m;\mathbb{R})$ are area-minimizing.