Area-minimizing cones over products of Grassmannian manifolds

Abstract

This paper is the continuation of the previous one Jiao and Cui (Area-Minimizing Cones Over Grassmannian Manifolds. J. Geom. Anal. 32, 224 (2022). https://doi.org/10.1007/s12220-022-00963-7), 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 \({\widetilde{G}}(n,m;{\mathbb {R}})\) by using Lawlor’s Curvature Criterion Lawlor (Mem Amer Math Soc 91(446), 1991). Here, 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 \({\mathbf {7}}\) Lawlor (Mem Amer Math Soc 91(446), 1991), 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 Ohno and Sakai (Josai Math Monogr 13:69–91, 2021), or isoparametric theory Tang and Zhang (J Differ Geom 115(2):367–393, 2020) recently, and the generic ones in our results are completely new. We also prove that the cones over minimal product of general \({\widetilde{G}}(n,m;{\mathbb {R}})\) are area-minimizing, it can be seen as generalized results for some \({\widetilde{G}}(2,m;{\mathbb {R}})\) shown in Ohno and Sakai (Josai Math Monogr 13:69–91, 2021), Tang and Zhang (J Differ Geom 115(2):367–393, 2020).