Area-minimizing cones over products of Grassmannian manifolds

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 ; F ) ( F = R , C , H ) , Cayley plane O P 2 from the point view of Hermitian orthogonal projectors, and gave area-minimizing cones associated to oriented real Grassmannians G ~ ( n , m ; 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 8 , over minimal products of G ( n , m ; 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 7 Lawlor (Mem Amer Math Soc 91(446), 1991), we gain more area-minimizing cones by carefully computing the Jacobian i n f v d e t ( I - t H ij v ) . 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 G ~ ( n , m ; R ) are area-minimizing, it can be seen as generalized results for some G ~ ( 2 , m ; R ) shown in Ohno and Sakai (Josai Math Monogr 13:69–91, 2021), Tang and Zhang (J Differ Geom 115(2):367–393, 2020).