Area-Minimizing Cones Over Grassmannian Manifolds

Abstract

It is a well-known fact that there exists a standard minimal embedding map for the Grassmannians of n-planes \(G(n,m;{\mathbb {F}})({\mathbb {F}}={\mathbb {R}},{\mathbb {C}},{\mathbb {H}})\) and Cayley plane \({\mathbb {O}}P^2\) into Euclidean spheres, then a natural question is that if the cones over these embedded Grassmannians are area-minimizing? In this paper, detailed descriptions for this embedding map are given from the point of view of Hermitian orthogonal projectors which can be seen as a direct generalization of Gary R. Lawlor’s( [23]) original considerations for the case of real projective spaces, then we re-prove the area-minimization of those cones which was gradually obtained in [18, 19], and [30] from the perspectives of isolated orbits of adjoint actions or canonical embedding of symmetric R-spaces, all based on the method of Gary R. Lawlor’s Curvature Criterion. Additionally, area-minimizing cones over almost all common Grassmannians have been given by Takahiro Kanno, except those cones over oriented real Grassmannians \({\widetilde{G}}(n,m;{\mathbb {R}})\) which are not Grassmannians of oriented 2-planes. The second part of this paper is devoted to complement this result, a natural and key observation is that the oriented real Grassmannians can be considered as unit simple vectors in the exterior vector spaces, we prove that all their cones are area-minimizing except \({\widetilde{G}}(2,4;{\mathbb {R}})\).