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 an 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 view of Hermitian orthogonal projectors which can be seen as an direct generalization of Gary R. Lawlor's(\cite{lawlor1991sufficient}) original considerations for the case of real projective spaces, then we re-prove the area-minimization of those cones which was gradually obtained in \cite{kerckhove1994isolated}, \cite{kanno2002area} and \cite{ohno2015area} 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 has 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 $\widetilde{G}(2,4;\mathbb{R})$.