Area-minimizing Cones over Grassmannian Manifolds

In this paper, we extend Gary R. Lawlor's original examples of area-minimizing cones over nonoriented manifolds\cite{lawlor1991sufficient}. From the point view of submanifolds themselves, by considering real nonoriented Grassmannians, complex Grassmannians, quaternion Grassmannians and complex, quaternion projective spaces and Cayley projective plane as the Hermitian orthogonal projection operators uniformly, we prove that there exists a family of opposite cones associated with them. These cones are shown area-minimizing by Lawlor's Curvature Criterion, it also can be seen as direct proofs for these cones being area-minimizing under the perspective of isolated orbits of adjoint action and the perspective of symmetric spaces (\cite{kerckhove1994isolated},\cite{hirohashi2000area},\cite{kanno2002area},\cite{ohno2015area}). Additionally, for the left oriented real Grassmannian manifolds-the double cover of the nonoriented cases, by embedding them in the exterior vector spaces as unit simple vectors, we prove these cones are area-minimizing except one.