Area-minimizing Cones over Grassmannian Manifolds

Abstract

In this paper, we extend Gary R. Lawlor's original examples of areaminimizing cones over nonoriented manifolds [Law91]. 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 ([Ker94],[HKT00],[Kan02],[OS15]). 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.

Area-minimizing Cones over Products of Grassmannian Manifolds

“…where P L = P * L ,P 2 L = P L , tr P L = n and L = {z ∈ F m |P L z = z}. This embedding is minimal, and it's cone was shown area-minimizing in [JC21], other perspectives for these cones being area-minimizing can be seen in [Ker94], [HKT00], [Kan02], [OS15].…”

“…Embedding of Grassmannian manifolds into spheres. This subsection is based on [Che14], a detailed discussion for embedding of Grassmannian into the Euclidean space consists of the Hermitian matrices is given in [JC21].…”

“…First, RP 2 is embedded in H(3, R)(or its hyperplane of trace zero,[JC21]), the normal vector ξ 1 is given by diag{0, a, −a}(to see this, first assume ξ 1 is an any given normal vector, if g * ξ 1 := gξ 1 g T is diagonal for an isotropy isometry action g, we could change the standard orthonormal tangent basis E α := E α1 + E 1α (2 ≤ α ≤ 3) given in [JC21] to (g −1 ) * E α , note the second fundamental forms is also equivariant, then h((g −1 ) * E α , (g −1 ) * E β ), ξ 1 = h(E α , E β ), g * ξ 1 , the computation is simplified), |ξ 1 | 2 = a 2 , note the image lies in a sphere of radius 1 √ 3 , then the value of H ξ 1 1 at the sphere of radius 1 is given by…”

“…First, G(2, 4; R) is embedded in H(4, R)(or its hyperplane of trace zero), the normal vector ξ 2 can be given by diag{a, −a, b, −b}(to see this, first assume ξ 2 is an any given normal vector, if g * ξ 2 := gξ 2 g T is diagonal for an isotropy isometry action g, we could change the standard orthonormal tangent basis E α a := E αa + E aα (1 ≤ a ≤ 2, 3 ≤ α ≤ 4) given in [JC21] to (g −1 ) * E α a , note the second fundamental forms is also equivariant, then the computation is simplified). The result H ξ 2 2 has value diag{−a + b, a + b, −a − b, a − b}.…”

“…In [JC21], we have studied the area-minimizing cones over Grassmannian manifold which considered uniformly by Hermitian orthogonal projectors, other perspectives for these cones being area-minimizing can be seen in [Ker94], [HKT00], [Kan02], [OS15]. In this paper, we give a class of area-minimizing cones over products of Grassmannian manifolds which inspired by the minimal product considered in [TZ20].…”