Area-minimizing of the cone over symmetric $R$-spaces

Hirohashi, Daigo; Kanno, Takahiro; Tasaki, Hiroyuki

doi:10.21099/tkbjm/1496164053

Cited by 10 publications

(5 citation statements)

References 3 publications

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“…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].…”

Section: The Upper Bound Of Second Fundamental Forms Of F Is Given By αmentioning

confidence: 92%

“…Other important researches on the area-minimizing cones can be found in [Mur91], [LM95],and [Che88], [Ker94], [HKT00], [Kan02], [OS15], [XYZ18], [TZ20], etc.…”

Section: Introductionmentioning

confidence: 99%

“…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].…”

Section: Introductionmentioning

confidence: 99%

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Area-minimizing Cones over Products of Grassmannian Manifolds

Jiao¹,

Cui²,

Jiao³

2021

Preprint

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The research on minimal cones over products of spheres is a very important problem with a long history, many people have contributed to this issue. It was given a complete answer in [Law91] by Gary R. Lawlor where he developed a general method for proving that a cone is area-minimizing, the so-called Curvature Criterion. In this paper, we generalize the case of products of spheres to the case of products of embedded Grassmannian manifolds into spheres where they are similar in many ways, and we give a class of area-minimizing cones associated with them by using Curvature Criterion.

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Section: The Upper Bound Of Second Fundamental Forms Of F Is Given By αmentioning

confidence: 92%

“…Other important researches on the area-minimizing cones can be found in [Mur91], [LM95],and [Che88], [Ker94], [HKT00], [Kan02], [OS15], [XYZ18], [TZ20], etc.…”

Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

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Area-minimizing Cones over Products of Grassmannian Manifolds

Jiao¹,

Cui²,

Jiao³

2021

Preprint

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“…Now we shall study an orbit AdðKÞH of the linear isotropy representation of ðG; KÞ through H 2 m. An orbit AdðKÞH is a submanifold of the hypersphere S of radius kHk in m. From [6],…”

Section: Orbits Of S-representationsmentioning

confidence: 99%

Weakly reflective submanifolds and austere submanifolds

Ikawa¹,

Sakai²,

Tasaki³

2009

J. Math. Soc. Japan

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An austere submanifold is a minimal submanifold where for each normal vector, the set of eigenvalues of its shape operator is invariant under the multiplication by À1. In the present paper, we introduce the notion of weakly reflective submanifold, which is an austere submanifold with a reflection for each normal direction, and study its fundamental properties. Using these, we determine weakly reflective orbits and austere orbits of linear isotropy representations of Riemannian symmetric spaces.

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“…In chapter 3, we prove that the Grassmannian manifold G(n, m; F) and the Grassmannian manifold G(m − n, m; F)(F = R, C, H) can be embedded into the Euclidean sphere as a pair of opposite minimal submanifolds simultaneously, and the associated cones are a pair of opposite cones, the nonoriented Grassmannian manifolds G(n, m; R) and its nonoriented cones extend Gary R. Lawlor's area-minimizing cones over nonoriented manifolds. Moreover, we note here these cones has arisen from different perspectives, like been as an isolated singular orbit of adjoint action of special orthogonal group, special unitary group or Symplectic group and been as an symmetric R-space then consider its canonical embedding ([Ker94], [HKT00], [Kan02]). Here we regard them as Hermitian orthogonal projectors uniformly, and prove these cones are area-minimizing from the point view of submanifolds themselves.…”

Section: Introductionmentioning

confidence: 99%

Area-minimizing Cones over Grassmannian Manifolds

Jiao¹,

Cui²

2021

Preprint

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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.

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Area-minimizing of the cone over symmetric $R$-spaces

Cited by 10 publications

References 3 publications

Area-minimizing Cones over Products of Grassmannian Manifolds

Area-minimizing Cones over Products of Grassmannian Manifolds

Weakly reflective submanifolds and austere submanifolds

Area-minimizing Cones over Grassmannian Manifolds

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