Area-minimizing Cones over Products of Grassmannian Manifolds
aa r X i v : . [ m a t h . DG ] F e b Area-minimizing Cones over Productsof Grassmannian Manifolds
Xiaoxiang Jiao, Jialin Xin and Hongbin Cui ∗ School of Mathematical Sciences, University of Chinese Academy ofSciences, Beijing 100049, China, ∗ Corresponding AuthorE-mail: [email protected]; [email protected];[email protected]
Abstract.
The research on minimal cones over products of spheres isa very important problem with a long history, many people have con-tributed to this issue. It was given a complete answer in [
Law91 ] byGary R. Lawlor where he developed a general method for proving thata cone is area-minimizing, the so-called Curvature Criterion. In thispaper, we generalize the case of products of spheres to the case of prod-ucts of embedded Grassmannian manifolds into spheres where they aresimilar in many ways, and we give a class of area-minimizing cones as-sociated with them by using Curvature Criterion.
Keywords
Area-minimizing cone, Hermitian matrix, Grassmannianmanifold, minimal product, Normal radius, Vanishing angle
Mathematics Subject Classification (2000).
Primary 49Q05; Sec-ondary 53C38, 53C40.
1. Introduction
Area-minimizing cones are a class of area-minimizing surfaces whoseits truncated part inside the unit ball owns the least area among allintegral currents with the same boundary in the sphere.An excellent work which link to the researches of area-minimizing sur-faces and cones is the calibrated geometry which had been developed ex-tensively by Harvey and Lawson([
HL82 ]), and numerous researches onthe calibrated geometry, calibrated area-minimizing surfaces and conescan be found in [
Har90 ], [
Che88 ],[
Law91 ],[
Mor87 ],[
Mur91 ], etc.Gary R. Lawlor[
Law91 ] has developed a general method for provingthat a cone is area-minimizing, the so-called Curvature Criterion. It wasexplained from two points of view, the method of vanishing calibrationand the construction of retractions, which are linked by the fact that thetangent space of retraction surface is just the orthogonal complement ofthe dual of the vanishing calibration. They both derive two equivalentordinary differential equations(in this paper ,we simple call it
ODE ).The area-minimizing tests include if the
ODE has solutions, and whatis the maximal existence interval of a solution then compare it with an important potential-normal radius of the cone. Gary R. Lawlor alsostudied those cones which his Curvature Criterion is both necessary andsufficient like the minimal, isoparametric hypercones and the cones overprincipal orbits of polar group actions. Based on this method, the clas-sification of minimizing cones over products of spheres is completed, healso proved some cones over unorientable manifolds, cones over compactmatrix groups under suitable chosen embeddings are area-minimizingand gave new proofs of a large class of homogeneous hypercones beingarea-minimizing.Other important researches on the area-minimizing cones can befound in [ Mur91 ],[
LM95 ],and [
Che88 ], [
Ker94 ],[
HKT00 ],[
Kan02 ],[
OS15 ],[
XYZ18 ],[
TZ20 ], etc.In [
JC21 ], we have studied the area-minimizing cones over Grass-mannian manifold which considered uniformly by Hermitian orthogonalprojectors, other perspectives for these cones being area-minimizing canbe seen in [
Ker94 ],[
HKT00 ],[
Kan02 ],[
OS15 ]. In this paper, we give aclass of area-minimizing cones over products of Grassmannian manifoldswhich inspired by the minimal product considered in [
TZ20 ].The product of Grassmannian manifolds is an product of minimal em-beddings which each factor is an embedding of Grassmannian into unitsphere contained in an Euclidean spacetheir associated cones can be seenas an generalization of the classical area-minimizing cones over productsof spheres which completely classified by Gary R. Lawlor. They are sim-ilar in many ways, like they share the same length of second fundamentalforms, the similar normal radius and they both have the same criticalsituation—the cones of dimension seven.This article is organized as follows:In chapter 2, we give illustration of minimal product and the concreteformula for length of the projection of second fundamental forms on anfixed normal vector. we review some known results for the standardembedding of Grassmannians into spheres and part of the work of GaryR. Lawlor in [
Law91 ].In chapter 3, we study the minimal product of Grassmannian man-ifolds, then we give the proof for area-minimizing of cones which ownthe dimension greater than 7.In chapter 4, we completely find all the cones of dimension 7 whichall the factors are Grassmannian submanifolds by computing for theminimum polynomials.In chapter 5, we give discussions for cones over the minimal productof Grassmannian manifolds which have reduced factors and an exampleof area-minimizing cone belonging to this type.
2. Preliminaries2.1. Minimal products.
Given a family of minimal immersion f i : M k i i ֒ → S a i (1)(1 ≤ i ≤ m ), the standard Euclidean inner product isdenoted by h , i , then h f i , f i i = 1, denote the second fundamental formof f i with respect to the normal vector v i by H v i i = −h df i , dv i i .Consider the product immersion,(2.1) f : M = M k × · · · × M k m m → S a + ··· + a m + m − (1)( x , . . . , x m ) ( λ f ( x ) , . . . , λ m f m ( x m )) , where P mi =1 λ i = 1, dim M = k + · · · + k m .Now, df = ( λ df , . . . , λ m df m ), set v = ( v , . . . , v m ), then v is a nor-mal vector of M if and only if h df, v i = 0 and h f, v i = 0, i.e. in everyEuclidean subspaces R a i +1 , v i = ξ i + b i x i , here x i denote the positionvector f i ( x i ), and it should satisfy(2.2) m X i =1 λ i b i = 0 . Let H v denote the second fundamental forms of f with respect to thenormal vector v , then(2.3) H v = −h df, dv i = − m X i =1 ( λ i h df i , dξ i i + λ i b i h dx i , dx i i )= − m X i =1 ( − λ i H ξ i i + λ i b i g i ) , where g i = h dx i , dx i i is the induced metric of immersion f i . Proposition 2.1.
Let λ i = q k i P mi =1 k i , then f is also a minimal im-mersion.Proof. Let { e i , . . . , e ik i } be an orthonormal basis of ( M i , g i ), then { . . . , e i λ i , . . . , e iki λ i , . . . } is an orthonormal basis of M , hence(2.4) H v ( e il λ i , e is λ i ) = − m X i =1 b i − H ξ i i ( e il , e is ) λ i , where l, s ∈ { , . . . , k i } , and under the above orthonormal basis, thematrix of H v is given by(2.5) H v = diag {− b λ I k + 1 λ H ξ , . . . , − b m λ m I k m + 1 λ m H ξ m m } , where H ξ i i is the matrix of second fundamental forms of f i on the direc-tion ξ i . Hence, f is a minimal immersion if and only if(2.6) tr H v = − m X i =1 b i k i λ i = 0for any chosen normal vector v, v i = ξ i + b i x i , since every H ξ i i has tracezero.The left proof is similar to [ Law91 ], if we choose λ i = q k i P mi =1 k i = q dimM i dimM , then (2.6) is holds. q.e.d.We continue to compute the norm of H v ,(2.7) | H v | = m X i =1 k i b i + | H ξ i i | − b i tr H ξ i i λ i = dim M ( X i b i + X i | ξ i | k i | H ξi | ξi | i | )= dim M (1 + X i | ξ i | ( 1 k i | H ξi | ξi | i | − dim M · max { , α k , . . . , α m k m } . where α i := sup ξ | H ξi | , ξ is a unit normal vector of M i , k i = dim M i . Theorem 2.2.
The upper bound of second fundamental forms of f is given by α := sup v | H v | = dim M · max { , α k , . . . , α m k m } . Thissubsection is based on [
Che14 ], a detailed discussion for embeddingof Grassmannian into the Euclidean space consists of the Hermitianmatrices is given in [
JC21 ].Let F denote the field of real numbers R , the field of complex numbers C , the normed quaternion associative algebra H , d = dim R F = { , , } .We use the following notations: M ( m ; F ) denote the space of all m × m matrices over F , H ( m ; F ) ⊂ M ( m ; F ) denote the space of all Hermitianmatrices over F ,(2.8) H ( m ; F ) = { A ∈ M ( m ; F ) | A ∗ = A } .H ( m ; F ) can be identified with real Euclidean space E N with theinner product:(2.9) g ( A, B ) = 12
Re tr F ( AB ) , where A, B ∈ H ( m ; F ), N = m + dm ( m − / Let U ( m ; F ) = { A ∈ M ( m ; F ) | AA ∗ = A ∗ A = I } denote the F -unitarygroup, U ( m ; F ) has an natural adjoint action ρ on H ( m ; F ), given by ρ ( Q, P ) =
QP Q ∗ , where Q ∈ U ( m ; F ) , P ∈ H ( m ; F ), this action isisometry and transitive.Let G ( n, m ; F ) denote the Grassmannian of n -plane in the right vectorspace F m , for every L ∈ G ( n, m ; F ), there exists an Hermitian orthog-onal projector P L associated with it, hence give an embedding ϕ of G ( n, m ; F ) into the hypersphere contained in H ( m ; F )(the dimension ofthe sphere can reduce to N − ϕ : G ( n, m ; F ) → H ( m ; F ) L P L , where P L = P ∗ L , P 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-minimizingin [ JC21 ], other perspectives for these cones being area-minimizing canbe seen in [
Ker94 ],[
HKT00 ], [
Kan02 ],[
OS15 ]. Now, we recallGary R.Lawlor’s work in [
Law91 ], for the following
ODE (see definition1.1.6 in [
Law91 ]):(2.11) ( drdθ = r q r k ( cosθ ) k − inf v ( det ( I − tan ( θ ) h vij )) − r (0) = 1 , where h vij is the matrix representation of the second fundamental formof an minimal submanifold M in sphere, v is an unit normal, k is thedimension of cone C = C ( M ), and r = r ( θ ) describe a projection curve,the ODE is build at a fixed point p ∈ M .Denote the real vanishing angle by θ (see Definition 1.1.7 in [ Law91 ]),Lawlor use the following estimates.Let θ ( k, α ) be the estimated vanishing angle function which replacinginf v det ( I − tan ( θ ) h vij ) by an smaller positive-valued function(2.12) F ( α, tan ( θ ) , k −
1) = (1 − αtan ( θ ) r k − k − αtan ( θ ) p ( k − k −
2) ) k − in (2.11), where the condition α tan ( θ ) ≤ k − k − should be satisfied,and F is an decreasing function of α when tan ( θ ) , k are fixed, it is alsodecreasing with respect to k when α, tan ( θ ) are fixed.Let θ ( k, α ) be the estimated vanishing angle function which replacinginf v det ( I − tan ( θ ) h vij ) by(2.13) lim k →∞ F ( α, tan ( θ ) , k −
1) = (1 − αtan ( θ )) e αtan ( θ ) in (2.11), where the condition α tan ( θ ) ≤ − αtan ( θ )) e αtan ( θ ) is also an decreasing function of α when tan ( θ ) arefixed. The three angles have the following relation:(2.14) θ ≤ θ ( k, α ) ≤ θ ( k, α ) , and Lawlor use the angle function θ for dim C = { , . . . , } , the anglefunction θ for dim C = 12 to gain ”The Table”(see section 1.4 in[ Law91 ]).
3. Cones of dimension bigger than i ∈ { , . . . , m } , consider the minimal embedding([ JC21 ])(3.1) G ( l i , k i ; F ) → S n i ( r i ) A A − l i k i I, where r i = q l i ( k i − l i )2 k i , dim G ( l i , k i ; F ) = dl i ( k i − l i ), n i = k i − k i ( k i − d , and we assume l i ≤ k i − l i for every i , it contain the fol-lowing three reduced cases: R P ≡ S ( ), C P ≡ S ( ) and H P ≡ S ( )([ JC21 ]).Denote f i be the minimal embedding f i : r i G ( l i , k i ; F ) → S n i (1), then f : M ≡ Q mi =1 λ i r i G ( l i , k i ; F ) → S P mi =1 n i + m − (1) is minimal if and onlyif a i = λ i r i = q dk i dim M , where dim M = P mi =1 dl i ( k i − l i ).Hence, Proposition 3.1.
The cone over Q mi =1 √ k i G ( l i , k i ; F ) is minimal. We will use Gary R. Lawlor’s Curvature Criterion to verify that ifthese cones are area-minimizing.
Proposition 3.2.
The upper bound of the second fundamental formsof the cone over Q mi =1 √ k i G ( l i , k i ; F ) at the points belong to the unitsphere is given by sup v | H v | = dim M .Proof. The product of embedded Grassmannians can be seen as anorbit of product of associated unitary group([
Ker94 ]), then we cancompute the second fundamental forms at an fixed point.Since sup v | H v | = dim M · max { , α dl ( k − l ) , . . . , α m dl m ( k m − l m ) } , and α i = dl i ( k i − l i ) k i see [ Ker94 ],[
JC21 ], then α i dl i ( k i − l i ) = k i − l i k i <
1. q.e.d.The computation of normal radius follows [
Ker94 ],[
JC21 ] directly, inthe i -factor of P = ( . . . , a i (1 − l i k i ) , . . . , a i (1 − l i k i ) , a i ( − l i k i ) , . . . , a i ( − l i k i ) , . . . ),we exchange one pair of a i (1 − l i k i ) and a i ( − l i k i ) in the same one factorto gain another point ˜ P , and(3.2) h P, ˜ P i = 1 − dk i dim M . Hence,
Proposition 3.3.
The normal radius of the cone over Q mi =1 √ k i G ( l i , k i ; F ) is arccos (1 − dk dim M ) , where k = min { k , . . . , k m } . The left work is to compare the normal radius and the estimatedvanishing angles, we have
Theorem 3.4.
The cone over M = Q mi =1 √ k i G ( l i , k i ; F ) is area-minimizing, if dimM ≥ .Proof. We first consider the cases: 7 ≤ dimM ≤ F = H , it is H P × H P . ( dimC, α ) = (9 , . ◦ by Lawlor’s table, and the normal radius is arccos (1 − dk dimM ) = π , it is area-minimizing;(2) F = C , all of the normal radius between different cases have theminimum value: arccos (1 − ) = arccos ( ) = 53 ◦ , and the maximalvalue of vanishing angle is no more than 12 . ◦ by Lawlor’s table, sothese cases are also area-minimizing;(3)) F = R , all of the normal radius between different cases have theminimum value: arccos (1 − ) = arccos ( ) = 35 ◦ , and the maximalvalue of vanishing angle is no more than 15 . ◦ by Lawlor’s table, sothese cases are also area-minimizing.For dimM ≥
12, let k = dimC = dimM + 1, we use the followingformula given by Gary R. Lawlor:(3.3) tan ( θ ( k, α )) < k tan ( θ (12 , k α )) . Now, α = √ k −
1, then k α ≤ √
13, hence(3.4) tan ( θ ( k, α )) < k tan ( θ (12 , √ < k . The normal radius of cone over M = Q mi =1 √ k i G ( l i , k i ; F ) is arccos (1 − dk i k − ), we compare 2 arctan k and arccos (1 − dk i k − ) as follows:Note 2 arctan k < k , 1 − cos k < k and dk ≤
2, then the proof isfollowed by the above relations. q.e.d.Ther are also analogies for the product embedding of Grassmannianmanifolds over different base fields, consider the minimal embedding f i : r i G ( l i , k i ; F i ) → S n i (1), set d i = dim R F i , then r i = q l i ( k i − l i )2 k i , dim G ( l i , k i ; F i ) = d i l i ( k i − l i ), n i = k i − k i ( k i − d i , and we assume l i ≤ k i − l i for every i .Let M be Q mi =1 λ i r i G ( l i , k i ; F i ), then f : M → S P mi =1 n i + m − (1) is mini-mal if and only if a i = λ i r i = q d i k i dim M , where dim M = P mi =1 d i l i ( k i − l i ).Hence, Proposition 3.5.
The cone over Q mi =1 √ d i k i G ( l i , k i ; F i ) is minimal. The upper bound of second fundamental forms of this cone is also dimM , its normal radius is arccos (1 − min { d i k i } dim M ). After an similar dis-cussion to theorem 3.4, we have Theorem 3.6.
The cone C over Q mi =1 √ d i k i G ( l i , k i ; F i ) is area-minimizingif dimC > .
4. Cones of dimension which all the factors areGrassmannian submanifolds In this section, we consider M as the product of embedded Grass-mannian manifolds of dimM = 6, just like the cases of products ofspheres(see chapter 4 and 5 in [ Law91 ]), the cone over M may still havevanishing angles, we should find the concrete expressions of inf v det ( I − tH vij ) in (2.11).Let A be an arbitrary m × m symmetric matrix whose trace is zero, α := || A || = qP a ij , A just has m real eigenvalues { a , . . . , a m } , then det ( I − tA ) attains its minimum for any symmetric matrix A whenthere are only two different values represented among all of the a i ((seeAppendix of [ Law91 ]), and the conditions P a i = 0 and P a i = α completely determine the solution once we know how many should bepositive. Denote the multiplicity of the positive eigenvalue by r , thesolution is α q m − rmr of multiplicities r and − α q rm ( m − r ) of multiplicities m − r .Let L ( α, t, m, r ) be the result function:(4.1) L ( α, t, m, r ) := (1 − tα r m − rmr ) r (1 + tα r rm ( m − r ) ) m − r , where r ∈ [1 , m −
1] is an integer, L is decreasing in r , the minimum of L with respect to r is denoted by F ( α, t, m ) := L ( α, t, m, r is, the bigger the angle radius of normal wedge willbe. Lawlor use the function F ( α, t, m ) to build ”The table” in [ Law91 ].There’s a critical situation that when α = √ , m = 6, from ”The Table”,the solution of (2.11) associated to inf v det ( I − tH vij ) = F ( √ , t,
6) existsin some finite interval [0 , θ ], the vanishing angle does not exist. Though,for concrete cones under critical situation, the multiplicity of the positiveeigenvalue may fail to attain the least number 1, the cone over M maystill have vanishing angles.The multiplicity of the positive eigenvalue played key roles in the com-plete classifcation of cones over products of spheres, for α = √ , m = 6,the function of multiplicity 1 is(4.2) F ( t ) := F ( √ , t,
6) = L ( √ , t, ,
2) = (1 − t √ t √ , it associate to the products of spheres which one of the spheres is acircle, their cones are stable, by the studying of cones for which theCurvature Criterion is Necessary and Sufficient, Lawlor conclude thatthese cones are stable, not area-minimizing.The function of multiplicity 2 is(4.3) E ( t ) := L ( √ , t, ,
2) = (1 − t √ (1 + t √ , associated to the cones over S × S and S × S × S , their cones havevanishing angles, hence be area-minimizing.And the function of multiplicity 3 is(4.4) G ( t ) := L ( √ , t, ,
3) = (1 − t ) (1 + t ) , associated to the cone over S × S , the cone is area-minimizing too.Now, we consider the cones C = C ( M ) over the product of em-bedded Grassmannian manifolds of dimension 7. Under the standardembedding, R P ≡ S ( ), C P ≡ S ( ) and H P ≡ S ( )([ JC21 ]), wecall their associated factors in the product—the reduced f actors sincethere doesn’t exist the terms of second fundamental forms in (2.5), oradditionally, the factors could be any dimensional spheres less than 6,i.e. one of the factors is S , S , S , S , S , so there exists some cases ofproducts of spheres, i.e. all the factors are reduced f actors .There exists some cones over the products of embedded Grassman-nian manifolds which part of its factors are reduced f actors , we willtalk about them in the last section.There are five cones which all its factors are not reduced f actors , i.e.the factors are all the embedded Grassmannian submanifolds in spheres,they are :(1): R P × R P × R P ;(2): R P × G (2 , R );(3): R P × R P ;(4): R P × R P ;(5): R P × C P .We will prove that these cones are all area-minimizing by comput-ing their minimum polynomials which none of them are multiplicity 1polynomial F ( t ).(1) R P × R P × R P :This is the cone over products of three Veronese maps, for this case,we will prove that on (0 , √ ), for any chosen unit normal vector v ,(4.5) inf v det ( I − tH vij ) = (1 − t √ (1 + t √ = E ( t ) . Note √ = 0 . tan . ◦ = 0 . First, R P is embedded in H (3 , R )(or its hyperplane of trace zero,[ JC21 ]),the normal vector ξ is given by diag { , a, − a } (to see this, first as-sume ξ is an any given normal vector, if g ∗ ξ := gξ g T is diagonalfor an isotropy isometry action g , we could change the standard or-thonormal tangent basis E α := E α + E α (2 ≤ α ≤
3) given in [
JC21 ]to ( g − ) ∗ E α , note the second fundamental forms is also equivariant,then h h (( g − ) ∗ E α , ( g − ) ∗ E β ) , ξ i = h h ( E α , E β ) , g ∗ ξ i , the computationis simplified), | ξ | = a , note the image lies in a sphere of radius √ ,then the value of H ξ at the sphere of radius 1 is given by ± a √ = ± | ξ |√ .Follow (2.5)(we replace b i to − b i here),(4.6) I − tH v = diag { (1 − b tλ ) I k − tλ H ξ , . . . , (1 − b m tλ m ) I k m − tλ m H ξ m m } , hence(4.7) det ( I − tH vij ) = Y i =1 ((1 − √ b i t ) − | ξ i | t ) , where P i =1 ( b i + | ξ i | ) = 1.In the next, t is restricted on (0 , √ ), this choice of t is enough forensuring that the normal wedge of angle radius arctant won’t meetsthe focal points of the cone. Denote | ξ i | = c i ≥
0, we claim that ifminimum value of det ( I − tH vij ) is attained for fixed b i , c i (1 ≤ i ≤ c , c , c are non-zeros. If so, assume c > , c >
0, thenfor fixed b , b , b , c , the sum of c and c is constant. We adjust c , c as follows: if (1 − √ b t ) ≥ (1 − √ b t ) , i.e. b ≤ b , then we let c beits possible maximal value and c be zero, the result will be more small,this is an contradiction.So, we can assume c = c = 0, then(4.8) det ( I − tH vij ) = ((1 − √ b t ) − c t )(1 − √ b t ) (1 − √ b t ) , subject to(4.9) ( b + b + b = 0 ,b + b + b + c = 1 , c ≥ . We express det ( I − tH vij ) as functions of t, b , c , for convenience, wereplace b by a , c by b , the above equations should have two real rootsfor b , b , then we can let the domain be(4.10) D ≡ { ( a, b ) ∈ R | a + 2 b ≤ } , and(4.11) g ( a, b ) := det ( I − tH vij ) = ((1 −√ at ) − b t )(1+ √ at +(3 a + 32 b −
32 ) t ) Hence(we get almost all the following computations by using
M athematica ),(4.12) ∂g∂a = − t (1+ √ at +(3 a + 32 b −
32 ) t )( − √ √ a +5 √ b +9 at − a t +3 ab t ) , and(4.13) ∂g∂b = − bt (1+ √ at +(3 a + 32 b −
32 ) t )( − √ at − t − a t +9 b t ) . Consider 1 + √ at + (3 a + b − ) t as a quadratic function of a and compute the discriminant, we have 1 + √ at + (3 a + b − ) t > < t < √ .Solve the equations: ∂g∂a = 0 and ∂g∂b = 0, we conclude that in theinterior of D , i.e. 3 a + 2 b < g ( a, b ) attains minimum value at a = − , b = 0(another critical point is a = − , b = 0 which is biggerthan this one). Note that the norm of the second fundamental formattains maximum value √ a, b ) = ( − ,
0) or ( q , det ( I − tH vij ).On boundary of D ,(4.14) f ( a ) := g ( a, r − a ) = − √ at ) ( − √ at + 2 t − a t ) , subject to a ≤ , and(4.15) f ′ ( a ) = 148 t (2 √ at ) (27 ta − √ a − t ) . By analyzing the quadratic function 27 ta − √ a − t in the interval a ≤ , we find that if and only if 0 < t < √ , the sign of f ′ ( a ) is+ , − (if t > √ , the sign is + , − , +). Additionally, we compute that: g ( − q ) > g ( q ), then g ( q ) is the minimum value on the boundary.The vanishing angle associated to E ( t ) is 19 . ◦ , and tan (19 . ◦ ) < √ ,since the normal radius is arccos ( ) = 60 ◦ by proposition 3.3, hence Theorem 4.1.
The cone over three Veronese embeddings R P × R P × R P is area-minimizing. (2) R P × G (2 , R ):This case is similar to R P × R P × R P , we will prove on [0 , √ ),for any chosen unit normal vector v ,(4.16) inf v det ( I − tH vij ) = (1 − t √ (1 + t √ = E ( t ) . First, G (2 , R ) is embedded in H (4 , R )(or its hyperplane of tracezero), the normal vector ξ can be given by diag { a, − a, b, − b } (to see this, first assume ξ is an any given normal vector, if g ∗ ξ := gξ g T isdiagonal for an isotropy isometry action g , we could change the standardorthonormal tangent basis E αa := E αa + E aα (1 ≤ a ≤ , ≤ α ≤ JC21 ] to ( g − ) ∗ E αa , note the second fundamental forms is alsoequivariant, then the computation is simplified). The result H ξ hasvalue diag {− a + b, a + b, − a − b, a − b } . Note the image lies in a sphereof radius √ , then the value of H ξ at the sphere of radius 1 is given by √ diag {− a + b, a + b, − a − b, a − b } , we let x = a − b, y = a + b . Thediscussion for the first factor R P is the same to R P × R P × R P , weassume | ξ | = c , since b + √ b = 0, then(4.17) det ( I − tH vij ) = ((1 −√ b t ) − c t )((1+ √ b t ) − x t )((1+ √ b t ) − y t ) , it subject to(4.18) 32 b + c + 12 ( x + y ) = 1 . Fix b , let A = (1 − √ b t ) , B = (1 + √ b t ) , β = c , β = x , β = y , then we want to get the minimum value of(4.19) f ( β , β , β ) = ( A − t β )( B − t β )( B − t β ) , subject to D = { ( β , β , β ) ∈ R | β + β + β = 1 − b , β i ≥ i =1 , , } .We can use the method of Lagrange Multiplier to show that there areno minimum points in the interior of D , let g ( β , β , β ) = β + β + β − c , where c is a positive number. The gradient of g : ∇ g = (1 , , ),the gradient of f is given by ∇ f = ( ∂f∂β , ∂f∂β , ∂f∂β ), where(4.20) ∂f∂β = − t ( B − t β )( B − t β ) < , ∂f∂β = − t ( A − t β )( B − t β ) < , ∂f∂β = − t ( A − t β )( B − t β ) < . At a critical point ( β , β , β ),(4.21) ∇ f // ∇ g ⇔ ∂f∂β = ∂f∂β = 12 ∂f∂β , then β = β , B − t β = B − t β =: λ > A − t β = ( B − t β ) = λ .The Hessian matrix Hess ( f ) is of the form(4.22) c cc c c c , where c = t λ .A tangent vector v of the algebraic manifold g = 0 is given by { v =( x, y, z ) ∈ R | x + y + z = 0 } , then the quadratic form vHess ( f ) v T = − c ( y + yz + z ) < D .On boundary ∂D , assume β = 0, then(4.23) fA = B − t ( β + β ) + 916 t β β , attains minimum when at least one of β , β is zero.assume β = 0, then(4.24) fB = AB − t ( Bβ + 3 A β t β β , attains minimum when β = 0 if B ≥ A , when β = 0 if B ≤ A .So, f ( β , β , β ) attains minimum when at least two of β , β , β arezeros.( a ): β = β = 0,(4.25) det ( I − tH vij ) = (1 − √ b t + 92 b t − t )(1 + √ b t ) , and b ≤ .This equation is just (4.14), so on 0 < t < √ , the minimum is E ( t )and it is attained when b = q , c = 0.( b ): β = β = 0, i.e. c = x = 0, (the same for β = β = 0),(4.26) f = AB ( B − y t ) , subject to b + y = 1, i.e. y = 2 − b , then we need to find theminimum of(4.27) f ( b ) = (1 − √ b t ) (1 + √ b t ) (1 + √ b t + 3 b t − t ) , where b ≤ .And(4.28) f ′ ( b ) = 34 t ( √ tb )( − √ tb )(2 + √ tb )( − t + √ b + 3 tb ) . When 0 < t < √ , the sign of f ′ ( b ) is + , − , we need to compare f ( − q ) and f ( q ), the minimum point is f ( q ) = E ( t ).When √ < t < √ , the sign of f ′ ( b ) is − , + , − (we note here it is alittle complicated, the signs of √ tb and − t + √ b + 3 tb are all depending on t ), the smaller interior critical point is b = x = − √ t ,we compute that(4.29) f ( x ) − f ( r
23 ) = ( √ − t )( √ t ) ( − √ t + 48 √ t + 16 t )1024 > . Hence, for case( b ), when 0 < t < √ , f ( b ) attains the minimum E ( t )when b = q .In summary, when 0 < t < √ , the minimum of det ( I − tH vij ) is E ( t ).The vanishing angle associated to E ( t ) is 19 . ◦ , and tan (19 . ◦ ) ≤ √ ,the normal radius is arccos ( ) = 60 ◦ , then we have Theorem 4.2.
The cone over R P × G (2 , R ) is area-minimizing. (3) R P × R P : R P is embedded in H (5 , R ), the normal vector ξ can be given by diag { , c , c , c , c } where P i =1 c i = 0. Note the image lies in a sphereof radius √ √ , then the value of H ξ at the sphere of radius 1 is given by √ √ diag { c , c , c , c } . Denote | ξ | = d , let α = P i =1 c i , then | ξ | = α , hence α + b + d = 1. Since λ = √ , so b + √ b = 0, and(4.30) det ( I − tH vij ) = ((1 − √ b t ) − d t ) Y i =1 (1 + √ b t − √ √ c i t ) . Follow [
Law91 ], for fixed b , d , on t ∈ (0 , √ ), the minimum of Q i =1 (1 + √ b t − √ √ c i t ) is attained when c = c = c = − c areall negative, and it is (1 + √ b t − √ √ c t )(1 + √ b t + √ c t ) , forconvenience, we let b be √ √ c , let a be b , then we need to find theminimum of(4.31) g ( a, b ) = ((1 −√ at ) − (1 − a − b ) t )(1+ √ at − √ bt )(1+ √ at + 1 √ bt ) in the domain D = { ( a, b ) ∈ R | a + 2 b ≤ , b ≥ } .There are no critical points in the interior of D when t restricted on(0 , √ ) by using M athematica .The boundary of D are divided into two parts,( a ): b = 0 , a ≤ ,(4.32) g ( a,
0) = 132 (2 + √ at ) (2 − √ at − t + 9 a t ) , this is just the function 4.14 for the boundary case of R P × R P × R P ,so for this case, when 0 < t < √ , the minimum is E ( t ).( b ): on the ellipse b = q − a , define f ( a ) = g ( a, q − a ), then(4.33) f ( a ) = (1 − √ at ) (10 + 5 √ at − √ √ − a t )(10 + 5 √ at + √ √ − a t ) , and(4.34) f ′ ( a ) = 3 t ( √ at − √ at + √ p − a t ) (12 a −√ t +12 √ a t − √ a p − a t ) . Let h ( a ) = 12 a − √ t + 12 √ a t − √ a √ − a t , we will show that h is increasing in a . Since in the following we can see h ′ ( a ) → + ∞ when a → ± q , it suffices to prove that the minimum value of h ′ ( a ) is noless than zero.By using M athematica ,(4.35) h ′ ( a ) = 12 + 24 √ at + 9 √ a t √ − a − √ p − a t, and(4.36) h ′′ ( a ) = 6 t (9 √ a − √ a + 4 √ − a ) )(2 − a ) . It happens that the minimum points of h ′ ( a ), i.e. the roots of h ′′ ( a ),is independent of t .The numerical solution of h ′′ ( a ) = 0 is given by:(4.37) a → − . , a → . − . i, a → . . i. the unique real root x = − . h ′ ( a ).When a → x , the minimum value of h ′ ( a ) is 12 − . t which isdecreasing in t , and when t = √ , it is 2 . h ( a ) is increasing in a .Additionally, h ( − q ) = 7 √ t − √ < h ( q ) = 7 √ t +4 √ >
0, it tells us that the sign of f ′ ( a ) is + , − .Therefore the minimum value of f ( a ) is the smaller of f ( − q ) and f ( − q ), which is f ( − q ) = E ( t ) if 0 < t < √ .The vanishing angle associated to E ( t ) is 19 . ◦ , and tan (19 . ◦ ) ≤ √ ,the normal radius is arccos ( ) = 60 ◦ , hence Theorem 4.3.
The cone over R P × R P is area-minimizing. (4) R P × R P : R P is embedded in H (4 , R ), the normal vector ξ can be given by diag { , c , c , c } , respectively, ξ = diag { , d , d , d } , where P i =1 c i =0 and P i =1 d i = 0. Note the image lies in a sphere of radius √ √ , thenthe value of H ξ at the sphere of radius 1 is given by √ √ diag { c , c , c } ,the similar results for H ξ . | ξ | = P i =1 c i , | ξ | = P i =1 d i . Now λ = λ = √ , then b + b = 0, P i =1 c i + P i =1 d i + 2 b = 1, and(4.38) det ( I − tH vij ) = Y i =1 (1 − √ b t − √ c i t )(1 + √ b t − √ d i t ) . For fixed b , d i ( i = 1 , , Q i =1 (1 − √ b t − √ c i t )is attained when c is positive and c = c = − c , similar for thecase considering b , c i ( i = 1 , ,
3) fixed, all these can attained on t ∈ (0 , √ )([ Law91 ]). Then we can consider these normal vectors v givenas follows: let v = ( v , v ), v = b x + ξ , v = − b x + ξ , where x , x isthe position vectors, ξ = diag { , c , − c , − c } , ξ = diag { , d , − d , − d } ,and c , d ≥
0, 8 b + 3 c + 3 d = 4. For convenience, we set b = a , c = b ≥ d = c ≥
0, then det ( a, b, c ) := det ( I − tH vij ) = (1 − √ at − √ bt )(1 − √ at + √ bt ) (4.39) × (1 + √ at − √ ct )(1 + √ at + √ ct ) , and 8 a + 3 b + 3 c = 4 , b ≥ , c ≥ b, c are in the symmetric positions, we can further assume a ≥ f ( a, b ) = det ( a, b, q − a − b + ). First, we fix a and deter-mine the sign of f ′ ( b ), here a ≥ , ≤ b ≤ q − a + .By using M athematica , f ′ ( b ) = 9512 bt (cid:16) − √ at + √ bt + 4 (cid:17) (cid:16) t p − a − b + 4 + 4 √ at + 4 (cid:17) (4.40) × (cid:16) √ at p − a − b + 4 + p − a − b + 4 − a t + √ abt − √ a − b t − √ b + 2 t (cid:17) The last factor in 4.40 is denoted by g ( a, b ), then easy to see it hasthe same sign of f ′ ( b ).Now(4.41) g ( a, r − a (cid:16) a t + √ p − a at − p − a − √ a − t (cid:17) , it is less than zero, and(4.42) g ′ ( b ) = − √ abt √− a − b + 4 − b √− a − b + 4 + √ at − bt −√ g ( b )(hence that of f ′ ( b )) is + , − or − , wecompute that(4.43) f ( a, − f a, r − a ! = a (cid:0) − a (cid:1) t (cid:16) a √ − a t − a t + 3 √ − a t + 6 (cid:17) √ , it is obviously no less than zero which shows that when a is fixed, f ( a, b )attains minimum value h ( a ) := g ( q − a )(i.e. c is zero).We compute that(4.44) h ′ ( a ) = 34 t (cid:16) √ at + 1 (cid:17) (cid:16)p − a t − √ at + 2 (cid:17) (cid:16) √ a t + 4 p − a at − a − √ t (cid:17) . Let q ( a ) := 4 a √ − a t − (cid:0) − √ a t + 6 a + √ t (cid:1) , 4 a √ − a t ≤√ t , where the equal sign holds iff a = . Next, consider r ( a ) = − √ a t + 6 a + √ t as a quadratic function of a , then its minimum is r (0) = √ t . Therefore q ( a ) <
0, and consequently h ′ ( a ) < h ′ ( a ) < h is decreasing in a . Therefore(4.45) min ( h ) = min ( f ) = h ( 1 √ f ( 1 √ ,
0) = det ( 1 √ , ,
0) = (1+ t ) (1 − t ) = G ( t )when t ∈ (0 , √ ).The vanishing angle associated to G ( t ) is 19 ◦ , and tan (19 ◦ ) < √ ,the normal radius is arccos ( ) > ◦ , then we have Theorem 4.4.
The cone over R P × R P is area-minimizing. (5) R P × C P : R P is embedded in H (3 , R ), the normal vector ξ can be given by diag { , c, − c } , then | ξ | = c , the image lies in a sphere of radius √ ,then the value of H ξ at the sphere of radius 1 is given by diag { c, − c } . C P is embedded in H (3 , C ), the normal vector ξ can be given by diag { , b, − b } , then | ξ | = b . Note the image lies in a sphere of ra-dius √ , then the value of H ξ at the sphere of radius 1 is given by √ diag { b, b, − b, − b } ([ JC21 ]). Now λ = √ , λ = √ √ , then b + √ b = 0, b + c + b = 1, and(4.46) det ( I − tH vij ) = (cid:16) (1 − √ b t ) − c t (cid:17) (1 + √ b t ) − b t ! . Let b be a , let D = { ( a, b ) ∈ R | a + 2 b ≤ } , denote(4.47) g ( a, b ) = (cid:18) (1 − √ at ) − (1 − a − b ) t (cid:19) (1 + √ at ) − b t ! . There are no critical points in the interior of D when t restricted on(0 , √ ) by using M athematica .On boundary of D , i.e. c = 0, let(4.48) f ( a ) := g ( a, ± r − a −√ at ) (1+ √ at + bt √ (1+ √ at − bt √ , where a + b = 1.Let(4.49) α ≡ ( b λ ) + ( √ a + b √ + ( √ a − b √ = 3( 32 a + b ) − b ≤ . So, follow Lemma 3.2 and the Appendix in [
Law91 ], on (0 , √ ) ⊂ (0 , √ ] ⊂ (0 , α q ], we have(4.50) f ( a ) ≥ F ( α, t, ≥ F ( √ , t,
3) = (1 − t √ (1+ t √ = E ( t ) , the minimum E ( t ) of det ( I − tH vij ) can be attained by setting b = 0 , c =0, and a = q .The vanishing angle is 19 . ◦ , and tan (19 . ◦ ) < √ , since the normalradius is arccos ( ) = 60 ◦ , hence Theorem 4.5.
The cone over R P × C P is area-minimizing.
5. Cones of dimension which have reduced factors There are many cones of dimension 7 which have reduced factors, oradditionally, the factors could be any dimensional spheres less than 6,i.e. one of the factors is S , S , S , S , S , we won’t talk all of them likelast section. A special class of cones we note here is, if the cone is not the productof spheres and one of the factors are S ( ∼ = R P ), additionally, if theminimum value of det ( I − tH vij ) attains F ( t ) = (1 − t √ t √ ) , thenwe can only conclude these cones are stable, different like the case ofproducts of spheres, the embedding of an Grassmannian manifold intosphere is an isolated singular orbit(i.e. it is not an principal orbit of somepolar group action), its cone is not Lawlor’s cones which his CurvatureCriterion is sufficient and necessary, i.e. it’s still of possibility for thesecones being area-minimizing.In the next, we give an example of area-minimizing cone which havean reduced factor. It is the cone over C P × C P , the unique nontrivialcone of dimension 6 which all its factors are Grassmannians over C , thisproduct has an reduced factor C P , for its cone, we have Theorem 5.1.
The cone over C P × C P is area-minimizing.Proof. Since C P is the 2-sphere of radius , then we can let v =( v , v ) be an unit normal where v = b x , v = b x + ξ , x , x are theunit position vectors.Follow (2.5)(we replace b i to − b i here),(5.1) H v = diag { b λ I , b λ I + 1 λ H ξ } , where H ξ is the matrix of second fundamental forms of f : √ C P ֒ → S (1) projected on the direction ξ .Then,(5.2) det ( I − tH vij ) = (1 − b λ t ) (1 − b + | ξ |√ λ t ) (1 − b − | ξ |√ λ t ) , where λ = √ , λ = √ √ .We want to search minimum of (1 − b λ t )(1 − b + | ξ |√ λ t )(1 − b − | ξ |√ λ t ), let(5.3) α ≡ ( √ a ) + ( b + | ξ |√ λ ) + ( b − | ξ |√ λ ) = 3( b + b + | ξ | ≤ . So, follow Lemma 3.2 and the Appendix in [
Law91 ], on (0 , √ ] ⊂ (0 , α q ], we have(5.4) det ( I − tH vij ) ≥ F ( α, t, ≥ F ( √ , t,
3) = (1 − t √ (1+ t √ = E ( t ) , the minimum E ( t ) of det ( I − tH vij ) can be attained by setting ξ = 0,and b = λ , b = − λ .The vanishing angle is 19 . ◦ , and tan (19 . ◦ ) < √ , since the nor-mal radius is arccos ( ) > ◦ by proposition 3.3, so this cone is area-minimizing. q.e.d. Remark 5.2.
The minimal product is suitable for various mixedcases which the factors could be any concrete embedded minimal sub-manifolds in spheres or the spheres themselves. For the cases of prod-ucts of Grassmannians, there is a very challenging question here on thecomplete classification of the associated cones, we list some other coneswhich deserve further considerations:(1)Whether the cones of dimension less than are area-minimizing,stable or unstable, similar to the cones over products of spheres? Exceptthe products of spheres(i.e. all the factors are belong to R P , C P , H P ),all the others cannot be proved area-minimizing directly by CurvatureCriterion in [ Law91 ] , such examples are: S × R P , S × G (2 , R ) ,the product of two Veronese maps R P × R P , and R P × R P , etc.(2)Cones of dimension which having reduced factors, it seems easyto analyze them than those cones which all the factors are embeddedGrassmannian submanifolds, by finding those cones which their mini-mum polynomials are not F ( t ) , it seems that there exists some morearea-minimizing cones.(3)Cones of dimension which have reduced factors(not all), and theminimum polynomials are F ( t ) , could these cones be area-minimizing?(4)There are some other cones of dimension bigger than whichsome of the factors could be O P , or the cone is an product of P l ¨ ucker embedded oriented real Grassmannians considered in [ JC21 ] . Acknowledgments . This work is supported by NSFC No.11871450.
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