Minimal two-spheres with constant curvature in the complex hyperquadric

JID:MATPUR

AID:2817 /FLA

[m3L; v1.175; Prn:25/04/2016; 14:29] P.1 (1-24) J. Math. Pures Appl. ••• (••••) •••–•••

Contents lists available at ScienceDirect

Journal de Mathématiques Pures et Appliquées www.elsevier.com/locate/matpur

Minimal two-spheres with constant curvature in the complex hyperquadric Chiakuei Peng a , Jun Wang b,c , Xiaowei Xu d,e,∗ a

School of Mathematical Sciences, University of Chinese Academy of Sciences, Beijing 100049, China School of Mathematics Sciences, Nanjing Normal University, Nanjing 210023, China c Institute of Mathematics, Nanjing Normal University, Nanjing 210023, China d School of Mathematical Sciences, University of Science and Technology of China, Hefei, 230026, Anhui province, China e Wu Wen-Tsun Key Laboratory of Mathematics, USTC, Chinese Academy of Sciences, Hefei, 230026, Anhui, China b

a r t i c l e

i n f o

Article history: Received 1 December 2015 Available online xxxx MSC: 53C42 53C55 53E20 Keywords: Minimal two-spheres Constant curvature Kähler angle Totally geodesic

a b s t r a c t In this paper, various constant curved minimal two-spheres in the complex hyperquadric Qn are obtained, which exhaust all the minimal homogeneous ones in Qn . Their geometric quantities of Gauss curvature, Kähler angle and the length of the second fundamental form are expressed explicitly. As an application, the totally geodesic, constant curved holomorphic and antiholomorphic two-spheres in Qn are completely classified. © 2016 Elsevier Masson SAS. All rights reserved.

r é s u m é Dans cet article on obtient des résultats pour différentes deux-sphères minimales à courbure constante dans le complexe hyperquadrique Qn , ces résultats épuisent la question de toutes celles qui sont homogènes et minimales dans Qn . Leurs caractéristiques gémétriques comme la courbure de Gauss, l’angle de Kälher et la longueur de la seconde forme fondamentale sont calculées explicitement. Comme application on obtient une classification complète des deux-sphères totalement géodésiques holomorphes et antiholomorhes et de celles qui sont à courbure constante dans Qn . © 2016 Elsevier Masson SAS. All rights reserved.

1. Introduction It is a long history of studying minimal two-spheres in symmetric spaces. For their geometry, one can refer to J.L.M. Barbosa [1], J. Bolton, G.R. Jensen, M. Rigoli and L.M. Woodward [3], S. Bando, Y. Ohnita [4], * Corresponding author at: School of Mathematical Sciences, University of Science and Technology of China, Hefei, 230026, Anhui province, China. E-mail addresses: [email protected] (C. Peng), [email protected] (J. Wang), [email protected] (X. Xu). http://dx.doi.org/10.1016/j.matpur.2016.02.017 0021-7824/© 2016 Elsevier Masson SAS. All rights reserved.

JID:MATPUR 2

AID:2817 /FLA

[m3L; v1.175; Prn:25/04/2016; 14:29] P.2 (1-24) C. Peng et al. / J. Math. Pures Appl. ••• (••••) •••–•••

E. Calabi [8], M.P. do Carmo, N. Wallach [9], Q. Chi, Y.B. Zheng [10], L. Delisle, V. Hussin and W.J. Zakrzewski [11,12], Zh.Q. Li, Zh.H. Yu [17], X.X. Jiao, J.G. Peng [18,19], C.K. Peng, X.W. Xu [24] and Y.B. Zheng [28] etc. For construction of harmonic (a generalization of minimal) two-spheres in symmetric spaces, one can refer to A. Bahy-El-Dien, J.C. Wood [5], S.S. Chern, J. Wolfson [7], J. Eells, J.C. Wood [14], L. Fernandez [15], J.G. Wolfson [25], K. Uhlenbeck [26] and J.C. Wood [27]. When the symmetric spaces are space forms, the results are very rich. In this paper, we will consider minimal two-spheres in another symmetric space, the complex hyperquadric Qn , which is isomorphic to the homogeneous space O(n + 2)/(SO(2) × O(n)). On the other hand, although Qn is a complex algebraic submanifold of CPn+1 , its geometry structure is more complicated than that of space forms. For example, Qn does not have constant holomorphic sectional curvature; the E. Calabi’s rigidity for holomorphic curves in complex projective space is not true for the ones in the complex hyperquadric; the construction of harmonic two-spheres (see [3,5,14,15,25]) in Qn is much more sophisticated than the ones in space forms. Thus, it is an interesting problem to study the minimal two-spheres in Qn . So far, very little is known except for [20–22] etc. It is well known that constant curved minimal two-spheres in S n (1) and CPn are homogeneous (see [3,4,8,9]). They also determined the values distribution of the constant curvature completely. X.X. Jiao and the second author [21] proved that constant curved minimal two-spheres in Q2 are also homogeneous. Naturally, one may ask the questions: Does the minimal two-spheres with constant curvature in Qn (n ≥ 3) must be homogeneous? What numbers can be realized as the curvatures of minimal two-spheres with constant curvature in Qn ? In this paper, as the first step, we completely characterize the minimal homogeneous two-spheres in the complex hyperquadric Qn by using the real representation of SU (2), which plays an important role in determining all the constant curved minimal two-spheres. Their geometric properties are obtained in Theorem 5.4. As a consequence, we give a classification of totally geodesic, constant curved holomorphic and antiholomorphic (without the assumption “homogeneous”) two-spheres in Qn , see the Theorem 5.5 and Theorem 5.6. Some new features of minimal two-spheres in Qn have been found in the Remarks of Theorem 5.4. Furthermore, some remaining problems are proposed. Rigidity is an important property of the geometry of submanifolds. For instance, J.L.M. Barbosa [1] proved that a minimal two-sphere in S n (1) is determined by its first fundamental form; J. Bolton, G.R. Jensen, M. Rigoli and L.M. Woodward [3] proved that a minimal two-sphere in the complex projective space CPn is determined by its first fundamental form and the Kähler angle. These phenomenons are not true for the minimal ones in Qn . We find that there exist families of minimal two-spheres that share the same Gauss curvature and Kähler angle, but they are not congruent in Qn , see Theorem 5.4 and Theorem 5.6. Our paper is organized as follows. Some known results on the irreducible unitary representations and real representations of the special unitary group SU (2) are reviewed in Section 2. The image of a homogeneous immersion from S 2 into Qn is a SU (2)-orbit (Theorem 3.3) is proved in Section 3, and the best base point in each orbit (Theorem 3.5) is found. According to these base points, the orbits are divided into three different types. The geometries of these orbits are studied in Section 4. The minimal orbits of these types are determined in Section 5, and four tables of all linearly full minimal homogeneous twospheres in Q2 , Q3 , Q4 and Q5 are presented. Some interesting problems for further study are proposed in Section 6. 2. The representations of SU (2) In this section, for completeness, we review known results on unitary and real representations of the special unitary group SU (2).

JID:MATPUR

AID:2817 /FLA

[m3L; v1.175; Prn:25/04/2016; 14:29] P.3 (1-24) C. Peng et al. / J. Math. Pures Appl. ••• (••••) •••–•••

3

The special unitary group SU (2) is defined by  SU (2) =



 a b −¯b a ¯

g=

   2 2  |a| + |b| = 1, a, b ∈ C .

The Lie algebra su(2) of SU (2) is given by 

 ix y −¯ y −ix

su(2) =

   x ∈ R, y ∈ C ,

where i2 = −1. We defined a basis {ε1 , ε2 , ε3 } of su(2) by  ε1 =



 i 0

0 −i

, ε2 =



 0 1 −1 0

, ε3 =

 0 i

i 0

,

satisfying [ε1 , ε2 ] = 2ε3 , [ε3 , ε1 ] = 2ε2 , [ε2 , ε3 ] = 2ε1 .

(2.1)

The Maurer–Cartan forms of SU (2) are given by  Θ := dgg

−1

=

 iω ϕ −ϕ¯ −iω

,

(2.2)

where ω, ϕ are real and complex one-forms respectively. The Maurer–Cartan equation is dΘ = Θ ∧ Θ which gives d(iω) = −ϕ ∧ ϕ, ¯

dϕ = (2iω) ∧ ϕ.

(2.3)

2.1. The irreducible unitary representation of SU (2) We first recall the irreducible unitary representations of SU (2). Let Vn be the (n+1)-dimensional complex vector space of all complex homogeneous polynomial of degree n w.r.t. the two complex variables z0 and z1 . We define a Hermitian inner product  ,  on Vn by f1 , f2  :=

n 

ak ¯bk k!(n − k)!,

k=0

for f1 =

n  k=0

ak z0k z1n−k , f2 =

n  k=0

  bk z0k z1n−k ∈ Vn . So, vk,n = z0k z1n−k / k!(n − k)!  0 ≤ k ≤ n is a unitary

basis for Vn . A unitary representation ρn of SU (2) on Vn is defined by ρn (g)f (z0 , z1 ) := f ((z0 , z1 )g −1 ) = f (¯ az0 + ¯bz1 , −bz0 + az1 )   for g ∈ SU (2) and f ∈ Vn . Under the basis vk,n  0 ≤ k ≤ n , we obtain a matrix representation of ρn : SU (2) −→ U (n + 1), g → ρn (g). The matrix ρ(g) is described by

JID:MATPUR

AID:2817 /FLA

[m3L; v1.175; Prn:25/04/2016; 14:29] P.4 (1-24) C. Peng et al. / J. Math. Pures Appl. ••• (••••) •••–•••

4

vk,n ρn (g) := ρn (g)(vk,n ) =

n 

Λkl vl,n ,

l=0

where Λkl =

  k n−k−p p q ¯ k−q l!(n − l)!  n−k a b a ¯ (−b) k!(n − k)! p q

(2.4)

p+q=l

¯ n−k n−l . which satisfies Λkl = (−1)k+l Λ The action of su(2) on Vn is described as follows: vk,n dρn (ε) :=

 d ρn (exp tε)(vk,n ) t=0 dt

(2.5)

for 0 ≤ k ≤ n and any element ε ∈ su(2). dρn (ε) will be viewed as an element in End(Vn ). In particular, when ε takes ε1 , ε2 , ε3 respectively, we have

for 0 ≤ k ≤ n, and ak,n = Set



vk,n dρn (ε1 ) = (n − 2k)ivk,n ,

(2.6)

vk,n dρn (ε2 ) = ak−1,n vk−1,n − ak,n vk+1,n ,

(2.7)

vk,n dρn (ε3 ) = ak−1,n ivk−1,n + ak,n ivk+1,n ,

(2.8)

(k + 1)(n − k).

λk := n − 2k, vλk ,n := vk,n , aλk ,n := ak,n =



 [(n + 1)2 − (λk − 1)2 ] 2

(2.9)

for 0 ≤ k ≤ n. An element in Δn = {λ0 , λ1 , . . . , λn } is called a weight of the unitary representation ρn , and λ0 is called the highest weight. Then the representation space Vn has a decomposition, i.e., Vn = Vλ0 ,n ⊕ Vλ1 ,n ⊕ · · · ⊕ Vλn ,n , where dimC Vλk ,n = 1 and Vλk ,n = SpanC {vλk ,n } is called the weight space w.r.t. the weight λk . The weight spaces are pairwise orthogonal w.r.t. the Hermitian inner product  , . We identify Vn with Cn+1 naturally in the unitary basis {vλk ,n | 0 ≤ k ≤ n}. Then the action of ρn (SU (2)) on Cn+1 induces an action on the complex projective space CPn as follows: SU (2) × CPn −→ CPn , (g, [v]) → [vρn (g)]. It is known that the ρn (SU (2))-orbit through the point [vλk ,n ] ∈ CPn (0 ≤ k ≤ n) determines an immersion from S 2 into CPn , called the k-th Veronese surface in CPn . In terms of homogeneous coordinates, the k-th Veronese surface can be denoted by ψλk ,n : S 2 −→ CPn , [a, b] → [Λk0 , . . . , Λkn ]. For more details about the Veronese surfaces, one can refer to references [3] and [4]. 2.2. The real representation of SU (2) Let J be a conjugate-linear automorphism on C2 defined by J (z0 , z1 ) := (−¯ z1 , z¯0 ),

z0 , z1 ∈ C.

Extending J to an automorphism on Vn by (J f )(z0 , z1 ) := f (J (z0 , z1 )),

f ∈ Vn .

JID:MATPUR

AID:2817 /FLA

[m3L; v1.175; Prn:25/04/2016; 14:29] P.5 (1-24) C. Peng et al. / J. Math. Pures Appl. ••• (••••) •••–•••

5

Then J is a structure map on Vn with J 2 = (−1)n 1. So, (Vn , ρn ) is a self-conjugate representation of index (−1)n . Here, a complex irreducible representation (V, ρ) of a compact Lie group G is said to be self-conjugate if V has a structure map J s.t.   J ρ(g)v = ρ(g)J (v), g ∈ G, v ∈ V, ¯ J (v) + μ J (λv + μw) = λ ¯J (w), λ, μ ∈ C, v, w ∈ V, J 2 = ±1,

  and (V, ρ) is said to be of index 1 (resp. −1) if J 2 = 1 (resp. J 2 = −1). We will denote by (Vn )R , ρn the representation of SU (2) over R obtained by the restriction of the coefficient field from C to R, and we   always equip an inner product ( , ) on (Vn )R by ( , ) := the real part of  , . It is known that (Vn )R , ρn   is real irreducible if n is an odd, and (Vn )R , ρn is real reducible if n is an even. Furthermore, when n is an even, (1 + J )(Vn )R and (1 − J )(Vn )R are mutually equivalent real irreducible representations of SU (2), and (Vn )R = (1 + J )(Vn )R ⊕ (1 − J )(Vn )R . It’s well known that {(Vn , ρn ) | n = 0, 1, 2, · · · } are all inequivalent irreducible unitary representations     of SU (2). So, by the Theorem 6.3 in [2], we know that (V2n +1 )R , ρ2n +1 , (1 + J )(V2n )R , ρ2n  n =  0, 1, · · · are all inequivalent irreducible real representations of SU (2). We need to mention that the real dimensions of (V2n +1 )R , (1 + J )(V2n )R are 4n + 4, 2n + 1 respectively.  There is an orthonormal basis vλk ,n , uλk ,n := ivλk ,n  k = 0, 1, . . . , n of (Vn )R w.r.t. the inner product ( , ). So, the representation space (Vn )R has the orthogonal decomposition n n   (Vλk ,n )R = (VλRk ,n ⊕ UλRk ,n ),

(Vn )R =

k=0

(2.10)

k=0

where VλRk ,n = SpanR {vλk ,n }, UλRk ,n = SpanR {uλk ,n }. By (2.5)–(2.8), the Lie algebra su(2) acts on (Vn )R as described by vλk ,n dρn (ε1 ) = λk uλk ,n ,

(2.11)

uλk ,n dρn (ε1 ) = −λk vλk ,n ,

(2.12)

vλk ,n dρn (ε2 ) = aλk +2,n vλk +2,n − aλk ,n vλk −2,n ,

(2.13)

uλk ,n dρn (ε2 ) = aλk +2,n uλk +2,n − aλk ,n uλk −2,n ,

(2.14)

vλk ,n dρn (ε3 ) = aλk +2,n uλk +2,n + aλk ,n uλk −2,n ,

(2.15)

uλk ,n dρn (ε3 ) = −aλk +2,n vλk +2,n − aλk ,n vλk −2,n .

(2.16)

Let n = 2n be an even number, we will introduce a real irreducible representation (V˜n , ρ˜n ) of SU (2),   which is equivalent to (1 + J )(Vn )R , ρn . Set V˜n := SpanR wλk ,n :=

√

  n−λk 2 vλk ,n + (−1) 2 u−λk ,n  k = 0, 1, . . . , n , 2

which is a (n + 1)-dimensional real subspace of (Vn )R . It is easy to see that V˜n is invariant under ρn , and we denote the restriction of ρn to V˜n by ρ˜n . In fact, by using the formula above (2.4), we have ρ˜n (g)(wλk ,n ) = n  Ξkl wλl ,n , where Ξkl = Λkl + (−1)k+l Λm−k m−l + i((−1)k Λm−k l − (−1)l Λk m−l ) is a real number from l=0

JID:MATPUR 6

AID:2817 /FLA

[m3L; v1.175; Prn:25/04/2016; 14:29] P.6 (1-24) C. Peng et al. / J. Math. Pures Appl. ••• (••••) •••–•••

¯ n−k n−l . On the other hand, let I be the identity element in Aut(Vn ), one can check the fact Λkl = (−1)k+l Λ √ −1 ρ˜n = Sρn S , where S = 22 (I + iρn (ε2 )) ∈ Aut(Vn ). Notice that ρn is complex irreducible and hence ρ˜n   is real irreducible, so we obtain that (V˜n , ρ˜n ) is equivalent to (1 + J )(Vn )R , ρn by the Theorem 6.3 in [2] again.  Since wλk ,n k = 0, 1, . . . , n} is an orthonormal basis of V˜n w.r.t. the inner product ( , ), we have orthogonal decomposition: V˜n =

n 

 WλRk ,n , WλRk ,n = SpanR wλk ,n .

(2.17)

k=0

By (2.5)–(2.8), the Lie algebra su(2) acts on V˜n can be described as wλk ,n d˜ ρn (ε1 ) = (−1)

n−λk 2

λk w−λk ,n ,

(2.18)

ρn (ε2 ) = aλk +2,n wλk +2,n − aλk ,n wλk −2,n , wλk ,n d˜ wλk ,n d˜ ρn (ε3 ) = (−1)

n−λk −2 2

aλk +2,n w−λk −2,n + (−1)

n−λk +2 2

(2.19)

aλk ,n w−λk +2,n .

(2.20)

Since any real representation (V, ρ) of SU (2) is completely reducible, up to an isomorphism, we can write

ρ=

s 

ρnα ⊕

α=1

t 

ρ˜mβ , V =

s 

(Vnα )R ⊕

α=1

β=1

t 

V˜mβ ,

(2.21)

β=1

where nα are odd numbers and mβ are even numbers. We always equip V with the natural inner product induced from Vnα and V˜mβ . In terms of the weights, we also have the orthogonal decomposition V =



R R (Vλ,n ⊕ Uλ,n )⊕ α α

λ,α



R Wμ,m . β

(2.22)

μ,β

Notice that dρ is a real representation of su(2), we can extend naturally dρ to be a complex representation of sl(2; C), which will also be denoted by dρ. Set  σ1 =



 1 0

0 −1

, σ2 =



 0 1 0 0

, σ3 =

 0 0 1 0

,

 then σ1 , σ2 , σ3 is a basis of sl(2; C). Denote the complexification of V by V C . The elements dρn (σ1 ), dρn (σ2 ), −dρn (σ3 ) ∈ End(V C ) will be denoted by H, A and B respectively in the sequel. Alternatively, they can also be viewed as H = −idρ(ε1 ), A =

dρ(ε2 ) + idρ(ε3 ) dρ(ε2 ) − idρ(ε3 ) , B=− . 2 2

(2.23)

Then the pull-back of the Maurer–Cartan forms of Aut(V C ) is dρρ−1 = H(iω) + Aϕ − B ϕ, ¯

(2.24)

where ω and ϕ are given in (2.3). Some properties of H, A and B will be presented in Proposition 3.6.

JID:MATPUR

AID:2817 /FLA

[m3L; v1.175; Prn:25/04/2016; 14:29] P.7 (1-24) C. Peng et al. / J. Math. Pures Appl. ••• (••••) •••–•••

7

3. Homogeneous two-spheres in Qn In this section we first prove that the image of any homogeneous immersion from S 2 into Qn is a two-dimensional ρ(SU (2))-orbit in Qn , ρ is a real representation of SU (2), and then we find the best base point in such orbit.  Let Qn = [z] ∈ CPn+1 | zzt = 0 be the complex hyperquadric. It is a complex algebraic submanifold in the complex projective space CPn+1 , and it is also isomorphic to the homogeneous space O(n + 2)/(SO(2) × O(n)). Let f : S 2 −→ Qn be a homogeneous isometric immersion, which means that f (S 2 ) is an orbit of a subgroup of the isometry group O(n +2) of Qn . Notice that f is isometric and homogeneous, so the isometry group of S 2 w.r.t. the induced metric is SO(3). Let x1 , x2 be two points in S 2 and yi := f (xi ) ∈ Qn , i = 1, 2. Since f is homogeneous, by Theorem 2 in [13], there exists a neighborhood V of y1 in f (S 2 ) and an isometry σ in O(n +2) such that σ(y1 ) = y2 and σ ·V ⊂ f (S 2 ). Therefore, there exist neighborhoods Ui of xi and Vi of yi such that V2 = σ · V1 and f |Ui : Ui −→ Vi are diffeomorphisms by the fact that f is an immersion. Notice that the induced metric on S 2 has constant curvature, we know that there exists gσ ∈ SO(3) such that gσ (x1 ) = x2 and f ◦gσ (x) = σ◦f (x) for all x ∈ U1 . It is clear that the set M := {x ∈ S 2 | f ◦gσ (x) = σ◦f (x)} is closed in S 2 . Suppose x0 ∈ M , then we take x0 , gσ (x0 ) instead of x1 , x2 and f (x0 ), σ·f (x0 ) instead of y1 , y2 respectively, by the same arguments, we obtain an open neighborhood U of x0 such that f ◦ gσ (x) = σ ◦ f (x) for all x ∈ U , i.e., M is also open in S 2 . Therefore, we have proved: Lemma 3.1. Let f : S 2 −→ Qn be a homogeneous immersion, then for any two points x1 , x2 ∈ S 2 there exists an isometry σ ∈ O(n + 2) of Qn and an isometry gσ ∈ SO(3) of S 2 such that gσ (x1 ) = x2 and f ◦ gσ = σ ◦ f . Let g be an element in SO(3) and f ◦ g = f , then g has the eigenvalue 1. So, g has a fixed point x0 ∈ S 2 . The differential maps of f and g at x0 satisfy f∗x0 ◦ g∗x0 = f∗x0 . Since f∗x0 is injective, it follows that g∗x0 = id, i.e., g = id. Therefore, we have Lemma 3.2. If g ∈ SO(3) such that f ◦ g = f , then g = id. Notice that f (S 2 ) is compact, the set of all σ in Lemma 3.1 is a compact Lie subgroup of O(n + 2), which acts transitively on f (S 2 ) and will be denoted by H. According to Lemma 3.1 and Lemma 3.2, we obtain a natural Lie group homomorphism π : H −→ SO(3), σ → gσ .

(3.1)

Let h be the Lie algebra of H. We obtain a Lie algebra homomorphism π∗ : h −→ so(3) from (3.1). Denote the kernel of π and π∗ by K, k respectively. So K is a normal Lie subgroup of H and its Lie algebra is k. We can equip an Ad H -invariant inner product  ,  on h since H is compact. Denote the orthogonal complement of k in h w.r.t.  ,  by k⊥ . Notice that k is an ideal of h, so k⊥ is also an ideal of h. Therefore, we obtain the following Lie algebra isomorphisms: π∗ (h) = π∗ (k⊥ ) ∼ = h/k ∼ = k⊥ .

(3.2)

By the fundamental theorem of Lie group, there exists a unique connected Lie subgroup G of H with its Lie algebra k⊥ . Since K keeps the points in f (S 2 ) fixed, we know that f (S 2 ) is also a G-orbit in Qn . So we obtain dim π∗ (k⊥ ) = dim k⊥ = dim G ≥ 2 by (3.2). By the known fact that there is no two-dimensional subalgebra of so(3), we have k⊥ ∼ = so(3). Therefore, we obtain a covering homomorphism π|G : G −→ SO(3), which implies

JID:MATPUR

AID:2817 /FLA

8

[m3L; v1.175; Prn:25/04/2016; 14:29] P.8 (1-24) C. Peng et al. / J. Math. Pures Appl. ••• (••••) •••–•••

∼ i = that G is isomorphic to SU (2) or SO(3). If G ∼ = SU (2), we obtain the homomorphism: ρ : SU (2) −→ G → ∼ i Ad = O(n +2). If G ∼ = SO(3), we also obtain a homomorphism: ρ : SU (2) −→ SO(3) −→ G → O(n +2), where Ad

is the adjoint representation of SU (2). Therefore, f is the ρ(SU (2))-orbit in Qn , i.e., f is SU (2)-equivariant. That is Theorem 3.3. Let f : S 2 −→ Qn be a homogeneous immersion, then there exists a real representation ρ of SU (2) such that f (S 2 ) is a two-dimensional ρ(SU (2))-orbit in Qn . Remark. The idea of the proof of this theorem is inspired from [16]. In general, a ρ(SU (2))-orbit in Qn is a principal orbit, i.e., it is a three-dimensional orbit. For the two-dimensional ρ(SU (2))-orbit, we have Lemma 3.4. Let M be a ρ(SU (2))-orbit in Qn , where ρ is a nontrivial representation of SU (2), then M is two-dimensional if and only if there exists a point [z0 ] ∈ M s.t. [z0 ] is invariant under the ρ(T )-action. Proof. Let [z] be a point in M . Since dim M = 2, the isotropy group H at [z] is a 1-dimensional subgroup of SU (2). Therefore, there exists a nonzero element ε ∈ su(2) s.t. TI2 H = SpanR {ε}. Notice that ε ∈ su(2), there exists a fixed g ∈ SU (2) s.t. g −1 εg = cε1 for some nonzero real number c. It follows that the isotropy group at [z0 ] = [zρ(g)] ∈ M contains the maximal torus subgroup T , i.e., [z0 ] is invariant under ρ(T )-action. The sufficiency follows from the fact that there is no two-dimensional subgroup in SU (2). This completes the proof. 2 Remarks. (1) The ρ(SU (2))-action on Qn is defined by SU (2) × Qn −→ Qn , (g, [z]) → [zρ(g)].

(3.3)

On the other hand, for a given two-dimensional ρ(SU (2))-orbit M in Qn , Lemma 3.4 implies that we can obtain an immersion f from S 2 into Qn as follows: f : S 2  SU (2)/T −→ Qn , [g] → [z0 ρ(g)],

(3.4)

 where [z0 ] is a fixed point in M and T := diag{eit , e−it } | t ∈ R is the maximal torus subgroup of SU (2). So, by Theorem 3.3, we only need to study the two-dimensional ρ(SU (2))-orbits in Qn if we want to study the homogeneous immersions from S 2 into Qn . (2) The point [z] ∈ M is invariant under the ρ(T )-action if and only if z dρ(ε1 ) = cz,

(3.5)

for some constant c ∈ C. For the sake of clarity, we first consider the case that the associated representation ρ is irreducible. Firstly, let the associated representation ρ = ρn , n is an odd, and [z] ∈ M be the point obtained in Lemma 3.4. So, by the decomposition (2.10), we can write z=

 (zλ vλ,n + zλ uλ,n ),

zλ , zλ ∈ C.

λ

By (2.11) and (2.12), in coordinates zλ and zλ , (3.5) is equivalent to

(3.6)

JID:MATPUR

AID:2817 /FLA

[m3L; v1.175; Prn:25/04/2016; 14:29] P.9 (1-24) C. Peng et al. / J. Math. Pures Appl. ••• (••••) •••–•••

λzλ − czλ = 0, czλ + λzλ = 0,

9

(3.7)

for all λ ∈ Δn . Since z = 0, there is a λ0 such that the pair (zλ0 , zλ 0 ) is nonzero. This implies c = ±iλ0 by (3.7), without loss of generality, we can assume c = −iλ0 . By using (3.7) again, we have z = zλ0 vλ0 ,n + izλ0 uλ0 ,n , zλ0 ∈ C.

(3.8)

Here the complex constant zλ0 can be reduced to a real number. Secondly, let the associated representation ρ = ρm , m is an even, and [z] ∈ M be the point obtained in Lemma 3.4. So, by the decomposition (2.17), we can write z=



zμ wμ,m ,

zμ ∈ C.

(3.9)

μ

By (2.18), in coordinates zμ , (3.5) is equivalent to czμ + (−1)

m−μ 2

 μz−μ = 0, (−1)

m−μ 2

 μzμ − cz−μ = 0, for μ = 0,

(3.10)

and czμ = 0, for μ = 0.

(3.11)

 Since z = 0, there is a μ0 such that the pair (zμ0 , z−μ ) is nonzero. This implies c = ±iμ0 by (3.10) and 0 (3.11), without loss of generality, we can assume c = −iμ0 when μ = 0. By using (3.10) again and zzt = 0, we have

z = zμ0 wμ0 ,m + i(−1)

m−μ0 2

zμ0 w−μ0 ,m , zμ0 ∈ C.

(3.12)

Here the complex constant zμ0 can be reduced to be a real number. s t   ρnα ⊕ ρ˜mβ , nα , mβ are odd and even numbers Generally, let the associated representation ρ = α=1

β=1

respectively, and [z] ∈ M be the point obtained in Lemma 3.4. So, by the decomposition (2.22), we can write z=



 (zλ,α vλ,α + zλ,α uλ,α ) +

λ,α



 zμ,β wμ,β ,

  zλ,α , zλ,α zμ,β ∈ C.

(3.13)

μ,β

By using (2.11), (2.12) and (2.18), the coordinates form of (3.5) is   λzλ,α − czλ,α = 0, czλ,α + λzλ,α = 0, for all λ, α,  + (−1) czμ,β

mβ −μ 2

 μz−μ,β = 0, (−1)

mβ −μ 2

  μzμ,β − cz−μ,β = 0, for all β and μ = 0,

(3.14) (3.15)

and  cz0,β = 0, for all β.

(3.16)

Notice that λ, μ are odd and even numbers respectively, by using the same arguments of the irreducible case, from (3.14)–(3.16) we obtain c = −iλ0 or −iμ0 for some fixed λ0 , μ0 . This tells us that the associated representation

JID:MATPUR

AID:2817 /FLA

[m3L; v1.175; Prn:25/04/2016; 14:29] P.10 (1-24) C. Peng et al. / J. Math. Pures Appl. ••• (••••) •••–•••

10

ρ=

s 

ρnα or

t 

α=1

ρ˜mβ ,

(3.17)

β=1

and hence z can be written as z=



(zλ0 ,α vλ0 ,α + izλ0 ,α uλ0 ,α ), zλ0 ,α ∈ C,

(3.18)

α

or z=



zμ0 ,β wμ0 ,β + i(−1)

mβ −μ0 2

zμ0 ,β w−μ0 ,β , zμ0 ,β ∈ C,

(3.19)

β

or z=



  z0,β w0,β , z0,β ∈ C.

(3.20)

β

In summary, we have Theorem 3.5. Let M be a two-dimensional ρ(SU (2))-orbit in Qn , then one of the following holds: s 

ρnα , nα are odd numbers, and there exists a point [z1 ] ∈ M and a fixed α=1  weight λ such that z1 = (zλ,α vλ,α + izλ,α uλ,α ), zλ,α ∈ C are nonzero;

(I) The representation ρ =

α

(II) The representation ρ =

t 

ρ˜mβ , mβ are even numbers, and there exists a point [z2 ] ∈ M and a fixed

β=1

weight μ = 0 such that z2 =

 β

 zμ,β wμ,β + i(−1)

mβ −μ 2

  zμ,β w−μ,β , zμ,β ∈ C are nonzero;

t  (III) The representation ρ = ρ˜mβ , mβ are even numbers, and there is a point [z3 ] ∈ M such that     β=1  z3 = z0,β w0,β , (z0,β )2 = 0 and z0,β ∈ C are nonzero. β

β

Remark. Based on this theorem, the two-dimensional ρ(SU (2))-orbits in Qn will be divided into three types, which correspond to the (I), (II) and (III) in Theorem 3.5. To end this section, we give some properties of the operators H, A, B (introduced in the end of Section 2), that is Proposition 3.6. Let zi be the points obtained in Theorem 3.5, then we have

z1 A = −



z1 H = −λz1 ,   aλ,α zλ,α vλ−2,α + izλ,α uλ−2,α ,

(3.21) (3.22)

α

z1 B =



  aλ+2,α zλ,α vλ+2,α + izλ,α uλ+2,α ,

(3.23)

α

z2 A = −

 β

z2 H = −μz2 , mβ −μ+2     aμ,β zμ,β wμ−2,β + (−1) 2 izμ,β w−μ+2,β ,

(3.24) (3.25)

JID:MATPUR

AID:2817 /FLA

[m3L; v1.175; Prn:25/04/2016; 14:29] P.11 (1-24) C. Peng et al. / J. Math. Pures Appl. ••• (••••) •••–•••



z2 B =

11

mβ −μ−2     aμ+2,β zμ,β wμ+2,β + (−1) 2 izμ,β w−μ−2,β ,

(3.26)

β

z3 H = 0, z3 A =



a0,β

β

z3 B =



 1 + (−1) 2

a2,β

β

(3.27)   z0,β w−2,β ,

(3.28)

 1 − (−1)mβ /2 i   1 + (−1)mβ /2 i  z0,β w2,β − z0,β w−2,β . 2 2

(3.29)

mβ /2

i

 z0,β w2,β −

1 − (−1) 2

mβ /2

i

Proof. We only prove the identity (3.22), the others are similar. By (2.13) and (2.14), we have z1 dρ(ε2 ) =



zλ,α (aλ+2,α vλ+2,α − aλ,α vλ−2,α )

α

+ izλ,α (aλ+2,α uλ+2,α − aλ,α uλ−2,α )    aλ,α zλ,α vλ−2,α + izλ,α uλ−2,α =− α

+



  aλ+2,α zλ,α vλ+2,α + izλ,α uλ+2,α .

(3.30)

α

Similarly, by (2.15) and (2.16), we have z1 dρ(ε3 ) =

 α

+

  aλ,α zλ,α uλ−2,α − izλ,α vλ−2,α



  aλ+2,α zλ,α uλ+2,α − izλ,α vλ+2,α .

(3.31)

α

Substituting (3.30) and (3.31) into the second identity of (2.23), we obtain z1 A = −



  aλ,α zλ,α vλ−2,α + izλ,α uλ−2,α .

α

This completes the proof. 2 Remark. The operator H acting on zi is just a dilation. The operator A sends the vector z1 (resp. z2 , z3 )  R C  R C  R R R to (Vλ−2,α ⊕ Uλ−2,α ) (resp. (Wμ−2,α ⊕ W−(μ−2),α ) ), the complexification space of (Vλ−2,α ⊕ α α β  R  R R R Uλ−2,α ) (resp. (Wμ−2,α ⊕ W−(μ−2),α )). The operator B sends the vector z1 (resp. z2 , z3 ) to (Vλ+2,α ⊕ α

β

C  R C  R R R R ) (resp. (Wμ+2,α ⊕ W−(μ+2),α ) ), the complexification space of (Vλ+2,α ⊕ Uλ+2,α ) (resp. Uλ+2,α α β  R R (Wμ+2,α ⊕ W−(μ+2),α )). β

4. Geometry of the two-dimensional ρ(SU (2))-orbits in Qn In this section we study the geometry of the two-dimensional ρ(SU (2))-orbits in Qn , where ρ is a real representation of SU (2). Let (V, ρ) be a real representation of SU (2), and V C be the complexification space of V . We can equip V C with a natural Hermitian inner product  ,  induced from the inner product ( , ) on V . For example,  let V = (Vn )R , an element z ∈ (Vn )C (zλ vλ,n + zλ uλ,n ), zλ , zλ ∈ C. Then the R can be written as z = Hermitian inner product  ,  on (Vn )C R as described by

λ

JID:MATPUR 12

AID:2817 /FLA

[m3L; v1.175; Prn:25/04/2016; 14:29] P.12 (1-24) C. Peng et al. / J. Math. Pures Appl. ••• (••••) •••–•••

z, w =

 (zλ w ¯λ + zλ w ¯λ ), λ

for z, w ∈ (Vn )C R.  Let [z] be the point obtained in Theorem 3.5, and the orbit M := [zρ(g)]  g ∈ SU (2)}. One can further assume that z is unitary, i.e., z¯ zt = |z|2 = 1. So, we obtain a field of unitary frame: Z = z ρ.

(4.1)

Taking the exterior derivative of (4.1), by using (2.24), we have dZ = z(H(iω) + Aϕ − B ϕ)ρ ¯ = z(−λ(iω) + Aϕ − B ϕ)ρ. ¯

(4.2)

Here λ also stands for μ when z taking z2 or z3 . Set X = zAρ, Y = −zBρ,

(4.3)

by using Proposition 3.6 and the orthogonality of weight spaces, we have dZ = Xϕ + Y ϕ¯

mod Z.

(4.4)

Here, X and Y are Cn+2 -valued functions defined on S 2 , which satisfy X, Y  = 0

(4.5)

by Proposition 3.6. The identity (4.5) tells us that the orbit is conformal in Qn . Therefore, we obtain the induced metric ds2 = (|X|2 + |Y |2 )ϕϕ. ¯

(4.6)

Notice that |X|2 + |Y |2 = |zA|2 + |zB|2 is a constant and the metric ϕϕ¯ has the curvature 4 from the structure equation (2.3), we know that the metric ds2 has constant curvature K=

4 . |X|2 + |Y |2

(4.7)

The Kähler angle θ (cf. [6]) is given by cos θ =

|X|2 − |Y |2 , |X|2 + |Y |2

(4.8)

which gives a measure of the failure of the orbit to be a holomorphic one in Qn. Explicitly, here, the orbit is holomorphic (totally real, antiholomorphic) if |Y |2 = 0 (|X|2 = |Y |2 , |X|2 = 0). The function |X, Y¯ |2 is also an important invariant on the orbit M , which measures the minimality of ι(M ) ⊂ CPn+1 , ι : Qn → CPn+1 is the inclusion mapping. More explicitly, ι(M ) is minimal (see Theorem 3.1 in [29]) in CPn+1 if and only if |X, Y¯ |2 = 0.

(4.9)

JID:MATPUR

AID:2817 /FLA

[m3L; v1.175; Prn:25/04/2016; 14:29] P.13 (1-24) C. Peng et al. / J. Math. Pures Appl. ••• (••••) •••–•••

13

From the structure equation (2.3), we know the connection 1-form of the metric ds2 is −2(iω). Set  = dZ, Z.

(4.10)

Then, similar as S.S. Chern and J. Wolfson did in [7], we define the covariant differentials of X and Y by DX = dX − X − 2(iω)X,

(4.11)

DY = dY − Y + 2(iω)Y.

(4.12)

Set DX ≡ pϕ + qϕ¯

¯ mod Z, Z,

(4.13)

DY ≡ qϕ + rϕ¯

¯ mod Z, Z.

(4.14)

Here, p, q, r are local Cn+1 -valued functions. Then, the quadratic differential form pϕ2 + 2qϕϕ¯ + rϕ¯2 is the “second fundamental form” of the orbit in Qn . By using these covariant differentials, we can give a criterion to measure the minimality of the orbit M in Qn , that is: Proposition 4.1. The orbit M is minimal in Qn if and only if one of the following holds: (a) DX ≡ 0,

¯ ϕ, mod Z, Z,

(b) DY ≡ 0,

¯ ϕ. mod Z, Z, ¯

Proof. It is very similar to the proof of Theorem 2.1 in [7], we omit the details. This completes the proof. 2 We will give an explicit expression of the criterion (a) and (b) in Proposition 4.1. By using the Proposition 3.6 and identity (4.2), one has  = dZ, Z = −λ(iω). So, by (4.3) and (4.11), we have DX = dX − X − 2(iω)X   = zA H(iω) + Aϕ − B ϕ¯ ρ + λ(iω)X − 2(iω)X   = −(λ − 2)(iω)X + z A2 ϕ − AB ϕ¯ ρ + λ(iω)X − 2(iω)X   = z A2 ϕ − AB ϕ¯ ρ.

(4.15)

Therefore, by Proposition 4.1, modulo ρ, the orbit M is minimal if and only if zAB ≡ 0,

¯, mod z, z

(4.16)

or equivalently, zAB = pz + q¯ z,

(4.17)

for some constants p, q ∈ C. Similarly, we have   DY = z − BAϕ + B 2 ϕ¯ ρ.

(4.18)

JID:MATPUR

AID:2817 /FLA

[m3L; v1.175; Prn:25/04/2016; 14:29] P.14 (1-24) C. Peng et al. / J. Math. Pures Appl. ••• (••••) •••–•••

14

The equivalence between (a) and (b) follows from the fact that [A, B] = H. By Proposition 4.1, if the orbit is minimal, then the square of the length of its second fundamental form can be expressed as S=

4(|zA2 |2 + |zB 2 |2 ) . |X|2 + |Y |2

(4.19)

5. Classification of minimal homogeneous two-spheres in Qn In this section we will determine all minimal homogeneous two-spheres in Qn , and we also calculate their geometric quantities: the Gauss curvature K, the cosine of Kähler angle θ and the length of their second fundamental form S. Throughout this section and below, we will agree on the following conventions: φλ,α := uλ,nα ρnα (g),

φλ,α := vλ,nα ρnα (g),

φ˜μ,β := wμ,mβ ρ˜mβ (g), 

where g ∈ SU (2). More explicitly, let λ = λk , nα = n, μ = μk and mβ = m, in terms of g =

 a b −¯b a ¯

∈

SU (2) (or a and b), we have   ¯ k0 , . . . , Λkn + Λ ¯ kn , −i(Λk0 − Λ ¯ k0 ), . . . , −i(Λkn − Λ ¯ kn ) /2, φλk ,n = Λk0 + Λ   ¯ k0 ), . . . , i(Λkn − Λ ¯ kn ), . . . , Λk0 + Λ ¯ k0 , . . . , Λkn + Λ ¯ kn , /2, φλk ,n = i(Λk0 − Λ and   φ˜μk ,m = Ξk0 , . . . , Ξkl , . . . , Ξkm /2, where Λkl is given by (2.4) and Ξkl = Λkl + (−1)k+l Λm−k m−l + i((−1)k Λm−k l − (−1)l Λk m−l ). Based on the discussions in the last paragraph in Subsection 2.1, we find that φλ,α, φλ,α and φ˜μ,β are make up of the Veronese surfaces through suitable combinations. 5.1. Type (I) s  In this subsection, we consider the first case in Theorem 3.5, that is ρ = ρnα , nα are odd numbers, α=1  z = (zλ,α vλ,α + izλ,α uλ,α ), zλ,α ∈ C are nonzero constants. We will always agree on that z is unitary, α

i.e., |z|2 = z¯ zt = 1. By using Proposition 3.6 repeatedly, we have zAB = −



aλ,α (zλ,α vλ−2,α + izλ,α uλ−2,α )B

α

=−



a2λ,α (zλ,α vλ,α + izλ,α uλ,α ).

(5.1)

α

So, by the Proposition 4.1 and (4.17), the orbit M through the base point [z] is minimal if and only if there are constants p, q ∈ C s.t. a2λ,α zλ,α = pzλ,α + q¯ zλ,α , which imply

a2λ,α zλ,α = pzλ,α − q¯ zλ,α , for all α,

(5.2)

JID:MATPUR

AID:2817 /FLA

[m3L; v1.175; Prn:25/04/2016; 14:29] P.15 (1-24) C. Peng et al. / J. Math. Pures Appl. ••• (••••) •••–•••

15

a2λ,α = p, for all α.

(5.3)

Notice that the weight λ is fixed, so by the expression of aλ,α in (2.9), we obtain n1 = n2 = · · · = ns := n0 .

(5.4)

Therefore, in terms of the homogeneous coordinate, the orbit M determine a minimal immersion from S 2 into Qn by [a, b] → [z1 (φλ,n0 + iφλ,n0 ), . . . , zs (φλ,n0 + iφλ,n0 )].

(5.5)

Proposition 5.1. The immersion defined by (5.5) is congruent to √ [a, b] →

2 [cos τ (φλ,n0 + iφλ,n0 ), i sin τ (φλ,n0 + iφλ,n0 ), 0, . . . , 0], 2

(5.6)

up to a rigidity of Qn , where the parameter τ ∈ [0, π/2].  Proof. We write zα := xα + iyα , where xα , yα ∈ R, z := (z1 , . . . , zs ) and A :=

x1 · · · xs y1 · · · ys

 . Clearly,

|z|2 = 12 . Notice that z is determined up to a transformation of ξ → eiξ , and such transformation induces a transformation of (xα , yα ) → (cos ξ xα − sin ξ yα , sin ξ xα + cos ξ yα ). By the knowledge of linear algebra, there exists a ξ, and O2 ∈ O(s) s.t. √  2 cos τ O1 AO2 , or I1,−1 O1 AO2 = 2 0

0 sin τ

0 ··· 0 0 ··· 0

 ,



   cos ξ − sin ξ 1 0 where O1 = , I1,−1 = , and the parameter τ ∈ [0, π/2]. So, the immersion (5.5) sin ξ cos ξ 0 −1 is congruent to (5.6) by choosing the isometry O2 ⊗ I2(n0 +1) ∈ O(n + 2), where I2(n0 +1) is the identity matrix of order 2(n0 + 1) and ⊗ is the tensor product of two matrices. This completes the proof. 2 Next, we calculate the geometry quantities of immersion (5.6) in the following. By (3.22), (3.23) and (4.3), we obtain |X|2 + |Y |2 = |zA|2 + |zB|2 = a2λ,n0 + a2λ+2,n0 . So, by (4.7) and (4.8) respectively, the Gauss curvature is K=

a2λ,n0

4 , + a2λ+2,n0

(5.7)

and the Kähler angle is cos θ =

a2λ,n0 − a2λ+2,n0 . a2λ,n0 + a2λ+2,n0

(5.8)

By using (3.22) and (3.23) twice, we obtain the square of the length of second fundamental form S=

4(a2λ−2,n0 a2λ,n0 + a2λ+2,n0 a2λ+4,n0 ) . a2λ,n0 + a2λ+2,n0

(5.9)

JID:MATPUR

AID:2817 /FLA

[m3L; v1.175; Prn:25/04/2016; 14:29] P.16 (1-24) C. Peng et al. / J. Math. Pures Appl. ••• (••••) •••–•••

16

5.2. Type (II) In this subsection, we consider the second case in Theorem 3.5, that is ρ = z=

 β

t 

ρ˜mβ , mβ are even numbers,

β=1  zμ,β wμ,β + i(−1)

mβ −μ 2

  zμ,β w−μ,β , μ = 0 and zμ,β ∈ C are nonzero constants.

By using (3.25) and (3.26) in Proposition 3.6 repeatedly, we have zAB = −



 aμ,β (zμ,β wμ−2,β + (−1)

mβ −μ 2

 izμ,β w−μ+2,β )B

β

=−



 a2μ,β (zμ,β wμ,α + (−1)

mβ −μ 2

 izμ,β w−μ,β ).

(5.10)

β

So, by the Proposition 4.1 and (4.17), the orbit M through the base point [z] is minimal if and only if    a2μ,β zμ,β = pzμ,β + q¯ zμ,β ,

   a2μ,β zμ,β = pzμ,β − q¯ zμ,β , for all β,

(5.11)

which imply a2μ,β = p, for all β.

(5.12)

Notice that the weight μ is fixed, so by the expression of aμ,β in (2.9), we obtain m1 = m2 = · · · = ms := m0 .

(5.13)

Therefore, in terms of the homogeneous coordinate, the orbit M determines a minimal immersion from S 2 into Qn by [a, b] → [z1 (φ˜μ,m0 + (−1)

m0 −μ 2

iφ˜−μ,m0 ), . . . , zt (φ˜μ,m0 + (−1)

m0 −μ 2

iφ˜−μ,m0 )].

(5.14)

Proposition 5.2. The immersion defined by (5.14) is congruent to √ [a, b] →

m0 −μ m0 −μ 2 [cos τ (φ˜μ,m0 + (−1) 2 iφ˜−μ,m0 ), i sin τ (φ˜μ,m0 + (−1) 2 iφ˜−μ,m0 ), 0, . . . , 0], 2

(5.15)

up to a rigidity of Qn , where the parameter τ ∈ [0, π/2]. Proof. The proof is the same as the one of Proposition 5.1, we omit the details. This completes the proof. 2 Next, we calculate the geometry quantities of immersion (5.15). By (3.25), (3.26) and (4.3), we obtain |X|2 + |Y |2 = |zA|2 + |zB|2 = a2λ,n0 + a2λ+2,n0 . So, by (4.7) and (4.8) respectively, the Gauss curvature is K=

4 , a2μ,m0 + a2μ+2,m0

(5.16)

and the Kähler angle is cos θ =

a2μ,m0 − a2μ+2,m0 . a2μ,m0 + a2μ+2,m0

(5.17)

JID:MATPUR

AID:2817 /FLA

[m3L; v1.175; Prn:25/04/2016; 14:29] P.17 (1-24) C. Peng et al. / J. Math. Pures Appl. ••• (••••) •••–•••

17

By using (3.25) and (3.26) twice, we obtain the square of the length of second fundamental form

S=

4(a2μ−2,m0 a2μ,m0 + a2μ+2,m0 a2μ+4,m0 ) . a2μ,m0 + a2μ+2,m0

(5.18)

5.3. Type (III) t  In this subsection, we consider the third case in Theorem 3.5, that is ρ = ρ˜mβ , mβ are even numbers, β=1    z = z0,β w0,β , and z0,β ∈ C are nonzero constants. β

By using (3.28) first and later (3.26), together with the facts a0,β = a2,β and a4,β = a−2,β , we have zAB =

 β

=−

a0,β



1 + (−1)mβ /2 i   (z0,β w2,β + (−1)mβ /2 iz0,β w−2,β )B 2

 a20,β z0,β w0,β .

(5.19)

β

So, by the Proposition 4.1 and (4.17), the orbit M through the base point [z] is minimal if and only if    a20,β z0,β = pz0,β + q¯ z0,β , for all β.

(5.20)

This implies that a20,β satisfies the equation x2 − (p + p¯)x + (|p|2 − |q|2 ) = 0.

(5.21)

Therefore, we have two distinct a0,β at the most, or equivalently, we have two distinct mβ at the most. Then the associated representation takes the form of ρ = ρ˜m0 ⊕ · · · ⊕ ρ˜m0 or ρ˜m1 ⊕ · · · ⊕ ρ˜m1 ⊕ ρ˜m2 ⊕ · · · ⊕ ρ˜m2 ,          t

t1

t2

where m0 , m1 , m2 are even numbers and m1 > m2 . (III.I) If the associated representation ρ = ρ˜m0 ⊕ · · · ⊕ ρ˜m0 , in terms of homogeneous coordinates, the    t

orbit M determines a minimal immersion from S 2 into Qn by [a, b] → [z1 φ˜0,m0 , . . . , zt φ˜0,m0 ].

(5.22)

Notice that z = (z1 , . . . , zt ) satisfies zz t = 0, by the same arguments in Proposition 5.1, the immersion defined by (5.22) is congruent to √ [a, b] →

2 ˜ [φ0,m0 , iφ˜0,m0 , 0, . . . , 0]. 2

By using (3.28), (3.29) and (4.3), we obtain |X|2 = a20,m0 , |Y |2 = a22,m0 .

(5.23)

JID:MATPUR

AID:2817 /FLA

[m3L; v1.175; Prn:25/04/2016; 14:29] P.18 (1-24) C. Peng et al. / J. Math. Pures Appl. ••• (••••) •••–•••

18

So, by (4.7) and (4.8), we obtain the Gauss curvature K=

a20,m0

4 , + a22,m0

(5.24)

and the Kähler angle is cos θ = 0,

(5.25)

by using the fact a0,m0 = a2,m0 . Through direct calculations, from (4.19), we obtain the square of the length of the second fundamental form S=

4(a20,m0 a2−2,m0 + a22,m0 a24,m0 ) . a20,m0 + a22,m0

(5.26)

(III.II) If the associated representation ρ = ρ˜m1 ⊕ · · · ⊕ ρ˜m1 ⊕ ρ˜m2 ⊕ · · · ⊕ ρ˜m2 , m1 > m2 , in terms of       t1

t2

homogeneous coordinates, the orbit M determines a minimal immersion from S 2 into Qn by [a, b] → [z1 φ˜0,m1 , . . . , zt1 φ˜0,m1 , z1 φ˜0,m2 , . . . , zt2 φ˜0,m2 ].

(5.27)

Proposition 5.3. The immersion defined by (5.27) is congruent to √ [a, b] →

2 ˜ [φ0,m1 , 0, . . . , 0, iφ˜0,m2 , 0, . . . , 0]. 2

(5.28)

Proof. We write z = (z1 , · · · , zt1 ), z  = (z1 , · · · , zt2 ). By the same arguments in Proposition 5.1, we know there exists a ξ, O1 ∈ O(t1 ), O2 ∈ O(t2 ) s.t. eiξ zO1 = (x1 , iy2 , 0, · · · , 0), eiξ z  O2 = (x1 + iy1 , iy2 , 0, · · · , 0), where x1 , y2 , x1 , y1 and y2 ∈ R. By using the fact zzt = 0, we have x1 y1 = 0, (x1 )2 + (x1 )2 = (y2 )2 + (y1 )2 + (y2 )2 =

1 . 2

If x1 = 0 and y1 = 0, then the immersion defined by (5.27) is congruent to [a, b] → [x1 φ˜0,m1 , iy2 φ˜0,m1 , 0, . . . , 0, x1 φ˜0,m2 , iy2 φ˜0,m2 , 0, . . . , 0],

(5.29)

by choosing the isometry diag{O1 ⊗ Im1 +1 , O2 ⊗ Im2 +1 } of Qn . If x1 = 0 and y1 = 0, by choosing suitable O2 s.t. eiξ z  O2 = (0, iy2 , 0, · · · , 0), then the immersion (5.27) is congruent to [a, b] → [x1 φ˜0,m1 , iy2 φ˜0,m1 , 0, . . . , 0, 0, iy2 φ˜0,m2 , 0, . . . , 0].

(5.30)

Clearly, the immersion (5.27) is also congruent to (5.30) if x1 = y1 = 0. Anyway, we show that the immersion (5.27) is congruent to the form of (5.29). Without loss of generality, we can assume that one of x1 , y1 (resp. x1 , y1 ) is not equal to zero at least.

JID:MATPUR

AID:2817 /FLA

[m3L; v1.175; Prn:25/04/2016; 14:29] P.19 (1-24) C. Peng et al. / J. Math. Pures Appl. ••• (••••) •••–•••

19

Notice that the base point of (5.27) can be written as z = (x1 , iy2 , 0, . . . , 0, x1 , iy2 , 0, . . . , 0). Then, by using the minimality condition (4.17), we have −a20,m1 x1 = px1 + qx1 ,

−a20,m1 y2 = py2 − qy2 ,

(5.31)

−a20,m2 x1 = px1 + qx1 ,

−a20,m2 y2 = py2 − qy2 ,

(5.32)

for some constants p, q. If x1 y2 = 0 or x1 y2 = 0, then (5.31) or (5.32) implies that q = 0, and hence a20,m1 = a20,m2 = −p, i.e., m1 = m2 . It is a contradiction with m1 > m2 . So, x1 y2 = x1 y2 = 0. By using the facts |z|2 = 1 and zzt = 0, without loss of generality, the base point can be written as √ z=

2 (1, 0, . . . , 0, i, 0, . . . , 0). 2

(5.33)

Therefore, the immersion (5.27) is congruent to (5.28). This completes the proof. 2 Notice that the base point of immersion (5.28) is given by (5.33), by using (3.28), (3.29) and (4.3), we obtain |X|2 =

a20,m1 + a20,m2 , 2

|Y |2 =

a22,m1 + a22,m2 . 2

(5.34)

So, by (4.7) and (4.8), we obtain the Gauss curvature K=

a20,m1

4 , + a20,m2

(5.35)

and the Kähler angle is cos θ = 0,

(5.36)

by using the fact a0,m = a2,m . Through direct calculations, from (4.19), we obtain the square of the length of the second fundamental form S=

4(a20,m1 a2−2,m1 + a20,m2 a2−2,m2 ) , a20,m1 + a22,m2

(5.37)

by using a0,m = a2,m and a4,m = a−2,m . In summary, we have Theorem 5.4. Let f : S 2 −→ Qn be a linearly full minimal homogeneous immersion, then one of the followings holds: (I) The representation associated to f is ρ = ρn0 ⊕ ρn0 , n0 is an odd, 4n0 + 2 = n, and f is congruent to √ [a, b] →

2 [cos τ (φλ,n0 + iφλ,n0 ), i sin τ (φλ,n0 + iφλ,n0 )], 2

where τ ∈ [0, π/2] and λ ∈ Δn0 . Moreover, the Gauss curvature K= the Kähler angle

8 , n20 + 2n0 − λ2

JID:MATPUR

AID:2817 /FLA

[m3L; v1.175; Prn:25/04/2016; 14:29] P.20 (1-24) C. Peng et al. / J. Math. Pures Appl. ••• (••••) •••–•••

20

cos θ =

2λ , n20 + 2n0 − λ2

and the square of the length of the second fundamental form

S=

(n0 + 1)4 − 2(λ2 + 5)(n0 + 1)2 + λ4 + 22λ2 + 9 . n20 + 2n0 − λ2

(II) The representation associated to f is ρ˜ = ρ˜m0 ⊕ ρ˜m0 , m0 is an even, 2m0 = n, and f is congruent to √ [a, b] →

m0 −μ m0 −μ 2 [cos τ (φ˜μ,m0 + (−1) 2 iφ˜−μ,m0 ), i sin τ (φ˜μ,m0 + (−1) 2 iφ˜−μ,m0 )], 2

where τ ∈ [0, π/2] and μ ∈ Δm0 − {0}. Moreover, the Gauss curvature K=

8 , m20 + 2m0 − μ2

the Kähler angle cos θ =

m20

2μ , + 2m0 − μ2

and the square of the length of the second fundamental form

S=

(m0 + 1)4 − 2(μ2 + 5)(m0 + 1)2 + μ4 + 22μ2 + 9 . m20 + 2m0 − μ2

(III.I) The representation associated to f is ρ˜ = ρ˜m ⊕ ρ˜m , m is an even number, 2m = n, and f is congruent to √ [a, b] →

2 ˜ [φ0,m , iφ˜0,m ]. 2

Moreover, the Gauss curvature K=

8 , m2 + 2m

the Kähler angle cos θ = 0, and the square of the length of the second fundamental form

S=

(m + 1)4 − 10(m + 1)2 + 9 . m2 + 2m

JID:MATPUR

AID:2817 /FLA

[m3L; v1.175; Prn:25/04/2016; 14:29] P.21 (1-24) C. Peng et al. / J. Math. Pures Appl. ••• (••••) •••–•••

21

(III.II) The representation associated to f is ρ˜ = ρ˜m1 ⊕ ρ˜m2 , m1 , m2 are an even numbers, m1 > m2 , m1 + m2 = n, and f is congruent to √ [a, b] →

2 ˜ [φ0,m1 , iφ˜0,m2 ]. 2

Moreover, the Gauss curvature K=

16 , m21 + m22 + 2(m1 + m2 )

the Kähler angle cos θ = 0, and the square of the length of the second fundamental form S=

(m1 + 1)4 + (m2 + 1)4 − 10(m1 + 1)2 − 10(m2 + 1)2 + 18 . m21 + m22 + 2(m1 + m2 )

Remarks. (1) When the parameter τ takes the value 0 or π/2 in the Type (I) orbit, we agree on the associated representation ρ = ρn0 is irreducible, 2n0 = n, and f is congruent to √ [a, b] →

2 [φλ,n0 + iφλ,n0 ]. 2

Similar conventions will be agreed on for the Type (II) and (III) orbits. (2) When the weight λ = n0 (or −n0 ), μ = m0 (or −m0 ) in the Type (I) and (II) orbits respectively, then f is holomorphic or antiholomorphic. The Type (III) orbits always totally real in Qn . (3) There is no linearly full totally real minimal homogeneous two-spheres in Q2n+1 . (4) By using the identity (4.9), one can check that the Type (I) and (II) orbits are also minimal in the complex projective space CPn+1 , and the Type (III) orbits are not minimal in CPn+1 . (5) The first fundamental form and Kähler angle can’t determine the minimal two-spheres in Qn completely. According to this theorem, one can find that families of minimal two-spheres in Qn share the common curvature and Kähler angle. Subsequently, we have Theorem 5.5. Let f : S 2 −→ Qn be a linearly full totally geodesic immersion, then one of the following holds: √ √ (i) n = 2 and f is congruent to [a, b] → 22 [φ1,1 + iφ1,1 ] or [a, b] → 22 [φ−1,1 + iφ−1,1 ] or [a, b] → √ 2 ˜ 2 [φ0,2 , i]; √ (ii) n = 4 and f is congruent to [a, b] → 2 [φ˜0,2 , iφ˜0,2 ]. 2

Proof. By the well known fact that the totally geodesic submanifolds in a homogeneous space are also homogeneous, see [23] for details. This completes the proof. 2 Theorem 5.6. Let f : S 2 −→ Qn be a linearly full holomorphic or antiholomorphic immersion with constant curvature, then one of the followings holds: (i) n = 2n0 for an odd number n0 , f is congruent to

JID:MATPUR

AID:2817 /FLA

[m3L; v1.175; Prn:25/04/2016; 14:29] P.22 (1-24) C. Peng et al. / J. Math. Pures Appl. ••• (••••) •••–•••

22

√

√ 2 2  [φn0 ,n0 + iφn0 ,n0 ], or [φ−n0 ,n0 + iφ−n0 ,n0 ], [a, b] → 2 2 and its curvature K = 4/n0 ; (ii) n = 4n0 + 2 for an odd number n0 , f is congruent to √

[a, b] →

2 [cos τ (φn0 ,n0 + iφn0 ,n0 ), i sin τ (φn0 ,n0 + iφn0 ,n0 )], 2 √ 2 [cos τ (φ−n0 ,n0 + iφ−n0 ,n0 ), i sin τ (φ−n0 ,n0 + iφ−n0 ,n0 )], τ ∈ (0, π/2), or 2

and its curvature K = 4/n0 ; (iii) n = m0 − 1 for an even number m0 , f is congruent to √

√ 2 ˜ 2 ˜ ˜ [φm0 ,m0 + iφ−m0 ,m0 ], or [φ−m0 ,m0 + iφ˜m0 ,m0 ], [a, b] → 2 2 and its curvature K = 4/m0 ; (iv) n = 2m0 for an even number m0 , f is congruent to √

[a, b] →

2 [cos τ (φ˜m0 ,m0 + iφ˜−m0 ,m0 ), i sin τ (φ˜m0 ,m0 + iφ˜−m0 ,m0 )], 2 √ 2 [cos τ (φ˜−m0 ,m0 + iφ˜m0 ,m0 ), i sin τ (φ˜−m0 ,m0 + iφ˜m0 ,m0 )], τ ∈ (0, π/2), or 2

and its curvature K = 4/m0 . Remark. This gives a complete classification of the constant curved holomorphic and antiholomorphic twospheres in the complex hyperquadric. Proof. Since Qn is a complex submanifold in CPn+1 , it follows that any holomorphic and antiholomorphic submanifolds in Qn are also holomorphic and antiholomorphic in CPn+1 . So, by the known results of J. Bolton, G.R. Jensen, M. Rigoli, L.M. Woodward [3] and S. Bando and Y. Ohnita [4], f (S 2 ) is homogeneous in CPn+1 and hence is homogeneous in Qn . This completes the proof. 2 To end this section, we present four tables to list all linearly full minimal homogeneous two-spheres in Q2 , Q3 , Q4 and Q5 respectively. Table 1 Minimal homogeneous two-spheres in Q2 . Representations

Orbits

K

cos θ

S

ρ1

[φ1,1 + iφ1,1 ] [φ−1,1 + iφ−1,1 ] ˜0,2 , i] [φ

4 4 2

1 −1 0

0 0 0

ρ˜2 ⊕ ρ˜0

Table 2 Minimal homogeneous two-spheres in Q3 . Representations ρ˜4

Orbits ˜4,4 + iφ ˜−4,4 ] [φ ˜−2,4 ] ˜2,4 + iφ [φ ˜−2,4 + iφ ˜2,4 ] [φ ˜−4,4 + iφ ˜4,4 ] [φ

K

cos θ

S

1 2/5 2/5 1

1 1/5 −1/5 −1

24 72/5 72/5 24

JID:MATPUR

AID:2817 /FLA

[m3L; v1.175; Prn:25/04/2016; 14:29] P.23 (1-24) C. Peng et al. / J. Math. Pures Appl. ••• (••••) •••–•••

23

Table 3 Minimal homogeneous two-spheres in Q4 . Representations

Orbits

K

cos θ

S

ρ˜2 ⊕ ρ˜2

˜2,2 + iφ ˜−2,2 ), i sin τ (φ ˜2,2 + iφ ˜−2,2 )] [cos τ (φ ˜0,2 , iφ ˜0,2 ] [φ ˜−2,2 + iφ ˜2,2 ), i sin τ (φ ˜−2,2 + iφ ˜2,2 )] [cos τ (φ ˜0,4 , i] [φ

2 1 2 2/3

1 0 −1 0

8 0 8 16

ρ˜4 ⊕ ρ˜0

Table 4 Minimal homogeneous two-spheres in Q5 . Representations ρ˜6

Orbits ˜6,6 + iφ ˜−6,6 ] [φ ˜4,6 + iφ ˜−4,6 ] [φ ˜2,6 + iφ ˜−2,6 ] [φ ˜−2,6 + iφ ˜2,6 ] [φ ˜−4,6 + iφ ˜4,6 ] [φ ˜6,6 ] ˜−6,6 + iφ [φ

K

cos θ

S

2/3 1/4 2/11 2/11 1/4 2/3

1 1/4 1/11 −1/11 −1/4 −1

40 30 408/11 408/11 30 40

Remark. The parameter τ in Table 3 belongs to (0, π/2). 6. Further discussions and problems In this section we propose some problems of minimal two-spheres in the complex hyperquadric. To the authors’ knowledge, there is only some fragmentary results about the minimal two-spheres in complex hyperquadric. X.X. Jiao and J. Wang [21] completely determined the linearly full constant curved minimal two-spheres in Q2 , which are homogeneous and just the ones listed in Table 1. They [22] also gave a method of constructing minimal totally real surfaces in Qn , and they obtained some totally real constant curved minimal two-spheres in Qn , which are contained in Type (III) orbits. According to the Theorem 5.4, its remarks and the Tables 1, 2, 3, 4, we can find: the curvature of a minimal homogeneous two-sphere can be written as 4/N for some positive integer N ; the Type (I) and (II) orbits are not only minimal in Qn but also are minimal in CPn+1 , however, the Type (III) orbits are only minimal in Qn ; the linearly full totally real minimal homogeneous two-spheres are only contained in Q2n . Based on the results mentioned above, we present the following problems: Problem 1. Does these minimal two-spheres obtained in Theorem 5.4 exhaust all the constant curved minimal two-spheres in Qn for n ≥ 3? If not, how to construct a non-homogeneous constant curved minimal two-sphere in Qn for n ≥ 3? Problem 2. Does the curvature of a constant curved minimal two-sphere in Qn can be written as 4/N for some positive integer N ? Problem 3. Does there exist a general minimal (not holomorphic, antiholomorphic and totally real) twosphere in Qn , but it is not minimal in CPn+1 ? Problem 4. Does there exist a linearly full totally real minimal two-sphere with constant curvature in Q2n+1 (maybe doesn’t)? Problem 5. Does there exist a totally real minimal two-sphere in Qn , which is also minimal in CPn+1 ? Acknowledgements This work was supported by NSFC (Grants Nos. 11271343, 11301273, 11331002, 11471299).

JID:MATPUR 24

AID:2817 /FLA

[m3L; v1.175; Prn:25/04/2016; 14:29] P.24 (1-24) C. Peng et al. / J. Math. Pures Appl. ••• (••••) •••–•••

References [1] J.L.M. Barbosa, On minimal immersions of S 2 into S 2m , Trans. Am. Math. Soc. 210 (1975) 75–106. [2] T. Bröcker, T. tom Dieck, Representations of Compact Lie Groups, GTM, vol. 98, Springer-Verlag, New York, 1985. [3] J. Bolton, G.R. Jensen, M. Rigoli, L.M. Woodward, On conformal minimal immersions of S 2 into CP n , Math. Ann. 279 (1988) 599–620. [4] S. Bando, Y. Ohnita, Minimal 2-spheres with constant curvature in CP n , J. Math. Soc. Jpn. 3 (1987) 477–487. [5] A. Bahy-El-Dien, J.C. Wood, The explicit construction of all harmonic two-spheres in G2 (Rn ), J. Reine Angew. Math. 389 (1989) 36–66. [6] S.S. Chern, J.G. Wolfson, Minimal surfaces by moving frames, Am. J. Math. 105 (1983) 59–83. [7] S.S. Chern, J.G. Wolfson, Harmonic maps of the two-sphere into a complex Grassmann manifold II, Ann. Math. 125 (1987) 301–335. [8] E. Calabi, Minimal immersions of surfaces in Euclidean spheres, J. Differ. Geom. 1 (1967) 111–126. [9] M.P. do Carmo, N.R. Wallach, Minimal immersion of spheres into spheres, Ann. Math. 93 (1971) 43–62. [10] Q. Chi, Y.B. Zheng, Rigidity of pseudo-holomorphic curves of constant curvature in Grassmann manifolds, Trans. Am. Math. Soc. 313 (1989) 393–406. [11] L. Delisle, V. Hussin, W.J. Zakrzewski, Constant curvature solutions of Grassmannian sigma models: (1) holomorphic solutions, J. Geom. Phys. 66 (2013) 24–36. [12] L. Delisle, V. Hussin, W.J. Zakrzewski, Constant curvature solutions of Grassmannian sigma models: (2) non-holomorphic solutions, J. Geom. Phys. 71 (2013) 1–10. [13] B. Doubrov, B. Komrakov, Classification of homogeneous submanifolds in homogeneous spaces, Lobachevskii J. Math. 3 (1999) 19–38. [14] J. Eells, J.C. Wood, Harmonic maps from surfaces to the complex projective spaces, Adv. Math. 49 (1983) 217–263. [15] L. Fernandez, The dimension and structure of the space of harmonic 2-spheres in the m-sphere, Ann. Math. 175 (2012) 1093–1125. [16] J. Fei, X.X. Jiao, L. Xiao, Xiaowei Xu, On the classification of homogeneous 2-spheres in complex Grassmannians, Osaka J. Math. 50 (2013) 135–152. [17] Zh.Q. Li, Zh.H. Yu, Constant curved minimal 2-spheres in G(2, 4), Manuscr. Math. 100 (1999) 305–316. [18] X.X. Jiao, J.G. Peng, Pseudo-holomorphic curves in complex Grassmann manifolds, Trans. Am. Math. Soc. 355 (2003) 3715–3726. [19] X.X. Jiao, J.G. Peng, Classification of holomorphic two-spheres with constant curvature in the complex Grassmannians G2,5 , Differ. Geom. Appl. 20 (2004) 267–277. [20] X.X. Jiao, J. Wang, Conformal minimal two-spheres in Qn , Sci. China Math. 54 (4) (2011) 817–830. [21] X.X. Jiao, J. Wang, Minimal surfaces in a complex hyperquadric Q2 , Manuscr. Math. 140 (2013) 597–611. [22] X.X. Jiao, J. Wang, Totally real minimal surfaces in the complex hyperquadrics, Differ. Geom. Appl. 31 (2013) 540–555. [23] S. Kobayashi, K. Nomizu, Foundations of Differential Geometry, vol. 2, John Wiley and Sons, New York, 1969. [24] C.K. Peng, X.W. Xu, Minimal two-spheres with constant curvature in the Complex Grassmannians, Isr. J. Math. 202 (1) (2014) 1–20. [25] J.G. Wolfson, Harmonic maps of two-sphere into the complex hyperquadric, J. Differ. Geom. 24 (1986) 141–152. [26] K. Uhlenbeck, Harmonic maps into Lie groups (classical solution of the chiral model), J. Differ. Geom. 30 (1989) 1–50. [27] J.C. Wood, The explicit construction and parametrization of all harmonic maps from the two-sphere to a complex Grassmannian, J. Reine Angew. Math. 386 (1988) 1–31. [28] Y.B. Zheng, Quantization of curvature of harmonic two-spheres in Grassmann manifolds, Trans. Am. Math. Soc. 316 (1989) 193–214. [29] X. Zhong, J. Wang, X.X. Jiao, Totally real conformal minimal tori in the hyperquadric Q2 , Sci. China Math. 54 (10) (2013) 2015–2023.