Contact angle for immersed surfaces in S

Differential Geometry and its Applications 25 (2007) 92–100 www.elsevier.com/locate/difgeo

Contact angle for immersed surfaces in S 2n+1 Rodrigo Ristow Montes a,∗ , Jose A. Verderesi b a Departamento de Matemática, Universidade Federal da Paraíba, BR-58.051-900 João Pessoa, P.B., Brazil b Departamento de Matemática Pura, Instituto de Matemática e Estatística, Universidade de São Paulo, Caixa Postal 66281,

BR-05315-970 São Paulo, S.P., Brazil Received 21 April 2004; received in revised form 11 October 2005 Available online 5 June 2006 Communicated by J. Stasheff

Abstract In this paper we introduce the notions of contact angle and of holomorphic angle for immersed surfaces in odd dimensional spheres. We deduce formulas for the Laplacian and for the Gaussian curvature, and we classify minimal surfaces in S 5 with the two angles constant. This classification gives a 2-parameter family of minimal tori of S 5 . Also, we give an alternative proof of the classification of minimal Legendrian surfaces of S 5 with constant Gaussian curvature. © 2006 Elsevier B.V. All rights reserved. MSC: 53C42; 53D10; 53D35 Keywords: Contact angle; Holomorphic angle; Clifford torus; Legendrian surfaces

1. Introduction The notion of Kähler angle was introduced by Chern and Wolfson in [3] and [9]; it is a fundamental invariant for minimal surfaces in complex manifolds. Using the technique of moving frames, Wolfson obtained equations for the Laplacian and Gaussian curvature for an immersed minimal surface in CPn . Later, Kenmotsu in [5], Ohnita in [7] and Ogata in [8] classified minimal surfaces with constant Gaussian curvature and constant Kähler angle. A few years ago, Li in [11] gave a counterexample to the conjecture of Bolton, Jensen and Rigoli (see [2]), according to which a minimal immersion (non-holomorphic, non-anti-holomorphic, non-totally real) of a two-sphere in CPn with constant Kähler angle would have constant Gaussian curvature. In [6] we introduced the notion of contact angle, that can be considered as a new geometric invariant useful to investigate the geometry of immersed surfaces in S 3 . Geometrically, the contact angle (β) is the complementary angle between the contact distribution and the tangent space of the surface. Also in [6], we deduced formulas for the Gaussian curvature and the Laplacian of an immersed minimal surface in S 3 , and we gave a characterization of the Clifford Torus as the only minimal surface in S 3 with constant contact angle. * Corresponding author.

E-mail addresses: [email protected] (R.R. Montes), [email protected] (J.A. Verderesi). 0926-2245/$ – see front matter © 2006 Elsevier B.V. All rights reserved. doi:10.1016/j.difgeo.2006.05.004

R.R. Montes, J.A. Verderesi / Differential Geometry and its Applications 25 (2007) 92–100

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We define α to be the angle given by cos α = ie1 , v, where e1 and v are defined in Section 2. The holomorphic angle α is the analogue of the Kähler angle introduced by Chern and Wolfson in [3]. We obtain the following formula for the Gaussian curvature of an immersed minimal surface in S 5 : K = −(1 + tan2 β)|∇β|2 − tan ββ − 2 cos α(1 + 2 tan2 β)β1 + 2 tan β sin αα1 − 4 tan2 β cos2 α. Also, we obtain the following equation for the Laplacian: tan ββ = (1 + csc2 β)(a 2 + b2 ) + 2b csc β(α1 − sin α cot β) − 2a csc βα2 2 − tan2 β |∇β + 2 cos αe1 |2 − cot β∇α + sin α(1 − cot2 β)e1 + sin2 α(1 − tan2 β). Using the equations of Gauss, Codazzi and Ricci, we construct a family of minimal tori in S 5 with constant contact and holomorphic angle. Theorem 1. Compact, orientable, minimal surfaces in S 5 with constant contact angle (β) and constant holomorphic angle (α) are flat tori. These tori are parametrized by the following circle equation 2 cos β sin4 β a2 + b − (1) = 2 , 1 + sin2 β (1 + sin2 β)2 where a and b are given in Section 4 (Eq. (25)). Remark 1. In particular, when a = 0 in (1), we recover the examples found by Kenmotsu, in [4]. These examples are defined for 0 < β < π2 . Also, when b = 0 in (1), we find a new family of minimal tori in S 5 , and these tori are defined for π4 < β < π2 . When the holomorphic angle is zero, we have an interesting characterization of the Clifford torus without the hypothesis that the contact angle is constant. Proposition 1. The Clifford torus is the only non-Legendrian minimal surface in S 5 with contact angle 0 β < and null holomorphic angle.

π 2

Blair in [1], and Yamaguchi, Kon and Miyahara in [10] classified Legendrian minimal surfaces in S 5 with constant Gaussian curvature. As a particular case of Theorem 1, that is when β = π2 , we give an alternative proof of this classification using moving frames and contact structure equations. Corollary 1. The Clifford torus and the totally geodesic spheres are the only Legendrian minimal surfaces immersed in S 5 with constant Gaussian curvature. 2. Contact angle for immersed surfaces in S 2n+1 Consider in Cn+1 the following objects: • • • • •

the Hermitian product: (z, w) = nj=0 zj w¯ j ; the inner product: z, w = Re(z, w); the unit sphere: S 2n+1 = {z ∈ Cn+1 |(z, z) = 1}; the Reeb vector field in S 2n+1 , given by: ξ(z) = iz; the contact distribution in S 2n+1 , which is orthogonal to ξ : z = v ∈ Tz S 2n+1 | ξ, v = 0 .

We observe that is invariant by the complex structure of Cn+1 . Let now S be an immersed orientable surface in S 2n+1 .

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Definition 1. The contact angle β is the complementary angle between the contact distribution and the tangent space T S of the surface. Let (e1 , e2 ) be a local frame of T S, where e1 ∈ T S ∩ . Then cos β = ξ, e2 . Finally, let v be the unit vector in the direction of the orthogonal projection of e2 on , defined by the following relation e2 = sin βv + cos βξ.

(2)

3. Equations for curvature and Laplacian of a minimal surface in S 5 In this section, we deduce the equations for the Gaussian curvature and for the Laplacian of a minimal surface in S 5 in terms of the contact angle and the holomorphic angle. Consider the normal vector fields e3 = i csc αe1 − cot αv, e4 = cot αe1 + i csc αv, e5 = csc βξ − cot βe2 ,

(3)

where β = 0, π and α = 0, π . We will call (ej )1 j 5 an adapted frame. Using (2) and (3), we get v = sin βe2 − cos βe5 ,

iv = sin αe4 − cos αe1 ,

ξ = cos βe2 + sin βe5 .

(4)

It follows from (3) and (4) that ie1 = cos α sin βe2 + sin αe3 − cos α cos βe5 , ie2 = − cos βz − cos α sin βe1 + sin α sin βe4 .

(5) j

Consider now the dual basis (θ j ) of (ej ). The connection forms (θk ) are given by Dej = θjk ek , and the second fundamental form with respect to this frame are given by j

j

II j = θ1 θ 1 + θ2 θ 2 ;

j = 3, . . . , 5.

Using (5) and differentiating v and ξ on the surface S, we get Dξ = − cos α sin βθ 2 e1 + cos α sin βθ 1 e2 + sin αθ 1 e3 + sin α sin βθ 2 e4 − cos α cos βθ 1 e5 , Dv = (sin βθ21 − cos βθ51 )e1 + cos β(dβ − θ52 )e2 + (sin βθ23 − cos βθ53 )e3 + (sin βθ42 − cos βθ54 )e4 + sin β(dβ + θ25 )e5 . Differentiating e3 , e4 and e5 , we have θ31 = −θ13 , θ32 = sin β(dα + θ41 ) − cos β sin αθ 1 , θ34 = csc βθ12 − cot α(θ13 + csc βθ24 ), θ35 = cot βθ23 − csc β sin αθ 1 , θ41 = −dα − csc βθ23 + sin α cot βθ 1 , θ42 = −θ24 , θ43 = csc βθ21 + cot α(θ13 + csc βθ24 ), θ45 = cot βθ24 − sin αθ 2 , θ51 = − cos αθ 2 − cot βθ21 ,

(6)


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θ52 = dβ + cos αθ 1 , θ53 = − cot βθ23 + csc β sin αθ 1 , θ54 = − cot βθ24 + sin αθ 2 .

(7)

The conditions of minimality and of symmetry are equivalent to the following equations: θ1λ ∧ θ 1 + θ2λ ∧ θ 2 = 0 = θ1λ ∧ θ 2 − θ2λ ∧ θ 1 .

(8)

On the surface S, we consider θ13 = aθ 1 + bθ 2 . It follows from (8) that θ13 = aθ 1 + bθ 2 , θ23 = bθ 1 − aθ 2 , θ14 = dα + (b csc β − sin α cot β)θ 1 − a csc βθ 2 , θ24 = dα ◦ J − a csc βθ 1 − (b csc β − sin α cot β)θ 2 , θ15 = dβ ◦ J − cos αθ 2 , θ25 = −dβ − cos αθ 1 ,

(9)

where J is the complex structure of S is given by J e1 = e2 and J e2 = −e1 . Moreover, the normal connection forms are given by: θ34 = − sec β dβ ◦ J − cot α csc β dα ◦ J + a cot α cot2 βθ 1 + (b cot α cot2 β − cos α cot β csc β + 2 sec β cos α)θ 2 , θ35 = (b cot β − csc β sin α)θ 1 − a cot βθ 2 , θ45 = cot β(dα ◦ J ) − a cot β csc βθ 1 + −b csc β cot β + sin α(cot2 β − 1) θ 2 ,

(10)

while the Gauss equation is equivalent to the equation: dθ21 + θk1 ∧ θ2k = θ 1 ∧ θ 2 .

(11)

Therefore, using Eqs. (9) and (11), we have K = 1 − |∇β|2 − 2 cos αβ1 − cos2 α − (1 + csc2 β)(a 2 + b2 ) + 2b sin α csc β cot β + 2 sin α cot βα1 − |∇α|2 + 2a csc βα2 − 2b csc βα1 − sin2 α cot2 β = 1 − (1 + csc2 β)(a 2 + b2 ) − 2b csc β(α1 − sin α cot β) + 2a csc βα2 − |∇β + cos αe1 |2 − |∇α − sin α cot βe1 |2 .

(12)

Using (7) and the complex structure of S, we get θ21 = tan β(dβ ◦ J − 2 cos αθ 2 ).

(13)

Differentiating (13), we conclude that dθ21 = −(1 + tan2 β)|∇β|2 − tan ββ − 2 cos α(1 + 2 tan2 β)β1 + 2 tan β sin αα1 − 4 tan2 β cos2 α θ 1 ∧ θ 2 , where = tr ∇ 2 is the Laplacian of S. The Gaussian curvature is therefore given by: K = −(1 + tan2 β)|∇β|2 − tan ββ − 2 cos α(1 + 2 tan2 β)β1 + 2 tan β sin αα1 − 4 tan2 β cos2 α.

(14)

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From (12) and (14), we obtain the following formula for the Laplacian of S: tan ββ = (1 + csc2 β)(a 2 + b2 ) + 2b csc β(α1 − sin α cot β) − 2a csc βα2 2 − tan2 β |∇β + 2 cos αe1 |2 − cot β∇α + sin α(1 − cot2 β)e1 + sin2 α(1 − tan2 β).

(15)

3.1. Proof of Proposition 1 When α = 0, we have v = ie1 , hence the adapted frame is e2 = sin β(ie1 ) + cos βξ, e3 ∈ , e4 = ie3 , e5 = csc βξ − cot βe2 .

(16)

Differentiating e2 , we get De2 = θ21 e1 + θ23 e3 + θ24 e4 + θ25 e5 ;

(17)

on the other hand, we have De2 = tan β(θ15 − θ 2 )e1 − sin βθ14 e3 + sin βθ13 e4 − (dβ + θ 1 )e5 .

(18)

It follows from (17) and (18) that θ21 = tan β(θ15 − θ 2 ),

θ23 = − sin βθ14 ,

θ24 = sin βθ13 ,

θ25 = −dβθ 1 .

Finally, from (8) we get: (1 − sin2 β)θ13 = 0. The surface S is non-Legendrian so we have θ13 = 0. Then Eq. (15) becomes β = − tan(β)|∇β + 2e1 |2 .

(19)

then, by Hopf’s Lemma, we have that β is constant and using Eq. (14), we have K = 0. Assume that 0 β < Therefore S is the Clifford torus, and this proves Proposition 1. π 2;

4. Proof of Theorem 1 In this section we will give a complete characterization of minimal surfaces in S 5 with constant contact and holomorphic angle. 4.1. The case β = 0 The holomorphic angle α is not defined when β = 0, then the adapted frame to the surface S is given by e2 = ξ, e3 = ie1 , e4 ∈ , e4 ⊥ (e1 , ie1 ), e5 = ie4 .

(20)

Differentiating e2 , we get De2 = θ21 e1 + θ23 e3 + θ24 e4 + θ25 e5 .

(21)


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Differentiating ξ and using (20), we obtain Dξ = θ 1 (ie1 ) + θ 2 (ie2 ) + θ 3 (ie3 ) + θ 4 (ie4 ) + θ 5 (ie5 ) = −θ 3 e1 + θ 1 e3 − θ 5 e4 + θ 4 e5 .

(22)

The dual forms θ 3 , θ 4 and θ 5 on the surface S vanish, and using (8), (21) and (22), we have that also the connection forms θ21 , θ14 , θ24 , θ15 , θ25 vanish. Hence, we get θ13 = θ 2 ,

θ23 = θ 1 .

(23)

Differentiating the equation e3 = ie1 we get (23), and θ34 and θ35 vanish. It follows from (23) that De1 = θ 2 e3 . Thus, De1 e1 = 0 and so e1 is a geodesic field. Since e1 is in , the surface S is a torus, and it is parametrized by a product of two circles, one circle at the distribution and the other circle at the Hopf fibration. 4.1.1. Example: contact angle of standard Clifford torus Consider the torus in S 5 defined by

1 1 T 2 = (z1 , z2 , 0) ∈ C 3 | z1 z¯ 1 = , z2 z¯ 2 = 2 2 and the immersion

√ 2 iu1 iu2 f (u1 , u2 ) = (e , e , 0). 2 The contact vector field is i ξ = √ (eiu1 , eiu2 , 0); 2 the following vector fields are tangent to the surface: i e1 = √ (eiu1 , −eiu2 , 0), 2 Therefore

i e2 = √ (eiu1 , eiu2 , 0). 2

cos β = e2 , ξ = 1, hence β = 0. The following vector fields are orthogonal to the surface: 1 e3 = √ (−eiu1 , eiu2 , 0), 2 4.2. The case 0 < β