2 Contact Angle for Immersed Surfaces in S

with parallel normal vector field

Rodrigo Ristow Montes

Abstract.In this paper we proof that the Holomorphic angle for compact minimal surfaces in the sphereS⁵ with constant Contact angle and with a parallel normal vector field must be constant.

M.S.C. 2000: 53C42, 53D10, 53D35.

Key words: contact angle, holomorphic angle, Clifford torus, parallel field.

1 Introduction

The notion of K¨ahler angle was introduced by Chern and Wolfson in [3] and [12]; it is a fundamental invariant for minimal surfaces in complex manifolds. Using the tech- nique of moving frames, Wolfson obtained equations for the Laplacian and Gaussian curvature for an immersed minimal surface inCPⁿ. Later, Kenmotsu in [7], Ohnita in [10] and Ogata in [11] classified minimal surfaces with constant Gaussian curvature and constant K¨ahler angle.

A few years ago, Li in [14] 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 inCPⁿwith constant K¨ahler angle would have constant Gaussian curvature.

In [8] 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 S³. Geometrically, the Contact angle (β) is the complementary angle between the con- tact distribution and the tangent space of the surface. Also in [8], we deduced formulas for the Gaussian curvature and the Laplacian of an immersed minimal surface inS³, and we gave a characterization of the Clifford Torus as the only minimal surface in S³with constant Contact angle.

We defineαto be the angle given by cosα=hie₁, vi, wheree₁ andv are defined in section 2. The Holomorphic angleαis the analogue of the K¨ahler angle introduced by Chern and Wolfson in [3].

Recently, in [9], we construct a family of minimal tori inS⁵ with constant Contact and Holomorphic angle. These tori are parametrized by the following circle equation

Balkan Journal of Geometry and Its Applications, Vol.12, No.1, 2007, pp. 100-106.∗

c

°Balkan Society of Geometers, Geometry Balkan Press 2007.

a²+ µ

b− cosβ 1 + sin²β

¶₂

= 2 sin⁴β (1 + sin²β)², (1.1)

where a and b are given in Section 3 (equation (3.7)). In particular, when a = 0 in (1.1), we recover the examples found by Kenmotsu, in [6]. These examples are defined for 0< β < ^π₂. Also, whenb = 0 in (1.1), we find a new family of minimal tori inS⁵, and these tori are defined for ^π₄ < β < ^π₂. Also, in [9], whenβ = ^π₂, we give an alternative proof of this classification of a Theorem from Blair in [1], and Yamaguchi, Kon and Miyahara in [13] for Legendrian minimal surfaces inS⁵ with constant Gaussian curvature.

In this paper, we will classify minimal surfaces in S⁵ with constant Contact angle and with a parallel normal vector field. We suppose thate₃ (in equation (3.1)) is a parallel normal vector field, and we get the following

Theorem 1. The Holomorphic angle (0< α < ^π₂)is constant for compact minimal surfaces inS⁵ with constant Contact angle β and null principal curvatures a, b Remark 1. The Theorem 1 implies a more general classification in [9] that gives a family of minimal flat tori inS⁵ with constant Contact angle and constant Holomor- phic angle

2 Contact Angle for Immersed Surfaces in S

Consider inCⁿ⁺¹the following objects:

• the Hermitian product: (z, w) =P_n

j=0z^jw¯^j;

• the inner product:hz, wi=Re(z, w);

• the unit sphere:S²ⁿ⁺¹=©

z∈Cⁿ⁺¹|(z, z) = 1ª

;

• theReebvector field inS²ⁿ⁺¹, given by:ξ(z) =iz;

• the contact distribution inS²ⁿ⁺¹, which is orthogonal toξ:

∆z=©

v∈TzS²ⁿ⁺¹|hξ, vi= 0ª . We observe that ∆ is invariant by the complex structure ofCⁿ⁺¹.

Let nowS be an immersed orientable surface inS²ⁿ⁺¹.

Definition 1. TheContact angleβ is the complementary angle between the contact distribution ∆ and the tangent spaceT Sof the surface.

Let (e₁, e₂) be a local frame of T S, where e₁ ∈ T S∩∆. Then cosβ = hξ, e₂i.

Finally, letv be the unit vector in the direction of the orthogonal projection ofe2 on

∆, defined by the following relation

e2= sinβv+ cosβξ.

(2.1)

3 Equations for Gaussian curvature and Laplacian of a minimal surface in S

In this section, we deduce the equations for the Gaussian curvature and for the Lapla- cian of a minimal surface inS⁵ in terms of the Contact angle and the Holomorphic angle. Consider the normal vector fields

e3 = icscαe1−cotαv e4 = cotαe1+icscαv (3.1)

e₅ = cscβξ−cotβe₂

whereβ6= 0, π andα6= 0, π. We will call (ej)1≤j≤5 anadapted frame.

Using (2.1) and (3.1), we get

v= sinβe2−cosβe5, iv= sinαe4−cosαe1

(3.2)

ξ= cosβe2+ sinβe5

It follows from (3.1) and (3.2) that

ie1 = cosαsinβe2+ sinαe3−cosαcosβe5

(3.3)

ie2 = −cosβz−cosαsinβe1+ sinαsinβe4

Consider now the dual basis (θ^j) of (ej). The connection forms (θ^j_k) are given by

Dej =θ_j^kek,

and the second fundamental form with respect to this frame are given by

II^j = θ^j₁θ¹+θ^j₂θ²; j = 3, ...,5.

Using (3.3) and differentiatingvandξ on the surfaceS, we get

Dξ = −cosαsinβθ²e1+ cosαsinβθ¹e2+ sinαθ¹e3+ sinαsinβθ²e4

−cosαcosβθ¹e5, (3.4)

Dv = (sinβθ¹₂−cosβθ¹₅)e1+ cosβ(dβ−θ²₅)e2+ (sinβθ₂³−cosβθ₅³)e3

+(sinβθ₄²−cosβθ⁴₅)e4+ sinβ(dβ+θ⁵₂)e5.

Differentiatinge3,e4ande5, we have

θ¹₃ = −θ³₁

θ²₃ = sinβ(dα+θ¹₄)−cosβsinαθ¹ θ⁴₃ = cscβθ₁²−cotα(θ³₁+ cscβθ⁴₂) θ⁵₃ = cotβθ₂³−cscβsinαθ¹ θ¹₄ = −dα−cscβθ³₂+ sinαcotβθ¹ θ²₄ = −θ⁴₂

θ³₄ = cscβθ₂¹+ cotα(θ³₁+ cscβθ⁴₂) θ⁵₄ = cotβθ₂⁴−sinαθ²

θ¹₅ = −cosαθ²−cotβθ¹₂ θ²₅ = dβ+ cosαθ¹ (3.5)

θ³₅ = −cotβθ₂³+ cscβsinαθ¹ θ⁴₅ = −cotβθ₂⁴+ sinαθ²

The conditions of minimality and of symmetry are equivalent to the following equa- tions:

θ^λ₁ ∧θ¹+θ₂^λ∧θ²= 0 =θ^λ₁∧θ²−θ^λ₂∧θ¹. (3.6)

On the surfaceS, we consider

θ³₁ = aθ¹+bθ² It follows from (3.6) that

θ₁³ = aθ¹+bθ² θ₂³ = bθ¹−aθ²

θ₁⁴ = dα+ (bcscβ−sinαcotβ)θ¹−acscβθ² θ₂⁴ = dα◦J−acscβθ¹−(bcscβ−sinαcotβ)θ² (3.7)

θ₁⁵ = dβ◦J−cosαθ² θ₂⁵ = −dβ−cosαθ¹

whereJ is the complex structure ofS is given byJe1=e2andJe2=−e1. Moreover, the normal connection forms are given by:

θ⁴₃ = −secβdβ◦J −cotαcscβdα◦J+acotαcot²βθ¹ +(bcotαcot²β−cosαcotβcscβ+ 2 secβcosα)θ² θ⁵₃ = (bcotβ−cscβsinα)θ¹−acotβθ²

(3.8)

θ⁵₄ = cotβ(dα◦J)−acotβcscβθ¹+ (−bcscβcotβ+ sinα(cot²β−1))θ², while the Gauss equation is equivalent to the equation:

dθ₂¹+θ¹_k∧θ^k₂ = θ¹∧θ². (3.9)

Therefore, using equations (3.7) and (3.9), we have

K = 1− |∇β|²−2 cosαβ1−cos²α−(1 + csc²β)(a²+b²) +2bsinαcscβcotβ+ 2 sinαcotβα1− |∇α|²

+2acscβα2−2bcscβα1−sin²αcot²β

= 1−(1 +csc²β)(a²+b²)−2bcscβ(α1−sinαcotβ) + 2acscβα2

(3.10)

−|∇β+ cosαe1|²− |∇α−sinαcotβe1|² Using (3.5) and the complex structure ofS, we get

θ¹₂ = tanβ(dβ◦J−2 cosαθ²) (3.11)

Differentiating (3.11), we conclude that

dθ₂¹ = (−(1 + tan²β)|∇β|²−tanβ∆β−2 cosα(1 + 2 tan²β)β1

+2 tanβsinαα1−4 tan²βcos²α)θ¹∧θ²

where ∆ =tr∇²is the Laplacian ofS. The Gaussian curvature is therefore given by:

K = −(1 + tan²β)|∇β|²−tanβ∆β−2 cosα(1 + 2 tan²β)β1

+2 tanβsinαα1−4 tan²βcos²α.

(3.12)

From (3.10) and (3.12), we obtain the following formula for the Laplacian ofS:

tanβ∆β = (1 + csc²β)(a²+b²) + 2bcscβ(α1−sinαcotβ)−2acscβα2

−tan²β(|∇β+ 2 cosαe1|²− |cotβ∇α+ sinα(1−cot²β)e1|²) + sin²α(1−tan²β)

(3.13)

4 Gauss-Codazzi-Ricci equations for a minimal sur- face in S

with constant Contact angle β

In this section, we will compute Gauss-Codazzi-Ricci equations for a minimal surface inS⁵ with constant Contact angleβ.

Using the connection form (3.7) and (3.8) in the Codazzi-Ricci equations, we have dθ³₁+θ³₂∧θ²₁+θ³₄∧θ⁴₁+θ₅³∧θ⁵₁ = 0

This implies that

(b1−a2) + (a²+b²) cotαcscβcot²β−acotα(csc²β+ cot²β)α2

(4.1)

+b(cotα(csc²β+ cot²β)α1−cosαcotβ(csc²β+ cot²β−3 sec²β(1 + sin²β)))

−cosαcscβ(2(cotβ−tanβ)α1−sinα(cot²β−3)) + cotαcscβ|∇α|²= 0 Replacing the following (3.8) in the Codazzi-Ricci equations

dθ³₂+θ³₁∧θ¹₂+θ³₄∧θ⁴₂+θ₅³∧θ⁵₂ = 0 dθ⁴₁+θ⁴₂∧θ²₁+θ⁴₃∧θ³₁+θ₅⁴∧θ⁵₁ = 0 dθ⁵₃+θ⁵₁∧θ¹₃+θ⁵₂∧θ²₃+θ₄⁵∧θ⁴₃ = 0

We get

(a1+b2) +bcotαα2+a(cotαα1+ 6 tanβcosα)

−2 secβcosαα2= 0 (4.2)

Using the connection form (3.8) in the Codazzi-Ricci equations dθ⁴₂+θ⁴₁∧θ¹₂+θ⁴₃∧θ³₂+θ₅⁴∧θ⁵₂ = 0 dθ⁵₄+θ⁵₁∧θ¹₄+θ⁵₂∧θ²₄+θ₃⁵∧θ³₄ = 0 dθ⁴₃+θ⁴₁∧θ¹₃+θ⁴₂∧θ²₃+θ₅⁴∧θ⁵₃ = 0 We have

(a2−b1)−(a²+b²) cotαsinβcot²β+acotαα2

(4.3)

+b(−cotαα1+ 2 cosα(cotβ−3 tanβ)) + 2 cosαsinβ(cotβ−tanβ)α1

+ sinαcosαsinβ(5−cot²β) + sinβ∆α= 0 Codazzi-Ricci equations

dθ²₁+θ₃²∧θ³₁+θ₄²∧θ₁⁴+θ²₅∧θ₁⁵ = θ²∧θ¹ dθ⁵₁+θ₂⁵∧θ²₁+θ₃⁵∧θ₁³+θ⁵₄∧θ₁⁴ = 0 give the following equation

(a²+b²)(1 + csc²β) + 2bcscβ(α1−cotβsinα)−2acscβα2

+|∇α|²+ 2 sinα(tanβ−cotβ)α1−4 tan²βcos²α

−sin²α(1−cot²β) = 0 (4.4)

The following Codazzi equation is automatically verified dθ⁵₂+θ⁵₁∧θ¹₂+θ⁵₃∧θ³₂+θ₄⁵∧θ⁴₂ = 0

5 Proof of the Theorem 1

In this section, we will give a proof of the theorem, using Gauss-Codazzi-Ricci equations for a minimal surface inS⁵ with constant Contact angle and null principal curvaturesa, b.

Suppose thata, bare nulls and the Contact angleβis constant, then using the Codazzi equation (4.1), we have

cosα(2(cotβ−tanβ)α1−sinα(cot²β−3))−cotα|∇α|²= 0 (5.1)

On the other hand, Codazzi equation (4.3) witha, bnulls and constant Contact angle implies

2 cosα(cotβ−tanβ)α1+ sinαcosα(5−cot²β) + ∆α= 0 (5.2)

Using equations (5.1) and (5.2), we obtain a new Laplacian equation ofα

∆α = −sin(2α)−cotα|∇α|² (5.3)

Now suppose that (0< α < ^π₂). Using the Hopf’s Lemma, we get the Theorem 1.¤ Acknowledgement:I want to express my sincere thanks to department of mathe- matics at Washington University in Saint Louis for the hospitality during my Post- Doc. Also, I want to thanks the brasilian agency CNPq for the financial support.

References

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Author’s address:

Rodrigo Ristow Montes

Departamento de Matem´atica, Universidade Federal da Para´iba BR– 58.051-900 Jo˜ao Pessoa, P.B., Brazil.

email: [email protected], [email protected]