A Weierstrass Representation for Minimal Surfaces ... - Semantic Scholar

Results. Math. 60 (2011), 311–323 c 2011 Springer Basel AG 1422-6383/11/010311-13 published online June 28, 2011 DOI 10.1007/s00025-011-0169-y

Results in Mathematics

A Weierstrass Representation for Minimal Surfaces in 3-Dimensional Manifolds J. H. Lira, M. Melo and F. Mercuri Dedicated to Prof. Keti Tenenblat, a very special friend and a mark of Brazilian mathematics, on the occasion of her 65th anniversary

Abstract. In this paper we will discuss a Weierstrass type representation for minimal surfaces in Riemannian and Lorentzian 3-dimensional manifolds. Mathematics Subject Classification (2010). Primary 53C42; Secondary 53C50. Keywords. Minimal surfaces, Weierstrass representation.

1. Introduction In the classical theory of minimal surfaces in R3 a basic tool is the Weierstrass representation. Its local version may be stated as follows Theorem 1.1 (Weierstrass representation). Let Ω ⊆ C be an open set endowed with a complex coordinate z = u + iv. Let f : Ω → R3 be a conformal minimal immersion. We define φ ∈ Γ(f ∗ T M ⊗ C) as the complex tangent vector 1 ∂f ∂f ∂f := −i . φ= ∂z 2 ∂u ∂v Then, the Euclidean components φa , a = 1, 2, 3, of φ, defined by φ=

3 a=1

φa

∂ , ∂xa

Work partially supported by CNPq, FAPESP and FUNCAP, Brazil.

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satisfy the following conditions: i. |φ1 |2 + |φ2 |2 + |φ3 |2 > 0, ii. φ21 + φ22 + φ23 = 0, a iii. ∂φ ∂ z¯ = 0. Conversely, if Ω is simply connected and φa : Ω → C, a = 1, 2, 3, are functions satisfying the above conditions, the map f = 2 Real φ dz is a well-defined conformal minimal immersion. Remark 1.2. The first condition tells us that f is an immersion, the second one that f is conformal and the third one that f is minimal. The importance of Theorem 1.1 is twofold: on the one hand it is a very powerful tool for producing examples and on the other hand permits to use the theory of holomorphic functions for investigating structural properties of minimal surfaces. We state below the Björling Theorem which is a typical example of a general result derived from Weierstrass representation. Theorem 1.3. Given a real analytic curve β : (0, 1) → R3 and a unitary analytic vector field ξ along β, there exist > 0 and a conformal minimal immersion f : (0, 1) × (−, ) → R3 such that f (t, 0) = β(t) and ξ is normal to f along β. Moreover two such immersions coincide along the intersection of their domains. Remark 1.4. This result is a local existence and unicity theorem for solutions of a kind of initial value problem with analytic data. This problem was firstly proposed by Bj¨ orling in 1844 (see [2]) and solved by Schwarz in 1890 (see [11]). It is worthwhile to observe that Schwartz gave an “explicit” formula for the solution in terms of holomorphic extensions of β and ξ and integration. The aim of this paper is to discuss extensions of Theorem 1.1 for minimal surfaces in more general spaces, as Riemannian and Lorentzian 3-manifolds.

2. Minimal Surfaces in Lorentz–Minkowski 3-Space We consider the Lorentz–Minkowski space L3 , i.e., the affine three space R3 endowed with the Lorentzian metric g = dx21 + dx22 − dx23 . If Ω is an open subset in R2 and f : Ω → L3 is an immersion we will say that f is spacelike if the induced metric f ∗ g on Ω is Riemannian. If f ∗ g is a symmetric non-degenerate form of index 1, that is, if it is a Lorentzian metric, then we say that f is timelike.

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For the case of spacelike minimal immersions in the Lorentz–Minkowski space L3 a Weierstrass representation type theorem was proved by Kobayashi in [4] and can be stated as follows Theorem 2.1. Let Ω ⊆ C be an open set endowed with a complex coordinate z = u + iv and f : Ω → L3 be a conformal spacelike minimal immersion. Then the complex tangent vector 3

φ=

∂ ∂f = φa ∂z ∂xa a=1

satisfies the following conditions i. |φ1 |2 + |φ2 |2 − |φ3 |2 > 0, ii. φ21 + φ22 − φ23 = 0, a iii. ∂φ ∂ z¯ = 0.

Conversely, if Ω is simply connected and φa : Ω → C, a = 1, 2, 3, are functions satisfying the above conditions, the map f = 2 Real φ dz defines a conformal spacelike minimal immersion into L3 . Remark 2.2. In [1] Al´ıas et al. proved a version of the Bj¨ orling Theorem in the context of spacelike minimal surfaces in L3 . Using integral expressions similar to those ones in the classical case, these authors were able to obtain explicit formulae describing a series of interesting examples of spacelike minimal surfaces in L3 . The case of timelike minimal surfaces in L3 requires that the local parameters belong to a Lorentz structure in those surfaces. This structure is modeled after the algebra of paracomplex numbers that we will describe below. This approach permits to deduce a Weierstrass representation type result for timelike surfaces. We recall that the algebra L of paracomplex numbers is the algebra of formal linear combinations a + τ b where a, b ∈ R and τ is an imaginary unit with τ 2 = 1. The operations are the obvious ones. It turns out that L is the Clifford algebra of R with respect to the canonical quadratic form and it is isomorphic to the algebra R ⊕ R, via the “rotation” ρ(a + τ b) =

1 (u + v, u − v). 2

Conjugation in L is defined as in the complex case, that is, a + τ b = a − τ b. We set K = {a + τ b ∈ L : a = ±b}. Then K\{0} is the set of zero divisors of L. The norm of z = a + τ b ∈ L is denoted by |z| and defined by |z| = |z z¯|.

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The set L has a natural topology as a 2-dimensional real vector space. Given an open set Ω ⊆ L the derivative of a function f : Ω → L at a point z0 ∈ Ω is defined by f (z0 ) =

f (z) − f (z0 ) z − z0

lim

z → z0 , z − z0 ∈ K

if this limit exists. Remark 2.3. If f is L-differentiable at z0 then it is not necessarily continuous at z0 . However, if it is L-differentiable in the whole Ω, it is differentiable in the usual sense. We observe that there are L-differentiable functions of any class of (usual) differentiability. The condition of L-differentiability for a function f (u, v) := a(u, v) + τ b(u, v), u + τ v ∈ Ω, is equivalent to the system of PDEs ∂a ∂b = , ∂v ∂u

∂b ∂a = . ∂v ∂u

According with the algebraic structure fixed above, we define dz = du + τ dv, ∂ 1 ∂ ∂ = +τ , ∂z 2 ∂u ∂v

d¯ z = du − τ dv, ∂ 1 ∂ ∂ = −τ . ∂ z¯ 2 ∂u ∂v

(2.1) (2.2)

With these notations at hand the condition of L-differentiability for a function f : Ω → L becomes ∂f = 0. ∂ z¯ Again, for timelike minimal surfaces, we have a Weierstrass representation type theorem. This theorem was proved by Konderak in [5] (see also [8]) and can be stated as follows Theorem 2.4. Let Ω ⊆ L be an open set and f : Ω → L3 a conformal timelike minimal immersion. Then the tangent vector 3

∂ ∂f := φa ∂z ∂x a a=1

φ := satisfies the following conditions i. |φ1 |2 + |φ2 |2 − |φ3 |2 > 0, ii. φ21 + φ22 − φ23 = 0, a iii. ∂φ ∂ z¯ = 0. where the operators

∂ ∂z

and

∂ ∂ z¯

are defined in (2.2).

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Conversely, if Ω is simply connected and φa : Ω → L, a = 1, 2, 3, are functions satisfying the above conditions, then the map f = 2 Real φ dz defines a conformal timelike minimal immersion into L3 .1 Remark 2.5. The Björling problem for timelike minimal surfaces has been studied in a recent paper by Chaves, Dussan and Magid (see [3]). In this case it is necessary to assume that the initial curve does not have light-like tangent vectors. In fact the light-like curves are characteristics for the corresponding differential equation and the solution may not be unique.

3. Minimal Surfaces in 3-Dimensional Manifolds In recent times there has been a growing interest in the study of the geometry of 3-manifolds, in particular the Thurston geometries, i.e., simply connected homogeneous 3-manifolds that admit compact quotients. In particular a very prolific research has been devoted to the study of minimal surfaces in these ambients. In this section we extend Theorems 1.1, 2.1 and 2.4 to the case of minimal surfaces in either Lorentzian or Riemannian 3-dimensional manifolds. We will treat the three cases at the same time in a unified approach. For that we will denote by K either the complex numbers C or the paracomplex numbers L. We now describe the general setting we will work with. Let M be a Lorentzian or Riemannian 3-manifold, Ω ⊆ K an open set and f : Ω → M a conformal immersion. Consider the (para)complexified tangent bundle T M ⊗ K and the pull back bundle E = f ∗ T M ⊗ K. The metric on M induces on T M ⊗ K and on E two structures, namely • a K-bilinear symmetric form, denoted by (·, ·), • a (para) Hermitian metric, that we will keep denoting with ·, ·. Endowing Ω with the induced metric makes f an isometric immersion. We the Levi–Civita connections on Ω and M respectively. will denote by ∇ and ∇ Then ∇ induces a connection on E, compatible with the two forms above, that we will denote also by ∇. We observe that Lorentzian surfaces might be endowed with paracomplex isothermic coordinates and, as in the Riemannian case, they are locally described by complex isothermic charts. Thus, we will denote by z = u + iv (resp. z = u + τ v) a complex (resp. paracomplex) isothermal coordinate in Ω. Let N be a unit normal vector field along f . If f is a timelike immersion, this means that N, N = 1. We denote the first and second fundamental forms of f respectively by I and II. They are given by I = df, df 1

Observe that the L-differentiability implies that φ has no real periods in Ω.

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and II = −dN, df . From the choice of isothermic parameters it follows that there exists a positive function ω : Ω → R such that I = exp2ω (du2 ± dv 2 ),

(3.1)

where we will use the sign + (respectively −) when Ω is endowed with a Riemannian (respectively Lorentzian) metric. The mean curvature H of f is defined by H = traceI II.

(3.2)

We then prove the following result Theorem 3.1. Let f : Ω → M be a conformal minimal immersion where Ω ⊆ K is an open set. Given a (para)complex isothermic coordinate z, define the (para)complex tangent vector φ ∈ Γ(f ∗ T M ⊗ K) by ∂f . φ(z) = ∂z f (z) Then φ satisfies the following conditions i. φ, φ > 0, ii. (φ, φ) = 0, ∂ φ = 0. iii. ∇ ∂z ¯

Conversely, if Ω is convex and φ ∈ Γ(f ∗ T M ⊗K) satisfies the conditions above, then the immersion f is represented by the line integral f = 2 Real φ dz, where the integration is performed in paths contained in Ω. Proof. We will prove the Theorem only for the case of timelike immersions since that the other two remaining cases might be treated similarly. Since we will work locally we may suppose that M is covered by a single chart with local coordinates labeled by x1 , x2 , x3 . In this case, we identify M with an open set of R3 . The local components of the ambient metric and the c respectively. corresponding Christoffel symbols are denoted by gãb and Γ ab ∂f In order to prove (iii), we write φ = ∂z as φ = X1 +τ X2 , for some sections X1 , X2 of f ∗ T M . Then 1 X X2 ) + τ (∇ X X1 − ∇ X X2 )] [(∇X1 X1 − ∇ 2 2 1 2 from what follows that X X1 − ∇ ∂ φ = 1 [∇ X X2 ]. ∇ 1 2 ∂z ¯ 2 ∇

∂ ∂z ¯

φ=

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Let hij and Γkij be local components of II and the Christoffel symbols asso in ciated to the induced connection ∇ in Ω, respectively. Thus, splitting ∇ tangential and normal components one obtains ∂ φ = 1 (∇X X1 − ∇X X2 ) + 1 (h11 − h22 )N. ∇ 1 2 ∂z ¯ 2 2 Thus it follows that ∂ φ = 1 (Γ111 − Γ122 )X1 + 1 (Γ211 − Γ222 )X2 + 1 (h11 − h22 )N, ∇ ∂z ¯ 2 2 2 ∂ω 1 1 2 However, in view of (3.1), we have Γ11 = Γ22 = ∂u , Γ11 = Γ222 = ∂ω ∂v . Hence f ∂ φ = 0. This proves (iii). is minimal if and only if ∇ ∂z ¯ Conditions (i) and (ii) may be deduced directly from (3.1) and from the ∂f + τ ∂f fact that φ = 12 ( ∂u ∂v ). Now, we prove the converse statement. Let φ be a vector field along f , i.e. a section φ of E = f ∗ T M ⊗ K, written in local coordinates as φ=

3 a=1

φa

∂ ∂xa

and satisfying (i), (ii) and (iii) above. Then ∂φa ∂ ∂φc ∂ φ= ∂ ∂ = cab φa φ¯b ∂ . ∇ + Γ + φa φ¯b ∇ ∂z ¯ ∂xb ∂x ∂ z¯ ∂xa ∂ z¯ ∂xc a a c a,b

b

c should be computed on Notice that in the expression above the functions Γ ab c ¯ f (z). Since the expression a,b Γab φa φb is real, (iii) implies that ∂φa ∈ R, a = 1, 2, 3 ∂ z¯ what implies that the forms Real φa dz, a = 1, 2, 3, are closed and then exact in Ω. It follows that it makes sense to define the line integrals z Real z0 φa dz, a = 1, 2, 3, where z0 , z ∈ Ω. Directly from our construction it

z follows that f = 2 Real z0 φ dz. Remark 3.2. In the Riemannian case, Theorem 3.1 was considered by various authors for special ambient spaces (see, for example [6] for the case of hyperbolic space) and the general version was obtained by Mercuri et al. in [9]. The spacelike minimal surfaces case was considered, for some particular ambient spaces, by Lee in [7].

4. The Case of Lie Groups In the most general case, equation (iii) in Theorem 3.1 is a quite complicated non-linear integro-differential equation. Therefore, it seems unreasonable to hope to find explicit solutions for it, and hence examples of conformal minimal

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immersions by this method. It is then natural to look for classes of (ambient) manifolds for which that equation becomes simpler. A natural class of manifolds to consider is that one of Lie groups. The fact that the Levi–Civita connection in Lie groups may be expressed in terms of the structure constants permits to express the Christoffel symbols in (iii), even if we do not know explicitly the coordinates of the immersion. Suppose from now on that M is a 3-dimensional Lie group endowed with a left-invariant metric (Riemannian or Lorentzian). As before, Ω ⊆ K denotes an open set and f : Ω ⊆ K → M stands for a conformal minimal immersion. Let {E1 , E2 , E3 } be a left invariant orthonormal frame field, with E1 , E2 spacelike and E3 timelike if the left-invariant metric is Lorentzian. Fixed a isothermic parameter z in Ω, we can write the (para)complex tangent field ∂f ∂z along f both in terms of local coordinates x1 , x2 , x3 in M and also using the left-invariant frame field. Hence, one has 3

3

∂ ∂f = φa = ψ a Ea , ∂z ∂xa a=1 a=1

(4.1)

where the functions φa and ψb , a, b = 1, 2, 3 are related by φa =

3

Aab ψb ,

a = 1, 2, 3,

b=1

where A : Ω → GL(3, R). In terms of the components ψa , a = 1, 2, 3, equation (iii) in Theorem 3.1 may be written as 3 ∂ψc + Lcab ψ¯a ψb = 0, ∂ z¯

c = 1, 2, 3,

(4.2)

a,b=1

where the symbols Lcab are defined by E Eb = ∇ a

3

Lcab Ec ,

a, b = 1, 2, 3.

c=1

It is easily verified that these symbols are constants given in terms of structure c constants Cab = Ec∗ ([Ea , Eb ]) of the Lie algebra by 1 c a b (C − Cbc εa εc − Cac εb εc ), (4.3) 2 ab where εa = Ea , Ea , a = 1, 2, 3. Theorem 3.1 may be rephrased in the case of 3-dimensional Lie groups as follows Lcab =

Theorem 4.1. Let M be a 3-dimensional Lie group endowed with a left invariant Riemannian or Lorentzian metric and {Ea }3a=1 a left invariant orthonormal frame field. Let Ω ⊆ K be an open set and f : Ω → M a conformal minimal

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immersion. We denote by φ the (para) complex tangent vector φ :=

∂f ∂z

Then, the components ψa , a = 1, 2, 3, of φ defined by φ(z) =

3

ψa Ea |f (z)

a=1

satisfy the following conditions i. |ψ1 |2 + |ψ2 |2 + ε|ψ3 |2 > 0, ii. ψ12 + ψ22 + εψ32 = 0, c ¯ c iii. ∂ψ a,b Lab ψa ψb = 0, ∂ z¯ + where ε = 1 (resp. −1) if M is Riemannian (resp. Lorentzian). Conversely, if Ω is simply connected and ψa : Ω → K, a = 1, 2, 3, are functions satisfying (i), (ii) and (iii), then there exists a map f : Ω → M which solves the equation f −1

3

∂f = ψa Ea |e , ∂z a=1

where e is the identity element of M . The components of f in terms of the coordinates x1 , x2 , x3 are then given by fa = 2 Real

3

Aab ψb dz,

a = 1, 2, 3,

(4.4)

b=1

where the matrix A is defined by Eb |f (z) =

a

Aab

∂ , ∂xa f (z)

z ∈ Ω.

Moreover, the map f defines a conformal minimal immersion. Remark 4.2. Equation 4.2 is a (non linear) PDE with constant coefficients and then it is much easier to handle with than the corresponding equation for the coordinate components φa , a = 1, 2, 3. However we are faced with another problem. Formula (4.4) does not give the coordinate components of the immersion f just by a direct integration, since the functions Aab , a, b = 1, 2, 3, in (4.4) have to be computed along f itself. Formula (4.4) in fact is an integral equation. However, for some particular Lie groups, it is possible to overcome this problem. We proceed describing some examples where ad hoc arguments permit to obtain integral representations.

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4.1. The Heisenberg Group H3 The Heisenberg group is the two-step nilpotent matrix group ⎧⎡ ⎫ ⎤ ⎨ 1 x1 x3 + 12 x1 x2 ⎬ ⎦ , (x1 , x2 , x3 ) ∈ R3 . x2 H3 = ⎣ 0 1 ⎩ ⎭ 0 0 1 We consider a left invariant metric H3 for which the left invariant frame field given by E1 =

∂ x2 ∂ − , ∂x1 2 ∂x3

E2 =

∂ x1 ∂ + , ∂x2 2 ∂x3

E3 =

∂ , ∂x3

is orthonormal with E3 , E3 = ε. The Lie bracket in H3 reduces the system of equations (iii) in Theorem 4.1 to ∂ψ1 ε + (ψ2 ψ¯3 + ψ¯2 ψ3 ) = 0, ∂ z¯ 2 ε ∂ψ2 − (ψ1 ψ¯3 + ψ¯1 ψ3 ) = 0, ∂ z¯ 2 1 ∂ψ3 − (ψ1 ψ¯2 − ψ¯1 ψ2 ) = 0. ∂ z¯ 2 Thus the map f : Ω → H3 with coordinates f1 = 2 Re ψ1 dz, f2 = 2 Re ψ2 dz, f2 f1 f3 = 2 Re ψ3 − ψ1 + ψ2 dz 2 2

(4.5) (4.6) (4.7)

defines a minimal immersion in H3 . These integral expressions are obtained from the fact that f2 f1 φ1 = ψ1 , φ2 = ψ2 , φ3 = ψ3 − ψ1 + ψ2 . 2 2 Some particular solutions of (4.5)–(4.7) yield Lorentz minimal surfaces in H3 invariant under a one-parameter subgroup of isometries. Let ( , θ, h), > 0, be cylindrical coordinates in H3 defined by x1 = cos θ,

x2 = sin θ,

x3 = h.

(4.8)

Given a constant a, we define a variable ζ by ζ = h − aθ. R2+

(4.9)

In the half-plane = {( , ζ) : > 0} we consider the graph ζ → ( (ζ), ζ) of a solution of the ODE k 2 − 2 2 d = (4.10) dζ k 4 2 − ( 2 − 2a)2

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where k is a given constant. This graph is a minimal immersion invariant under screw motions around the x3 -axis. In the particular case when a = 0, we have a minimal surface of revolution. This family of examples also comprise minimal cylinders given by = k. 4.2. The Hyperbolic Space H3 and de Sitter Space S31

The hyperbolic space H3 and de Sitter space S31 might be modeled as the halfspace R3+ = {(x1 , x2 , x3 ) ∈ R3 : x3 > 0} endowed with the metric 1 (dx21 + dx22 + εdx23 ), x23

(4.11)

where ε = 1 (resp. ε = −1) corresponds to the metric in H3 (resp. S31 ). It turns out that there is a natural Lie group structure both in H3 and 3 S1 given by the identification of R3+ with the subgroup of GL(4, R) consisting of matrices of the type ⎡ ⎤ 1 0 0 log x3 ⎢ 0 x3 0 x1 ⎥ ⎢ ⎥ , (x1 , x2 , x3 ) ∈ R3+ . ⎣ 0 0 x3 x2 ⎦ 0 0 0 1 The constant curvature metric (4.11) is left invariant with respect to this structure and the left invariant frame field given by ∂ ∂ ∂ , E2 = x3 , E3 = x3 E1 = x3 ∂x1 ∂x2 ∂x3 is orthonormal with respect to (4.11) with E3 , E3 = ε. In this case, system (iii) in Theorem 4.1 is written as ∂ψ1 + ε(ψ2 ψ¯2 + ψ3 ψ¯3 ) = 0, ∂ z¯ ∂ψ2 − ψ1 ψ¯2 = 0, ∂ z¯ ∂ψ3 − ψ1 ψ¯3 = 0. ∂ z¯

It is easy to see that φa = x3 ψa , a = 1, 2, 3. In particular, since f3 = 2Real f3 ψ3 one obtains f3 = exp2 Real

ψ3

.

Then the immersion f , if expressed in terms of the components ψa , a = 1, 2, 3, is given by a direct integration f (z) = 2 Real f3 ψ1 , 2 Real f3 ψ2 , f3 .

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4.3. The Product Group H2 × R We consider the half plane model for the hyperbolic plane given by H2 = {(x1 , x2 ) ∈ R2 : x2 > 0} and let s be the natural coordinate in R. Then x1 , x2 , x3 = s are global parameters in H2 × R. This space has a Lie group structure for which the frame ∂ ∂ ∂ E1 = x2 , E2 = x2 , E3 = ∂x1 ∂x2 ∂x3 is left invariant and orthonormal. Fixed this frame, it holds that φ1 = x2 ψ1 , φ2 = x2 ψ2 , φ3 = ψ3 .

In particular the second equation admits the solution f2 = exp2 Real ψ2 and the immersion f , in terms of the components ψa , is given by a direct integration f (z) = 2 Real f2 ψ1 , f2 , 2 Real ψ3 . The system (iii) in Theorem 4.1 is written as ∂ψ1 − ψ¯1 ψ2 = 0, (4.12) ∂ z¯ ∂ψ2 + εψ¯1 ψ1 = 0, (4.13) ∂ z¯ ∂ψ3 = 0. (4.14) ∂ z¯ We observe that if ψ2 is a L-differentiable function then it follows from (4.12)– 1 (4.13) that ψ1 ψ¯1 = 0 and ψ1 ∂ψ ∂ z¯ = 0. We may consider for example ψ1 = 0 what corresponds to planes x1 = cte.

References [1] Al´ıas, L.J., Chaves, R.M.B., Mira, P.: Bj¨ orling problem for maximal surfaces in Lorentz-Minkowski space. Math. Proc. Camb. Philos. Soc. 134(2), 289–316 (2003) [2] Bj¨ orling, E.G.: In integrazionem aequationis derivatarum partialum superfici cujus inpuncto uniquoque principales ambos radii curvedinis aequales sunt signoque contrario. Arch. Math. Phys. 4(1), 290–315 (1844) [3] Chaves, R.M.B., Dussan, M.P., Magid, M.: Bj¨ orling problem for timelike surfaces in the Lorentz-Minkowski space. J. Math. Anal. Appl. 377(2), 481–494 (2010) [4] Kobayashi, O.: Maximal surfaces in 3-dimensional Lorentz space L3 . Tokyo J. Math. 6, 297–309 (1983) [5] Konderak, J.: A Weierstrass representation theorem for Lorentz surfaces. Complex Var. Theory Appl. 50(5), 319–332 (2005) [6] Kokubu, M.: Weierstrass representation for minimal surfaces in hyperbolic space. Tˆ ohoku Math. J. 49, 367–377 (1997)

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[7] Lee, S.: Maximal surfaces in a certain 3-dimensional homogeneous spacetime. Differ. Geom. Appl. 26(5), 536–543 (2008) [8] Lawn, M.A.: Immersions of Lorentzian surfaces in R2,1 . J. Geom. Phys. 58, 683– 700 (2008) [9] Mercuri, F., Montaldo, S., Piu, P.: Weierstrass representation formulae of minimal surfaces in H3 and H2 × R. Acta Math. Sinica (English Series) 22(6), 1603–1612 (2006) [10] Mercuri, F., Onnis, I.: On the Bj¨ orling problem in a three-dimensional Lie group. Ill. Math. J. 53(2), 431–440 (2009) [11] Schwarz, H.A.: Gesammelte Mathematische Abhandlungen. Springer, Berlin (1890) [12] Weinstein, T.: An Introduction to Lorentz Surfaces. de Gruyter Expositions in Mathematics 22. Walter de Gruyter, Berlin-New York (1996) J. H. Lira, M. Melo and F. Mercuri Departamento de Matem´ atica Universidade Federal do Cear´ a Campus do Pici 60455-900 Fortaleza, CE, Brazil e-mail: [email protected]; [email protected] F. Mercuri Departamento de Matem´ atica Universidade Estadual de Campinas 13081-970 Campinas, SP, Brazil e-mail: [email protected]

Received: January 12, 2011. Revised: May 7, 2011. Accepted: May 13, 2011.