1 Introduction SPACELIKE SURFACES IN L 4 WITH PRESCRIBED GAUSS MAP AND NONZERO MEAN CURVATURE. A. C. Asperti J. A. M. Vilhena

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1 Matemática Contemporânea, Vol 33, c 2007, Sociedade Brasileira de Matemática SPACELIKE SURFACES IN L 4 WITH PRESCRIBED GAUSS MAP AND NONZERO MEAN CURVATURE A. C. Asperti J. A. M. Vilhena Dedicated to Francesco Mercuri & Renato Tribuy on their sixtieth birthdays Abstract In this work we study the generalied Gauss map of spacelike surfaces in the Lorent-Minkowski space L n, with emphasis to the case n = 4. We present necessary and sufficient conditions for a complex map to be a Gauss map of a spacelike surface in L 4 and a representation formula of Kenmotsu type, and this gives a method to construct spacelike surfaces with prescribed nonvanishing scalar mean curvature and Gauss map. We also present the extension to L 4 of the complete integrability conditions for existence of surface in R 3 and L 3 of Kenmotsu and Akutagawa-Nishikawa. 1 Introduction Let {e i : 1 i n} be the canonical basis of the real vector space R n, n 2, and define a symmetric bilinear form, of index 1 in R n by u, v = u 1 v 1 + +u n 1 v n 1 u n v n, where u = u i e i and v = v i e i are vectors of L n. The form,, is the Lorent scalar product and the n dimensional Lorent Minkowski space L n is the space R n endowed with this form. A spacelike surface in L n, n 3, is the image S = XM 2 of a connected, oriented abstract surface M 2 through a locally conformal immersion X : M 2 L n such that the induced form ds 2 = X, is a Riemannian metric in M Mathematics Subject Classification. Primary 53A10; Secondary 53C42, 53C50. Key words and phrases. Lorent Minkowski space, Riemann surfaces, spacelike surfaces, Gauss map The first author was benefitted by Projeto Temático FAPESP 99/ The second author was partially supported by CNPq, proc /

2 56 A. C. ASPERTI J. A. M. VILHENA The classical Gauss map for surfaces in R 3, was introduced by Gauss in his fundamental work on the theory of surfaces and he used it to define what today is known as Gauss curvature. For surfaces of higher codimension in Euclidian space, we have a natural generaliation of the Gauss map, where the image space is the Grassmannian G 2,n of the oriented 2-planes in R n, which may be identified with the complex quadric Q n 2 of the complex projective space CP n 1. D. Hoffman and R. Osserman in [6] studied in detail the properties of the generalied Gauss map of a surface S immersed in R n. For connected, oriented spacelike surfaces S the n dimensional Lorent-Minkowski space, it is also natural to think in the generalied Gauss map. This concept was introduced by F. J. M. Estudillo and A. Romero [4] as follows: let G + 2,n be the set of all oriented spacelike 2 planes in Ln. It is known that G + 2,n is an open subset of the classical Grassmannian of 2-planes in Ln. Also, we may identify G + 2,n with the complex quadric Qn 2 1 := {[] CP n 1 1 : n 1 2 n 2 = 0}, where CP n 1 1 := { C n \ {0} :, > 0}/C and, w = 1 w n w n is the indefinite hermitian form in C n, see [3, 12]. Therefore, if we associate to each point p M 2, the tangent plane T p M in G + 2,n, we can define a map G : M 2 Q n 2 1 which is called the generalied Gauss map of the spacelike surface S = XM 2 in L n. Motivated by these results, in this work we firstly study what conditions a generalied Gauss map G : M 2 Q n 2 1 of a spacelike surface in L n must satisfy by the virtue of being such a map. This is done in section 2, where the main result is Theorem 2.2. The similar problem in the Euclidian space R n was studied by D. Hoffman and R. Osserman in [7, 8]. In section 3 we study the generalied Gauss map of spacelike surfaces S = XM 2 in L 4 and prove that if it is given locally by a function Φ, that is, X = µφ, with Φ =

3 SPACELIKE SURFACES IN THE LORENTZ-MINKOWSKI SPACE ab, i1 ab, a b, a + b [2, 5], then F1 F 2 = F 1 F 2 and Im{T 1 + T 2 } = 0 see Theorem 3.3. This result is very important in the study of a special class of the spacelike surfaces in L 4, the surfaces with degenerate Gauss map [15]. In section 4 we relate our work with the work of K. Kenmotsu, and K. Akutagawa and S. Nishikawa. In [9], Kenmotsu proved that the Gauss map and the mean curvature of an arbitrary surface in R 3 satisfy a second order differential equation, and later he extended this result for surfaces in R 4, see [10]. In [1] Akutagawa and Nishikawa, showed that a same type of equation is satisfied for arbitrary spacelike surfaces in L 3. In Theorem 4.1, we show that the generalied Gauss map and the mean curvature vector of a spacelike surface in L 4 satisfy a second order differential equation which generalies the above mentioned equations. In Proposition 4.3, we introduce a complex representation formula of Kenmotsu s type for simply connected spacelike surfaces conformally immersed in L 4 with nonero mean curvature h = H, H ; we emphasie that this is equivalent to say that the mean curvature vector H is nonero nor lightlike. Finally in section 5 we prove our main result Theorem 5.3, which gives sufficient conditions for a complex map G : M 2 Q 2 1 to be the generalied Gauss map of a conformal spacelike immersion X of M 2 onto S in L 4 with nonero mean curvature h. We observe that besides the Weierstrass type representation for spacelike surfaces in L 4 given in [2, 5], there exists another Weierstrass type representation that is discussed in [11].

4 58 A. C. ASPERTI J. A. M. VILHENA 2 The generalied Gauss map of spacelike surfaces in L n The generalied Gauss map G of a spacelike surface S in L n, given by a conformal immersion X : M 2 L n, is locally defined by G = [X ], 1 where = u + iv is a conformal parameter for M 2. The local conformality of S is expressed by λ 2 = X u, X u = X v, X v, X u, X v = 0. Thus ds 2 = λ 2 d 2, λ 2 = 2 X, X. 2 Moreover, M X = 2H, X = λ2 H, 3 2 where M X is the Laplacia-Beltrami operator on M 2 and H is the mean curvature vector field. Now given a map of M 2 in the quadric Q n 2 1 of CP n 1 1, we can represent it locally in the form [Φ], where Φ = φ 1,..., φ n satisfies φ φ φ n 1 2 φ n 2 = 0. 4 This map describes the Gauss map of a spacelike surface S in L n, given by X : M 2 L n, if X = µφ, 5 for some map µ : M 2 C. Note that S is regular where µ is nonvanishing. Therefore, from 2, 3 and 5 it follows that λ 2 = 2 X, X = 2 µ 2 Φ, Φ, λ 2 2 H = µφ = µ Φ + µφ.

5 SPACELIKE SURFACES IN THE LORENTZ-MINKOWSKI SPACE 59 Thus, whenever µ 0. µ Φ, Φ H = Φlog µ + Φ, 6 Lemma 2.1. Let W be a vector of C n 1 in the form W = A + ib where A and B are spacelike vectors of L n satisfying A, A = B, B and A, B = 0. Let Π 2 be the spacelike 2 plane spanned by A and B. For any pair of vectors C and D of L n, let Z = C + id C n 1 and define Z := C + id, where C and D are the projections of C and D on Π 2. Then, Z = Z, W W, W W + Z, W W, W W, 7 where C n 1 := C n,,, with, w := n 1 k=1 k w k n w n,, w C n. Proof: Since C = C, A C, B A + A, A B, B B, D = are the projections of C, D on Π 2 and A = W + W 2 the result follows., B = W W 2i, C = Z + Z 2 D, A D, B A + A, A B, B B, D = Z Z, 2i We can apply Lemma 2.1 for W = Φ, in such a way that Π 2 is the tangent plane of a surface S, given by X such that X = µφ and Z = Φ. Since [Φ] Q n 2 1, then Φ, Φ = 0. Hence, Φ = ηφ, η := Φ, Φ Φ, Φ. 8

6 60 A. C. ASPERTI J. A. M. VILHENA Denoting by V the component of Φ orthogonal to the plane Π 2 = T p S, that is, V := Φ Φ, we have V = Φ ηφ. 9 The mean curvature vector H is orthogonal to the tangent plane of S, thus from µ Φ, Φ H = Φ ηφ + log µ + ηφ, it follows that log µ + η = 0, 10 and by 9 we have V = µ Φ, Φ H. 11 The proofs of Theorem 2.2 and of Lemma 2.3 below are similar to the proofs given by Hoffman and Osserman to Theorem 2.3 and to Lemma 2.4 in the case R n [7]. For this reason these proofs will be omitted here. Theorem 2.2. Let S be a spacelike surface in L n locally given by a conformal map X : Ω L n. Let Φ be the Gauss map of S, that is, X = µφ. Then, for every Ω, V is of the form V = e iα R, 12 where R is a vector of L n. Furthermore, on the set { Ω : V 0}, the function α : Ω R is uniquely defined modulo 2π, and satisfies α = Imη. 13 Note that V and η given in 12 and 13, are expressed explicitly by 8 and 9 in terms of the local representation Φ of the Gauss map G. By Lemma 2.3 below, V and η are independent of the particular representation of G.

7 SPACELIKE SURFACES IN THE LORENTZ-MINKOWSKI SPACE 61 Lemma 2.3. Given a map Φ : Ω C n \ {0}, set Φ = fφ where f is a smooth nonvanishing complex function. Let η, V be defined in terms of Φ as in 8 and 9 and the correspondents η, V in terms of Φ. Then V = fv and in the set where V and V are nonero, the functions α, α defined by 12 satisfy α Im η = α Imη. 3 The generalied Gauss map for spacelike surfaces in L 4 In this section we shall obtain explicit conditions that the generalied Gauss map of a spacelike surface S in L 4 must satisfy. Let S be a spacelike surface, immersed in L 4 by X : M 2 L 4 with generalied Gauss map G : M 2 Q 2 1, given locally by G = [X ] where U, = u+iv are local isothermal coordinates of M 2. It follows from 4 that we may express G by a pair of complex functions a := φ3 + φ 4 φ 1 iφ 2, b := φ3 + φ 4 φ 1 iφ 2 Since λ2 = 4 µ 2 1 ab 2, then ab 1. Hence, we can write G = [Φ] where Φ = 1 + ab, i1 ab, a b, a + b. 14 We now introduce certain auxiliary functions derived from the functions a and b describing the Gauss map, as follows: F 1 = F 1 a, b := a 1 ab, F 2 = F 2 a, b := b 1 ab, 15 F 1 = F 1 a, b := a 1 ab, F2 = F 2 a, b := b 1 ab,

8 62 A. C. ASPERTI J. A. M. VILHENA S 1 a, b := a a + 2b T 1 a, b := a 1 ab, S 2a, b := b b + 2a b 1 ab, 16 a + 2b a F b 1, T 2 a, b := + 2a b F 2, 17 where T 1 and T 2 are defined on the set { : a 0, b 0}. Lemma 3.1. Let Rw = αw + β βw + α, α 2 β 2 = 1, be a Möbius transformation and Y any of the auxiliary functions F 1 F 2, F 1 F 2, F1 F2, F1 F2, S k and T k, k = 1, 2. Then Y is invariant by Ra, Rb, that is, Y Ra, Rb = Y a, b. Proof: We will indicate the proof only for the function S 1 and the proof for the other functions is analogous. By a straightforward calculation, we obtain that Ra = a βa + α, Ra a 2 = βa + α, 2 Ra = a βa + α 2a a β βa + α 3. Since the same is true for Rb, we obtain that 1 RaRb = 1 ab βa + αβb + α. Thus, S 1 Ra, Rb = Ra Ra + 2RbRa 1 RaRb, S 1 Ra, Rb = a a 2a β βa + α + 2αb + β βa + α a 1 ab = a a + 2ba 1 ab = S 1a, b.

9 SPACELIKE SURFACES IN THE LORENTZ-MINKOWSKI SPACE 63 Clearly the functions F k and T k above are smooth whenever w k w 1 = a, w 2 = b are finite. Now if a = or b =, we may apply a Möbius transformation as above in a way that the functions a and b stay both finite in a neighborhood of a point. This corresponds to a Lorent transformation in the surface S of L 4. In fact, consider the group SU1, 1 = U1, 1 SL2, C, that is, {[ ] } α β SU1, 1 = : α 2 β 2 = 1. β α [ ] α β Therefore, for A = in SU1, 1, we have that the matrix β α Reα 2 + β 2 Imα 2 β 2 0 2Reαβ Λ A = Imα 2 + β 2 Reα 2 β 2 0 2Imαβ Reαβ 2Imαβ 0 α 2 + β 2 is in O ++ 3, 1, the group of Lorent transformations of L 4 which preserve space and time orientation. We conclude that to each change of conformal coordinates in the spacelike surface S of L 4, through a Möbius transformation Rw in such a way a and b are finite, corresponds a Lorent transformation Λ A O ++ 3, 1 of L 4 for more details see [14]. Lemma 3.2. Let F k and F k, k = 1, 2, defined in 15. Then { bf1 Im + } { b af 2 = Im F1 + a F } 2. Proof: By a direct calculation, we obtain that bf1 b F 1 = a b a b 1 ab 2,

10 64 A. C. ASPERTI J. A. M. VILHENA since b = b and b = b. Analogously, Therefore, On the other hand, { bf1 2iIm + } af 2 This proves the lemma. af2 a F 2 = a b a b 1 ab 2. af2 bf 1 = a F 2 b F 1 18 =. 18 af2 = bf 1 a F2 b F 1 b = F1 + a F b 2 F1 + a F 2 { b = 2iIm F1 + a F } 2. af2 bf 1 a F2 b F 1 Now we present the necessary conditions for a map in Q 2 1 to be a generalied Gauss map of a spacelike surface in L 4. Theorem 3.3. Let S be an oriented spacelike surface given by the immersion X : M 2 L 4, with generalied Gauss map G locally given by 14 via the pair of functions a and b, where is a local conformal parameter on M 2. Then, F 1 F 2 = F 1 F 2, 19 Im{T 1 + T 2 } = 0 whenever a 0, b Proof: We apply Theorem 2.2 to show that 12 and 13 imply respectively 19 and 20. The first step is to express the functions η and V, defined

11 SPACELIKE SURFACES IN THE LORENTZ-MINKOWSKI SPACE 65 in 8 and 9, in terms of the functions a, b and F k. From 14 we have Φ = a b, ib, 1, 1 + b a, ia, 1, 1, 21 Φ, Φ = 2 1 ab 2 > 0, 22 because ab 1. It follows that Φ, Φ = 2ba ab 1 + 2ab b 1. Hence, η = ba 1 ab ab 1 ab η = bf 1 + af Now, substituting 14, 21 and 23 in V = Φ ηφ, yields V = + a 2Reb, 2Imb, 1 b 2, 1 + b ab b 2Rea, 2Ima, a 2 1, a ab = F 1 B + F 2 A, 24 where A : = 2Rea, 2Ima, a 2 1, a 2 + 1, 25 B : = 2Reb, 2Imb, 1 b 2, 1 + b Note that A and B are nonero, lightlike and future directed vectors of L 4, such that A, B = 2 1 ab 2 < 0. So, they are linearly independent vectors of the light cone of L 4. From 24, we conclude that V = 0 if and only if F 1 = F 2 = 0, which is equivalent to a = b = 0. In particular, wherever V = 0, the condition 19 is trivially satisfied. Letting V = V 1, V 2, V 3, V 4, A = A 1, A 2, A 3, A 4 and B = B 1, B 2, B 3, B 4, the condition 12 of Theorem 2.2, implies that V t V = R t R wherever V 0.

12 66 A. C. ASPERTI J. A. M. VILHENA Therefore, the entries V j V k, 1 j, k 4 of the matrix V t V must be real. On the other hand, from 24 we have V j V k = F 1 B j + F 2 A j F 1 B k + F 2 A k = = F 1 2 B j B k + F 2 2 A j A k + F 1 F 2 A j B k + F 1 F 2 A k B j. Thus, V j V k R if and only if F 1 F 2 A j B k + F 1 F 2 A k B j R because F 1 2 B j B k + F 2 2 A j A k is a real number. Hence, This implies that V j V k R F 1 F 2 F 1 F 2 A j B k A k B j = 0. F 1F 2 F 1F 2 = 0 or A j B k A k B j = 0, 1 j < k We will show that A j B k A k B j is nonero for all 0 U, where U, = u+iv are local isothermal coordinates of S. In fact, since S is a spacelike surface, the metric ds 2 = 4 µ 2 1 ab 2 d 2 induced by X : M 2 L 4 is Riemannian, hence ab 1 on U. Let 0 be any point in U such that V 0 0 and A j 0 B k 0 A k 0 B j 0 = 0, 1 j < k 4. These six equations imply that 1. ab = ab, 2. Rea + b1 ab = 0, 3. Rea b1 ab = 0, 4. Ima + b1 ab = 0, 5. Ima b1 ab = 0, 6. a 2 b 2 = 1. By the equations 1 and 6 above we have that a 2 b 2 = 1 abab = 1 ab 2 = 1 a 0 b 0 = ±1. 28 Now using equations 2, 3, 4 and 5 we may easily conclude that a 0 = b 0 = 0, which contradicts 28. Hence, V t V is a real hermitian matrix if and only if F 1 F 2 = F 1 F 2, which proves 19.

13 SPACELIKE SURFACES IN THE LORENTZ-MINKOWSKI SPACE 67 To prove 20, we need first to establish a relation between the argument of the coordinates of V and the argument of the functions F 1 and F 2. From 19 and 24 we have that F 1 V = F 1 F 1 B + F 2 A = F 1 F 1 B + F 1 F 2 A = F 1 F 1 B + F 1 F 2 A = F 1 F 1 B + F 2 A = F 1 V. In a similar way, we prove that F 2 V = F 2 V. Since we are assuming that a 0 and b 0, we have F 1 0 and F 2 0. So V 0 and there exists some nonero component V j of V which satisfies F 1 V j = F 1 V j, F 2 V j = F 2 V j. 29 Then 29 implies that argf 1 V j = 0 mod π and argf 2 V j = 0 mod π, which are equivalent to argf k + argv j = 0 mod π, k = 1, 2. Now from 12 we have V = e iα R, then argv j = α mod 2π. Therefore, α = argf k mod π, k = 1, 2. Hence α = 1 2 argf1 + argf 2 α = 1 2 argf1 F 2 mod π Now 15 yields { } a argf 1 F 2 = arg 1 ab b a b = Im log. 1 ab 1 ab 2 Therefore, Thus, α = 1 2 Im log a + log b + 2nπ, n Z. α = 1 { 2 Im a + a b b }. 31

14 68 A. C. ASPERTI J. A. M. VILHENA From 31 we have that α = 1 { 2 Im a + 2b F b a F 2 a b α = 1 } b {T 2 Im 1 a, b + T 2 a, b Im F1 + a F 2 b 2 F1 + a F } 2,. 32 On the other hand, we have seen from 13 that α = Imη. Hence from 23 we have that bf1 α = Im + af Therefore, comparing 32 with 33 and using the Lemma 3.2, we prove 20. Finally, observe that 19 and 20 are independent of the choice of the conformal coordinates of M 2. This concludes the proof of the theorem. In the following corollary we will be using the notation of Akutagawa and Nishigawa for spacelike surfaces in L 3 [1]. Corollary 3.4. Let S be a spacelike surface in L 3 defined by the immersion X : M 2 L 3, and let w = f be the local representation of the Gauss map, that is N = 1 1 f 2 2Ref, 2Imf, 1+ f 2 and f 1, in local isothermal coordinates U, = u + iv of M 2. following two conditions must hold: Then at every point of M 2, one of the { f i f = 0 or ii f 0 and Im + 2ff } f 1 f 2 = 0. Remark 3.5. We can give an example of a map [Φ] : U Q 2 1 which cannot represent the Gauss map of any spacelike surface in L 4. In fact, for instance setting a = + and b = i in 14, we have a = 1 and b = i.

15 SPACELIKE SURFACES IN THE LORENTZ-MINKOWSKI SPACE 69 Hence, F 1 F 2 = i 1 4ReIm 2, F i 1F 2 = 1 4ReIm 2. This tells us that the condition 19 does not hold. 4 The Kenmotsu s type formula for spacelike surfaces in L 4 In this section we prove that the mean curvature vector field H and the components a, b of the Gauss map of a spacelike surface in L 4 must satisfy a second order partial differential equation. We also write the integration factor µ, defined in 5, explicitly in terms of a, b and H. This allows us to give a representation formula for spacelike surfaces in L 4 in terms of the Gauss map and of mean curvature vector H. In [1] it was proved that the Gauss map f and the mean curvature h of a spacelike surface in L 3 need to satisfy the equation h f + 2ff f 1 f 2 = h f S 1 f, f = log h 34 wherever f 0. Since h is real, we have that log h is a real number. Hence, Im{S 1 f, f } = 0 Im{T 1 f, f} = 0. Our goal now is to generalie the equation 34 for spacelike surfaces in L 4, in such a way that the necessary condition Im{T 1 + T 2 } = 0, given in 20, becomes a consequence of this generalied equation. More precisely, we have the following. Theorem 4.1. Let S be a spacelike surface immersed in L 4 by X : M 2 L 4, with generalied Gauss map G given locally by 14 via the pair of functions

16 70 A. C. ASPERTI J. A. M. VILHENA a and b, where is a local conformal parameter on M 2. Then the mean curvature vector H and the functions a and b satisfy the second order PDE H, H b a + 2ba a + a b + 2ab b = a b H, H ab 1 ab Proof: The Gauss map of S is locally defined by G = [X ] and given by 14, that is, X = µφ for some function µ : M 2 C. From 8 and 9 we obtain We can easily verify that µ Φ, Φ H = Φ ηφ, η := Φ, Φ Φ, Φ. µ Φ, Φ H, µ Φ, Φ H = = Φ, Φ 2η Φ, Φ +η 2 Φ, Φ. Since [Φ] Q 2 1, that is Φ, Φ = 0, we have that Φ, Φ = 0. Thus, the above equation is reduced to Φ, Φ 2 H, H µ 2 = Φ.Φ. 36 Let us now explicitly calculate Φ, Φ in function of a and b. From 21 we have that Therefore, Φ, Φ = 4a b. 37 Φ, Φ 2 H, H µ 2 = 4a b. 38 Now differentiating µ Φ, Φ H = Φ ηφ with respect to, we obtain Φ, Φ µh+ Φ, Φ µh = Φ η Φ ηφ.

17 SPACELIKE SURFACES IN THE LORENTZ-MINKOWSKI SPACE 71 On the other hand, Φ, Φ µh+ Φ, Φ µh = Thus, = Φ, Φ + Φ, Φ µh+ Φ, Φ µ H + µh. Φ, Φ µh = Φ η Φ ηφ Φ, Φ + Φ, Φ µh Moreover, note that µh = Φ, Φ Hµ. Φ ηφ Φ, Φ, Φ, Φ H = Φ ηφ. Therefore, µ Φ, Φ µh = Φ ηφ Φ, Φ Φ, Φ + Φ, Φ Φ, Φ + µ Φ + µ Φ, Φ + Φ, Φ + Φ, Φ Φ, Φ + µ η η Φ. µ From 10 we know that µ µ = η. Also, η = Φ, Φ, so that Φ, Φ Φ, Φ µh = Φ ηφ Φ, Φ Φ, Φ, Φ Φ Φ + Φ, Φ η η Φ. 39 Calculating the symmetric product,, between 39 and µ Φ, Φ H = Φ ηφ, we have that Φ, Φ 2 H, H µ 2 = Φ, Φ η Φ, Φ + Φ, Φ Φ, Φ Φ, Φ Φ, Φ. Now using that 0 = Φ, Φ = Φ, Φ + Φ, Φ in the above equation, we get Φ, Φ 2 H, H µ 2 = 1 2 Φ, Φ Φ, Φ Φ, Φ Φ.Φ. 40 From 21 and Φ, Φ = 21 ab1 ab, we can verify that Φ, Φ Φ, Φ = b F 1 a, b + a F 2 a, b, 41

18 72 A. C. ASPERTI J. A. M. VILHENA where F i is given in 15. Finally, substituting 37 and 41 into 40, we obtain Φ, Φ 2 µ 2 H, H = 2 a b + a b + 2 b F 1 + a F 2 get = 2 b a + 2ba a 1 ab 2a b = + a b + 2ab b 1 ab Taking the product of the above equation with H, H and using 38, we b a + 2ba a + a b + 2ab b H, H = 2a b H, H, 1 ab 1 ab which proves the theorem. Let S be a spacelike surface in L 3. We may identify L 3 with the subset of L 4 given by {x 1, x 2, x 3, x 4 L 4 : x 3 = 0}. So S can be considered as a spacelike surface in L 4 and, in this case, we denote it by Š and we may assume that b = a in 14. In analogous way, any surface S in R 3 can be viewed as a surface Ŝ in R3 {x 1, x 2, x 3, x 4 L 4 : x 4 = 0} and in this case we may assume that b = a in 14. Now we will use formula 35 to obtain, in a new way, the integrability conditions for a spacelike surface in L 3 and a surface in R 3 with prescribed nonvanishing mean curvature h and Gauss map a. The vectors A and B given in 25 and 26, are future directed lightlike vectors such that A, B = 2 1 ab 2 < 0, therefore are linearly independent wherever a and b describe the Gauss map Φ of a spacelike surface S in L 4. It is easy to see that Φ, A = 0 and Φ, B = 0. Thus, {A, B} is a basis.

19 SPACELIKE SURFACES IN THE LORENTZ-MINKOWSKI SPACE 73 for T p S. Moreover, H T p S and then we can write H in the form Hence H = H, H = H, B A + H, A B. A, B 2 H, A H, B. 42 A, B 1 st case: If b = a, we know that Φ = 1 + a 2, i1 a 2, 0, 2a locally represents the Gauss map of a spacelike surface Š in L3 L 4. In this case, the mean curvature vector H = Ȟ is timelike and parallel to the classical Gauss map Ň : M 2 H of Š, where H2 0 1 = {x L 3 : x, x = 1}. From 42 we have that Ǎ Ȟ, ˇB Ȟ, Ȟ = Ȟ, 1 a 2 2, with Ǎ = ˇB = 2Rea, 2Ima, 0, a It is well-known that Ň = Hence from 43, we have that 1 1 a 2 2Rea, 2Ima, 0, 1 + a Ǎ = 1 a 2 Ň, ˇB = 1 a 2 Ň. Therefore, Ȟ, Ȟ = Ȟ, Ň 2 = h 2, where h is the mean curvature of Š. Substituting b = a in 35, we have 2h 2 a a + 2aa a 1 a 2 = 2hh a 2 ha h a + 2aa a 1 a 2 h a = 0. Since h = 0 if and only if a = 0, the above equation implies that h a + 2aa a 1 a 2 = h a.

20 74 A. C. ASPERTI J. A. M. VILHENA This equation is a complete integrability condition for the existence of a spacelike surface with prescribed nonvanishing mean curvature and Gauss map in L 3 and was firstly obtained by Akutagawa and Nishikawa [1]. 2 nd case: If b = a, we know that Φ = 1 a 2, i1+a 2, 2a, 0 locally represents the Gauss map of spacelike surfaces Ŝ in R3 L 4. In this case, the mean curvature vector H = Ĥ is spacelike and parallel to the classical Gauss map ˆN : M 2 S 2 1 of Ŝ. It is well-known that ˆN = Now from 42 we have that a 2 2Rea, 2Ima, a 2 1,  Ĥ, ˆB Ĥ, Ĥ = Ĥ, 1 + a 2 2, with  = ˆB = 2Rea, 2Ima, a 2 1, 0. Hence from 44 it follows that  = 1 + a 2 ˆN, ˆB = 1 + a 2 ˆN. Therefore, Ĥ, Ĥ = Ĥ, ˆN 2 = h 2, where h is the mean curvature of Ŝ. Substituting b = a in 35, we get 2h 2 a a 2aa a 1 + a 2 = 2hh a 2 ha h a 2aa a 1 + a 2 h a = 0. Since h = 0 if and only if a = 0, the above equation implies that h a 2aa a 1 + a 2 = h a.

21 SPACELIKE SURFACES IN THE LORENTZ-MINKOWSKI SPACE 75 This is a complete integrability condition for the existence of a surface with prescribed nonvanishing mean curvature and Gauss map in R 3 and was firstly obtained by Kenmotsu in [9]. As an immediate consequence of Theorem 4.1 we have the following Corollary 4.2. Under the hypothesis of Theorem 4.1, if the mean curvature h = H, H is nonvanishing for every point of M 2, then S 1 a, b + S 2 a, b = log H, H, 45 where is a local conformal parameter on M 2. Observe that since H, H is a real function, then log H, H R. Hence, 45 implies that Im{S 1 a, b + S 2 a, b } = 0 Im{T 1 a, b + T 2 a, b} = 0. Therefore, the equation 45 implies the equation 20 of Theorem 3.3. We can give another interesting consequence of the proof Theorem 4.1. From equation 38 it follows that µ 2 = 4a b H, H Φ, Φ 2 wherever H, H 0. In other words, the integration factor µ can be given explicitly by µ 2 = a b H, H 1 ab Proposition 4.3. Let S be a spacelike surface in L 4 whose Gauss map is given locally by 14. If the mean curvature vector H is nonvanishing nor lightlike, then the surface S can be obtained explicitly by X = 2Re 0 µ 1 + ab, i1 ab, a b, a + b dw + X 0, 47

22 76 A. C. ASPERTI J. A. M. VILHENA where µ is given by Sufficient conditions for a map to be a Gauss map We will show now that the conditions 19 and45 are also sufficient for the existence for spacelike surface with prescribed Gauss map and nonero mean curvature. Therefore, it will also holds the converse of Proposition 4.3. That is, given an arbitrary map of a simply connected Riemann surface into the quadric Q 2 1, locally represented by Φ, as in 14, via the pair of functions a, b, such that they satisfy the conditions F 1 F 2 = F 1 F 2, V, V 0 and S 1 +S 2 = log h, where h : M 2 R\{0} is a smooth function, then 47 defines a spacelike surface immersed in L 4 with locally Gauss map given by Φ, H, H = h and induced metric ds 2 = 4 µ 2 1 ab 2 d 2 given, using 46, by ds 2 4 a b = d 2. H, H 1 ab 2 To prove these results we need two lemmas. Lemma 5.1. Let U be a simply connected open set of C and let Φ : U C 4 be a C 2 map. Then there exists a spacelike surface S given by X : U L 4 such that X = Φ if, and only if ImΦ = 0. Proof: We know that X = λ2 2 H R4. Hence, if there exists X : U L 4 such that X = Φ it follows that ImΦ = 0. Conversely, the condition ImΦ = 0 implies that the system { Xu = 2ReΦ X v = 2ImΦ

23 SPACELIKE SURFACES IN THE LORENTZ-MINKOWSKI SPACE 77 is integrable and the solution is given by X = 2 Φdw. Lemma 5.2. If the system X = µφ, with µ 2 = a b, Φ = 1 + ab, i1 ab, a b, a + b, h : C R \ {0} h 1 ab 4 satisfies S 1 a, b + S 2 a, b = log h, then X = µv, 48 where V = F 1 B + F 2 A. Proof: Putting P = h1 ab 2 1 ab 2, we can write µ 2 = a b P. Thus, a b 2µ µ = P a b P P 2, and since a = a and h = h we have that a b + a b P a b P 2µ µ = P 2, 49 P = h 1 ab 4 + h 2 1 ab 2 1 ab a b ab ab 2 1 ab a b ab. Substituting 50 in 49 we have that 2µ µp 2 = 1 ab h 4 b a + 2b a a b 1 ab 1 ab 2 + a b + 2a b b a 1 ab 1 ab 2 + a b 2hba 1 ab 1 ab a b 2hab 1 ab 1 ab 2, + a b h + 51

24 78 A. C. ASPERTI J. A. M. VILHENA 2µ µp 2 = 1 ab h b 4 a + 2b a a + 1 ab + a b + 2a b b a b h ab +a b 2hba 1 ab 1 ab 2 + a b 2hab 1 ab 1 ab 2. By hypothesis S 1 a, b + S 2 a, b = log h hence, 1 µµ = b b 1 ab a h 1 ab a b 1 ab b On the other hand, µx = µµ Φ + µ 2 Φ, 54 Φ = a b, ib, 1, 1 + b a, ia, 1, Substituting 53 and 55 in 54, yields µx 1 = a b a h 1 ab 4 1 ab 2Reb + b 1 ab 2Rea, µx 2 = a b h 1 ab 4 µx 3 = a b a h 1 ab 4 µx 4 = a b h 1 ab 4 a 1 ab 2Imb + b 1 ab 2Ima, 1 ab 1 b b ab a 2 1, a 1 ab 1 + b b ab a Thus, in agreement with the definitions of F 1, F 2, A, B and V we have that that is, µx = µ 2 a 1 ab B + b 1 ab A, X = µv.

25 SPACELIKE SURFACES IN THE LORENTZ-MINKOWSKI SPACE 79 Theorem 5.3. Let M 2 be a simply connected Riemann surface and G : M 2 Q 2 1 be a smooth map given locally by Φ = Φa, b as in 14 and let h : M 2 R \ {0} be a smooth function. Then there exists a spacelike surface S given by a conformal immersion X : M 2 L 4 with generalied Gauss map G and mean curvature vector H satisfying H, H = h, if and only if Φ and h satisfy: 1 F 1 F 2 = F 1 F 2, a b 0; 2 S 1 a, b + S 2 a, b = log h. Moreover, S is given explicitly by X = 2Re with µ 2 = a b h 1 ab. 4 0 µ 1 + ab, i1 ab, a b, a + b dw + X o, 56 Proof: The necessity of conditions 1 and 2 is an immediate consequence of Theorem 3.3 and Corollary 4.2. Conversely, suppose that Φ = Φa, b and h satisfy 1 and 2. To prove that Φ locally represent the generalied Gauss map of some spacelike surface S given by the immersion X : M 2 L 4 we need to show that X = µφ for some function µ : M 2 C. Therefore, we need to prove that the system is integrable. X = µ1 + ab, i1 ab, a b, a + b, µ 2 = a b h 1 ab 4 57 Now by Lemma 5.2, it follows that X = µv On the other hand, the equation F 1 F 2 = F 1 F 2 implies that V j V k R, 1 j, k 4. 58

26 80 A. C. ASPERTI J. A. M. VILHENA Set V j = ρ j e iθj, V k = ρ k e iθ k, hence V j V k = ρ j ρ k e iθj θ k is a real number if, and only if, θ j θ k modπ. Putting θ = θ j0, for some j 0 between 1 and 4, we have that V = e iθ R, with R = R 1, R 2, R 3, R 4 L 4, so argv j = θmodπ, j. 59 Since F k V j = F k V j for all j = 1, 2, 3, 4 and k = 1, 2 we also have that argv j = argf k modπ. 60 From 59 and 60 it follows that θ = argf k modπ, and this implies that 2θ = argf 1 F 2 modπ. 61 On the other hand, from expression of µ given in 57 we get that 2argµ = argf 1 F 2 modπ. 62 Suppose that µ is written in the form µ = ρe iα, that is argµ = α. Then from 61 and 62 we conclude that θ αmodπ. Then V = e iα R. 63 Thus, X = µv X = ρr. Therefore, by Lemma 5.1 the system 57 is integrable and has the solution: X = 2Re µ1 + ab, i1 ab, a b, a + bdw + X 0, 0 µ 2 = a b h 1 ab 4. Let H be the mean curvature vector of the spacelike surface S given by X above. It remains to show that H, H = h. From Corollary 4.2, it follows that S 1 a, b + S 2 a, b = log H, H.

27 SPACELIKE SURFACES IN THE LORENTZ-MINKOWSKI SPACE 81 Since H, H and h are nonero real numbers we have that log H, H = log h log H, H = 0 h H, H = ch, where c is a nonero constant. By a homothety of L 4, we obtain H, H = h. This completes the proof of the Theorem. Example 5.4. Consider M 2 = C, a = ke + 2t, b = k k e + 2t, h = 1 t2 4t 4, where = u + iv C and k = 1 t 1 + t, with t 1, 1 \ {0}. Since a = a 2t, b = b and 1 ab 4 = 1 k 4, from 57 we have that 2t µ 2 = k 4t 2 h t = 2t k1 + t4 2 4 t 2 1 t 2 µ = 1 + t. 4t Moreover, Φ = t, 2t k i, a 1 + t a, a + k. a It follows that Φ = 2t 1 + t 1 t, i, 1 t 2 t sinh +, 2t 1 t 2 t cosh + 2t. Thus, X = t, i, 1 t 2 t sinh +, 2t 1 t 2 t cosh + 2t. Therefore, u X t u, v = t, v, 1 t 2 cosh u t, 1 t 2 sinh u t.

28 82 A. C. ASPERTI J. A. M. VILHENA Letting e 1 := X u and e 2 := X v, from H = t 2 we conclude that H = 2t 2 0, 0, cosh u t, sinh u. Hence, t H, H = 1 t2 4t 4. e1 e 1 = 1 2 X uu = 1 2 X uu, Acknowledgment. The authors thank the referee for the valuable suggestions and comments. References [1] Akutagawa, K.; Nishikawa, S., The Gauss map and spacelike surfaces with prescribed mean curvature in Minkowski 3-space, Tôhoku Math. J , [2] Asperti, A. C.; Vilhena, J. A. M., Björling problem for spacelike, ero mean curvature surfaces in L 4, J. Geom. Phys. 2, , [3] Barros, M.; Romero, A., Indefinite Kählerian manifolds, Math. Ann , [4] Estudillo, F. J. M.; Romero, A., On maximal surfaces in the n dimensional Lorent Minkowski space, Geom. Dedicata , [5] Franco Filho, A. P.; Simões, P. A., The Gauss map of spacelike surfaces of the Minkowski space, preprint. [6] Hoffman, D. A.; Osserman, R., The geometry of the generalied Gauss map, Mem. Amer. Math. Soc. No. 236, vol. 28, [7] Hoffman, D. A.; Osserman, R., The Gauss map of surfaces in R n, J. Diff. Geom ,

29 SPACELIKE SURFACES IN THE LORENTZ-MINKOWSKI SPACE 83 [8] Hoffman, D. A.; Osserman, R., The Gauss map of surfaces in R 3 and R 4, Proc. London Math. Soc. 3, , [9] Kenmotsu, K., Weierstrass formula for surfaces of prescribed mean curvature, Math. Ann , [10] Kenmotsu, K., The mean curvature vector of surfaces in R 4, Bull. London Math. Soc , [11] Konopelchenko, B. G., Weierstrass representations for surfaces in 4D spaces and their integrable deformations via DS hierarchy, arxiv:dg/ [12] Montiel, S,; Romero, A., Complex Einstein hypersufaces of indefinite complex space forms, Math. Proc. Camb. Phil. Soc , [13] O Neill, B., Semi Riemannian geometry, Academic press, [14] Penrose, R.; Wolfgang, R., Spinors and space-time. Vol. 1. Two-spinors calculus and relativistic fields, Cambridge Monographs on Math. Physics, Cambridge University Press, [15] Vilhena, J. A. M., A Aplicação de Gauss de superfícies tipo espaço em L 3 e L 4, Doctor Thesis, IME- USP, São Paulo Universidade de São Paulo - IME Universidade Federal do Pará - CCEN Departamento de Matemática Departamento de Matemática Caixa Postal Rua Augusto Corrêa , São Paulo, Brasil , Belém, Pará, Brasil asperti@ime.usp.br vilhena@ufpa.br javilhena@yahoo.com.br