Abstracts
In this paper, we introduce the fourth fundamental forms for hypersurfaces in Hn+1 and spacelike hypersurfaces in S1n+1, and discuss the conformality of the normal Gauss map of the hypersurfaces in Hn+1 and S1n+1. Particularly, we discuss the surfaces with conformal normal Gauss map in H³ and S³1, and prove a duality property. We give a Weierstrass representation formula for spacelike surfaces in S³1 with conformal normal Gauss map. We also state the similar results for timelike surfaces in S³1. Some examples of surfaces in S³1 with conformal normal Gauss map are given and a fully nonlinear equation of MongeAmpère type for the graphs in S³1 with conformal normal Gauss map is derived.
fourth fundamental form; conformal normal Gauss map; generalized Gauss map; duality property; de Sitter Gauss map; MongeAmpère equation
Neste artigo, introduzimos a quarta forma fundamental de uma hipersuperfície em Hn+1 de uma hipersuperfície tipoespaço em S1n+1, e discutimos a conformalidade da aplicação normal de Gauss de tais hipersuperfícies. Em particular, investigamos o caso de superfícies com aplicação normal de Gauss conforme em H³ e S³1, e provamos um teorema de dualidade. Apresentamos uma representação de Weierstrass para superfícies tipoespaço em S³1 com aplicação de Gauss conforme. Enunciamos também resultados semelhantes para superfícies tipotempo em S³1. São dados alguns exemplos de superfícies em S³1 com aplicações de Gauss conformes, e é deduzida uma equação totalmente nãolinear do tipo MongeAmpère para gráficos em S³1 com aplicações de Gauss conformes.
quarta forma fundamental; aplicação normal de Gauss conforme; aplicação de Gauss generalizada; propriedade de dualidade; aplicação de Gauss de Sitter; equação de MongeAmpère
MATHEMATICAL SCIENCES
The hypersurfaces with conformal normal Gauss map in H^{n}^{+1} and S_{1}^{n+1}
Shuguo Shi
School of Mathematics and System Sciences, Shandong University, Jinan 250100, P.R. China
ABSTRACT
In this paper, we introduce the fourth fundamental forms for hypersurfaces in H^{n}^{+1} and spacelike hypersurfaces in S_{1}^{n+1}, and discuss the conformality of the normal Gauss map of the hypersurfaces in H^{n}^{+1} and S_{1}^{n+1}. Particularly, we discuss the surfaces with conformal normal Gauss map in H^{3} and S^{3}_{1}, and prove a duality property. We give a Weierstrass representation formula for spacelike surfaces in S^{3}_{1} with conformal normal Gauss map. We also state the similar results for timelike surfaces in S^{3}_{1}. Some examples of surfaces in S^{3}_{1} with conformal normal Gauss map are given and a fully nonlinear equation of MongeAmpère type for the graphs in S^{3}_{1} with conformal normal Gauss map is derived.
Key words: fourth fundamental form, conformal normal Gauss map, generalized Gauss map, duality property, de Sitter Gauss map, MongeAmpère equation.
RESUMO
Neste artigo, introduzimos a quarta forma fundamental de uma hipersuperfície em H^{n}^{+1} de uma hipersuperfície tipoespaço em S_{1}^{n+1}, e discutimos a conformalidade da aplicação normal de Gauss de tais hipersuperfícies. Em particular, investigamos o caso de superfícies com aplicação normal de Gauss conforme em H^{3} e S^{3}_{1}, e provamos um teorema de dualidade. Apresentamos uma representação de Weierstrass para superfícies tipoespaço em S^{3}_{1} com aplicação de Gauss conforme. Enunciamos também resultados semelhantes para superfícies tipotempo em S^{3}_{1}. São dados alguns exemplos de superfícies em S^{3}_{1} com aplicações de Gauss conformes, e é deduzida uma equação totalmente nãolinear do tipo MongeAmpère para gráficos em S^{3}_{1} com aplicações de Gauss conformes.
Palavraschave: quarta forma fundamental, aplicação normal de Gauss conforme, aplicação de Gauss generalizada, propriedade de dualidade, aplicação de Gauss de Sitter, equação de MongeAmpère.
1 INTRODUCTION
It is well known that the classical Gauss map has played an important role in the study of the surface theory in R^{3} and has been generalized to the submanifold of arbitrary dimension and codimension immersed into the space forms with constant sectional curvature (see Osserman 1980).
Particularly, for the ndimensional submanifold x: M ® V in space V with constant sectional curvature, Obata (Obata 1968) introduced the generalized Gauss map which assigns each point p of M to the totally geodesic nsubspace of V tangent to x(M) at x(p). He defined the third fundamental form of the submanifold in constant curvature space as the pullback of the metric of the set of all totally geodesic nsubspaces in V under the generalized Gauss map. He derived a relationship among the Ricci tensor of the immersed submanifold and the first, the second and the third fundamental forms of the immersion. Meanwhile, Lawson (Lawson 1970) discussed the generalized Gauss map of the immersed surfaces in S^{3}, and prove a duality property between the minimal surfaces in S^{3} and their generalized Gauss map images.
Epstein (Epstein 1986) and Bryant (Bryant 1987) defined the hyperbolic Gauss map for surfaces in H^{3}, and Bryant (Bryant 1987) obtained a Weierstrass representation formula for constant mean curvature one surfaces with conformal hyperbolic Gauss map. Using the Weierstrass representation formula, Bryant also studied the properties of constant mean curvature one surfaces. Using the hyperbolic Gauss map, Gálvez and Martínez and Milán (Gálvez et al. 2000) studied the flat surfaces in H^{3} with conformal hyperbolic Gauss map with respect to the second conformal structure on surfaces (see (Klotz 1963) for the definition), and obtained a Weierstrass representation formula for such surfaces.
Kokubu (Kokubu 1997) considered the ndimensional hyperbolic space H^{n} as a Lie group G with a leftinvariant metric, and defined the normal Gauss map of the surfaces which assigns each point of the surface to the tangent plane left translated to the Lie algebra of G. He also gave a Weierstrass representation formula for minimal surfaces in H^{n}. On the other hand, Gálvez and Martínez (Gálvez and Martínez 2000) studied the properties of the Gauss map of a surface S immersed into the Euclidean 3space R^{3} by using the second conformal structure on surface, and obtained a Weierstrass representation formula for surfaces with prescribed Gauss map. Motivated by their work, the author (Shi 2004) gave a Weierstrass representation formula for surfaces with prescribed normal Gauss map and Gauss curvature in H^{3} by using the second conformal structure on surfaces. From this, the surfaces whose normal Gauss maps are conformal have been found, and the translational surfaces with conformal normal Gauss map locally are given. In (Shi 2006), the author classified locally the ruled surfaces with conformal normal Gauss map within the Euclidean ruled surfaces, and studied some global properties of the ruled surfaces and translational surfaces with conformal normal Gauss map.
Aiyama and Akutagawa (Aiyama and Akutagawa 2000) defined the normal Gauss map for spacelike surfaces in the de Sitter 3space , and gave a Weierstrass representation formula for spacelike surfaces in with prescribed mean curvature and normal Gauss map.
The purpose of this paper is to study the conformality of the normal Gauss map for hypersurfaces in H^{n}^{+1} and spacelike hypersurfaces in , and to prove a duality property between the surfaces in H^{3} and the spacelike surfaces in with conformal normal Gauss map. The rest of this paper is organized as follows. In the second section, we describe the generalized definition of the normal Gauss map for hypersurfaces in H^{n}^{+1} and spacelike hypersurfaces in (cf. Kokubu 1997, Aiyama and Akutagawa 2000). The third section introduces the fourth fundamental forms for hypersurfaces in H^{n}^{+1} and , and obtains a relation among the first, the second, the third and the fourth fundamental forms of the hypersurfaces. As an application, we discuss the conformality of the normal Gauss map for hypersurfaces in H^{n}^{+1} and spacelike hypersurfaces in . By means of the generalized Gauss map of the surfaces in H^{3} and , the fourth one proves a duality property between the surfaces in H^{3} and the spacelike surfaces in with conformal normal Gauss map. The fifth one gives a Weierstrass representation formula for spacelike surfaces in with conformal normal Gauss map, and the sixth one derives the fully nonlinear equation of MongeAmpère type for spacelike graphs in with conformal normal Gauss map and classifies locally the translational surfaces and the Euclidean ruled surfaces in with conformal normal Gauss map. In the last section, we state the similar results for timelike surfaces in with conformal normal Gauss map.
2 PRELIMINARIES
Take upper halfspace models of hyperbolic space H^{n}^{+1}(1) and de Sitter space (1)
with respectively Riemannian metric
and Lorentz metric
(see Aiyama and Akutagawa 2000 or section 4).
Let M be a ndimensional Riemannian manifold and x: M^{n} ® H^{n}^{+1} (resp. x: M^{n} ® ) be an immersed hypersurface (resp. spacelike hypersurface) with local coordinates u_{1},u_{2},...,u_{n}. In this paper,we agree with the following ranges of indices: 1 < i,j,k,... < n and 1 < A,B,C,... < n + 1. The first and the second fundamental forms are given, respectively, by I = g_{ij}du_{i}du_{j} and II = h_{ij}du_{i}du_{j}. The unit normal vector ( resp. timelike unit normal vector) of x(M) is
where = 1 (resp. = 1).
We have the Weingarten formula
Identitying H^{n}^{+1} and with Lie group (Kokubu 1997)
the multiplication is defined as matrix multiplication and the identity is e = (0,0,¼,0,1). The Riemannian metric of H^{n}^{+1} and the Lorentz metric of are leftinvariant, and
are the leftinvariant unit orthonormal vector fields. Now, the unit normal vector (resp. timelike unit normal vector) field of x(M) can be written as N = h1 1 + h2 2 + ... + h_{n}+1 _{n}_{+1}. Left translating N to T_{e}(), we obtain
Call the normal Gauss map of the immersed hypersurface x: M ® H^{n}^{+1} (resp. spacelike hypersurface x: M ® ) (Kokubu 1997, Aiyama and Akutagawa 2000).
3 THE FOURTH FUNDAMENTAL FORM
DEFINITION. Let M be a ndimensional Riemannian manifold. Call IV = ád, dñ the fourth fundamental form of the immersed hypersurface x: M ® (resp. spacelike hypersurface x: M ® ), where the scalar product á·,·ñ is induced by the Euclidean metric of R^{n}^{+1} (resp. the LorentzMinkowski metric of L^{n}^{+1}).
THEOREM 3.1. Let M be a ndimensional Riemannian manifold with Ricci tensor Ric. Let x: M ® H^{n}^{+1} (resp. x: M ® ) be an immersed hypersurface (resp. spacelike hypersurface) with mean curvature H =
tr(II). Then
where III = nHII  (n  1)I  Ric (resp. III = nHII  (n  1)I + Ric) is Obata's third fundamental form of x(M) (see Obata 1968).
PROOF. At first we prove the Theorem for H^{n}^{+1}. Choose the normal coordinates u_{1},u_{2},...,u_{n} near p Î M. By the Weingarten formula, we get
III = h_{ik}h_{jk}du_{i}du_{j} is the third fundamental form (Obata 1968) and by the Gauss equation, III = nHII  (n  1)I  Ric. (3.1) is proved.
Next, similar to the above proof, for , we have
Similar to the proof of (3.1), we can prove (3.2).
Next, we consider the applications of these formulas (3.1)(3.4). In the following of this paper, that the normal Gauss map is conformal means that the fourth fundamental form is proportional to the second fundamental form, i.e. IV = rII for some smooth function r on M (Shi 2004, 2006).
THEOREM 3.2. Let M be a ndimensional Riemannian manifold and x: M ® H^{n}^{+1} (resp. x: M ® ) be an immersed hypersurface ( resp. spacelike hypersurface) without umbilics. Then the normal Gauss map of x(M) is conformal if and only if at each point of M, there exists exactly two distinct principal curvatures and the sectional curvature R(X Ù Y) = 1 +
( resp. R(X Ù Y) = 1 ), where the vectors X and Y belong to different principal direction spaces.PROOF. The case of H^{n}^{+1}. For any point p Î M, let {e_{1},e_{2},...,e_{n}} be a local frame field so that (h_{ij}) is diagonalized at this point, i.e. h_{ij}(p) = l_{i}d_{ij}. By IV = rII and (3.3), we get, for i = 1,2,...,n, that
i.e.
Because x(M) has no umbilics, the equation (3.6) with respect to l_{i} has exactly two distinct solutions l and µ and lm = . By the Gauss equation, one may prove R(X Ù Y) = 1 + lm = 1 + .
Conversely, choose the local tangent frame {e_{1},e_{2},..., e_{n}} and the dual frame {w_{1},w_{2}, ...,w_{n}} near p, such that h_{ij} = 0, i ¹ j and h_{11} = h_{22} = ... = h_{rr} = l ¹ m = h_{r}_{+1r+1} = ... = h_{nn} at p.Then = lm. By (3.3),
The sufficiency has been proved for H^{n}^{+1}. Similarly, we can prove Theorem 3.2 for .
REMARK. By (3.5), we know that the normal Gauss map of all totally umbilical hypersurfaces except totally geodesic hyperspheres in H^{n}^{+1} are conformal. Similarly, for spacelike hypersurfaces in , since h_{n}_{+1}¹ 0, the normal Gauss maps of all totally umbilical spacelike hypersurfaces except totally geodesic spacelike hypersurfaces are conformal.
For H^{3} and , by Theorem 3.2, we immediately get
THEOREM 3.3. Let M be a 2dimensional Riemannian manifold and x: M ® H^{3} (resp. x: M ® ) be an immersed surface (resp. spacelike surface) without umbilics. Then the normal Gauss map of x(M) is conformal if and only if the Gauss curvature K = 1 +
(resp. K = 1  ).REMARK. In (Shi 2004, 2006), we assume that the second fundamental form is positive definite and induces the conformal structure on the surfaces in H^{3}. Here, the assumption with respect to the positive definite second fundamental form is dropped.
THEOREM 3.4. Let M be a ndimensional Einstein manifold and x: M ® H^{n}^{+1} (resp. x: M ® ) be an immersed hypersurface (resp. spacelike hypersurface) with nondegenerate second fundamental form and without umbilics. If the normal Gauss map of x(M) is conformal map, i.e. IV = rII, then n = 2 and r = 2(H  h_{3}) (resp. r = 2(H + h_{3})).
PROOF. We only prove the Theorem for H^{n}^{+1}. M is an Einstein manifold, so Ric = I, where S is the scalar curvature of M. (3.1) becomes
Because x(M) has no umbilics, we have
nH = 2h_{n+1} + r.
By Theorem 3.2 and its proof, we assume that l_{1} = ... = l_{r} = l ¹ m = l_{r+1} = ... = l_{n}, then
rl + (n  r)µ = 2h_{n+1} + r.
By (3.6),
l + µ = 2h_{n+1} + r.
So (r  1)l + (n  r  1)µ = 0. By Theorem 3.2, l and µ have same signature. So r = 1 and n = 2. Hence r = 2H  2h_{3}.
4 A DUALITY FOR THE SURFACES IN H^{3} AND WITH CONFORMAL NORMAL GAUSS MAPS
Let L^{4} be the Minkowski 4space with canonical coordinates X_{0},X_{1}, X_{2},X_{3} and LorentzMinkowski scalar product . The Minkowski model of H^{3} is given by
H^{3} = {(X_{0}, X_{1}, X_{2}, X_{3})  = 1, X_{0} > 0}
and is identified with the upper halfspace model of H^{3} by
Accordingly, the spacelike normal vector of the surface in the Minkowski model of H^{3} is
where
We get
The Minkowski model of the de Sitter 3space is defined as
and can be divided into three components as follows (cf. Aiyama and Akutagawa 2000),
Identify S_{} and S_{+} with upper halfspace model of the de Sitter 3space by (cf. Aiyama and Akutagawa 2000)
For spacelike surface X: M ® , let U_{} = X^{1}(S_{}) and U_{+} = X^{1}(S_{+}), then U_{}È U_{+} is the open dense subset of M. On U_{}È U_{+}, the timelike unit normal vector is
where
We get
REMARK. In (Aiyama and Akutagawa 2000), the normal Gauss map of the spacelike surface X: M ® is defined globally on M. Because of the density of U_{} and U_{+} in M, in this paper, we may consider that the normal Gauss map of the spacelike surface X: M ® is defined on U_{} and U_{+}.
Let X: M ® H^{3} (resp. X: M ® ) be an immersed surface (resp. spacelike surface). Parallel translating the spacelike ( resp. timelike) unit normal vector N to the origin of L^{4}, one gets the map N: M ® ( resp. N: M ® H^{3}) which is usually called generalized Gauss map of X: M ® H^{3} (resp. X: M ® ). The generalized Gauss map image can be considered as the surface in (resp. H^{3}).
THEOREM 4.1. (Kokubu 2005, Prop. 3.5). (1) Let X: M ® H^{3}be a 2dimensional immersed surface. Then its generalized Gauss map N: M ® is a branched spacelike immersion into with branch points where K = 1. And, when K ¹ 1, the curvature of N: M ® is K^{*} = and the volume element is dV_{N} = K + 1dV_{X}.
(2) Let X: M ® be a 2dimensional spacelike immersed surface. Then its generalized Gauss map N: M ® H^{3}is a branched immersion into H^{3}with branch points where K = 1. And, when K ¹ 1, the curvature of N: M ® H^{3}is K^{*} = and the volume element is dV_{N} = 1  KdV_{X}.
PROOF. In the context of this paper, we prove (2). For any p Î M, let {e_{0},e_{1},e_{2},e_{3}} be the orthonormal frame near p, such that e_{3} = X, e_{0} = N. Let {w_{0},w_{1},w_{2},w_{3}} be the dual frame. The connection 1forms is , a,b = 0,1,2,3. The coefficients of the second fundamental form of X: M ® is given by = h_{ij}w_{j}, h_{ij} = h_{ji}, i,j = 1,2. The induced metric of N : M ® H^{3} is = ádN, dNñ = h_{ik}h_{jk}w_{i}w_{j}. Choose the local tangent frame {e_{1},e_{2}} near p, such that h_{ij} = l_{i}d_{ij}. Then = . So, when l_{1}l_{2}¹ = 0, i.e. K ¹ 1, N(M) is an immersed surface into H^{3}. Its spacelike unit normal vector is X and the second fundamental form is II = ádX, dNñ = l_{1}(w_{1})^{2} l_{2}(w_{2})^{2}. By the Gauss equation, K^{*} = 1 + = .
By Theorem 3.3, (4.1), (4.2) and Theorem 4.1, we get the following duality.
THEOREM 4.2. Let M be a connected 2dimensional manifold. Let X: M ® H^{3} be an immersed surface with K ¹ 1 and without umbilics and let N: M ® be a spacelike surface with K ¹ 1 and without umbilics. Suppose that N: M ® is the generalized Gauss map of X: M ® H^{3} and vice versa. Then, the normal Gauss map of X: M ® H^{3} is conformal if and only if one of N: M ® is conformal. And, at this time, dV_{N} =
dV_{X}.REMARK. Like (Lawson 1970) for minimal surfaces in S^{3}, we call the generalized Gauss map N: M ® the polar variety of the immersed surface X: M ® H^{3} with conformal normal Gauss map and vice versa.
5 WEIERSTRASS REPRESENTATION FORMULA
In this section, we give a Weierstrass representation formula for spacelike surfaces in with conformal normal Gauss map. At first, we describe the normal Gauss map and the de Sitter Gauss map of the spacelike surfaces in . Take upper halfspace model of .
The normal Gauss map of the spacelike surface x: M ® is given by
By means of the stereographic projection from the north pole (0,0,1) of H^{2}(1) to the (x_{1}, x_{2})plane identified with C, we get
which is also called the normal Gauss map of the spacelike surface x: M ® · can be written as
Next, we describe the definition of the de Sitter Gauss map for spacelike surfaces in (in (Lee 2005), it is still called hyperbolic Gauss map), which is the analogue of Epstein and Bryant's hyperbolic Gauss map for surfaces in H^{3} (Epstein 1986, Bryant 1987, Shi 2004). The timelike geodesic is either the Euclidean equilateral halfhyperbola consisting of two branches which is orthonormal to the coordinate plane {(x_{1}, x_{2},0)(x_{1},x_{2}) Î R^{2}} or the Euclidean straight line which is orthonormal to the above coordinate plane. For the spacelike surface x = (x_{1},x_{2},x_{3})\colon M ® , at each point x Î M, the oriented timelike geodesic in passing through x with timelike tangent vector N meets {(x_{1},x_{2},0)(x_{1},x_{2}) Î R^{2}}È{¥} at two points. Since the geodesic is oriented, we may speak of one of the two points as the initial point and the other one as the final point. Call the final point the image of the de Sitter Gauss map for x(M) at the point x. Denote the de Sitter Gauss map by G^{S}. On the coordinate plane {(x_{1},x_{2},0)(x_{1},x_{2}) Î R^{2}}, we introduce the natural complex coordinate z = x_{1} + _{2}. Using the Euclidean geometry, as similar as done in the Theorem 5.1 of (Shi 2004), we get
Let x = (x_{1},x_{2},x_{3}) : M ® H^{3} be an immersed surface with unit normal vector
By the duality given in section 4, the generalized Gauss map of x: M ® H^{3} is given, when h_{3} > 0, by
and when h_{3} < 0, by
and in the Minkowski model of the de Sitter 3space, their timelike unit normal vector is X: M ® H^{3}. Again by the duality given in section 4, a straightforward computation shows us that the normal Gauss map of N: M ® is given by
So,
where g^{H}: M ® C È{¥} is exactly the normal Gauss map of x: M ® H^{3} (Kokubu 1997, Shi 2004, 2006). From this, we also prove the Theorem 4.2.
By (5.1)(5.4) and the Theorem 5.1 of (Shi 2004), we get that when h_{3} > 0, i.e. g^{S} > 1,
and when h_{3} < 0, i.e. g^{S} < 1,
where G^{H} is exactly the hyperbolic Gauss map of x: M ® H^{3} (Epstein 1986, Bryant 1987, Shi 2004).
In the following, we write respectively g^{S} and G^{S} as g and G.
By (5.2)(5.6) and the Weierstrass representation for surfaces in H^{3} with conformal normal Gauss map (Shi 2004), we get the Weierstrass representation formula for spacelike surfaces in with conformal normal Gauss map.
THEOREM 5.1. Let M be a simply connected Riemann surface. Given the map G: M ® C È{¥} and the nonconstant conformal map g: M ® C È{¥}\{z = 1}.
(1) When the holomorphic map g: M ® CÈ{¥}\{z = 1} satisfies g > 1 and
put
Then x = (x_{1},x_{2},x_{3}): M ® is a spacelike surface with de Sitter Gauss map G and holomorphic normal Gauss map g and Gauss curvature K satisfying
. And the conformal structure on M is induced by the negative definite second fundamental form. Conversely, any surface x: M ® with ( = h_{3}) can be given by (5.10), (5.11), (5.12), and the de Sitter Gauss map G and the normal Gauss map g must satisfy (5.7), (5.8), (5.9), where the conformal structure on M is induced by the negative definite second fundamental form.(2) When the antiholomorphic map g: M ® C È{¥}\{z = 1} without holomorphic points satisfies g < 1 and
put
Then x = (x_{1},x_{2},x_{3}): M ® is a spacelike surface with de Sitter Gauss map G and antiholomorphic normal Gauss map g and Gauss curvature K satisfying
. And the conformal structure on M is induced by the negative definite second fundamental form. Conversely, any surface x: M ® with ( = h_{3}) can be given by (5.16), (5.17), (5.18), and the de Sitter Gauss map G and the normal Gauss map g must satisfy (5.13), (5.14), (5.15), where the conformal structure on M is induced by the negative definite second fundamental form.6 GRAPHS AND EXAMPLES
In this section,we give the examples of surfaces in with conformal normal Gauss map within the translational surfaces and the Euclidean ruled surfaces.
In H^{3}, the graph (u,v,f(u,v)) with conformal normal Gauss map satisfies the following fully nonlinear equation of MongeAmpère type (Shi 2004, 2006)
Take upper halfspace model of . Consider the spacelike graph (u,v,f(u,v)) in with < 1. Its Gauss curvature is given by
So K = 1  is equivalent to
where < 1. This is the fully nonlinear equation of MongeAmpère type which the spacelike graph in with K = 1  must satisfy.
REMARK. There exists a nice duality between the solutions of minimal surface equation
in R^{3} and the ones of maximal surface equation
in LorentzMinkowski 3space L^{3} (Alías and Palmer 2001). Here, by the duality given by (5.2) (or (5.3)), we know that if f(u,v) is a solution of (6.1), then the local graph of the spacelike surface (ff_{u} u, ff_{v }v,f) (or (ff_{u }+ u, ff_{v }+ v,f)) in satisfies (6.2). Conversely, if f(u,v) is a solution of (6.2) with < 1, then the local graph of the surface (ff_{u }u,ff_{v }v,f) (or (u ff_{u}, v ff_{v}, f)) in H^{3} satisfies (6.1).
Next, as similar as done in section 6 of (Shi 2004), we get the following Theorem.
THEOREM 6.1. The nontrivial translational spacelike surfaces with form f(u,v) = f(u) + y(v) in
with conformal normal Gauss map are given, up to a linear translation of variables, by
with
< 1, where a and b are nonzero constants. The parameter form of these translational surfaces are locally given by
REMARK. We may check that the isometric transformation
preserves the concept of the ruled surfaces and the conformality of the normal Gauss map of the spacelike surface in .
Considered as surfaces in 3dimensional Minkowski space L^{3}, the spacelike ruled surfaces in can be represented as x(u,v) = a(v) + ub(v): D ® , where D(Ì R^{2}) is a parameter domain and a(v) and b(v) are two vector valued functions into L^{3} corresponding to two curves in L^{3}. When b is locally nonconstant, without loss of generality we can assume that either áb,bñ = 1, áb',b'ñ = ±1, and áa',b'ñ = 0 or áb,bñ = 1, áb',b'ñ = 0, and áa',bñ = 0, where á·,·ñ is the scalar product in L^{3}. As similar as done in Theorem 2 of (Shi 2006), we have
THEOREM 6.2. Up to an isometric transformation (6.5) in
, every spacelike ruled surface in with conformal normal Gauss map is locally a part of one of the following,(1) ordinary Euclidean spacelike planes in
,(2) (u cosh v, c · sinh v, u sinh v), for a constant c ¹ 0,
(3) (c_{2 }sinh v + u cosh v, c_{1 }sinh v, c_{2 }cosh v + u sinh v), for constants c_{1}¹ 0 and c_{2}¹ 0.
We should note that in the proof of Theorem 6.2, only when áb',b'ñ = 1, we may get the nontrivial cases (2) and (3).
Locally, the ruled surfaces (2) and (3) in Theorem 6.2 can be represented as the graph (u,v,f(u,v)) as follows,
COROLLARY. f(u,v) =
is a solution of equation (6.2), where c_{1}¹ 0 and c_{2} are constants.REMARK. In H^{3}, the translational surfaces
and the ruled surfaces
and
with conformal normal Gauss map have been obtained (Shi 2004, 2006), where a,b,c,c_{1} and c_{2} are nonzero constants. Using (5.2) (or (5.3)) and Theorem 4.2, we may check that up to a isometric transformation (6.5) in (q = ±), (6.4) in Theorem 6.1 and (2) and (3) in Theorem 6.2 are, respectively, the polar varieties of (6.6), (6.7) and (6.8) and vice versa.
REMARK. Every geodesic of H^{3}, corresponding respectively to u = 0, u = p, v = 0 and v = p on surfaces (6.6) and to v = on surfaces (6.7) and to v = ± on surfaces (6.8) follow which K = 1 is mapped to a single point in S_{0} by the generalized Gauss map.
7 TIMELIKE SURFACES IN WITH CONFORMAL NORMAL GAUSS MAP
In this section, we state the similar results as above for timelike surfaces in without proofs.
Take upperhalf space model of . Let M be a 2dimensional Lorentz surface and x: M ® be the timelike immersion with local coordinates u_{1},u_{2}. The first and the second fundamental forms are given, respectively, by I = g_{ij}du_{i}du_{j} and II = h_{ij}du_{i}du_{j}. The spacelike unit normal vector is
where = 1. We have the Weingarten formula
Lefttranslating N to T_{e}(), we obtain
which is called the normal Gauss map of timelike surface x: M ® (Aiyama and Akutagawa 2000). Call IV = ád,dñ the fourth fundamental form of the timelike surface x: M ® . We have
IV =
g_{ij}  2h_{3}h_{ij} + g^{kl} h_{ik}h_{jl}) du_{i}du_{j}.Of course, we may also define the highdimensional version of the fourth fundamental form for timelike hypersurfaces in (1).
THEOREM 7.1. Let M be a 2dimensional Lorentz surface and x: M ® be a timelike immersed surface without umbilics. Then the normal Gauss map of x(M) is conformal if and only if the Gauss curvature K = 1 +
.In the Minkowski model of the de Sitter 3space , the generalized Gauss map N: M ® of the timelike surface X: M ® is a branched timelike immersion with branch points where K = 1.
THEOREM 7.2. Let M be a connected 2dimensional Lorentz surface. Let X: M ® be a timelike surface with K ¹ 1 and without umbilics. If the normal Gauss map of X: M ® is conformal, then the normal Gauss map of its generalized Gauss map N: M ® is also conformal and vice versa.
The timelike graph (u,v,f(u,v)) in with conformal normal Gauss map also satisfies the fully nonlinear equation of MongeAmpère type (6.2) with > 1.
THEOREM 7.3. The nontrivial translational timelike surfaces with form f(u,v) = f(u) + y(v) in
with conformal normal Gauss map are given, up to a linear translation of variables, by
Flaherty's timelike surfaces in (Milnor 1987) f(u,v) = ±u + y(v) and f(u,v) = ±v + j(u), where a and b are nonzero constants and y'(v) ¹ 0 and j'(u) ¹ 0.
REMARK. We may check that the isometric transformation (6.5) preserves the concept of the ruled surfaces and the conformality of the normal Gauss map of the timelike surfaces in .
We may prove that the normal Gauss map of the timelike surfaces (2) and (3) in Theorem 6.2 are also conformal. In addition, for the timelike ruled surface x(u,v) = a(v) + ub(v) in , we may also assume the remained four cases:
(i) áb, bñ = 1, áb^{¢}, b^{¢}ñ = 1, and áa^{¢}, b^{¢}ñ = 0,
(ii) b is constant null vector,
(iii) b is constant and áb, bñ = 1, áa^{¢}, bñ = 0,
(iv) áb, bñ = 0, áb^{¢}, b^{¢}ñ = 1, and áa^{¢}, b^{¢}ñ = 0, where á·,·ñ is the scalar product in L^{3}. Hence, we have
THEOREM 7.4. Up to an isometric transformation (6.5) in
, every timelike ruled surface in with conformal normal Gauss map is locally a part of one of the following,(1) ordinary Euclidean timelike planes in
,(2) ordinary Euclidean generalized cylinder x(u,v) = a(v) + ub, where b = (0,0,1) and a(v) is arbitrary curve in L^{3} with áa^{¢},a^{¢}ñ > 0 and áa^{¢},bñ = 0,
(3) (u cosh v, c · sinh v, u sinh v), for a constant c ¹ 0,
(4) (c_{2 }sinh v + u cosh v, c_{1 }sinh v, c_{2 }cosh v + u sinh v), for constants c_{1}¹ 0 and c_{2}¹ 0,
(5) (u sinh v, c · cosh v, u cosh v), for a constant c ¹ 0,
(6) (c_{2 }cosh v + u sinh v, c_{1 }cosh v, c_{2} sinh v + u cosh v), for constants c_{1}¹ 0 and c_{2}¹ 0,
(7) Flaherty's timelike surfaces in (Milnor 1987), x(u,v) = a(v) + ub, where b = (1,0,1) and a(v) is arbitrary curve in L^{3}with áa^{¢}, bñ ¹ 0.
We should note that in the proof of Theorem 7.4, only for case (i) and (ii), we may get the surfaces (5), (6), (7) in Theorem 7.4. For case (iv), we may assume b(v) = (r(v) cos q(v), r(v) sin q(v), r(v)) with r^{2}(q^{¢})^{2} = 1. Next, we get a contradictory system of equations.
Locally, the ruled surfaces (5) and (6) in Theorem 7.4 can be represented as the graph (u,v,f(u,v)) as follows,
COROLLARY. f(u,v) =
is a solution of equation (6.2), where c_{1}¹ 0 and c_{2} are constants.REMARK. The totally umbilical timelike surfaces in given by (x_{1}  a)^{2 }+ (x_{2 } b)^{2 } (x_{3 } c)^{2} = R^{2}, where constants c ¹ 0 and R > 0, are Euclidean ruled surfaces but K ¹ 1 + . Their normal Gauss maps are also conformal.
REMARK. Up to a isometric transformation (6.5) in (q = ±), the timelike surfaces (3) and (4) in Theorem 7.4 are, respectively, the polar varieties of the timelike surfaces (5) and (6) in Theorem 7.4 and vice versa. The similar result also holds for the timelike surfaces in Theorem 7.3. Generally, if f(u,v) is a solution of (6.2) with > 1, then the local graph of the timelike surface (ff_{u }u,ff_{v} v,f) (or (u  ff_{u}, v ff_{v}, f)) in also satisfies (6.2).
REMARK. When we do not assume that f > 0, (6.3) and
f(u, v) = ± and f(u, v) = ±u + y(v) and f(u, v) = ±v + j(u)
with y^{¢}(v) ¹ 0 and j^{¢}(u) ¹ 0 are all nontrivial entire solutions of the equation (6.2) defined on R^{2}. In addition, the cone f(u,v) = is also the special solution of the equation (6.2), but its graph is the lightlike surface. By OmoriYau's Maximum Principle (Omori 1967, Yau 1975), there exist no entire solution f(u,v) of (6.2) satisfying > 1 and f > 0 on R^{2}. Does there exist nontrivial entire solutions of equation (6.2) defined on R^{2} satisfying < 1 and f > 0?
ACKNOWLEDGMENTS
The author would like to express his sincere gratitude to Professor Detang Zhou for his enthusiastic concern, encouragement, support and valuable help, as well as for his significant suggestions and heuristic discussions with the author, and for his providing the author with Omori and Yau's papers (Omori 1967, Yau 1975). The author would like to express his sincere gratitude to the referees for their interest, careful examination and valuable suggestions.
This work is supported by the starting foundation of science research of Shandong University.
Manuscript received on April 4, 2007; accepted for publication on October 1, 2007; presented by MANFREDO DO CARMO
AMS Classification: 53C42, 53A10.
Email: shishuguo@hotmail.com
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Publication Dates

Publication in this collection
10 Mar 2008 
Date of issue
Mar 2008
History

Accepted
01 Oct 2007 
Received
04 Apr 2007