On Bernstein-Type Theorems in Semi-Riemannian Warped Products

In the present paper, we are interested in the study of the complete spacelike hypersurfaces immersed in semi-Riemannian warped product manifolds, in particular, in steady state-type spacetime and hyperbolic-type space. Before giving details of ourmain results, we first present a brief outline of some recent papers containing theorems related to ours. By using a restriction on the height function of a complete spacelike hypersurface, Caminha and de Lima [1] obtained someunique results concerning complete spacelike hypersurfaces with constant mean curvature immersed in steady state space and hyperbolic space, respectively. Later, Albujer and Aĺıas [2] proved that on a complete spacelike hypersurface the constant mean curvature must be identically 1, provided that such a hypersurface is bounded away from the infinity of the steady state space. For some other Bernstein-type results concerning constant mean curvature, we refer the reader to some recent papers by Albujer et al. [3, 4] and Aquino and de Lima [5]. Also, by using the well known result according to Yau [6], Camargo et al. in [7] obtained some Bernstein-type results concerning complete spacelike hypersurfaces, in steady statetype and hyperbolic-type space. Noticing that in their paper the mean curvature of the complete spacelike hypersurface need not be a constant. de Lima in [8] obtained a new Bernstein-type theorem concerning complete spacelike hypersurfaces in hyperbolic space with the bounded mean curvature (not necessarily constant) and a restriction on the normal angle. In this paper, following [9, 10] we shall consider the Laplacian of the integral of the warping function. In fact, by using a technique provided by Yau in [6] and supposing an appropriate restriction on the mean curvature, we obtain the following Bernstein-type theorems.


Introduction
In the present paper, we are interested in the study of the complete spacelike hypersurfaces immersed in semi-Riemannian warped product manifolds, in particular, in steady state-type spacetime and hyperbolic-type space.Before giving details of our main results, we first present a brief outline of some recent papers containing theorems related to ours.
By using a restriction on the height function of a complete spacelike hypersurface, Caminha and de Lima [1] obtained some unique results concerning complete spacelike hypersurfaces with constant mean curvature immersed in steady state space and hyperbolic space, respectively.
Later, Albujer and Alías [2] proved that on a complete spacelike hypersurface the constant mean curvature must be identically 1, provided that such a hypersurface is bounded away from the infinity of the steady state space.For some other Bernstein-type results concerning constant mean curvature, we refer the reader to some recent papers by Albujer et al. [3,4] and Aquino and de Lima [5].Also, by using the well known result according to Yau [6], Camargo et al. in [7] obtained some Bernstein-type results concerning complete spacelike hypersurfaces, in steady statetype and hyperbolic-type space.Noticing that in their paper the mean curvature of the complete spacelike hypersurface need not be a constant.de Lima in [8] obtained a new Bernstein-type theorem concerning complete spacelike hypersurfaces in hyperbolic space with the bounded mean curvature (not necessarily constant) and a restriction on the normal angle.
In this paper, following [9,10] we shall consider the Laplacian of the integral of the warping function.In fact, by using a technique provided by Yau in [6] and supposing an appropriate restriction on the mean curvature, we obtain the following Bernstein-type theorems.
The Riemannian version of Theorem 1 is also presented as follows.
Theorem 2. Let  +1 = ×    be a Riemannian warped product whose fiber   is a complete Riemannian manifold.
Let  : Σ  →  +1 be a complete and connected spacelike hypersurface with the mean curvature  satisfying If ∇ℎ has integrable norm on Σ  , then Σ  is a slice of ×    .
Suppose that the warping function  in both Theorems 1 and 2 is given by  =   for  ∈ , then the above two theorems are just the corresponding results shown in [7].Finally, letting the warping function  in Theorem 2 be  =   for  ∈ R and the fiber   an -dimensional Euclidean space, then Theorem 2 gives a Bernstein-type theorem for spacelike hypersurfaces immersed in the hyperbolic space (see Section 4 for details).
This paper is organized as follows.We shall first recall some notations and collect some basic facts in a preliminaries section, and some key lemmas used to prove our main theorems are also given in this section.Section 3 is devoted to proving some unique theorems concerning spacelike hypersurfaces in semi-Riemannian warped products.Finally, in Section 4, some applications of our main theorems in steady state-type spacetimes and hyperbolic-type spaces are obtained respectively.

Preliminaries
In this section, from [11,12] we shall recall some basic notations and facts that will appear along this paper.
Let   be a connected, -dimensional ( ≥ 2) oriented Riemannian manifold,  ⊆ R an interval, and  :  → R a positive smooth function.We consider the product differential manifold  ×   and denote by   and   the projections onto the base  and fiber   , respectively.A particular class of semi-Riemannian manifolds is the one obtained by furnishing product manifolds  ×   with the metric for any  ∈  +1 and any V,  ∈    +1 , where  = ±1.
We call such a space warped product manifold, and  is known as the warping function and we denote the space by  +1 = ×    .Note that −×    is called a generalized Robertson-Walker spacetime [11], in particular, −×    is called a Robertson-Walker spacetime if the fiber   has constant sectional curvature.From [13] we know that a generalized Robertson-Walker spacetime has constant sectional curvature  if and only if the Riemannian fiber   has constant sectional curvature  and the warping function  satisfies the following differential equation: It follows from [14,15] that the vector field ( ∘   )  is conformal and closed (in this sense that its dual 1-form is closed) with conformal factor  =   ∘   , where the prime denotes differentiation with respect to  ∈ .For  0 ∈ , we orient the slice Σ   0 := { 0 } ×   by the unit normal vector field   , then from [9] we know that Σ   0 has constant th mean curvature   = −(  ( 0 )/( 0 )) with respect to   for 0 ≤  ≤ .
A smooth immersion  : Σ  → ×    of an -dimensional connected manifold Σ  is said to be a spacelike hypersurface if the induced metric via  is a Riemannian metric on Σ  .If Σ  is oriented by the unit vector field , one obviously has  =    =   .
We consider two particular functions naturally attached to complete spacelike hypersurfaces, namely, the vertical (height) function ℎ = (  )| Σ  and the support function ⟨,   ⟩.We denote by ∇ and ∇ the gradients with respect to the metrics of ×    and Σ  , respectively.Thus, by a simple computation we present the gradient of   on ×    as follows: Moreover, the gradient of ℎ on Σ  is given by We denote by | ⋅ | the norm of a vector field on Σ  , then we get According to [2,16], a spacelike hypersurface  : Σ  → ×    is said to be bounded away from the future infinity of ×    if there exists  ∈  such that Analogously, a spacelike hypersurface  : Σ  → ×    is said to be bounded away from the past infinity of ×    if there exists  ∈  such that Finally, Σ  is said to be bounded away from the infinity of ×    if it is both bounded away from the past and future infinity of ×    .Setting  = 0 in Lemma 4.1 of [9], we may obtain the Laplacian of the integral of the warping function in a generalized Robertson-Walker spacetime.By using the technique according to Alías and Colares [9], the second author and Wang in [10] generalize this result in a semi-Riemannian warped product as follows.
Lemma 3 (see [9,10]).Let  : Σ  → ×    be a spacelike hypersurface immersed in a semi-Riemannian warped product.If Advances in Mathematical Physics where Δ denotes the Laplacian operator and ℎ is the height function of Σ  .
Furthermore, we also need the following well known lemma according to Yau [6].
Lemma 4 (corollary on page 660 of [6]).Let Σ  be an dimensional complete Riemannian manifold.If  : Σ  → R is a smooth subharmonic or superharmonic function whose gradient norm is integrable on Σ  , then  must be actually harmonic.

Proofs of Main Theorems
Based on the above arguments in Section 2, we may present the proof of Theorem 1 as follows.
Proof of Theorem 1.Since   is a unitary timelike vector field globally defined on the ambient spacetime, then there exists a unique timelike unitary normal vector filed  globally defined on the spacelike hypersurface Σ  which is the same time orientation as   .By using the reverse Cauchy-Schwarz inequality we have Letting  = −1, then it follows from Lemma 3 that Δ (ℎ) = − (  (ℎ) +  (ℎ) ⟨,   ⟩ ) .
Noticing that the warping function is positive on , then, by using the above inequality and (1) in (13) we have the following inequality: this means that (ℎ) is a subharmonic function on Σ  .On the other hand, since the spacelike hypersurface Σ  is bounded away from the infinity of −×    , then the height function ℎ is bounded on Σ  .Also, we have Since |∇ℎ| is integrable on Σ  , from (16) we know that |∇(ℎ)| is also integrable on Σ  .The above arguments assure that Lemma 4 is applicable, then applying Lemma 4 on subharmonic function (ℎ) on Σ  we have Putting the above equation into (15) and noting that the warping function  is a smooth positive function on , we obtain (  /)(ℎ) = −⟨,   ⟩, thus, using inequalities ( 1) and (12) in this equation we have sup The hypothesis (1) implies that sup Σ  (  /)(ℎ) > 0 on Σ  , then it follows from the above inequality that −⟨,   ⟩ ≤ 1; comparing this inequality with inequality (12) we obtain an identity ⟨,   ⟩ = −1.Finally, using  = −1 and ⟨,   ⟩ = −1 in (7) gives that which means that ℎ is a constant on Σ  .Then we prove that Σ  is a slice of −×    .
Next we give the Riemannian version of the proof of Theorem 1 on a Riemannian warped product space as follows.
Proof of Theorem 2. In this context, we may consider  being the orientation of the hypersurface Σ  such that its angle function satisfies Now, letting  = 1, then it follows from Lemma 3 that The assumption (2) implies that  is positive, then, from (20) we have that Noticing that the warping function is positive on , then, using the above inequality and ( 2) in ( 21) we obtain this means that (ℎ) is a subharmonic function on Σ  .As the spacelike hypersurface Σ  is bounded away from the infinity of ×    , then the height function ℎ is also bounded on Σ  .Also, (16) holds in this context.As |∇ℎ| is integrable on Σ  , then (16) assures that |∇(ℎ)| is also Advances in Mathematical Physics integrable.From the above arguments we see that Lemma 4 is applicable; applying Lemma 4 on subharmonic function (ℎ) we obtain (17).Thus, putting ( 17) into (23) and noting that the warping function  is positive on , we obtain (  /)(ℎ) = −⟨,   ⟩, using inequality (2) in this equation yields that It follows from (24) that −⟨,   ⟩ ≥ 1, comparing this inequality with inequality (20) we obtain identity ⟨,   ⟩ = −1.Finally, using  = 1 and ⟨,   ⟩ = −1 in (7) gives which means that ℎ is a constant on Σ  .Then we prove that Σ  is a slice of ×    .
Remark 5.It is worth to point out that Colares and de Lima [17] obtained some rigidity theorems in semi-Riemannian warped products which are similar to ours.However, in the assumptions of Theorems 4.2 and 4.6 of [17], the warping function  is assumed to has convex logarithm.On the other hand, we refer the reader to [12,15] for some examples of semi-Riemannian warped products whose warping functions are not necessarily to have convex logarithm.That is, without requiring the assumption that ln  is convex, our Theorems 1 and 2 attain the same conclusions as the corresponding results proved in [17].

Applications
In this section, we apply our main theorems on some physical models, in particular, on steady state-type space spacetime and hyperbolic-type space.
According to [2], −R×     is said to be a steady statetype spacetime, where   is an -dimensional complete and connected Riemannian manifold.In particular, −R×   R  is called the ( + 1)-dimensional steady state spacetime, which is isometric to an open subset of the de Sitter space S +1 .The importance of studying the steady state spacetime comes from the fact that, in cosmology, −R×   R 3 is the steady state model of the universe proposed by Bondi and Gold [18], and Hoyle [19].
Suppose that the warping function is given by  =   , then the following result follows from Theorem 1. Corollary 6.Let  : Σ  → −R×     be a complete spacelike hypersurface in steady state-type spacetime.Suppose that Σ  is bounded away from the infinity of −R×     and that the mean curvature  satisfies  ≥ 1.If ∇ℎ has integrable norm on Σ  , then  = 1 and Σ  is a slice of −R×    .
We remark that Corollary 6 was proved in [7] by using a different method from ours.
The hyperbolic-type space is defined by R×     , where   is a complete connected Riemannian manifold.The motivation for investigating the hyperbolic-type space R×     comes from the fact that the ( + 1)-dimensional hyperbolic space H +1 is isometric to R×   R  .Noting that an explicit isometry between the half-space model and this hyperbolictype model has been pointed out by Alías and Dajczer in [20].Now letting the warping function be  =   for  ∈ R, then the following result follows from Theorem 2. Corollary 7. Let  : Σ  → R×     be a complete spacelike hypersurface in hyperbolic-type space.Suppose that Σ  is bounded away from the infinity of R×     and that the mean curvature  satisfies 0 <  ≤ 1.If ∇ℎ has integrable norm on Σ  , then  = 1 and Σ  is a slice of R×     .Also, letting the warping function be  =   and   an -dimensional Euclidean space, then the following result follows from Corollary 7.
Corollary 8. Let  : Σ  → R×   R  be a complete spacelike hypersurface in a hyperbolic space.Suppose that Σ  is bounded away from the infinity of R×   R  and that the mean curvature  satisfies 0 <  ≤ 1.If ∇ℎ has integrable norm on Σ  , then  = 1 and Σ  is a horosphere of R×   R  .Remark 9. De Lima in [8] proved that a complete spacelike hypersurface immersed in hyperbolic space with the mean curvature 0 ≤  ≤ 1 and −⟨,   ⟩ ≥ sup Σ   is a horosphere, provided that Σ  is under a horosphere of the hyperbolic space and the second fundamental from is bounded.