Spacelike Hypersurfaces in Weighted Generalized Robertson-Walker SpaceTimes

In recent years, spacelike hypersurfaces in Lorentzian manifolds have been deeply studied not only from their mathematical interest, but also from their importance in general relativity. Particularly, there are many articles that study spacelike hypersurfaces in weighted warped product space-times. A weighted manifold is a Riemannian manifold with a measure that has a smooth positive density with respect to the Riemannian one. More precisely, the weighted manifold Mf associated with a complete n-dimensional Riemannian manifold (Mn, g) and a smooth function f on Mn is the triple (Mn, g, dμ = e−fdM), where dM stands for the volume element of Mn. In this setting, we will take into account the so-called Bakry-Émery Ricci tensor (see [1]) which as an extension of the standard Ricci tensor Ric, which is defined by Ricf = Ric +Hessf. (1)


Introduction
In recent years, spacelike hypersurfaces in Lorentzian manifolds have been deeply studied not only from their mathematical interest, but also from their importance in general relativity.
Particularly, there are many articles that study spacelike hypersurfaces in weighted warped product space-times.A weighted manifold is a Riemannian manifold with a measure that has a smooth positive density with respect to the Riemannian one.More precisely, the weighted manifold   associated with a complete -dimensional Riemannian manifold (  , ) and a smooth function  on   is the triple (  , ,  =  − ), where  stands for the volume element of   .In this setting, we will take into account the so-called Bakry-Émery Ricci tensor (see [1]) which as an extension of the standard Ricci tensor Ric, which is defined by Ric  = Ric + Hess . ( Therefore, it is natural to extend some results of the Ricci curvature to analogous results for the Bakry-Émery Ricci tensor.Before giving more details on our work we present a brief outline of some recent results related to our one.
In [2], Wei and Wylie considered the complete -dimensional weighted Riemannian manifold and proved mean curvature and volume comparison results on the assumption that the ∞-Bakry-Émery Ricci tensor is bounded from below and  or |∇| is bounded.Later, Cavalcante et al. [3] researched the Bernstein-type properties concerning complete twosided hypersurfaces immersed in a weighted warped product space using the appropriated generalized maximum principles.Moreover, [4] obtained new Calabi-Bernstein's type results related to complete spacelike hypersurfaces in a weighted GRW space-time.More recently, some rigidity results of complete spacelike hypersurfaces immersed into a weighted static GRW space-time are given in [5].
In this paper we study spacelike hypersurfaces in a weighted generalized Robertson-Walker (GRW) space-times.Moreover, a GRW space-time is a space-time regarding a warped product of a negative definite interval as a base, a Riemannian manifold as a fiber, and a positive smooth function as a warped function.Furthermore, there exists a distinguished family of spacelike hypersurfaces in a GRW space-time, that is, the so-called slices, which are defined as level hypersurfaces of the time coordinate of the space-time.Notice that any slice is totally umbilical and has constant mean curvature.
We have organized this paper as follows.In Section 2, we introduce some basic notions to be used for spacelike hypersurfaces immersed in weighted GRW space-times.In Section 3, we prove some uniqueness results of spacelike hypersurface in a weighted GRW space-time under appropriate conditions on the weighted mean curvature and the 2 Advances in Mathematical Physics weighted function by using the generalized Omori-Yau maximum principle or the weak maximum principle.Finally, in Section 4, applying the weak maximum principle, we obtain some rigidity results for the special case when the ambient space is static.

Preliminaries
Let   be a connected -dimensional oriented Riemannian manifold and  be an open interval in R endowed with the metric − 2 .We let  :  → R + be a positive smooth function.Denote − ×    to be the warped product endowed with the Lorentzian metric where   and   are the projections onto  and , respectively.This space-time is a warped product in the sense of ([6], Chap.7), with fiber (, ⟨, ⟩), base (, − 2 ), and warping function .Furthermore, for a fixed point  0 ∈ , we say that    0 = −{ 0 } ×   is a slice of − ×    .Following the terminology used in [7], we will refer to − ×    as a generalized Robertson-Walker (GRW) space-time.Particularly, if the fiber   has constant section curvature, it is called a Robertson-Walker (RW) space-time.
Recall that a smooth immersion  : Σ  → − ×    of an -dimensional connected manifold Σ  is called a spacelike hypersurface if the induced metric via  is a Riemannian metric on Σ  , which will be also denoted for ⟨, ⟩.
In the following, we will deal with two particular functions naturally attached to spacelike hypersurface Σ  , namely, the angle (or support) function Θ = ⟨,   ⟩ and the height function ℎ = (  )| Σ , where   fl / is a (unitary) timelike vector field globally defined on  and  is a unitary timelike normal vector field globally defined on Σ.
Let ∇ and ∇ stand for gradients with respect to the metrics of − ×    and Σ  , respectively.By a simple computation, we have Therefore, the gradient of ℎ on Particularly, we have where | | denotes the norm of a vector field on Σ  .Now, we consider that a GRW space-time − ×    is endowed with a weighted function , which will be called a weighted GRW space-time − ×     .In this setting, for a spacelike hypersurface where  is a tangent vector field on Σ  .
For a smooth function  : Σ  → R, we define its drifting Laplacian by and we will also denote such an operator as the -Laplacian of Σ  .According to Gromov [8], the weighted mean curvature or -mean curvature   of Σ  is given by where  is the standard mean curvature of hypersurface Σ  with respect to the Gauss map .
It follows from a splitting theorem due to Case (see [9] Theorem 1.2) that if a weighted GRW space-time − ×     is endowed with a bounded weighted function  such that Ric  (, ) ≥ 0 for all timelike vector fields  on − ×     , then  must be constant along R. In the same spirit of this result, in the following we will consider weighted GRW spacetimes − ×     whose weighted function  does not depend on the parameter  ∈ ; that is, ⟨∇,   ⟩ = 0.Moreover, for simplicity, we will refer to them as  +1 fl − ×     .In the following, we give some technical lemmas that will be essential for the proofs of our main results in weighted GRW space-times  +1 = − ×     (for further details on the proof, see Lemma 1 in [4]).
Lemma 1.Let Σ  be a spacelike hypersurface immersed in a weighted GRW spacetime  +1 = − ×     , with height function ℎ.Then, If we denote L 1  as the space of the integrable functions on Σ  with respect to the weighted volume element  =  − Σ, using the relation of div  () =   div( − ) and Proposition 2.1 in [10], we can obtain the following extension of a result in [11].In the following, we will introduce the weak maximum principle for the drifted Laplacian.By the fact in [12], that is, the Riemannian manifold  satisfies the weak maximum principle if and only if  is stochastically complete, we can have the next lemma which extended a result of [13].
Lemma 3. Let (  , ⟨, ⟩,  − ) be an -dimensional stochastically complete weighted Riemannian manifold and  :  → R be a smooth function which is bounded from below on   .Then there is a sequence of points   ∈   such that Equivalently, for any smooth function  : Σ  → R which is bounded from above on   , there is a sequence of points   ∈   such that

Uniqueness Results in Weighted GRW Space-Times
In this section, we will state and prove our main results in weighted GRW space-times  +1 = − ×     .We point out that, to prove the following results, we do not require that the -mean curvature   of the spacelike hypersurface Σ  is constant.
Recall that a slab of a weighted GRW spacetime − ×     is a region of the type Theorem 4. Let  +1 = − ×     be a weighted GRW spacetime which obeys (log )  (ℎ) ≤ 0. Let  : Σ  →  +1 be a complete spacelike hypersurface that lies in a slab of Proof.From (10), we have By the hypotheses, we have Δ  (ℎ) ≤ 0.Moreover, since Σ  lies in a slab, there is a positive constant  such that Therefore, we can apply Lemma 2 to get Δ  (ℎ) = 0; that is, (ℎ) is constant.Therefore Σ  is a slice.
Theorem 5. Let  +1 = − ×     be a weighted GRW spacetime which obeys   (ℎ) ≤ 0. Let  : Σ  →  +1 be a complete spacelike hypersurface that lies in a slab of  +1 .If the -mean Proof.By a similar reasoning as in the proof of Theorem 4, we have where the last inequality is due to Θ 2 ≥ 1.
Taking into account the assumptions, we have Δ  (ℎ) ≤ 0. Now in the same argument as in Theorem 4, we have that Σ  is a slice.
Next, we will use the weak maximum principle to study the rigidity of the spacelike hypersurfaces in weighted GRW space-times.Theorem 6.Let  +1 = − ×     be a weighted GRW spacetime which satisfies (log )  ≤ 0 and there is a point ℎ 0 ∈  such that   (ℎ 0 ) = 0. Let  : Σ  →  +1 be a stochastically complete constant -mean curvature   spacelike hypersurface such that sup ℎ ≥ ℎ 0 , inf ℎ ≤ ℎ 0 which is contained in a slab; then Σ  is -maximal.In addition, if Σ  is complete and |∇ℎ| ∈ L 1  , then Σ  is a slice.
Proof.We take the Gauss map  of the hypersurface Σ  such that Θ > 0; from (7) we have Θ ≥ 1.By Lemma 3, the weak maximum principle for the drifted Laplacian holds on Σ  ; then there exist two sequences On the other hand, from (9), we have Since Σ  lies in a slab, if ℎ is bounded from below, then Moreover, if ℎ is bounded from above, we get Considering that the function −  / is increasing, then Hence,   = 0; that is, Σ  is a -maximal spacelike hypersurface.Using (10), we have In the following, by the same argument as in Theorem 4, we have that Σ  is a slice.

Weighted Static GRW Space-Times
In this section, we obtain some rigidity results of stochastically complete hypersurfaces in weighted static GRW spacetimes − ×    by the weak maximal principle.Firstly, we give the following technical result which extended the corresponding conclusion in [12].for  ∈ X(Σ).Moreover, we also have where   is the sectional curvature of the fiber   and  * =  + ⟨,   ⟩  and  *  =   + ⟨  ,   ⟩  are the projections of the tangent vector fields  and   onto   .
Theorem 9. Let  : Σ  → − ×    be a stochastically complete hypersurface with constant -mean curvature   in a weighted static GRW space-time − ×    .Assume that the sectional curvature   is nonnegative and the weighted function  is convex.If |∇ℎ| 2 is bounded from above, then Σ  is -maximal.
Proof.As in the proof of Theorem 8,taking Therefore Σ  is -maximal.
As a consequence of the proof of Theorem 8, we can get the following corollary.

Lemma 2 .
Let  be a smooth function on a complete weighted Riemannian manifold Σ  with weighted function  such that Δ   does not change sign on Σ  .If |∇| ∈ L 1  , then Δ   vanishes identically on Σ  .