# Spacelike Hypersurfaces in Weighted Generalized Robertson-Walker Space-Times

Authors: Ximin Liu; Ning Zhang
Journal: Advances in Mathematical Physics
 (2018)

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## Abstract

Applying generalized maximum principle and weak maximum principle, we obtain several uniqueness results for spacelike hypersurfaces immersed in a weighted generalized Robertson-Walker (GRW) space-time under suitable geometric assumptions. Furthermore, we also study the special case when the ambient space is static and provide some results by using Bochner’s formula.


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## Body

## 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 manifoldMf 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 (1)Ricf=Ric+Hessf. 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 n-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 f or ∇f is bounded. Later, Cavalcante et al. [3] researched the Bernstein-type properties concerning complete two-sided 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 weightedgeneralized 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 Section2, 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 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.


## 2. Preliminaries

LetMn be a connected n-dimensional oriented Riemannian manifold and I be an open interval in R endowed with the metric -dt2. We let ρ:I→R+ be a positive smooth function. Denote -I×ρMn to be the warped product endowed with the Lorentzian metric (2),=-πI∗dt2+ρπI2πM∗,M, where πI and πM are the projections onto I and M, respectively. This space-time is awarped product in the sense of ([6],Chap.7), withfiber(M,,),base(I,-dt2), andwarping functionρ. Furthermore, for a fixed point t0∈I, we say that Mt0n=-t0×Mn is aslice of -I×ρMn. Following the terminology used in [7], we will refer to -I×ρMn as ageneralized Robertson-Walker (GRW) space-time. Particularly, if the fiber Mn has constant section curvature, it is called aRobertson-Walker (RW) space-time.Recall that a smooth immersionψ:Σn→-I×ρMn of an n-dimensional connected manifold Σn is called aspacelike hypersurface if the induced metric via ψ is a Riemannian metric on Σn, which will be also denoted for ,.In the following, we will deal with two particular functions naturally attached to spacelike hypersurfaceΣn, namely, theangle (or support) functionΘ=N,∂t and theheight functionh=πIΣ, where ∂t≔∂/∂t is a (unitary) timelike vector field globally defined on M¯ and N is a unitary timelike normal vector field globally defined on Σ.Let∇¯ and ∇ stand for gradients with respect to the metrics of -I×ρMn and Σn, respectively. By a simple computation, we have (3)∇¯πI=-∇¯πI,∂t∂t=-∂t.Therefore, the gradient of h on Σn is (4)∇h=∇¯πI⊤=-∂t⊤=-∂t-ΘN.Particularly, we have (5)∇h2=-1+Θ2,where  denotes the norm of a vector field on Σn.Now, we consider that a GRW space-time-I×ρMn is endowed with a weighted function f, which will be called a weighted GRW space-time -I×ρMfn. In this setting, for a spacelike hypersurface Σn immersed into -I×ρMfn, the f-divergence operator on Σn is defined by (6)divf⁡X=efdive-fX,where X is a tangent vector field on Σn.For a smooth functionu:Σn→R, we define itsdrifting Laplacian by (7)Δfu=divf⁡∇u=Δu-∇u,∇f,and we will also denote such an operator as the f-Laplacian of Σn.According to Gromov [8], the weighted mean curvature or f-mean curvature Hf of Σn is given by (8)nHf=nH-∇¯f,N,where H is the standard mean curvature of hypersurface Σn with respect to the Gauss map N.It follows from a splitting theorem due to Case (see [9] Theorem 1.2) that if a weighted GRW space-time -I×ρMfn is endowed with a bounded weighted function f such that Ric¯fV,V≥0 for all timelike vector fields V on -I×ρMfn, then f must be constant along R. In the same spirit of this result, in the following we will consider weighted GRW space-times -I×ρMfn whose weighted function f does not depend on the parameter t∈I; that is, ∇¯f,∂t=0. Moreover, for simplicity, we will refer to them as M¯n+1≔-I×ρMfn.In the following, we give some technical lemmas that will be essential for the proofs of our main results in weighted GRW space-timesM¯n+1=-I×ρMfn (for further details on the proof, see Lemma 1 in [4]).Lemma 1.
LetΣn be a spacelike hypersurface immersed in a weighted GRW spacetime M¯n+1=-I×ρMfn, with height function h. Then, (9)Δfh=-log⁡ρ′hn+∇h2-nHfΘ,(10)Δfρh=-nρ′h2ρh+∇h2ρhlog⁡ρ′′h-nρ′hHfΘ.If we denoteLf1 as the space of the integrable functions on Σn with respect to the weighted volume element dμ=e-fdΣ, using the relation of divfX=efdive-fX and Proposition 2.1 in [10], we can obtain the following extension of a result in [11].Lemma 2.
Letu be a smooth function on a complete weighted Riemannian manifold Σn with weighted function f such that Δfu does not change sign on Σn. If ∇u∈Lf1, then Δfu vanishes identically on Σn.In the following, we will introduce the weak maximum principle for the drifted Laplacian. By the fact in [12], that is, the Riemannian manifold M satisfies the weak maximum principle if and only if M is stochastically complete, we can have the next lemma which extended a result of [13].Lemma 3.
LetMn,,,e-fdM be an n-dimensional stochastically complete weighted Riemannian manifold and u:M→R be a smooth function which is bounded from below on Mn. Then there is a sequence of points pk∈Mn such that(11)limk⁡upk=inf⁡u,limk⁡Δfupk≥0. Equivalently, for any smooth function u:Σn→R which is bounded from above on Mn, there is a sequence of points qk∈Mn such that(12)limk⁡uqk=sup⁡u,limk⁡Δfuqk≤0.


## 3. Uniqueness Results in Weighted GRW Space-Times

In this section, we will state and prove our main results in weighted GRW space-timesM¯n+1=-I×ρMfn. We point out that, to prove the following results, we do not require that the f-mean curvature Hf of the spacelike hypersurface Σn is constant.Recall that a slab of a weighted GRW spacetime-I×ρMfn is a region of the type (13)t1,t2×Mfn=t,p∈-I×ρMfn:t1≤t≤t2.Theorem 4.
LetM¯n+1=-I×ρMfn be a weighted GRW spacetime which obeys log⁡ρ′′h≤0. Let ψ:Σn→M¯n+1 be a complete spacelike hypersurface that lies in a slab of M¯n+1. If the f-mean curvature Hf satisfies Hf2≤1/Θ2ρ′h2/ρh2 and ∇h∈Lf1(Σ), then Σn is a slice of M¯n+1.Proof.
From (10), we have (14)1ρhΔfρh=-nρ′h2ρh2+log⁡ρ′′h∇h2-nρ′hρhHfΘ≤-nρ′h2ρh2+log⁡ρ′′h∇h2+n2ρ′h2ρh2+Hf2Θ2≤n2Hf2Θ2-ρ′h2ρh2+log⁡ρ′′h∇h2.
By the hypotheses, we haveΔfρ(h)≤0. Moreover, since Σn lies in a slab, there is a positive constant C such that (15)∇ρh≤Cρ′h∇h∈Lf1Σ. Therefore, we can apply Lemma 2 to get Δfρ(h)=0; that is, ρ(h) is constant. Therefore Σn is a slice.Theorem 5.
LetM¯n+1=-I×ρMfn be a weighted GRW spacetime which obeys ρ′′h≤0. Let ψ:Σn→M¯n+1 be a complete spacelike hypersurface that lies in a slab of M¯n+1. If the f-mean curvature Hf satisfies Hf2≤ρ′h2/ρh2 and ∇h∈Lf1Σ, then Σn is a slice of M¯n+1.Proof.
By a similar reasoning as in the proof of Theorem4, we have (16)1ρhΔfρh≤-nρ′h2ρh2+log⁡ρ′′h∇h2+n2ρ′h2ρh2+Hf2Θ2≤n2Hf2Θ2-n2ρ′h2ρh2+ρ′′hρh∇h2≤n2Hf2-ρ′h2ρh2Θ2+ρ′′hρh∇h2, where the last inequality is due to Θ2≥1.
Taking into account the assumptions, we haveΔfρ(h)≤0. Now in the same argument as in Theorem 4, we have that Σn 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.
LetM¯n+1=-I×ρMfn be a weighted GRW spacetime which satisfies log⁡ρ′′≤0 and there is a point h0∈I such that ρ′(h0)=0. Let ψ:Σn→M¯n+1 be a stochastically complete constant f-mean curvature Hf spacelike hypersurface such that sup⁡h≥h0, inf⁡h≤h0 which is contained in a slab; then Σn is f-maximal. In addition, if Σn is complete and ∇h∈Lf1, then Σn is a slice.Proof.
We take the Gauss mapN of the hypersurface Σn such that Θ>0; from (7) we have Θ≥1.
By Lemma3, the weak maximum principle for the drifted Laplacian holds on Σn; then there exist two sequences {pj},{qj}⊂Σn such that (17)limj⁡hpj=inf⁡h,limj⁡Δfhpj≥0.limj⁡hqj=sup⁡h,limj⁡Δfhqj≤0.
On the other hand, from (9), we have (18)Hf=-ρ′h/ρhn+∇h2-ΔfhnΘ. Since Σn lies in a slab, if h is bounded from below, then (19)Hf≤-ρ′inf⁡h/ρinf⁡hn+∇hpj2nΘpj. Moreover, if h is bounded from above, we get (20)Hf≥-ρ′sup⁡h/ρsup⁡hn+∇hqj2nΘqj.
Considering that the function-ρ′/ρ is increasing, then (21)Hf≤-ρ′inf⁡h/ρinf⁡hn+∇hpj2nΘpj≤-ρ′h0/ρh0n+∇hpj2nΘpj≤0,Hf≥-ρ′sup⁡h/ρsup⁡hn+∇hqj2nΘqj≥-ρ′h0/ρh0n+∇hqj2nΘqj≥0. Hence, Hf=0; that is, Σn is a f-maximal spacelike hypersurface. Using (10), we have (22)Δfρh=-nρ′h2ρh+∇h2ρhlog⁡ρ′′h≤0. In the following, by the same argument as in Theorem 4, we have that Σn is a slice.


## 4. Weighted Static GRW Space-Times

In this section, we obtain some rigidity results of stochastically complete hypersurfaces in weighted static GRW space-times-I×Mfn by the weak maximal principle. Firstly, we give the following technical result which extended the corresponding conclusion in [12].Lemma 7.
LetM be a stochastically complete Riemannian manifold and u:M→R be a nonnegative smooth function on M. If there exists a positive constant λ such that Δfu≥λu, then u=0.Theorem 8.
Letψ:Σn→-I×Mfn be a stochastically complete hypersurface with constant f-mean curvature Hf in a weighted static GRW spacetime -I×Mfn. Assume that KM≥-k for some positive constant k and the weighted function f is convex. If ∇h2≤αA2/(n-1)k for some constant 0<α<1, then Σn is a slice.Proof.
LetE1,…,En be a (local) orthonormal frame in X(Σ); using the Gauss equation, we have that (23)RicX,X=∑i=1nR¯X,EiX,Ei+nHAX,X+AX2,for X∈X(Σ). Moreover, we also have (24)R¯X,EiX,Ei=KMX∗,Ei∗X∗,X∗Ei∗,Ei∗-X∗,Ei∗2,where KM is the sectional curvature of the fiber Mn and X∗=X+X,∂t∂t and Ei∗=Ei+Ei,∂t∂t are the projections of the tangent vector fields X and Ei onto Mn.
By a direct computation and considering the hypothesisKM>-k, we get (25)∑i=1nR¯X,EiX,Ei≥-kn-1X2+n-2X,∇h2+X2∇h2. Substituting (25) into (23), (26)RicX,X≥-kn-1X2+n-2X,∇h2+X2∇h2+nHAX,X+AX2.
Furthermore, taking into account that the weighted functionf is convex, we have (27)HessfX,X=Hess¯fX,X-∇¯f,NAX,X≥-∇¯f,NAX,X.
Therefore,(28)RicfX,X≥-kn-1X2+n-2X,∇h2+X2∇h2+nHfAX,X+AX2.In particular, we have (29)Ricf∇h,∇h≥-kn-1∇h21+∇h2+nHfA∇h,∇h+A∇h2.
Now we recall the Bochner-Lichnerowicz formula (see [2]): (30)12Δf∇h2=Hessh2+Ricf∇h,∇h+∇Δfh,∇h. From the fact that Hf is a constant, we have (31)∇Δfh=-nHfA∇h. By [14], we get (32)Hessh2=A21+∇h2. Using (29), (31), and (32) in (30), we have (33)12Δf∇h2≥A2-n-1k∇h21+∇h2.
Finally, considering the hypothesis∇h2≤αA2/n-1k, we obtain (34)Δf∇h2≥21-αA2∇h2. Thus, there is a positive constant λ such that (35)Δf∇h2≥λ∇h2. Therefore, h is constant by Lemma 7.Theorem 9.
Letψ:Σn→-I×Mfn be a stochastically complete hypersurface with constant f-mean curvature Hf in a weighted static GRW space-time -I×Mfn. Assume that the sectional curvature KM is nonnegative and the weighted function f is convex. If ∇h2 is bounded from above, then Σn is f-maximal.Proof.
As in the proof of Theorem8, taking into account that the hypothesis KM is nonnegative, there is a constant k≥0 such that (36)12Δf∇h2≥Hessh2+n-1k∇h21+∇h2.
Moreover, considering the relationnHessh2≥Δfh2, we have (37)Δf∇h2≥2nΔfh2. Using (9) and (37), we obtain (38)Δf∇h2≥2nHf2Θ2≥2nHf2≥0. By the hypothesis that ∇h2 is bounded from above, applying Lemma 3, the weak maximum principle, we get (39)0≥limk⁡Δf∇h2qk≥2nHf2≥0.
ThereforeΣn is f-maximal.As a consequence of the proof of Theorem8, we can get the following corollary.Corollary 10.
Letψ:Σn→-I×Mfn be a stochastically complete hypersurface with constant f-mean curvature Hf in a weighted static GRW space-time -I×Mfn. Assume that KM≥-k and Hess¯f≥-β for some positive constants k and β. If ∇h2≤αA2/(n-1)k+β for some constant 0<α<1, then Σn is a slice.



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