Spacelike Hypersurfaces in Weighted Generalized Robertson ...downloads.hindawi.com/journals/amp/2018/4523512.pdf · Spacelike Hypersurfaces in Weighted Generalized Robertson-Walker

Research ArticleSpacelike Hypersurfaces in Weighted GeneralizedRobertson-Walker Space-Times

Ximin Liu and Ning Zhang

School of Mathematical Sciences Dalian University of Technology Dalian 116024 China

Correspondence should be addressed to Ximin Liu ximinliudluteducn

Received 28 October 2017 Accepted 6 February 2018 Published 5 March 2018

Academic Editor Remi Leandre

Copyright copy 2018 Ximin Liu and Ning Zhang This is an open access article distributed under the Creative Commons AttributionLicense which permits unrestricted use distribution and reproduction in any medium provided the original work is properlycited

Applying generalized maximum principle and weak maximum principle we obtain several uniqueness results for spacelikehypersurfaces immersed in a weighted generalized Robertson-Walker (GRW) space-time under suitable geometric assumptionsFurthermore we also study the special case when the ambient space is static and provide some results by using Bochnerrsquos formula

1 Introduction

In recent years spacelike hypersurfaces in Lorentzian man-ifolds have been deeply studied not only from their math-ematical interest but also from their importance in generalrelativity

Particularly there are many articles that study spacelikehypersurfaces in weighted warped product space-times Aweighted manifold is a Riemannian manifold with a measurethat has a smooth positive density with respect to the Rie-mannian one More precisely the weighted manifold 119872119891associated with a complete 119899-dimensional Riemannian man-ifold (119872119899 119892) and a smooth function 119891 on 119872119899 is the triple(119872119899 119892 119889120583 = 119890minus119891119889119872) where 119889119872 stands for the volumeelement of 119872119899 In this setting we will take into accountthe so-called Bakry-Emery Ricci tensor (see [1]) which as anextension of the standard Ricci tensor Ric which is definedby

Ric119891 = Ric +Hess119891 (1)

Therefore it is natural to extend some results of the Riccicurvature to analogous results for the Bakry-Emery Riccitensor Before giving more details on our work we present abrief outline of some recent results related to our one

In [2] Wei and Wylie considered the complete 119899-dimen-sional weighted Riemannian manifold and proved mean cur-vature and volume comparison results on the assumption that

theinfin-Bakry-Emery Ricci tensor is bounded from below and119891 or |nabla119891| is bounded Later Cavalcante et al [3] researchedthe Bernstein-type properties concerning complete two-sided hypersurfaces immersed in a weighted warped productspace using the appropriated generalized maximum prin-ciples Moreover [4] obtained new Calabi-Bernsteinrsquos typeresults related to complete spacelike hypersurfaces in aweighted GRW space-time More recently some rigidityresults of complete spacelike hypersurfaces immersed into aweighted static GRW space-time are given in [5]

In this paper we study spacelike hypersurfaces in aweighted generalized Robertson-Walker (GRW) space-timesMoreover a GRW space-time is a space-time regarding awarped product of a negative definite interval as a basea Riemannian manifold as a fiber and a positive smoothfunction as a warped function Furthermore there exists adistinguished family of spacelike hypersurfaces in a GRWspace-time that is the so-called slices which are defined aslevel hypersurfaces of the time coordinate of the space-timeNotice that any slice is totally umbilical and has constantmean curvature

We have organized this paper as follows In Section 2we introduce some basic notions to be used for spacelikehypersurfaces immersed in weighted GRW space-times InSection 3 we prove some uniqueness results of spacelikehypersurface in a weighted GRW space-time under appro-priate conditions on the weighted mean curvature and the

HindawiAdvances in Mathematical PhysicsVolume 2018 Article ID 4523512 6 pageshttpsdoiorg10115520184523512

2 Advances in Mathematical Physics

weighted function by using the generalized Omori-Yau max-imum principle or the weak maximum principle Finally inSection 4 applying the weak maximum principle we obtainsome rigidity results for the special case when the ambientspace is static

2 Preliminaries

Let 119872119899 be a connected 119899-dimensional oriented Riemannianmanifold and 119868 be an open interval in R endowed with themetricminus1198891199052We let120588 119868 rarr R+ be a positive smooth functionDenote minus119868 times120588 119872119899 to be the warped product endowed withthe Lorentzian metric

⟨ ⟩ = minus120587lowast119868 (1198891199052) + 120588 (120587119868)2 120587lowast119872 (⟨ ⟩119872) (2)

where 120587119868 and 120587119872 are the projections onto 119868 and 119872 respec-tively This space-time is a warped product in the sense of([6]Chap 7) with fiber (119872 ⟨ ⟩) base (119868 minus1198891199052) andwarpingfunction 120588 Furthermore for a fixed point 1199050 isin 119868 we saythat 1198721198991199050 = minus1199050 times 119872119899 is a slice of minus119868 times120588 119872119899 Following theterminology used in [7] we will refer to minus119868 times120588 119872119899 as ageneralized Robertson-Walker (GRW) space-time Particularlyif the fiber 119872119899 has constant section curvature it is called aRobertson-Walker (RW) space-time

Recall that a smooth immersion 120595 Σ119899 rarr minus119868 times120588 119872119899 ofan 119899-dimensional connected manifold Σ119899 is called a spacelikehypersurface if the induced metric via 120595 is a Riemannianmetric on Σ119899 which will be also denoted for ⟨ ⟩

In the following we will deal with two particular func-tions naturally attached to spacelike hypersurface Σ119899 namelythe angle (or support) function Θ = ⟨119873 120597119905⟩ and the heightfunction ℎ = (120587119868)|Σ where 120597119905 fl 120597120597119905 is a (unitary) timelikevector field globally defined on119872 and119873 is a unitary timelikenormal vector field globally defined on Σ

Letnabla andnabla stand for gradients with respect to themetricsof minus119868 times120588 119872119899 and Σ119899 respectively By a simple computationwe have

nabla120587119868 = minus⟨nabla120587119868 120597119905⟩ 120597119905 = minus120597119905 (3)

Therefore the gradient of ℎ on Σ119899 isnablaℎ = (nabla120587119868)⊤ = minus120597⊤119905 = minus120597119905 minus Θ119873 (4)

Particularly we have

|nablaℎ|2 = minus1 + Θ2 (5)

where | | denotes the norm of a vector field on Σ119899Now we consider that a GRW space-time minus119868 times120588 119872119899 is

endowed with a weighted function 119891 which will be called aweighted GRW space-time minus119868 times120588 119872119899119891 In this setting for aspacelike hypersurface Σ119899 immersed into minus119868 times120588 119872119899119891 the 119891-divergence operator on Σ119899 is defined by

div119891 (119883) = 119890119891div (119890minus119891119883) (6)

where119883 is a tangent vector field on Σ119899

For a smooth function 119906 Σ119899 rarr R we define its driftingLaplacian by

Δ119891119906 = div119891 (nabla119906) = Δ119906 minus ⟨nabla119906 nabla119891⟩ (7)

and we will also denote such an operator as the 119891-Laplacianof Σ119899

According to Gromov [8] the weighted mean curvatureor 119891-mean curvature119867119891 of Σ119899 is given by

119899119867119891 = 119899119867 minus ⟨nabla119891119873⟩ (8)

where 119867 is the standard mean curvature of hypersurface Σ119899with respect to the Gauss map119873

It follows from a splitting theorem due to Case (see [9]Theorem 12) that if a weighted GRW space-time minus119868 times120588 119872119899119891is endowed with a bounded weighted function 119891 such thatRic119891(119881 119881) ge 0 for all timelike vector fields 119881 on minus119868 times120588 119872119899119891then 119891 must be constant along R In the same spirit of thisresult in the followingwewill considerweightedGRWspace-timesminus119868 times120588 119872119899119891 whose weighted function119891 does not dependon the parameter 119905 isin 119868 that is ⟨nabla119891 120597119905⟩ = 0 Moreover forsimplicity we will refer to them as119872119899+1 fl minus119868 times120588 119872119899119891

In the following we give some technical lemmas that willbe essential for the proofs of our main results in weightedGRW space-times119872119899+1 = minus119868 times120588 119872119899119891 (for further details onthe proof see Lemma 1 in [4])

Lemma 1 Let Σ119899 be a spacelike hypersurface immersed in aweighted GRW spacetime 119872119899+1 = minus119868 times120588 119872119899119891 with heightfunction ℎ Then

Δ119891ℎ = minus (log 120588)1015840 (ℎ) (119899 + |nablaℎ|2) minus 119899119867119891Θ (9)

Δ119891120588 (ℎ) = minus1198991205881015840 (ℎ)2120588 (ℎ) + |nablaℎ|2 120588 (ℎ) (log 120588)10158401015840 (ℎ)minus 1198991205881015840 (ℎ)119867119891Θ

(10)

If we denote L1119891 as the space of the integrable functionson Σ119899 with respect to the weighted volume element 119889120583 =119890minus119891119889Σ using the relation of div119891(119883) = 119890119891div(119890minus119891119883) andProposition 21 in [10] we can obtain the following extensionof a result in [11]

Lemma 2 Let 119906 be a smooth function on a complete weightedRiemannian manifold Σ119899 with weighted function 119891 such thatΔ119891119906 does not change sign on Σ119899 If |nabla119906| isin L1119891 then Δ119891119906vanishes identically on Σ119899

In the following we will introduce the weak maximumprinciple for the drifted Laplacian By the fact in [12] thatis the Riemannian manifold 119872 satisfies the weak maxi-mum principle if and only if 119872 is stochastically completewe can have the next lemma which extended a result of[13]

Advances in Mathematical Physics 3

Lemma3 Let (119872119899 ⟨ ⟩ 119890minus119891119889119872) be an 119899-dimensional stochas-tically complete weighted Riemannian manifold and 119906 119872 rarrR be a smooth function which is bounded from below on119872119899Then there is a sequence of points 119901119896 isin 119872119899 such that

lim119896119906 (119901119896) = inf 119906

lim119896Δ119891119906 (119901119896) ge 0 (11)

Equivalently for any smooth function 119906 Σ119899 rarr R which isbounded from above on119872119899 there is a sequence of points 119902119896 isin119872119899 such that

lim119896119906 (119902119896) = sup 119906

lim119896Δ119891119906 (119902119896) le 0 (12)

3 Uniqueness Results in WeightedGRW Space-Times

In this section we will state and prove our main results inweighted GRW space-times119872119899+1 = minus119868 times120588 119872119899119891We point outthat to prove the following results we do not require thatthe 119891-mean curvature 119867119891 of the spacelike hypersurface Σ119899is constant

Recall that a slab of a weighted GRW spacetime minus119868 times120588119872119899119891 is a region of the type

[1199051 1199052] times 119872119899119891 = (119905 119901) isin minus119868 times120588 119872119899119891 1199051 le 119905 le 1199052 (13)

Theorem4 Let119872119899+1 = minus119868 times120588 119872119899119891 be aweightedGRWspace-time which obeys (log 120588)10158401015840(ℎ) le 0 Let 120595 Σ119899 rarr 119872119899+1 bea complete spacelike hypersurface that lies in a slab of119872119899+1 Ifthe119891-mean curvature119867119891 satisfies1198672119891 le (1Θ2)(1205881015840(ℎ)2120588(ℎ)2)and |nablaℎ| isin L1119891(Σ) then Σ119899 is a slice of119872119899+1Proof From (10) we have

1120588 (ℎ)Δ119891120588 (ℎ) = minus1198991205881015840 (ℎ)2120588 (ℎ)2 + (log 120588)10158401015840 (ℎ) |nablaℎ|2

minus 1198991205881015840 (ℎ)120588 (ℎ) 119867119891Θ

le minus1198991205881015840 (ℎ)2120588 (ℎ)2 + (log 120588)10158401015840 (ℎ) |nablaℎ|2

+ 1198992 (

1205881015840 (ℎ)2120588 (ℎ)2 + 1198672119891Θ2)

le 1198992 (1198672119891Θ2 minus

1205881015840 (ℎ)2120588 (ℎ)2 )

+ (log 120588)10158401015840 (ℎ) |nablaℎ|2

(14)

By the hypotheses we have Δ119891120588(ℎ) le 0 Moreover sinceΣ119899 lies in a slab there is a positive constant 119862 such that

1003816100381610038161003816nabla120588 (ℎ)1003816100381610038161003816 le 1198621205881015840 (ℎ) |nablaℎ| isin L1119891 (Σ) (15)

Therefore we can apply Lemma 2 to get Δ119891120588(ℎ) = 0 that is120588(ℎ) is constant Therefore Σ119899 is a sliceTheorem5 Let119872119899+1 = minus119868 times120588 119872119899119891 be aweightedGRWspace-time which obeys 12058810158401015840(ℎ) le 0 Let 120595 Σ119899 rarr 119872119899+1 be a completespacelike hypersurface that lies in a slab of119872119899+1 If the119891-meancurvature119867119891 satisfies1198672119891 le 1205881015840(ℎ)2120588(ℎ)2 and |nablaℎ| isin L1119891(Σ)then Σ119899 is a slice of119872119899+1Proof By a similar reasoning as in the proof of Theorem 4we have

1120588 (ℎ)Δ119891120588 (ℎ) le minus1198991205881015840 (ℎ)2120588 (ℎ)2 + (log 120588)10158401015840 (ℎ) |nablaℎ|2

+ 1198992 (

1205881015840 (ℎ)2120588 (ℎ)2 + 1198672119891Θ2)

le 11989921198672119891Θ2 minus

11989921205881015840 (ℎ)2120588 (ℎ)2 +

12058810158401015840 (ℎ)120588 (ℎ) |nablaℎ|2

le 1198992 (1198672119891 minus

1205881015840 (ℎ)2120588 (ℎ)2 )Θ2

+ 12058810158401015840 (ℎ)120588 (ℎ) |nablaℎ|2

(16)

where the last inequality is due to Θ2 ge 1Taking into account the assumptions we have Δ119891120588(ℎ) le0 Now in the same argument as in Theorem 4 we have thatΣ119899 is a sliceNext we will use the weak maximum principle to study

the rigidity of the spacelike hypersurfaces in weighted GRWspace-times

Theorem6 Let119872119899+1 = minus119868 times120588 119872119899119891 be aweightedGRWspace-time which satisfies (log 120588)10158401015840 le 0 and there is a point ℎ0 isin 119868such that 1205881015840(ℎ0) = 0 Let 120595 Σ119899 rarr 119872119899+1 be a stochasticallycomplete constant 119891-mean curvature 119867119891 spacelike hypersur-face such that sup ℎ ge ℎ0 inf ℎ le ℎ0 which is contained ina slab then Σ119899 is 119891-maximal In addition if Σ119899 is complete and|nablaℎ| isin L1119891 then Σ119899 is a sliceProof We take the Gauss map119873 of the hypersurface Σ119899 suchthat Θ gt 0 from (7) we have Θ ge 1

By Lemma 3 the weak maximum principle for thedrifted Laplacian holds on Σ119899 then there exist two sequences119901119895 119902119895 sub Σ119899 such that

lim119895ℎ (119901119895) = inf ℎ


lim119895Δ119891ℎ (119901119895) ge 0lim119895ℎ (119902119895) = sup ℎ

lim119895Δ119891ℎ (119902119895) le 0

(17)

On the other hand from (9) we have

119867119891 = (minus1205881015840 (ℎ) 120588 (ℎ)) 119899 + |nablaℎ|2 minus Δ119891ℎ119899Θ (18)

Since Σ119899 lies in a slab if ℎ is bounded from below then

119867119891 le (minus1205881015840 (inf ℎ) 120588 (inf ℎ)) 119899 + 10038161003816100381610038161003816nablaℎ (119901119895)100381610038161003816100381610038162119899Θ (119901119895) (19)

Moreover if ℎ is bounded from above we get

119867119891 ge (minus1205881015840 (sup ℎ) 120588 (sup ℎ)) 119899 + 10038161003816100381610038161003816nablaℎ (119902119895)100381610038161003816100381610038162119899Θ (119902119895) (20)

Considering that the function minus1205881015840120588 is increasing then

119867119891 le (minus1205881015840 (inf ℎ) 120588 (inf ℎ)) 119899 + 10038161003816100381610038161003816nablaℎ (119901119895)100381610038161003816100381610038162119899Θ (119901119895)

le (minus1205881015840 (ℎ0) 120588 (ℎ0)) 119899 + 10038161003816100381610038161003816nablaℎ (119901119895)100381610038161003816100381610038162119899Θ (119901119895) le 0

119867119891 ge (minus1205881015840 (sup ℎ) 120588 (sup ℎ)) 119899 + 10038161003816100381610038161003816nablaℎ (119902119895)100381610038161003816100381610038162119899Θ (119902119895)

ge (minus1205881015840 (ℎ0) 120588 (ℎ0)) 119899 + 10038161003816100381610038161003816nablaℎ (119902119895)100381610038161003816100381610038162119899Θ (119902119895) ge 0

(21)

Hence 119867119891 = 0 that is Σ119899 is a 119891-maximal spacelike hyper-surface Using (10) we have

Δ119891120588 (ℎ) = minus1198991205881015840 (ℎ)2120588 (ℎ) + |nablaℎ|2 120588 (ℎ) (log 120588)10158401015840 (ℎ) le 0 (22)

In the following by the same argument as in Theorem 4 wehave that Σ119899 is a slice4 Weighted Static GRW Space-Times

In this section we obtain some rigidity results of stochasti-cally complete hypersurfaces in weighted static GRW space-times minus119868 times 119872119899119891 by the weak maximal principle Firstlywe give the following technical result which extended thecorresponding conclusion in [12]

Lemma 7 Let 119872 be a stochastically complete Riemannianmanifold and 119906 119872 rarr R be a nonnegative smooth functionon119872 If there exists a positive constant 120582 such that Δ119891119906 ge 120582119906then 119906 = 0Theorem 8 Let 120595 Σ119899 rarr minus119868 times 119872119899119891 be a stochastically com-plete hypersurface with constant 119891-mean curvature 119867119891 in aweighted static GRW spacetime minus119868 times 119872119899119891 Assume that119870119872 geminus119896 for some positive constant 119896 and the weighted function 119891 isconvex If |nablaℎ|2 le 120572|119860|2(119899 minus 1)119896 for some constant 0 lt 120572 lt 1then Σ119899 is a sliceProof Let 1198641 119864119899 be a (local) orthonormal frame inX(Σ)using the Gauss equation we have that

Ric (119883119883) = 119899sum119894=1

⟨119877 (119883 119864119894)119883 119864119894⟩ + 119899119867 ⟨119860119883119883⟩+ |119860119883|2

(23)

for119883 isin X(Σ) Moreover we also have

⟨119877 (119883 119864119894)119883 119864119894⟩= 119870119872 (119883lowast 119864lowast119894 ) (⟨119883lowast 119883lowast⟩ ⟨119864lowast119894 119864lowast119894 ⟩ minus ⟨119883lowast 119864lowast119894 ⟩2)

(24)

where119870119872 is the sectional curvature of the fiber119872119899 and119883lowast =119883 + ⟨119883 120597119905⟩120597119905 and 119864lowast119894 = 119864119894 + ⟨119864119894 120597119905⟩120597119905 are the projections ofthe tangent vector fields119883 and 119864119894 onto119872119899

By a direct computation and considering the hypothesis119870119872 gt minus119896 we get119899sum119894=1

⟨119877 (119883 119864119894)119883 119864119894⟩ge minus119896 ((119899 minus 1) |119883|2 + (119899 minus 2) ⟨119883 nablaℎ⟩2 + |119883|2 |nablaℎ|2)

(25)

Substituting (25) into (23)

Ric (119883119883)ge minus119896 ((119899 minus 1) |119883|2 + (119899 minus 2) ⟨119883 nablaℎ⟩2 + |119883|2 |nablaℎ|2)

+ 119899119867 ⟨119860119883119883⟩ + |119860119883|2 (26)

Furthermore taking into account that the weighted func-tion 119891 is convex we have

Hess119891 (119883119883) = Hess119891 (119883119883) minus ⟨nabla119891119873⟩ ⟨119860119883119883⟩ge minus⟨nabla119891119873⟩ ⟨119860119883119883⟩ (27)

Therefore

Ric119891 (119883119883)ge minus119896 ((119899 minus 1) |119883|2 + (119899 minus 2) ⟨119883 nablaℎ⟩2 + |119883|2 |nablaℎ|2)

+ 119899119867119891 ⟨119860119883119883⟩ + |119860119883|2 (28)


In particular we have

Ric119891 (nablaℎ nablaℎ) ge minus119896 (119899 minus 1) |nablaℎ|2 (1 + |nablaℎ|2)+ 119899119867119891 ⟨119860 (nablaℎ) nablaℎ⟩ + |119860 (nablaℎ)|2 (29)

Now we recall the Bochner-Lichnerowicz formula (see[2])

12Δ119891 (|nablaℎ|2) = |Hess ℎ|2 + Ric119891 (nablaℎ nablaℎ)

+ ⟨nablaΔ119891ℎ nablaℎ) (30)

From the fact that119867119891 is a constant we havenablaΔ119891ℎ = minus119899119867119891119860 (nablaℎ) (31)

By [14] we get

|Hess ℎ|2 = |119860|2 (1 + |nablaℎ|2) (32)

Using (29) (31) and (32) in (30) we have12Δ119891 |nablaℎ|2 ge (|119860|2 minus (119899 minus 1) 119896 |nablaℎ|2) (1 + |nablaℎ|2) (33)

Finally considering the hypothesis |nablaℎ|2 le 120572|119860|2(119899minus1)119896we obtain

Δ119891 |nablaℎ|2 ge 2 (1 minus 120572) |119860|2 |nablaℎ|2 (34)

Thus there is a positive constant 120582 such that

Δ119891 |nablaℎ|2 ge 120582 |nablaℎ|2 (35)

Therefore ℎ is constant by Lemma 7

Theorem 9 Let 120595 Σ119899 rarr minus119868 times 119872119899119891 be a stochastically com-plete hypersurface with constant 119891-mean curvature 119867119891 in aweighted static GRW space-time minus119868 times 119872119899119891 Assume that thesectional curvature 119870119872 is nonnegative and the weighted func-tion 119891 is convex If |nablaℎ|2 is bounded from above then Σ119899 is119891-maximal

Proof As in the proof ofTheorem 8 taking into account thatthe hypothesis 119870119872 is nonnegative there is a constant 119896 ge 0such that

12Δ119891 |nablaℎ|2 ge |Hess ℎ|2 + (119899 minus 1) 119896 |nablaℎ|2 (1 + |nablaℎ|2) (36)

Moreover considering the relation 119899|Hess ℎ|2 ge (Δ119891ℎ)2we have

Δ119891 |nablaℎ|2 ge 2119899 (Δ119891ℎ)

2 (37)

Using (9) and (37) we obtain

Δ119891 |nablaℎ|2 ge 21198991198672119891Θ2 ge 21198991198672119891 ge 0 (38)

By the hypothesis that |nablaℎ|2 is bounded from above applyingLemma 3 the weak maximum principle we get

0 ge lim119896Δ119891 |nablaℎ|2 (119902119896) ge 21198991198672119891 ge 0 (39)

Therefore Σ119899 is 119891-maximal

As a consequence of the proof of Theorem 8 we can getthe following corollary

Corollary 10 Let 120595 Σ119899 rarr minus119868 times119872119899119891 be a stochastically com-plete hypersurface with constant 119891-mean curvature 119867119891 in aweighted static GRW space-time minus119868 times119872119899119891 Assume that 119870119872 geminus119896 and Hess119891 ge minus120573 for some positive constants 119896 and 120573 If|nablaℎ|2 le 120572|119860|2((119899minus1)119896+120573) for some constant 0 lt 120572 lt 1 thenΣ119899 is a sliceConflicts of Interest

The authors declare that they have no conflicts of interest

Acknowledgments

This work is supported by National Natural Science Founda-tion of China (no 11371076)

References

[1] D Bakry and M Mery ldquoDiffusions hypercontractivesrdquo inSminaire de probabilits XIX vol 1123 of Lecture Notes in Mathpp 177ndash206 Springer Berlin Germany 1983

[2] G Wei and W Wylie ldquoComparison geometry for the Bakry-Emery Ricci tensorrdquo Journal of Differential Geometry vol 83no 2 pp 377ndash405 2009

[3] M P Cavalcante H F de Lima and M S Santos ldquoOnBernstein-type properties of complete hypersurfaces in weight-ed warped productsrdquo Annali di Matematica Pura ed ApplicataSeries IV vol 195 no 2 pp 309ndash322 2016

[4] M P Cavalcante H F de Lima andM S Santos ldquoNew Calabi-Bernstein type results in weighted generalized Robertson-Walker spacetimesrdquo Acta Mathematica Hungarica vol 145 no2 pp 440ndash454 2015

[5] H F de Lima A M Oliveira and M S Santos ldquoRigidity ofcomplete spacelike hypersurfaces with constant weighted meancurvaturerdquoBeitrage zur Algebra undGeometrieContributions toAlgebra and Geometry vol 57 no 3 pp 623ndash635 2016

[6] B OrsquoNeill Semi-Riemannian Geometry with Applications toRelativity Academic Press San Diego Calif USA 1983

[7] L J Alas A Romero and M Sanchez ldquoUniqueness of com-plete spacelike hypersurfaces of constant mean curvature ingeneralized Robertson-Walker spacetimesrdquo General Relativityand Gravitation vol 27 no 1 pp 71ndash84 1995

[8] M Gromov ldquoIsoperimetry of waists and concentration ofmapsrdquoGeometric and Functional Analysis vol 13 no 1 pp 178ndash215 2003

[9] J S Case ldquoSingularity theorems and the Lorentzian splittingtheorem for the Bakry-Emery-Ricci tensorrdquo Journal of Geom-etry and Physics vol 60 no 3 pp 477ndash490 2010

[10] A Caminha ldquoThe geometry of closed conformal vector fieldson Riemannian spacesrdquo Bulletin of the Brazilian Mathemat-ical Society New Series Boletim da Sociedade Brasileira deMatematica vol 42 no 2 pp 277ndash300 2011

[11] S T Yau ldquoSome function-theoretic properties of complete Rie-mannian manifold and their applications to geometryrdquo IndianaUniversityMathematics Journal vol 25 no 7 pp 659ndash670 1976

[12] S Pigola M Rigoli and A G Setti ldquoA remark on the maxi-mum principle and stochastic completenessrdquo Proceedings of the


American Mathematical Society vol 131 no 4 pp 1283ndash12882003

[13] M Rimoldi Rigidity Results for Lichnerowicz Bakry-Emery RicciTensors [PhD thesis] Universita degli Studi di Milano MilanoItaly 2011

[14] J M Latorre and A Romero ldquoUniqueness of noncompactspacelike hypersurfaces of constant mean curvature in general-ized Robertson-Walker spacetimesrdquo Geometriae Dedicata vol93 pp 1ndash10 2002

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weighted function by using the generalized Omori-Yau max-imum principle or the weak maximum principle Finally inSection 4 applying the weak maximum principle we obtainsome rigidity results for the special case when the ambientspace is static

2 Preliminaries

Let 119872119899 be a connected 119899-dimensional oriented Riemannianmanifold and 119868 be an open interval in R endowed with themetricminus1198891199052We let120588 119868 rarr R+ be a positive smooth functionDenote minus119868 times120588 119872119899 to be the warped product endowed withthe Lorentzian metric

⟨ ⟩ = minus120587lowast119868 (1198891199052) + 120588 (120587119868)2 120587lowast119872 (⟨ ⟩119872) (2)

where 120587119868 and 120587119872 are the projections onto 119868 and 119872 respec-tively This space-time is a warped product in the sense of([6]Chap 7) with fiber (119872 ⟨ ⟩) base (119868 minus1198891199052) andwarpingfunction 120588 Furthermore for a fixed point 1199050 isin 119868 we saythat 1198721198991199050 = minus1199050 times 119872119899 is a slice of minus119868 times120588 119872119899 Following theterminology used in [7] we will refer to minus119868 times120588 119872119899 as ageneralized Robertson-Walker (GRW) space-time Particularlyif the fiber 119872119899 has constant section curvature it is called aRobertson-Walker (RW) space-time

Recall that a smooth immersion 120595 Σ119899 rarr minus119868 times120588 119872119899 ofan 119899-dimensional connected manifold Σ119899 is called a spacelikehypersurface if the induced metric via 120595 is a Riemannianmetric on Σ119899 which will be also denoted for ⟨ ⟩

In the following we will deal with two particular func-tions naturally attached to spacelike hypersurface Σ119899 namelythe angle (or support) function Θ = ⟨119873 120597119905⟩ and the heightfunction ℎ = (120587119868)|Σ where 120597119905 fl 120597120597119905 is a (unitary) timelikevector field globally defined on119872 and119873 is a unitary timelikenormal vector field globally defined on Σ

Letnabla andnabla stand for gradients with respect to themetricsof minus119868 times120588 119872119899 and Σ119899 respectively By a simple computationwe have

nabla120587119868 = minus⟨nabla120587119868 120597119905⟩ 120597119905 = minus120597119905 (3)

Therefore the gradient of ℎ on Σ119899 isnablaℎ = (nabla120587119868)⊤ = minus120597⊤119905 = minus120597119905 minus Θ119873 (4)

Particularly we have

|nablaℎ|2 = minus1 + Θ2 (5)

where | | denotes the norm of a vector field on Σ119899Now we consider that a GRW space-time minus119868 times120588 119872119899 is

endowed with a weighted function 119891 which will be called aweighted GRW space-time minus119868 times120588 119872119899119891 In this setting for aspacelike hypersurface Σ119899 immersed into minus119868 times120588 119872119899119891 the 119891-divergence operator on Σ119899 is defined by

div119891 (119883) = 119890119891div (119890minus119891119883) (6)

where119883 is a tangent vector field on Σ119899

For a smooth function 119906 Σ119899 rarr R we define its driftingLaplacian by

Δ119891119906 = div119891 (nabla119906) = Δ119906 minus ⟨nabla119906 nabla119891⟩ (7)

and we will also denote such an operator as the 119891-Laplacianof Σ119899

According to Gromov [8] the weighted mean curvatureor 119891-mean curvature119867119891 of Σ119899 is given by

119899119867119891 = 119899119867 minus ⟨nabla119891119873⟩ (8)

where 119867 is the standard mean curvature of hypersurface Σ119899with respect to the Gauss map119873

It follows from a splitting theorem due to Case (see [9]Theorem 12) that if a weighted GRW space-time minus119868 times120588 119872119899119891is endowed with a bounded weighted function 119891 such thatRic119891(119881 119881) ge 0 for all timelike vector fields 119881 on minus119868 times120588 119872119899119891then 119891 must be constant along R In the same spirit of thisresult in the followingwewill considerweightedGRWspace-timesminus119868 times120588 119872119899119891 whose weighted function119891 does not dependon the parameter 119905 isin 119868 that is ⟨nabla119891 120597119905⟩ = 0 Moreover forsimplicity we will refer to them as119872119899+1 fl minus119868 times120588 119872119899119891

In the following we give some technical lemmas that willbe essential for the proofs of our main results in weightedGRW space-times119872119899+1 = minus119868 times120588 119872119899119891 (for further details onthe proof see Lemma 1 in [4])

Lemma 1 Let Σ119899 be a spacelike hypersurface immersed in aweighted GRW spacetime 119872119899+1 = minus119868 times120588 119872119899119891 with heightfunction ℎ Then

Δ119891ℎ = minus (log 120588)1015840 (ℎ) (119899 + |nablaℎ|2) minus 119899119867119891Θ (9)

Δ119891120588 (ℎ) = minus1198991205881015840 (ℎ)2120588 (ℎ) + |nablaℎ|2 120588 (ℎ) (log 120588)10158401015840 (ℎ)minus 1198991205881015840 (ℎ)119867119891Θ

(10)

If we denote L1119891 as the space of the integrable functionson Σ119899 with respect to the weighted volume element 119889120583 =119890minus119891119889Σ using the relation of div119891(119883) = 119890119891div(119890minus119891119883) andProposition 21 in [10] we can obtain the following extensionof a result in [11]

Lemma 2 Let 119906 be a smooth function on a complete weightedRiemannian manifold Σ119899 with weighted function 119891 such thatΔ119891119906 does not change sign on Σ119899 If |nabla119906| isin L1119891 then Δ119891119906vanishes identically on Σ119899

In the following we will introduce the weak maximumprinciple for the drifted Laplacian By the fact in [12] thatis the Riemannian manifold 119872 satisfies the weak maxi-mum principle if and only if 119872 is stochastically completewe can have the next lemma which extended a result of[13]



lim119896119906 (119901119896) = inf 119906

lim119896Δ119891119906 (119901119896) ge 0 (11)


lim119896119906 (119902119896) = sup 119906

lim119896Δ119891119906 (119902119896) le 0 (12)







minus 1198991205881015840 (ℎ)120588 (ℎ) 119867119891Θ


+ 1198992 (

1205881015840 (ℎ)2120588 (ℎ)2 + 1198672119891Θ2)

le 1198992 (1198672119891Θ2 minus

1205881015840 (ℎ)2120588 (ℎ)2 )

+ (log 120588)10158401015840 (ℎ) |nablaℎ|2

(14)





+ 1198992 (

1205881015840 (ℎ)2120588 (ℎ)2 + 1198672119891Θ2)

le 11989921198672119891Θ2 minus

11989921205881015840 (ℎ)2120588 (ℎ)2 +

12058810158401015840 (ℎ)120588 (ℎ) |nablaℎ|2

le 1198992 (1198672119891 minus

1205881015840 (ℎ)2120588 (ℎ)2 )Θ2

+ 12058810158401015840 (ℎ)120588 (ℎ) |nablaℎ|2

(16)





lim119895ℎ (119901119895) = inf ℎ



lim119895Δ119891ℎ (119902119895) le 0

(17)












(21)






Ric (119883119883) = 119899sum119894=1

⟨119877 (119883 119864119894)119883 119864119894⟩ + 119899119867 ⟨119860119883119883⟩+ |119860119883|2

(23)



(24)




(25)



+ 119899119867 ⟨119860119883119883⟩ + |119860119883|2 (26)



Therefore


+ 119899119867119891 ⟨119860119883119883⟩ + |119860119883|2 (28)








By [14] we get

|Hess ℎ|2 = |119860|2 (1 + |nablaℎ|2) (32)











Δ119891 |nablaℎ|2 ge 2119899 (Δ119891ℎ)

2 (37)


Δ119891 |nablaℎ|2 ge 21198991198672119891Θ2 ge 21198991198672119891 ge 0 (38)







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lim119896119906 (119901119896) = inf 119906

lim119896Δ119891119906 (119901119896) ge 0 (11)


lim119896119906 (119902119896) = sup 119906

lim119896Δ119891119906 (119902119896) le 0 (12)







minus 1198991205881015840 (ℎ)120588 (ℎ) 119867119891Θ


+ 1198992 (

1205881015840 (ℎ)2120588 (ℎ)2 + 1198672119891Θ2)

le 1198992 (1198672119891Θ2 minus

1205881015840 (ℎ)2120588 (ℎ)2 )

+ (log 120588)10158401015840 (ℎ) |nablaℎ|2

(14)





+ 1198992 (

1205881015840 (ℎ)2120588 (ℎ)2 + 1198672119891Θ2)

le 11989921198672119891Θ2 minus

11989921205881015840 (ℎ)2120588 (ℎ)2 +

12058810158401015840 (ℎ)120588 (ℎ) |nablaℎ|2

le 1198992 (1198672119891 minus

1205881015840 (ℎ)2120588 (ℎ)2 )Θ2

+ 12058810158401015840 (ℎ)120588 (ℎ) |nablaℎ|2

(16)





lim119895ℎ (119901119895) = inf ℎ



lim119895Δ119891ℎ (119902119895) le 0

(17)












(21)






Ric (119883119883) = 119899sum119894=1

⟨119877 (119883 119864119894)119883 119864119894⟩ + 119899119867 ⟨119860119883119883⟩+ |119860119883|2

(23)



(24)




(25)



+ 119899119867 ⟨119860119883119883⟩ + |119860119883|2 (26)



Therefore


+ 119899119867119891 ⟨119860119883119883⟩ + |119860119883|2 (28)








By [14] we get

|Hess ℎ|2 = |119860|2 (1 + |nablaℎ|2) (32)











Δ119891 |nablaℎ|2 ge 2119899 (Δ119891ℎ)

2 (37)


Δ119891 |nablaℎ|2 ge 21198991198672119891Θ2 ge 21198991198672119891 ge 0 (38)







Acknowledgments


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lim119895Δ119891ℎ (119902119895) le 0

(17)












(21)






Ric (119883119883) = 119899sum119894=1

⟨119877 (119883 119864119894)119883 119864119894⟩ + 119899119867 ⟨119860119883119883⟩+ |119860119883|2

(23)



(24)




(25)



+ 119899119867 ⟨119860119883119883⟩ + |119860119883|2 (26)



Therefore


+ 119899119867119891 ⟨119860119883119883⟩ + |119860119883|2 (28)








By [14] we get

|Hess ℎ|2 = |119860|2 (1 + |nablaℎ|2) (32)











Δ119891 |nablaℎ|2 ge 2119899 (Δ119891ℎ)

2 (37)


Δ119891 |nablaℎ|2 ge 21198991198672119891Θ2 ge 21198991198672119891 ge 0 (38)







Acknowledgments


References
























Journal of












Journal of







Volume 2018






Volume 2018















By [14] we get

|Hess ℎ|2 = |119860|2 (1 + |nablaℎ|2) (32)











Δ119891 |nablaℎ|2 ge 2119899 (Δ119891ℎ)

2 (37)


Δ119891 |nablaℎ|2 ge 21198991198672119891Θ2 ge 21198991198672119891 ge 0 (38)







Acknowledgments


References
























Journal of












Journal of







Volume 2018






Volume 2018



















Journal of












Journal of







Volume 2018






Volume 2018















Journal of












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Volume 2018






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