Hindawi Publishing Corporation Journal of Applied Mathematics Volume 2013, Article ID 757041, 8 pages http://dx.doi.org/10.1155/2013/757041 Research Article On Complete Spacelike Hypersurfaces in a Semi-Riemannian Warped Product Yaning Wang1 and Ximin Liu2 1 2 Department of Mathematics, South China University of Technology, Guangzhou, Guangdong 510641, China School of Mathematical Sciences, Dalian University of Technology, Dalian, Liaoning 116024, China Correspondence should be addressed to Yaning Wang; wyn051@163.com Received 19 November 2012; Revised 26 January 2013; Accepted 26 January 2013 Academic Editor: P. G. L. Leach Copyright © 2013 Y. Wang and X. Liu. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. By applying Omori-Yau maximal principal theory and supposing an appropriate restriction on the norm of gradient of height function, we obtain some new Bernstein-type theorems for complete spacelike hypersurfaces with nonpositive constant mean curvature immersed in a semi-Riemannian warped product. Furthermore, some applications of our main theorems for entire vertical graphs in Robertson-Walker spacetime and for hypersurfaces in hyperbolic space are given. 1. Introduction The theory of spacelike hypersurfaces immersed in semiRiemannian warped products with constant mean curvature has got increasing interest both from geometers and physicists recently. One of the basic questions on this topic is the problem of uniqueness for this type of hypersurfaces. The aim of this paper is to study such type problems. Before giving details of our main results, we firstly present a brief outline of some recent papers containing theorems related to ours. By using a suitable application of well-known generalized maximal principal of Omori [1] and Yau [2], Albujer et al. [3] obtained uniqueness results concerning complete spacelike hypersurfaces with constant mean curvature immersed in a Robertson-Walker spacetime. By applying a maximal principal due to Akutagawa [4], Aquino and de Lima [5] obtained a new Bernstein-type theorem concerning constant mean curvature complete vertical graphs immersed in a Riemannian warped product, which is supposed to satisfy an appropriate convergence condition defined as the following: 0 β₯ ππ β₯ sup (πσΈ 2 β ππσΈ σΈ ) , πΌ (1) where ππ denotes the sectional curvature of the fiber ππ , which is very different from null convergence condition (defined by (2)). Replacing the null convergence condition by (ln π)σΈ σΈ β€ 0, AlΔ±Μas et al. obtained uniqueness theorems (see Section 2 of [6]) concerning compact spacelike hypersurface with higher constant mean curvature immersed in a spatially closed generalized Robertson-Walker spacetime. We also refer the reader to [7] for the other relevant results. In this paper, following [6, 8], we consider the Laplacian of integral of warping function. By using a suitable maximal principal of Omori and Yau and supposing an appropriate restriction on the norm of gradient of height function, we obtain some new Bernstein-type theorems as the following. π+1 = βπΌ ×π ππ be a Robertson-Walker Theorem 1. Let π spacetime whose Riemannian fiber ππ has constant sectional curvature π satisfying the null convergence condition defined as the following: π β₯ sup (ππσΈ σΈ β πσΈ 2 ) . πΌ (2) π+1 Let π : Ξ£π β π be a complete spacelike hypersurface with constant mean curvature π». Suppose that Ξ£π is bounded away π+1 from the infinity of π and that π» β€ min { infπ πβΞ£ πσΈ (β (π)) , 0} . π (3) 2 Journal of Applied Mathematics If the height function β of Ξ£π satisfies σ΅¨σ΅¨ σ΅¨σ΅¨π½ πσΈ σ΅¨ σ΅¨ |ββ|2 β€ πΌσ΅¨σ΅¨σ΅¨σ΅¨ infπ (β (π)) β π»σ΅¨σ΅¨σ΅¨σ΅¨ , σ΅¨σ΅¨πβΞ£ π σ΅¨σ΅¨ (4) for some positive constant πΌ and π½, then Ξ£π is a slice. π+1 Theorem 2. Let π = πΌ ×π ππ be a Riemannian warped product whose fiber ππ has constant sectional curvature π satisfying π β₯ sup (πσΈ 2 β ππσΈ σΈ ) . (5) πΌ π+1 Let π : Ξ£π β π be a complete spacelike hypersurface with constant mean curvature π» and π»2 is bounded from below. π+1 Suppose that Ξ£π is bounded away from the infinity of π and that πσΈ (β (π)) β€ π» β€ 0. πβΞ£π π sup (6) If the height function β of Ξ£π satisfies σ΅¨σ΅¨ σ΅¨σ΅¨π½ σ΅¨σ΅¨ σ΅¨σ΅¨ πσΈ σ΅¨ |ββ| β€ πΌσ΅¨σ΅¨π» β sup (β (π))σ΅¨σ΅¨σ΅¨ , σ΅¨σ΅¨ σ΅¨σ΅¨ π π πβΞ£ σ΅¨ σ΅¨ 2 (7) for some positive constant πΌ and π½, then Ξ£π is a slice. 2. Preliminaries In this section, we recall some basic notations and facts following from [3, 8] that will appear along this paper. Let ππ be a connected, π-dimensional (π β₯ 2) oriented Riemannian manifold, πΌ β R an interval, and π : πΌ β R a positive smooth function. We consider the product differential manifold πΌ × ππ and denote by ππΌ and ππ the projections onto the base πΌ and fiber ππ , respectively. A particular class of semi-Riemannian manifolds is the one obtained by furnishing πΌ × ππ with the metric β¨V, π€β©π = π β¨(ππΌ )β V, (ππΌ )β π€β© 2 + (π β ππΌ (π)) β¨(ππ)β V, (ππ)β π€β© , π+1 cosh π = β β¨π, ππ‘ β© . π+1 (9) (10) We denote by β and β the Levi-Civita connections in ππΌ ×π ππ and Ξ£π , respectively. Then the Gauss-Weingarten formulas for the spacelike hypersurface π : Ξ£π β ππΌ ×π ππ are given by βπ π = βπ π + π β¨π΄π, πβ© π, βπ π = βπ΄π, (11) for any π β Ξ(πΞ£π ), where π΄ : Ξ(πΞ£π ) β Ξ(πΞ£π ) be the shape operator of Ξ£π with respect to its Gauss map π, and Ξ(πΞ£π ) denotes the Lie algebra of all tangential vector fields on Ξ£π . The curvature tensor π of a spacelike hypersurface Ξ£π is given by [12] as the following: π (π, π) π = β[π,π] π β [βπ , βπ ] π, (12) where [, ] denotes the Lie bracket and π, π, π β Ξ(πΞ£π ). Let π and π be the curvature tensors of ππΌ ×π ππ and Ξ£π , β€ respectively. Denote by π the tangential component of a π+1 vector field π β Ξ(ππ ); thus, for any π, π, π β Ξ(πΞ£π ) we have the following Gauss equation: (8) for any π β π and any V, π€ β ππ π , where π = ±1. We call such a space warped product manifold; π is known as the warping function, and we denote the space by π+1 π = ππΌ ×π ππ . Note that βπΌ ×π ππ is called a generalized Robertson-Walker spacetime, in particular, βπΌ ×π ππ is called a Robertson-Walker spacetime if ππ has constant sectional curvature. From [9], we know that a generalized Robertson-Walker spacetime has constant sectional curvature π if and only if the Riemannian fiber ππ has constant sectional curvature π and the warping function π satisfies the following differential equation: πσΈ 2 + π πσΈ σΈ . =π= π π2 It follows from [10] that the vector field (π β ππΌ )ππ‘ is conformal and closed (in this sense that its dual 1-form is closed) with conformal factor π = πσΈ β ππΌ , where the prime denotes differentiation with respect to π‘ β πΌ. For π‘0 β πΌ, we orient the slice Ξ£ππ‘0 := {π‘0 } × ππ by using the unit normal vector field ππ‘ , then from [11] we know Ξ£ππ‘0 has constant πth π mean curvature π»π = βπ(πσΈ (π‘0 )/π(π‘0 )) with respect to ππ‘ . π A smooth immersion π : Ξ£ β ππΌ ×π ππ of an π-dimensional connected manifold Ξ£π is said to be a spacelike hypersurface if the induced metric via π is a Riemannian metric on Ξ£π . If Ξ£π is oriented by the unit vector field π, one obviously has π = πππ‘ = ππ. Moreover, when π = β1, we may define the normal hyperbolic angle π of Ξ£π as being the smooth function π : Ξ£π β [0, +β) given by β€ π (π, π) π = (π (π, π) π) + π β¨π΄π, πβ© π΄π β π β¨π΄π, πβ© π΄π. (13) Consider a local orthonormal frame {πΈ1 , . . . , πΈπ } and π β Ξ(πΞ£π ), then the Ricci curvature tensor of Ξ£π is given as the following: Ric (π, π) = β β¨π (π, πΈπ ) π, πΈπ β© + ππ» β¨π΄π, πβ© π (14) β π β¨π΄π, π΄πβ© . 3. Key Lemmas We consider two particular functions naturally attached to complete spacelike hypersurfaces, namely, the vertical (height) function β = (ππΌ )|Ξ£π and the support function Journal of Applied Mathematics 3 β¨π, ππ‘ β©. Denote by β and β the gradients with respect to the metrics of ππΌ ×π ππ and Ξ£π , respectively. Thus, by a simple computation, we have the gradient of ππΌ on ππΌ ×π ππ as the following: βππΌ = π β¨βππΌ , ππ‘ β© ππ‘ = πππ‘ . Denote by {π1 , . . . , ππ } a local orthonormal frame of Ξ(πΞ£π ). By the definition of Laplacian operator and using (22), then we have Ξπ (β) := div (βπ (β)) = div (π (β) ββ) = β β¨ππ , βππ (π (β) ββ)β© (15) π π Thus, the gradient of β on Ξ£ is given by = β β¨ππ , π (β) βππ βββ© + β β¨ππ , ππ (π (β)) βββ© π β€ β€ ββ = (βππΌ ) = π(ππ‘ ) = πππ‘ β β¨π, ππ‘ β© π. (16) = π (β) β β¨ππ , βππ βββ© + βππ (π (β)) β¨ππ , βββ© π We denote by | β | the norm of a vector field on Ξ£π , then we get 2 |ββ|2 = π (1 β β¨π, ππ‘ β© ) . π π (18) Analogously, a spacelike hypersurface π : Ξ£π β βπΌ ×π ππ is said to be bounded away from the past infinity of βπΌ ×π ππ if there exists π‘ β πΌ such that π (Ξ£π ) β {(π‘, π) β βπΌ ×π ππ : π‘ β₯ π‘} . (23) = π (β) Ξβ + πσΈ (β) β β¨ππ , βββ© β¨ππ , βββ© (17) According to [4], a spacelike hypersurface π : Ξ£π β βπΌ ×π ππ is said to be bounded away from the future infinity of βπΌ ×π ππ if there exists π‘ β πΌ such that π (Ξ£π ) β {(π‘, π) β βπΌ ×π ππ : π‘ β€ π‘} . π (19) Finally, Ξ£π is said to be bounded away from the infinity of βπΌ ×π ππ if it is both bounded away from the past and future infinity of βπΌ ×π ππ . In order to prove our main theorems, we will make use of the following computations. We also refer the reader to [6, 11] for a more generalized proposition which makes the following lemma as its trivial case. Lemma 3. Let π : Ξ£π β ππΌ ×π ππ be a spacelike hypersurface immersed in a semi-Riemannian warped product. If σΈ = π (β) Ξβ + π (β) |ββ|2 . By taking π = 1 in Lemma 2.2 of [13] (we also refer the reader to Lemma 4.1 of [11]), then we have σΈ Ξβ = (ln π) (β) (ππ β |ββ|2 ) + πππ» β¨π, ππ‘ β© . (24) By substituting (24) into (23), we complete the proof. We need another lemma proved by Albujer et al. in [3]. Lemma 4. Let π : Ξ£π β βπΌ ×π ππ be a spacelike hypersurface immersed in a Robertson-Walker spacetime whose Riemannian fiber ππ has constant sectional curvature π. Denote by β the height function on Ξ£π . If βπΌ ×π ππ obeys the null convergence condition, then β β¨π (π, πΈπ ) π, πΈπ β© β₯ (π β 1) π π + πσΈ 2 (β) 2 |π| , π2 (β) (25) where π is the Riemannian curvature tensor of βπΌ ×π ππ and π β Ξ(πΞ£π ). Remark 5. It follows from (14) and Lemma 4 that, if the Riemannian fiber has constant sectional curvature, the Ricci from below if Ξ£π is curvature tensor of Ξ£π is boundedπ+1 bounded away from the infinity of π . π‘ π (π‘) = β« π (π ) ππ , π‘0 (20) We also need another lemma shown by Aquino and de Lima in [5]. π+1 then Lemma 6. Let π σΈ Ξπ (β) = ππ (π (β) + π (β) β¨π, ππ‘ β© π») , (21) where Ξ denotes the Laplacian operator and β is the height function of Ξ£π . = πΌ ×π ππ be a warped product which π+1 satisfies convergence condition (5). Let π : Ξ£π β π be a complete hypersurface with both mean curvature π» and second fundamental form π΄ bounded. If πσΈ σΈ /π is bounded on Ξ£π , then the Ricci curvature of Ξ£π is bounded from below. Proof. Let β be the gradient with respect to the metric of Ξ£π π+1 induced from π . Thus, it is easy to see In order to prove our main theorems, we also need the well-known generalized maximal principal due to Omori [1] and Yau [2]. βπ (β) = π (β) ββ. Lemma 7. Let Ξ£π denote an π-dimensional complete Riemannian manifold whose Ricci curvature tensor is bounded from (22) 4 Journal of Applied Mathematics below. Then, for any C2 -function π’ : Ξ£π β R with π’β = supΞ£π π’ < β, there exists a sequence of points {ππ }πβπ in Ξ£π satisfying the following properties: (i) lim π’ (ππ ) = supπ’, πββ Ξ£π lim Ξπ (β) (ππ ) β€ βπ lim π (β) (ππ ) (infπ (ii) lim |βπ’| (ππ ) = 0, πββ (iii) lim Ξπ’ (ππ ) β€ 0. πββ Equivalently, for any C2 -function π’ : Ξ£π β R with π’β = inf Ξ£π π’ > ββ, there exists a sequence of points {ππ }πβπ in Ξ£π satisfying the following properties: Ξ£ πββ (iii) lim Ξπ’ (ππ ) β₯ 0. (27) πββ 4. Proofs of Main Theorems Proof of Theorem 1. Since ππ‘ is a unitary timelike vector field globally defined on the ambient spacetime, then there exists a unique timelike unitary normal vector filed π globally defined on the spacelike hypersurface Ξ£π which is the same time orientation as ππ‘ . By using the reverse Cauchy-Schwarz inequality, we have σ΅¨ β¨π, ππ‘ β©σ΅¨σ΅¨σ΅¨Ξ£π β€ β1 < 0. (28) We observe that, from Lemma 3, Ξπ (β) = βπ (πσΈ (β) + π (β) β¨π, ππ‘ β© π») . (29) By using the assumption (3), we have that π» β€ 0. Then, it follows from (28) that β¨π, ππ‘ β©π» β₯ βπ» β₯ 0. Notice that the warping function is positive on πΌ, then from (3) and (29), we have the following inequality: Ξπ (β) = βππ (β) ( β€ βππ (β) ( πσΈ (β) + β¨π, ππ‘ β© π») π (β) β€ βππ (β) (infπ Ξ£ (30) On the other hand, since the spacelike hypersurface is π+1 bounded away from the infinity of π , and the constant sectional curvature of the fiber satisfies the null convergence condition, then it follows from Lemma 4 and Remark 5 that the Ricci curvature tensor of Ξ£π is bounded from below. Then by the definition of π(β), that is, (20), it is easy to see inf Ξ£π π(β) < β. Thus, applying Lemma 7 to the smooth function π(β) on Ξ£π implies that there exists a sequence of points {ππ }πβπ in Ξ£π with the following properties: Lim π (β) (ππ ) = infπ π (β) > ββ, lim Ξπ (β) (ππ ) β₯ 0. πββ (33) Substituting the above equation into (4) implies that |ββ|2 β‘ 0, that is, β is a constant on Ξ£π ; thus Ξ£π is a slice of π+1 π . From (9), we see that every Robertson-Walker spacetime with constant sectional curvature trivially obeys the null convergence condition (2). Therefore, from Theorem 1, we obtain the following result. π+1 = βπΌ ×π ππ be a Robertson-Walker Corollary 8. Let π spacetime with constant sectional curvature. Let π : Ξ£π β π+1 π be a complete spacelike hypersurface with constant mean curvature π». Suppose that Ξ£π is bounded away from the infinity. If (3) and (4) hold, then Ξ£π is a slice. π+1 Theorem 9. Let π = βπΌ ×π ππ be a Robertson-Walker spacetime whose Riemannian fiber ππ has constant sectional curvature π satisfying the null convergence condition. Let π+1 be a complete spacelike hypersurface π : Ξ£π β π with constant mean curvature π» and bounded away from the infinity. If πσΈ (β (π)) , 0} , πβΞ£π π πσΈ (β) β π») π (β) Ξ£ πσΈ (β) β π» = 0. π (β) π» β₯ max {sup β€ 0. πββ infπ σΈ π (β) β π») π (β) πσΈ (β) β π») β€ 0. π (β) (32) Since the warping function is positive on Ξ£π and Ξ£π is bounded away from the infinity, from (31) and (32), we have limπ β β Ξπ(β)(ππ ) = 0, which means that Ξ£ (ii) lim |βπ’| (ππ ) = 0, πββ Ξ£ πββ (26) πββ (i) lim π’ (ππ ) = infπ π’, Note that π» is a nonpositive constant, then it follows from (30) that and the height function β of Ξ£π satisfies σ΅¨σ΅¨π½ σ΅¨σ΅¨σ΅¨ σ΅¨σ΅¨ πσΈ σ΅¨ |ββ|2 β€ πΌσ΅¨σ΅¨σ΅¨π» β sup (β (π))σ΅¨σ΅¨σ΅¨ , σ΅¨σ΅¨ σ΅¨σ΅¨ πβΞ£π π σ΅¨ σ΅¨ for some positive constant πΌ and π½, then Ξ£π is a slice. (34) (35) Proof. From (34), we see that the constant mean curvature π» is nonnegative, then it follows from (28) and (34) that β¨π, ππ‘ β©π» β€ βπ» β€ 0. Thus, from (29), we have Ξπ (β) = βππ (β) ( β₯ βππ (β) ( πσΈ (β) + β¨π, ππ‘ β© π») π (β) πσΈ (β) β π») π (β) β₯ βππ (β) (sup Ξ£π (31) β₯ 0. σΈ π (β) β π») π (β) (36) Journal of Applied Mathematics 5 Using the similar analysis with the proof of Theorem 1, applying the Lemma 7 to the smooth function π(β) on Ξ£π implies that there exists a sequences of points {ππ }πβπ in Ξ£π satisfying the following properties: Lim π (β) (ππ ) = supπ (β) < β, πββ Ξ£π lim Ξπ (β) (ππ ) β€ 0. (37) lim Ξπ (β) (ππ ) β₯ βπ lim π (β) (ππ ) (sup πββ Ξ£π πσΈ (β) β π») β₯ 0. π (β) (38) From (37) and (38), we have limπ β β Ξπ(β)(ππ ) = 0. Thus, from (36), we get Ξ£π πσΈ (β) β π» = 0. π (β) (40) Ξπ (β) = π (πσΈ (β) + π (β) β¨π, ππ‘ β© π») = ππ (β) ( It follows from (36) that sup β1 β€ β¨π, ππ‘ β© β€ 0. Then from Lemma 3, we have that πββ πββ Proof of Theorem 2. Initially, we consider that the orientation π of the hypersurface such that its angle function satisfies πσΈ (β) + β¨π, ππ‘ β© π») . π (β) By using the assumption (6) we see that the mean curvature π» is non-positive, then it follows from (40) that 0 β€ π»β¨π, ππ‘ β© β€ βπ». Notice that the warping function π is positive on πΌ, then from (41) we have the following inequality: Ξπ (β) = ππ (β) ( (39) By substituting the above equation into (35), we have |ββ|2 β‘ 0, that is, β is a constant on Ξ£π ; thus Ξ£π is a slice. β€ ππ (β) ( Noting that a Robertson-Walker spacetime with constant sectional curvature trivially obeys the null convergence condition; thus the proof of Corollary 10 follows from Theorem 9. It is well known that Robertson-Walker spacetime is called static Robertson-Walker spacetime if the warping function π is constant. Without losing the generality, we consider π = 1 in the following. In this case, the null convergence condition (2) implies that π β₯ 0. Also, (3) becomes that π» β€ 0. When π is a constant, (33) (or (39)) implies that π» = 0, that is, Ξ£π is maximal. On the other hand, letting the warping function π be a constant, we see from (34) that π» β₯ 0. Then a weaker assumption than (4) and (35), that is, the normal hyperbolic angle is bounded on Ξ£π , also guarantees Theorems 1 and 9. Thus, from Theorems 1 and 9, we have the following corollary. π+1 = βπΌ × ππ be a static RobertsonCorollary 11. Let π Walker spacetime whose Riemannian fiber ππ has nonπ+1 negative constant sectional curvature π. Let π : Ξ£π β π be a complete, connected spacelike hypersurface with constant mean curvature π». Suppose that the normal hyperbolic angle of Ξ£π is bounded, then Ξ£π is a maximal slice. We also refer the reader to [3, 14] for the similar results with Corollary 11 proved by using the different methods with ours. Next, we give the proof of Theorem 2, which is the Riemannian warped product version of our Theorem 1. πσΈ (β) + β¨π, ππ‘ β© π») π (β) πσΈ (β) β π») π (β) β€ ππ (β) (sup Ξ£π π+1 Corollary 10. Let π = βπΌ ×π ππ be a Robertson-Walker spacetime with constant sectional curvature. Let π : Ξ£π β π+1 π be a complete spacelike hypersurface with constant mean curvature π». Suppose that Ξ£π is bounded away from the infinity. If (34) and (35) hold, then Ξ£π is a slice. (41) (42) σΈ π (β) β π») π (β) β€ 0. Since the spacelike hypersurface is bounded away from the infinity, it follows that πσΈ σΈ /π is bounded on Ξ£π . Let π2 denote the second elementary symmetric function on the eigenvalues of the shape operator π΄, and π»2 = (2/π(π β 1))π2 denotes the mean value of π2 . It is easy to see |π΄|2 = π2 π»2 β π(π β 1)π»2 . Then the assumption that π»2 is bounded from below means that π΄ is bounded from above; from Lemma 6 we know that the Ricci curvature tensor of Ξ£π is bounded from below. Finally, by applying analogous arguments employed in the last part of the proof of Theorem 1, we conclude that Ξ£π is a slice of πΌ ×π ππ . 5. Application of Main Theorems In this section, we consider a particular model of Lorentzian warped product, namely, the steady state space, that is, the warped product Hπ+1 = βR×ππ‘ Rπ . (43) The importance of considering Hπ+1 comes from the fact that, in cosmology, H4 is the steady model of the universe proposed by Bondi and Gold [15] and Hoyle [16]. Moreover, in physical context the steady state space appears naturally as an exact solution for the Einstein equations, being a cosmological model where matter is supposed to travel along geodesic normal to horizontal hyperplanes. We also notice that Montiel [17] gave an alternative description of the steady state space Hπ+1 as follows. 6 Journal of Applied Mathematics Let Lπ+2 denote the (π + 2)-dimensional Lorentzian1 Minkowski space, that is, the real vector space Rπ+2 with a Lorentzian metric π+1 β¨V, π€β© = β Vπ π€π β Vπ+2 π€π+2 , (44) π=1 for all V, π€ β Rπ+2 . The (π + 1)-dimensional de Sitter space is defined as the following: Sπ+1 = {π β Lπ+2 : β¨π, πβ© = Sπ+1 1 1 1 π+2 1}. Let π β L1 be a nonzero null vector of the null cone with vertex in the origin, such that β¨π, ππ+2 β© > 0, where ππ+2 = (0, . . . , 0, 1). Then, it can be shown that the open region {π β Sπ+1 1 : β¨π, πβ© > 0} (45) is isometric to Hπ+1 (see [18] for of the de Sitter space Sπ+1 1 details). Recently, some uniqueness theorems for steady state space were obtained by [13, 19]. In fact, Caminha and de Lima [19] proved the following results. Theorem 12. Let π : Ξ£2 β H3 be a Riemannian immersion of a complete surface of non-negative Guassian curvature πΎΞ£π with constant mean curvature π» β₯ 1. If |ββ|2 β€ π»2 β 1, π In the last section of this paper, we investigate the applications of our main theorems for entire vertical graphs in a Robertson-Walker spacetime. We follow [3, 8, 20] for the basic notations and facts used in this section. Let Ξ© β ππ be a connected domain of ππ ; a vertical graph over Ξ© is defined by smooth function π’ β Cβ (Ξ©) and it is given by Ξ£π (π’) = {(π’ (π) , π) : π β Ξ© β ππ } β βπΌ ×π ππ . then π(Ξ£ ) is a slice of H . Suppose that the warping function π(π‘) = ππ‘ ; thus (34) becomes π» β₯ 1; meanwhile the inequality (35) becomes π (π) = holds for some positive constant πΌ and π½, then π(Ξ£π ) is a slice of H3 . Theorem 14. Let π : Ξ£2 β H3 be a Riemannian immersion of a complete surface with constant mean curvature π» β₯ 1. If 1 π·π’ (π)) , (π’ (π)) π β Ξ©. Moreover, the shape operator π΄ of Ξ£π (π’) with respect to π is given by π΄π = β (47) (48) π2 (51) β 1 π (π’) βπ2 (π’) β |π·π’|2ππ πσΈ (π’) βπ2 (π’) β |π·π’|2ππ +( Theorem 13. Let π : Ξ£2 β H3 be a Riemannian immersion of a complete surface with constant mean curvature π» β€ 0. If |ββ|2 β€ πΌ(π» β 1)π½ , π (π’ (π)) βπ2 (π’ (π)) β σ΅¨σ΅¨σ΅¨σ΅¨π·π’ (π)σ΅¨σ΅¨σ΅¨σ΅¨2ππ σ΅¨ × ( ππ‘ σ΅¨σ΅¨σ΅¨(π’(π),π) + where both πΌ and π½ are positive constants. Suppose that π½ = 2, then we have πΌ(π» β 1)π½ β₯ π»2 β 1 for all π» β₯ 1 if πΌ β₯ 1 + 2/(π» β 1) > 0. Thus, our Theorem 9 extends Theorem 4.5 in [19]. At last, we write the uniqueness theorems for surface in steady state space which follows from Theorems 1 and 9, respectively, as following. |ββ|2 β€ πΌ(1 β π»)π½ π·π π·π’ π (52) πσΈ (π’) β¨π·π’, πβ© 3/2 (π2 (π’) β |π·π’|2ππ ) β β¨π·π π·π’, π·π’β©ππ π (π’) (π2 (π’) β |π·π’|2ππ )) 3/2 ) π·π’, for any tangent vector field π on Ξ©, where π· denotes the Levi-Civita connection in Ξ© with respect to the metric β¨ , β©ππ . The mean curvature function of a spacelike graph Ξ£π (π’) with respect to π is given as the following: Div ( π·π’ βπ2 (π’) β |π·π’|2ππ ) (53) (49) holds for some positive constant πΌ and π½, then π(Ξ£π ) is a slice of H3 . (50) The metric induced on Ξ© from the Lorentzian metric on the ambient space via Ξ£π is β¨ , β© = βππ’2 + π2 (π’)β¨ , β©ππ . The graph is said to be entire if Ξ© = ππ . It is easy to see that a graph Ξ£(π’) is a spacelike hypersurface if and only if |π·π’|2ππ < π2 (π’), where π·π’ is the gradient of π’ in Ξ© and |π·π’|ππ its norm, both with respect to the metric β¨ , β©ππ in Ξ©. Let Ξ£π (π’) be a spacelike vertical graph over a domain Ξ©, its future-pointing Gauss map is given by the vector field (46) 3 |ββ|2 β€ πΌ|π» β 1|π½ , 6. Entire Vertical Graphs in Robertson-Walker Spacetime = ππ (π’) (π» (π’) β σΈ π (π’) β π2 (π’) β |π·π’|2ππ ), Journal of Applied Mathematics 7 where Div denotes the divergence operator on Ξ© with respect to the metric β¨ , β©ππ . Now, we give the following nonparametric version of Theorem 1. π+1 Theorem 15. Let π = βπΌ ×π ππ be a Robertson-Walker spacetime whose Riemannian fiber ππ has constant sectional π+1 obeys the null convergence curvature π. Suppose that π π condition (2). Let Ξ£ (π’) be an entire spacelike vertical graph π+1 in π with constant mean curvature π» and bounded away from the infinity. If π» β€ min { infπ πβΞ£ |π·π’|2ππ πσΈ (π’ (π)) , 0} , π (54) σ΅¨π½ σ΅¨σ΅¨ σ΅¨σ΅¨inf Ξ£π (π’) (πσΈ /π) (π’) β π»σ΅¨σ΅¨σ΅¨ σ΅¨ σ΅¨ β€ πΌ inf π (π’) σ΅¨σ΅¨ σ΅¨π½ , σΈ Ξ£π (π’) 1 + πΌσ΅¨σ΅¨inf Ξ£π (π’) (π /π) (π’) β π»σ΅¨σ΅¨σ΅¨ (55) 2 for some positive constant πΌ and π½, then Ξ£π (π’) is a slice. Proof. First of all and following the ideas of Albujer et al. in the proof of Theorem 4.1 in [20] and Theorem 4.1 in [3], it can be easily seen that the induced metric on the entire graph Ξ£π (π’) is complete. Then it follows from [8, 20] that |ββ|2 = |π·π’|2ππ π2 (π’) β |π·π’|2ππ . (56) Thus, together with (55) and (56) and by a straightforward computation, we know that (4) holds. Then the proof of Theorem 15 follows from Theorem 1. π+1 Let π = βπΌ ×π ππ be a static Robertson-Walker spacetime, that is, π = 1. Thus, the null convergence condition implies that π β₯ 0 and (54) becomes π» β€ 0. 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