On Complete Spacelike Hypersurfaces in a Semi-Riemannian Warped Product

Abstract

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 semi-Riemannian 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≥𝑘_𝑀≥sup_𝐼𝑓^′2−𝑓𝑓^′′, (1) where 𝑘_𝑀 denotes the sectional curvature of the fiber 𝑀^𝑛, which is very different from null convergence condition (defined by (2)).

Replacing the null convergence condition by (ln𝑓)^′′≤0, Alías 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.

Theorem 1. Let 𝑀^𝑛+1=−𝐼×_𝑓𝑀^𝑛 be a Robertson-Walker spacetime whose Riemannian fiber 𝑀^𝑛 has constant sectional curvature 𝑘 satisfying the null convergence condition defined as the following: 𝑘≥sup_𝐼𝑓𝑓^′′−𝑓^′2. (2) Let 𝜓∶Σ^𝑛→𝑀^𝑛+1 be a complete spacelike hypersurface with constant mean curvature 𝐻. Suppose that Σ^𝑛 is bounded away from the infinity of 𝑀^𝑛+1 and that 𝐻≤mininf_{𝑝∈Σ𝑛}𝑓^′𝑓ℎ𝑝,0. (3) If the height function ℎ of Σ^𝑛 satisfies |∇ℎ|²≤𝛼inf_{𝑝∈Σ𝑛}𝑓^′𝑓ℎ𝑝−𝐻^𝛽, (4) for some positive constant 𝛼 and 𝛽, then Σ^𝑛 is a slice.

Theorem 2. Let 𝑀^𝑛+1=𝐼×_𝑓𝑀^𝑛 be a Riemannian warped product whose fiber 𝑀^𝑛 has constant sectional curvature 𝑘 satisfying 𝑘≥sup_𝐼𝑓^′2−𝑓𝑓^′′. (5) Let 𝜓∶Σ^𝑛→𝑀^𝑛+1 be a complete spacelike hypersurface with constant mean curvature 𝐻 and 𝐻₂ is bounded from below. Suppose that Σ^𝑛 is bounded away from the infinity of 𝑀^𝑛+1 and that sup_{𝑝∈Σ𝑛}𝑓^′𝑓ℎ𝑝≤𝐻≤0. (6) If the height function ℎ of Σ^𝑛 satisfies |∇ℎ|²≤𝛼𝐻−sup_{𝑝∈Σ𝑛}𝑓^′𝑓ℎ𝑝^𝛽, (7) for some positive constant 𝛼 and 𝛽, then Σ^𝑛 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 𝑀^𝑛 be a connected, 𝑛-dimensional (𝑛≥2) oriented Riemannian manifold, 𝐼⊆ℝ an interval, and 𝑓∶𝐼→ℝ a positive smooth function. We consider the product differential manifold 𝐼×𝑀^𝑛 and denote by 𝜋_𝐼 and 𝜋_𝑀 the projections onto the base 𝐼 and fiber 𝑀^𝑛, respectively. A particular class of semi-Riemannian manifolds is the one obtained by furnishing 𝐼×𝑀^𝑛 with the metric ⟨𝑣,𝑤⟩_𝑝=𝜖𝜋_𝐼∗𝑣,𝜋_𝐼∗𝑤+𝑓∘𝜋_𝐼𝑝²𝜋_𝑀∗𝑣,𝜋_𝑀∗𝑤, (8) for any 𝑝∈𝑀^𝑛+1 and any 𝑣,𝑤∈𝑇_𝑝𝑀^𝑛+1, where 𝜖=±1. We call such a space warped product manifold; 𝑓 is known as the warping function, and we denote the space by 𝑀^𝑛+1=𝜖𝐼×_𝑓𝑀^𝑛. Note that −𝐼×_𝑓𝑀^𝑛 is called a generalized Robertson-Walker spacetime, in particular, −𝐼×_𝑓𝑀^𝑛 is called a Robertson-Walker spacetime if 𝑀^𝑛 has constant sectional curvature. From [9], we know that a generalized Robertson-Walker spacetime has constant sectional curvature 𝑘 if and only if the Riemannian fiber 𝑀^𝑛 has constant sectional curvature 𝑘 and the warping function 𝑓 satisfies the following differential equation: 𝑓^′′𝑓=𝑓𝑘=^′2+𝑘𝑓². (9)

It follows from [10] that the vector field (𝑓∘𝜋_𝐼)𝜕_𝑡 is conformal and closed (in this sense that its dual 1-form is closed) with conformal factor 𝜙=𝑓^′∘𝜋_𝐼, where the prime denotes differentiation with respect to 𝑡∈𝐼. For 𝑡₀∈𝐼, we orient the slice Σ^𝑛_𝑡0∶={𝑡₀}×𝑀^𝑛 by using the unit normal vector field 𝜕_𝑡, then from [11] we know Σ^𝑛_𝑡0 has constant 𝑟th mean curvature 𝐻_𝑟=−𝜖(𝑓^′(𝑡₀)/𝑓(𝑡₀))^𝑟 with respect to 𝜕_𝑡.

A smooth immersion 𝜓∶Σ^𝑛→𝜖𝐼×_𝑓𝑀^𝑛 of an 𝑛-dimensional connected manifold Σ^𝑛 is said to be a spacelike hypersurface if the induced metric via 𝜓 is a Riemannian metric on Σ^𝑛. If Σ^𝑛 is oriented by the unit vector field 𝑁, one obviously has 𝜖=𝜖_𝜕𝑡=𝜖_𝑁. Moreover, when 𝜖=−1, we may define the normal hyperbolic angle 𝜃 of Σ^𝑛 as being the smooth function 𝜃∶Σ^𝑛→[0,+∞) given by cosh𝜃=−𝑁,𝜕_𝑡. (10)

We denote by ∇ and ∇ the Levi-Civita connections in 𝜖𝐼×_𝑓𝑀^𝑛 and Σ^𝑛, respectively. Then the Gauss-Weingarten formulas for the spacelike hypersurface 𝜓∶Σ^𝑛→𝜖𝐼×_𝑓𝑀^𝑛 are given by ∇_𝑋𝑌=∇_𝑋𝑌+𝜖⟨𝐴𝑋,𝑌⟩𝑁,∇_𝑋𝑁=−𝐴𝑋, (11) for any 𝑋∈Γ(𝑇Σ^𝑛), where 𝐴∶Γ(𝑇Σ^𝑛)→Γ(𝑇Σ^𝑛) be the shape operator of Σ^𝑛 with respect to its Gauss map 𝑁, and Γ(𝑇Σ^𝑛) denotes the Lie algebra of all tangential vector fields on Σ^𝑛.

The curvature tensor 𝑅 of a spacelike hypersurface Σ^𝑛 is given by [12] as the following: 𝑅(𝑋,𝑌)𝑍=∇_[𝑋,𝑌]𝑍−∇_𝑋,∇_𝑌𝑍, (12) where [,] denotes the Lie bracket and 𝑋,𝑌,𝑍∈Γ(𝑇Σ^𝑛).

Let 𝑅 and 𝑅 be the curvature tensors of 𝜖𝐼×_𝑓𝑀^𝑛 and Σ^𝑛, respectively. Denote by 𝑋^⊤ the tangential component of a vector field 𝑋∈Γ(𝑇𝑀^𝑛+1); thus, for any 𝑋,𝑌,𝑍∈Γ(𝑇Σ^𝑛) we have the following Gauss equation: 𝑅(𝑋,𝑌)𝑍=𝑅(𝑋,𝑌)𝑍^⊤+𝜖⟨𝐴𝑋,𝑍⟩𝐴𝑌−𝜖⟨𝐴𝑌,𝑍⟩𝐴𝑋. (13) Consider a local orthonormal frame {𝐸₁,…,𝐸_𝑛} and 𝑋∈Γ(𝑇Σ^𝑛), then the Ricci curvature tensor of Σ^𝑛 is given as the following: Ric(𝑋,𝑋)=_𝑖𝑅𝑋,𝐸_𝑖𝑋,𝐸_𝑖⟨⟩⟨⟩.+𝑛𝐻𝐴𝑋,𝑋−𝜖𝐴𝑋,𝐴𝑋 (14)

3. Key Lemmas

We consider two particular functions naturally attached to complete spacelike hypersurfaces, namely, the vertical (height) function ℎ=(𝜋_𝐼)|_Σ^𝑛 and the support function ⟨𝑁,𝜕_𝑡⟩. Denote by ∇ and ∇ the gradients with respect to the metrics of 𝜖𝐼×_𝑓𝑀^𝑛 and Σ^𝑛, respectively. Thus, by a simple computation, we have the gradient of 𝜋_𝐼 on 𝜖𝐼×_𝑓𝑀^𝑛 as the following: ∇𝜋_𝐼=𝜖∇𝜋_𝐼,𝜕_𝑡𝜕_𝑡=𝜖𝜕_𝑡. (15) Thus, the gradient of ℎ on Σ^𝑛 is given by ∇ℎ=∇𝜋_𝐼^⊤=𝜖𝜕_𝑡^⊤=𝜖𝜕_𝑡−𝑁,𝜕_𝑡𝑁. (16) We denote by |⋅| the norm of a vector field on Σ^𝑛, then we get |∇ℎ|²=𝜖1−𝑁,𝜕_𝑡². (17)

According to [4], a spacelike hypersurface 𝜓∶Σ^𝑛→−𝐼×_𝑓𝑀^𝑛 is said to be bounded away from the future infinity of −𝐼×_𝑓𝑀^𝑛 if there exists 𝑡∈𝐼 such that 𝜓Σ^𝑛⊂𝑡,𝑝∈−𝐼×_𝑓𝑀^𝑛∶𝑡≤𝑡. (18) Analogously, a spacelike hypersurface 𝜓∶Σ^𝑛→−𝐼×_𝑓𝑀^𝑛 is said to be bounded away from the past infinity of −𝐼×_𝑓𝑀^𝑛 if there exists 𝑡∈𝐼 such that 𝜓Σ^𝑛⊂𝑡,𝑝∈−𝐼×_𝑓𝑀^𝑛∶𝑡≥𝑡. (19) Finally, Σ^𝑛 is said to be bounded away from the infinity of −𝐼×_𝑓𝑀^𝑛 if it is both bounded away from the past and future infinity of −𝐼×_𝑓𝑀^𝑛.

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 𝜓∶Σ^𝑛→𝜖𝐼×_𝑓𝑀^𝑛 be a spacelike hypersurface immersed in a semi-Riemannian warped product. If 𝜎(𝑡)=^𝑡_𝑡0𝑓(𝑠)𝑑𝑠, (20) then Δ𝜎(ℎ)=𝜖𝑛𝑓^′(ℎ)+𝑓(ℎ)𝑁,𝜕_𝑡𝐻, (21) where Δ denotes the Laplacian operator and ℎ is the height function of Σ^𝑛.

Proof. Let ∇ be the gradient with respect to the metric of Σ^𝑛 induced from 𝑀^𝑛+1. Thus, it is easy to see ∇𝜎(ℎ)=𝑓(ℎ)∇ℎ. (22) Denote by {𝑒₁,…,𝑒_𝑛} a local orthonormal frame of Γ(𝑇Σ^𝑛). By the definition of Laplacian operator and using (22), then we have (ℎ)((ℎ𝑓(ℎ)=Δ𝜎∶=div∇𝜎))=div∇ℎ_𝑖𝑒_𝑖,∇_𝑒𝑖𝑓(ℎ)∇ℎ=_𝑖𝑒_𝑖,𝑓(ℎ)∇_𝑒𝑖∇ℎ+_𝑖𝑒_𝑖,𝑒_𝑖𝑓(ℎ)(ℎ)∇ℎ=𝑓_𝑖𝑒_𝑖,∇_𝑒𝑖∇ℎ+_𝑖𝑒_𝑖𝑓(ℎ)𝑒_𝑖(ℎ),∇ℎ=𝑓Δℎ+𝑓^′(ℎ)_𝑖𝑒_𝑖,∇ℎ𝑒_𝑖(ℎ),∇ℎ=𝑓Δℎ+𝑓^′(ℎ)|∇ℎ|². (23)
By taking 𝑟=1 in Lemma 2.2 of [13] (we also refer the reader to Lemma 4.1 of [11]), then we have Δℎ=ln𝑓^′(ℎ)𝜖𝑛−|∇ℎ|²+𝜖𝑛𝐻𝑁,𝜕_𝑡. (24) By substituting (24) into (23), we complete the proof.

We need another lemma proved by Albujer et al. in [3].

Lemma 4. Let 𝜓∶Σ^𝑛→−𝐼×_𝑓𝑀^𝑛 be a spacelike hypersurface immersed in a Robertson-Walker spacetime whose Riemannian fiber 𝑀^𝑛 has constant sectional curvature 𝑘. Denote by ℎ the height function on Σ^𝑛. If −𝐼×_𝑓𝑀^𝑛 obeys the null convergence condition, then _𝑖𝑅𝑋,𝐸_𝑖𝑋,𝐸_𝑖≥(𝑛−1)𝑘+𝑓^′2(ℎ)𝑓²(ℎ)|𝑋|², (25) where 𝑅 is the Riemannian curvature tensor of −𝐼×_𝑓𝑀^𝑛 and 𝑋∈Γ(𝑇Σ^𝑛).

Remark 5. It follows from (14) and Lemma 4 that, if the Riemannian fiber has constant sectional curvature, the Ricci curvature tensor of Σ^𝑛 is bounded from below if Σ^𝑛 is bounded away from the infinity of 𝑀^𝑛+1.

We also need another lemma shown by Aquino and de Lima in [5].

Lemma 6. Let 𝑀^𝑛+1=𝐼×_𝑓𝑀^𝑛 be a warped product which satisfies convergence condition (5). Let 𝜓∶Σ^𝑛→𝑀^𝑛+1 be a complete hypersurface with both mean curvature 𝐻 and second fundamental form 𝐴 bounded. If 𝑓^′′/𝑓 is bounded on Σ^𝑛, then the Ricci curvature of Σ^𝑛 is bounded from below.

In order to prove our main theorems, we also need the well-known generalized maximal principal due to Omori [1] and Yau [2].

Lemma 7. Let Σ^𝑛 denote an 𝑛-dimensional complete Riemannian manifold whose Ricci curvature tensor is bounded from below. Then, for any 𝒞²-function 𝑢∶Σ^𝑛→ℝ with 𝑢^∗=sup_Σ𝑛𝑢<∞, there exists a sequence of points {𝑝_𝑘}_𝑘∈𝑁 in Σ^𝑛 satisfying the following properties: (i)lim_𝑘→∞𝑢𝑝_𝑘=sup_Σ𝑛𝑢,(ii)lim_𝑘→∞|∇𝑢|𝑝_𝑘()=0,iiilim_𝑘→∞Δ𝑢𝑝_𝑘≤0. (26) Equivalently, for any 𝒞²-function 𝑢∶Σ^𝑛→ℝ with 𝑢_∗=inf_Σ^𝑛𝑢>−∞, there exists a sequence of points {𝑞_𝑘}_𝑘∈𝑁 in Σ^𝑛 satisfying the following properties: (i)lim_𝑘→∞𝑢𝑞_𝑘=inf_Σ𝑛𝑢,(ii)lim_𝑘→∞|∇𝑢|𝑞_𝑘()=0,iiilim_𝑘→∞Δ𝑢𝑞_𝑘≥0. (27)

4. Proofs of Main Theorems

Proof of Theorem 1. Since 𝜕_𝑡 is a unitary timelike vector field globally defined on the ambient spacetime, then there exists a unique timelike unitary normal vector filed 𝑁 globally defined on the spacelike hypersurface Σ^𝑛 which is the same time orientation as 𝜕_𝑡. By using the reverse Cauchy-Schwarz inequality, we have 𝑁,𝜕_𝑡Σ𝑛≤−1<0. (28) We observe that, from Lemma 3, Δ𝜎(ℎ)=−𝑛𝑓^′(ℎ)+𝑓(ℎ)𝑁,𝜕_𝑡𝐻. (29)
By using the assumption (3), we have that 𝐻≤0. Then, it follows from (28) that ⟨𝑁,𝜕_𝑡⟩𝐻≥−𝐻≥0. Notice that the warping function is positive on 𝐼, then from (3) and (29), we have the following inequality: (ℎ)(ℎ)𝑓Δ𝜎=−𝑛𝑓^′(ℎ)𝑓(ℎ)+𝑁,𝜕_𝑡𝐻(ℎ)𝑓≤−𝑛𝑓^′(ℎ)𝑓(ℎ)(ℎ)−𝐻≤−𝑛𝑓inf_Σ𝑛𝑓^′(ℎ)𝑓(ℎ)−𝐻≤0. (30)
On the other hand, since the spacelike hypersurface is bounded away from the infinity of 𝑀^𝑛+1, 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 Σ^𝑛 is bounded from below. Then by the definition of 𝜎(ℎ), that is, (20), it is easy to see inf_Σ^𝑛𝜎(ℎ)<∞. Thus, applying Lemma 7 to the smooth function 𝜎(ℎ) on Σ^𝑛 implies that there exists a sequence of points {𝑞_𝑘}_𝑘∈𝑁 in Σ^𝑛 with the following properties: Lim_𝑘→∞𝜎(ℎ)𝑞_𝑘=inf_Σ𝑛𝜎(ℎ)>−∞,lim_𝑘→∞Δ𝜎(ℎ)𝑞_𝑘≥0. (31) Note that 𝐻 is a nonpositive constant, then it follows from (30) that lim_𝑘→∞Δ𝜎(ℎ)𝑞_𝑘≤−𝑛lim_𝑘→∞𝑓(ℎ)𝑞_𝑘inf_Σ𝑛𝑓^′(ℎ)𝑓(ℎ)−𝐻≤0. (32)
Since the warping function is positive on Σ^𝑛 and Σ^𝑛 is bounded away from the infinity, from (31) and (32), we have lim_𝑘→∞Δ𝜎(ℎ)(𝑞_𝑘)=0, which means that inf_Σ𝑛𝑓^′(ℎ)𝑓(ℎ)−𝐻=0. (33)
Substituting the above equation into (4) implies that |∇ℎ|²≡0, that is, ℎ is a constant on Σ^𝑛; thus Σ^𝑛 is a slice of 𝑀^𝑛+1.

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.

Corollary 8. Let 𝑀^𝑛+1=−𝐼×_𝑓𝑀^𝑛 be a Robertson-Walker spacetime with constant sectional curvature. Let 𝜓∶Σ^𝑛→𝑀^𝑛+1 be a complete spacelike hypersurface with constant mean curvature 𝐻. Suppose that Σ^𝑛 is bounded away from the infinity. If (3) and (4) hold, then Σ^𝑛 is a slice.

Theorem 9. Let 𝑀^𝑛+1=−𝐼×_𝑓𝑀^𝑛 be a Robertson-Walker spacetime whose Riemannian fiber 𝑀^𝑛 has constant sectional curvature 𝑘 satisfying the null convergence condition. Let 𝜓∶Σ^𝑛→𝑀^𝑛+1 be a complete spacelike hypersurface with constant mean curvature 𝐻 and bounded away from the infinity. If 𝐻≥maxsup_{𝑝∈Σ𝑛}𝑓^′𝑓ℎ𝑝,0, (34) and the height function ℎ of Σ^𝑛 satisfies |∇ℎ|²≤𝛼𝐻−sup_{𝑝∈Σ𝑛}𝑓^′𝑓ℎ𝑝^𝛽, (35) for some positive constant 𝛼 and 𝛽, then Σ^𝑛 is a slice.

Proof. From (34), we see that the constant mean curvature 𝐻 is nonnegative, then it follows from (28) and (34) that ⟨𝑁,𝜕_𝑡⟩𝐻≤−𝐻≤0. Thus, from (29), we have (ℎ)(ℎ)𝑓Δ𝜎=−𝑛𝑓^′(ℎ)𝑓(ℎ)+𝑁,𝜕_𝑡𝐻(ℎ)𝑓≥−𝑛𝑓^′(ℎ)𝑓(ℎ)(ℎ)−𝐻≥−𝑛𝑓sup_Σ𝑛𝑓^′(ℎ)𝑓(ℎ)−𝐻≥0. (36)
Using the similar analysis with the proof of Theorem 1, applying the Lemma 7 to the smooth function 𝜎(ℎ) on Σ^𝑛 implies that there exists a sequences of points {𝑝_𝑘}_𝑘∈𝑁 in Σ^𝑛 satisfying the following properties: Lim_𝑘→∞𝜎(ℎ)𝑝_𝑘=sup_Σ𝑛𝜎(ℎ)<∞,lim_𝑘→∞Δ𝜎(ℎ)𝑝_𝑘≤0. (37) It follows from (36) that lim_𝑘→∞Δ𝜎(ℎ)𝑝_𝑘≥−𝑛lim_𝑘→∞𝑓(ℎ)𝑝_𝑘sup_Σ𝑛𝑓^′(ℎ)𝑓(ℎ)−𝐻≥0. (38) From (37) and (38), we have lim_𝑘→∞Δ𝜎(ℎ)(𝑝_𝑘)=0. Thus, from (36), we get sup_Σ𝑛𝑓^′(ℎ)𝑓(ℎ)−𝐻=0. (39)
By substituting the above equation into (35), we have |∇ℎ|²≡0, that is, ℎ is a constant on Σ^𝑛; thus Σ^𝑛 is a slice.

Corollary 10. Let 𝑀^𝑛+1=−𝐼×_𝑓𝑀^𝑛 be a Robertson-Walker spacetime with constant sectional curvature. Let 𝜓∶Σ^𝑛→𝑀^𝑛+1 be a complete spacelike hypersurface with constant mean curvature 𝐻. Suppose that Σ^𝑛 is bounded away from the infinity. If (34) and (35) hold, then Σ^𝑛 is a slice.

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 𝑓 is constant. Without losing the generality, we consider 𝑓=1 in the following. In this case, the null convergence condition (2) implies that 𝑘≥0. Also, (3) becomes that 𝐻≤0. When 𝑓 is a constant, (33) (or (39)) implies that 𝐻=0, that is, Σ^𝑛 is maximal. On the other hand, letting the warping function 𝑓 be a constant, we see from (34) that 𝐻≥0. Then a weaker assumption than (4) and (35), that is, the normal hyperbolic angle is bounded on Σ^𝑛, also guarantees Theorems 1 and 9. Thus, from Theorems 1 and 9, we have the following corollary.

Corollary 11. Let 𝑀^𝑛+1=−𝐼×𝑀^𝑛 be a static Robertson-Walker spacetime whose Riemannian fiber 𝑀^𝑛 has non-negative constant sectional curvature 𝑘. Let 𝜓∶Σ^𝑛→𝑀^𝑛+1 be a complete, connected spacelike hypersurface with constant mean curvature 𝐻. Suppose that the normal hyperbolic angle of Σ^𝑛 is bounded, then Σ^𝑛 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.

Proof of Theorem 2. Initially, we consider that the orientation 𝑁 of the hypersurface such that its angle function satisfies −1≤𝑁,𝜕_𝑡≤0. (40) Then from Lemma 3, we have that Δ𝜎(ℎ)=𝑛𝑓^′(ℎ)+𝑓(ℎ)𝑁,𝜕_𝑡𝐻(ℎ)𝑓=𝑛𝑓^′(ℎ)𝑓(ℎ)+𝑁,𝜕_𝑡𝐻. (41)
By using the assumption (6) we see that the mean curvature 𝐻 is non-positive, then it follows from (40) that 0≤𝐻⟨𝑁,𝜕_𝑡⟩≤−𝐻. Notice that the warping function 𝑓 is positive on 𝐼, then from (41) we have the following inequality: (ℎ)(ℎ)𝑓Δ𝜎=𝑛𝑓^′(ℎ)𝑓(ℎ)+𝑁,𝜕_𝑡𝐻(ℎ)𝑓≤𝑛𝑓^′(ℎ)𝑓(ℎ)(ℎ)−𝐻≤𝑛𝑓sup_Σ𝑛𝑓^′(ℎ)𝑓(ℎ)−𝐻≤0. (42)
Since the spacelike hypersurface is bounded away from the infinity, it follows that 𝑓^′′/𝑓 is bounded on Σ^𝑛. Let 𝑆₂ denote the second elementary symmetric function on the eigenvalues of the shape operator 𝐴, and 𝐻₂=(2/𝑛(𝑛−1))𝑆₂ denotes the mean value of 𝑆₂. It is easy to see |𝐴|²=𝑛²𝐻²−𝑛(𝑛−1)𝐻₂. Then the assumption that 𝐻₂ is bounded from below means that 𝐴 is bounded from above; from Lemma 6 we know that the Ricci curvature tensor of Σ^𝑛 is bounded from below.
Finally, by applying analogous arguments employed in the last part of the proof of Theorem 1, we conclude that Σ^𝑛 is a slice of 𝐼×_𝑓𝑀^𝑛.

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 ℋ^𝑛+1=−ℝ×_𝑒𝑡ℝ^𝑛. (43)

The importance of considering ℋ^𝑛+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 ℋ^𝑛+1 as follows.

Let 𝕃₁^𝑛+2 denote the (𝑛+2)-dimensional Lorentzian-Minkowski space, that is, the real vector space ℝ^𝑛+2 with a Lorentzian metric ⟨𝑣,𝑤⟩=^𝑛+1_𝑖=1𝑣_𝑖𝑤_𝑖−𝑣_𝑛+2𝑤_𝑛+2, (44) for all 𝑣,𝑤∈ℝ^𝑛+2. The (𝑛+1)-dimensional de Sitter space 𝕊₁^𝑛+1 is defined as the following: 𝕊₁^𝑛+1={𝑝∈𝕃₁^𝑛+2∶⟨𝑝,𝑝⟩=1}. Let 𝑞∈𝕃₁^𝑛+2 be a nonzero null vector of the null cone with vertex in the origin, such that ⟨𝑞,𝑒_𝑛+2⟩>0, where 𝑒_𝑛+2=(0,…,0,1). Then, it can be shown that the open region 𝑝∈𝕊₁^𝑛+1∶⟨𝑝,𝑞⟩>0 (45) of the de Sitter space 𝕊₁^𝑛+1 is isometric to ℋ^𝑛+1 (see [18] for 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 𝜓∶Σ²→ℋ³ be a Riemannian immersion of a complete surface of non-negative Guassian curvature 𝐾_Σ^𝑛 with constant mean curvature 𝐻≥1. If |∇ℎ|²≤𝐻²−1, (46) then 𝜓(Σ^𝑛) is a slice of ℋ³.

Suppose that the warping function 𝑓(𝑡)=𝑒^𝑡; thus (34) becomes 𝐻≥1; meanwhile the inequality (35) becomes |∇ℎ|²≤𝛼|𝐻−1|^𝛽, (47) where both 𝛼 and 𝛽 are positive constants. Suppose that 𝛽=2, then we have 𝛼(𝐻−1)^𝛽≥𝐻²−1 for all 𝐻≥1 if 𝛼≥1+2/(𝐻−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.

Theorem 13. Let 𝜓∶Σ²→ℋ³ be a Riemannian immersion of a complete surface with constant mean curvature 𝐻≤0. If |∇ℎ|²≤𝛼(1−𝐻)^𝛽 (48) holds for some positive constant 𝛼 and 𝛽, then 𝜓(Σ^𝑛) is a slice of ℋ³.

Theorem 14. Let 𝜓∶Σ²→ℋ³ be a Riemannian immersion of a complete surface with constant mean curvature 𝐻≥1. If |∇ℎ|²≤𝛼(𝐻−1)^𝛽, (49) holds for some positive constant 𝛼 and 𝛽, then 𝜓(Σ^𝑛) is a slice of ℋ³.

6. Entire Vertical Graphs in Robertson-Walker Spacetime

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 Ω⊆𝑀^𝑛 be a connected domain of 𝑀^𝑛; a vertical graph over Ω is defined by smooth function 𝑢∈𝒞^∞(Ω) and it is given by Σ^𝑛(𝑢)=𝑢𝑝,𝑝∶𝑝∈Ω⊆𝑀^𝑛⊂−𝐼×_𝑓𝑀^𝑛. (50) The metric induced on Ω from the Lorentzian metric on the ambient space via Σ^𝑛 is ⟨,⟩=−𝑑𝑢²+𝑓²(𝑢)⟨,⟩_𝑀^𝑛. The graph is said to be entire if Ω=𝑀^𝑛. It is easy to see that a graph Σ(𝑢) is a spacelike hypersurface if and only if |𝐷𝑢|²_𝑀^𝑛<𝑓²(𝑢), where 𝐷𝑢 is the gradient of 𝑢 in Ω and |𝐷𝑢|_𝑀^𝑛 its norm, both with respect to the metric ⟨,⟩_𝑀^𝑛 in Ω.

Let Σ^𝑛(𝑢) be a spacelike vertical graph over a domain Ω, its future-pointing Gauss map is given by the vector field 𝑁𝑝=𝑓𝑢𝑝𝑓²𝑢𝑝−𝐷𝑢𝑝²_𝑀𝑛×𝜕_{𝑡(𝑢(𝑝),𝑝)}+1𝑓²𝑢𝑝𝐷𝑢𝑝,𝑝∈Ω. (51) Moreover, the shape operator 𝐴 of Σ^𝑛(𝑢) with respect to 𝑁 is given by 1𝐴𝑋=−𝑓(𝑢)𝑓²(𝑢)−|𝐷𝑢|²_𝑀^𝑛𝐷_𝑋−𝑓𝐷𝑢_′(𝑢)𝑓²(𝑢)−|𝐷𝑢|²_𝑀^𝑛𝑋+𝑓^′(𝑢)⟨𝐷𝑢,𝑋⟩𝑓²(𝑢)−|𝐷𝑢|²_𝑀^𝑛3/2−𝐷_𝑋𝐷𝑢,𝐷𝑢_𝑀𝑛𝑓(𝑢)𝑓²(𝑢)−|𝐷𝑢|²_𝑀^𝑛)^3/2𝐷𝑢, (52) for any tangent vector field 𝑋 on Ω, where 𝐷 denotes the Levi-Civita connection in Ω with respect to the metric ⟨,⟩_𝑀^𝑛. The mean curvature function of a spacelike graph Σ^𝑛(𝑢) with respect to 𝑁 is given as the following: Div𝐷𝑢𝑓²(𝑢)−|𝐷𝑢|²_𝑀^𝑛(𝑢)𝐻(𝑢)−𝑓=𝑛𝑓^′(𝑢)𝑓²(𝑢)−|𝐷𝑢|²_𝑀^𝑛, (53) where Div denotes the divergence operator on Ω with respect to the metric ⟨,⟩_𝑀^𝑛. Now, we give the following nonparametric version of Theorem 1.

Theorem 15. Let 𝑀^𝑛+1=−𝐼×_𝑓𝑀^𝑛 be a Robertson-Walker spacetime whose Riemannian fiber 𝑀^𝑛 has constant sectional curvature 𝑘. Suppose that 𝑀^𝑛+1 obeys the null convergence condition (2). Let Σ^𝑛(𝑢) be an entire spacelike vertical graph in 𝑀^𝑛+1 with constant mean curvature 𝐻 and bounded away from the infinity. If 𝐻≤mininf_{𝑝∈Σ𝑛}𝑓^′𝑓𝑢𝑝,||,0𝐷𝑢²_𝑀^𝑛≤𝛼inf_Σ𝑛(𝑢)𝑓²(𝑢)inf_Σ^𝑛_(𝑢)𝑓^′/𝑓(𝑢)−𝐻^𝛽1+𝛼inf_Σ^𝑛_(𝑢)𝑓^′/𝑓(𝑢)−𝐻^𝛽, (54)(55) for some positive constant 𝛼 and 𝛽, then Σ^𝑛(𝑢) 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 Σ^𝑛(𝑢) is complete. Then it follows from [8, 20] that |∇ℎ|²=|𝐷𝑢|²_𝑀^𝑛𝑓²(𝑢)−|𝐷𝑢|²_𝑀^𝑛. (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.

Let 𝑀^𝑛+1=−𝐼×_𝑓𝑀^𝑛 be a static Robertson-Walker spacetime, that is, 𝑓=1. Thus, the null convergence condition implies that 𝑘≥0 and (54) becomes 𝐻≤0. In particular, from (55), we get |𝐷𝑢|²_𝑀^𝑛|𝐻|≤𝛼^𝛽1+𝛼|𝐻|^𝛽. (57)

Following from Theorem 15, we get a uniqueness theorem for entire spacelike vertical graphs in static spacetime.

Theorem 16. Let 𝑀^𝑛+1=−𝐼×𝑀^𝑛 be a static Robertson-Walker spacetime whose Riemannian fiber 𝑀^𝑛 has non-negative constant sectional curvature 𝑘. Let Σ^𝑛(𝑢) be an entire spacelike vertical graph in 𝑀^𝑛+1 with non-positive constant mean curvature 𝐻 and bounded away from the infinity. If the condition (57) is satisfied for some positive constant 𝛼 and 𝛽, then Σ^𝑛(𝑢) is a slice.

Acknowledgments

The project is supported by Natural Science Foundation of China (no. 10931005) and Natural Science Foundation of Guangdong Province of China (no. S2011010000471).