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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.

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 [

By applying a maximal principal due to Akutagawa [

Replacing the null convergence condition by

In this paper, following [

Let

Let

In this section, we recall some basic notations and facts following from [

Let

It follows from [

A smooth immersion

We denote by

The curvature tensor

Let

We consider two particular functions naturally attached to complete spacelike hypersurfaces, namely, the vertical (height) function

According to [

In order to prove our main theorems, we will make use of the following computations. We also refer the reader to [

Let

Let

By taking

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

Let

It follows from (

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

Let

In order to prove our main theorems, we also need the well-known generalized maximal principal due to Omori [

Let

Since

By using the assumption (

On the other hand, since the spacelike hypersurface is bounded away from the infinity of

Since the warping function is positive on

Substituting the above equation into (

From (

Let

Let

From (

Using the similar analysis with the proof of Theorem

By substituting the above equation into (

Let

Noting that a Robertson-Walker spacetime with constant sectional curvature trivially obeys the null convergence condition; thus the proof of Corollary

It is well known that Robertson-Walker spacetime is called static Robertson-Walker spacetime if the warping function

Let

We also refer the reader to [

Initially, we consider that the orientation

By using the assumption (

Since the spacelike hypersurface is bounded away from the infinity, it follows that

Finally, by applying analogous arguments employed in the last part of the proof of Theorem

In this section, we consider a particular model of Lorentzian warped product, namely, the steady state space, that is, the warped product

The importance of considering

Let

Recently, some uniqueness theorems for steady state space were obtained by [

Let

Suppose that the warping function

Let

Let

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 [

Let

Let

Let

First of all and following the ideas of Albujer et al. in the proof of Theorem 4.1 in [

Let

Following from Theorem

Let

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