On Calabi–Bernstein’s Problem for Uniqueness of Complete Spacelike Hypersurfaces in Weighted Generalized Robertson–Walker Spacetimes

Abstract

In this paper, we study Calabi–Bernstein’s problem for the uniqueness of complete constant weighted mean curvature spacelike hypersurfaces ${\Sigma }^n$ in weighted generalized Robertson–Walker spacetimes $-I{\times }_{\rho }{M}_f^n$. Under appropriate geometric assumptions, we confirm that ${\Sigma }^n$ is a spacelike slice.

1. Introduction

Let $M_f^n$ be an n-dimensional weighted manifold, which is the Riemannian manifold $(M^n,g)$ with a smooth function $f$ on $M^n$. Moreover, $M_f^n$ can be viewed as a triple $(M^n,g,\mathrm{d}\mu =e^{-f}\mathrm{d}M)$, where $\mathrm{d}M$ denotes the volume element of $M^n$. Define the Bakry–Émery Ricci tensor [1] on $M_f^n$ as

$${\mathrm{Ric}}_f=\mathrm{Ric}+\mathrm{Hess}f.$$

(1)

In this paper, we will study spacelike hypersurfaces in weighted generalized Robertson–Walker (GRW) spacetimes using the weak maximum principle. The generalized Robertson–Walker spacetimes can be regarded as Lorentzian warped products of the interval with a negative definite metric and the Riemannian manifold with a positive smooth function. In fact, the spacelike hypersurfaces in a weighted generalized Robertson–Walker spacetime are an important kind of submanifold, which are of important theoretical significance and research value. There is an important problem which arises for Calabi–Bernstein’s problem on the study of weighted generalized Robertson–Walker spacetimes.

Wei and Wylie discovered the mean curvature and volume comparison results in complete weighted manifolds [2]. Afterwards, [3,4] researched Calabi–Bernstein’s results concerning complete spacelike hypersurfaces in a weighted GRW spacetime using generalized maximum principles. Moreover, Liu and the author of [5] showed some rigidity results in weighted GRW spacetime via the application of the generalized maximum principle and weak maximum principle. In particular, [6] studied the uniqueness of complete weighted maximal hypersurfaces in weighted GRW spacetimes with parabolic fiber. More recently, [7,8] obtained some Calabi–Bernstein-type results of complete spacelike hypersurfaces in a weighted static GRW spacetime. Although there are some partial answers to the problem, they are still generally open thus far. In this article, we will give a further answer to Calabi–Bernstein’s problem.

This paper is organized as follows. In Section 3, we investigate the Laplacian of the angle function to obtain the parametric uniqueness result concerning spacelike hypersurfaces in weighted generalized Robertson–Walker spacetimes based on the weak maximum principle under appropriate geometric assumptions. As an application, in Section 4, we prove a new Calabi–Bernstein-type result of spacelike graphs in a weighted generalized Robertson–Walker spacetime.

2. Materials and Methods

Let $M^n$ be a connected oriented Riemannian manifold. Denote by $-I{\times }_{\rho }{M}^n$ the warped product [9] manifold endowed with the Lorentzian metric.

$$〈,〉=-{\pi }_I^{\ast }(\mathrm{d}{t}^2)+\rho {({\pi }_I)}^2{\pi }_M^{\ast }\left(〈,〉M\right),$$

which is the family of generalized Robertson–Walker (GRW) spacetimes [10], where the base $I\subset ℝ$ is an open interval with metric $-\mathrm{d}{t}^2$, the warping function $\rho :I\to {ℝ}^{+}$ is a positive smooth function, and ${\pi }_I$ and ${\pi }_M$ are the projections onto the base I and the fiber $M^n$, respectively.

Denote by ${\partial }_t:=\partial /\partial t$ the (unitary) timelike coordinate vector field which is globally defined on $-I{\times }_{\rho }{M}^n$. Thus $-I{\times }_{\rho }{M}^n$ is time-orientable. Let $\psi :{\Sigma }^n\to -I{\times }_{\rho }{M}^n$ be a spacelike hypersurface in $-I{\times }_{\rho }{M}^n$. Moreover, the metric on ${\Sigma }^n$ will be also denoted by $〈,〉$. In particular, a spacelike slice of $-I{\times }_{\rho }{M}^n$ is a hypersurface given by a fiber $M_{t_0}^n=\left\{t_0\right\}\times M^n,t_0\in I$. On the other hand, there is a unique (unitary) timelike normal vector field $N\in$$\mathfrak{X}$${}^{\perp }({\Sigma }^n)$ globally defined on ${\Sigma }^n$ with the same time orientation of $-{\partial }_t$, in a sense that $〈N,-{\partial }_t〉<0$. Thus, $〈N,{\partial }_t〉\ge 1$, and the equality holds if and only if $N=-{\partial }_t$ on ${\Sigma }^n$ (see [9], Proposition 5.30).

Let $\overline{\nabla }$ and $\nabla$ denote the Levi-Civita connections on $-I{\times }_{\rho }{M}^n$ and ${\Sigma }^n$, respectively. Moreover, a generalized Robertson–Walker spacetime has a timelike vector field $K=\rho ({\pi }_I){\partial }_t$ such that

$${\overline{\nabla }}_{X}K={\rho }^{\prime }({\pi }_I)X,$$

(2)

for $X\in$$\mathfrak{X}$$(-I{\times }_{\rho }{M}^n)$. In particular, we have $\mathcal{L}$${}_K〈,〉=2{\rho }^{\prime }({\pi }_I)〈,〉$, where $\mathcal{L}$${}_K$ is the Lie derivative on $K$. Thus, $K$ is concircular, which is particularly conformal. As is known, concircular vector fields were introduced by K. Yano [11,12,13,14] and N.S. Sinyukov called the spaces in which they exist equidistant; see [15]. Many questions about the geometry of these spaces are presented in [15]; see also [16].

Consider two functions related to ${\Sigma }^n$, namely, the height function $\tau ={\pi }_I\circ \psi :{\Sigma }^n\to I$ and the angle function $\theta =〈N,{\partial }_t〉:{\Sigma }^n\to [1,+\infty )$. By direct computation, we have

$$\overline{\nabla }{\pi }_I=-〈\overline{\nabla }{\pi }_I,{\partial }_t〉{\partial }_t=-{\partial }_t,$$

$$\nabla \tau ={(\overline{\nabla }{\pi }_I)}^{{\rm T}}=-{\partial }_t^{{\rm T}}=-{\partial }_t-\theta N.$$

(3)

Particularly,

$$|\nabla \tau {|}^2={\theta }^2-1,$$

(4)

where $|\cdot |$ is the norm of a vector field on ${\Sigma }^n$.

Furthermore, the Gauss and Weingarten formulas are given, respectively, by

$${\overline{\nabla }}_{X}Y={\nabla }_{X}Y-〈AX,Y〉N,$$

(5)

and

$$AX={\overline{\nabla }}_{X}N,$$

(6)

for $X,Y\in$$\mathfrak{X}$$({\Sigma }^n)$ and $N\in$$\mathfrak{X}$${}^{\perp }({\Sigma }^n)$, where $A:$$\mathfrak{X}({\Sigma }^n)\to$$\mathfrak{X}$$({\Sigma }^n)$ is the shape operator. Taking the tangential part of (2), then combining (5) and (6), we have

$${\nabla }_X{K}^{{\rm T}}=-\rho (\tau )\theta AX+{\rho }^{\prime }(\tau )X,$$

(7)

where ${\rho }^{\prime }(\tau ):={\rho }^{\prime }\circ \tau$ and $K^{{\rm T}}:=\rho (\tau ){\partial }_t^{{\rm T}}=K+〈K,N〉N$ is the tangential component of $K$ along $\psi$. From (2), (5), and (6), we obtain the gradient of the angle function $\theta$ that

$$\nabla \theta =-A{\partial }_t^{{\rm T}}-\frac{{\rho }^{\prime }(\tau )}{\rho (\tau )}\theta \nabla \tau .$$

(8)

From (3) and (7),

$$\Delta \tau =-(\log \rho {)}^{\prime }(\tau )(n+|\nabla \tau {|}^2)-nH\theta ,$$

where $H$ is the mean curvature on ${\Sigma }^n$ with respect to $N$.

Combining $\Delta \rho (\tau )={\rho }^{\prime }(\tau )\Delta \tau +{\rho }^{″}(\tau )|\nabla \tau {|}^2$, we have

$$\Delta \rho (\tau )=-n\frac{{\rho }^{\prime }{(\tau )}^2}{\rho (\tau )}+|\nabla \tau {|}^2\rho (\tau )(\log \rho {)}^{″}(\tau )-n{\rho }^{\prime }(\tau )H\theta .$$

(9)

A generalized Robertson–Walker spacetime $-I{\times }_{\rho }{M}^n$ with a weight function $f$ is called a weighted generalized Robertson–Walker spacetime $-I{\times }_{\rho }{M}_f^n$. Moreover, in a spacelike hypersurface ${\Sigma }^n$ in $-I{\times }_{\rho }{M}_f^n$, the $f$-divergence operator on ${\Sigma }^n$ satisfies

$${\mathrm{div}}_f(X)=e^f\mathrm{div}(e^{-f}X),$$

for any tangent vector field $X$ on ${\Sigma }^n$.

If $u:{\Sigma }^n\to ℝ$ is a smooth function, then the drifting Laplacian or f-Laplacian of $u$ is defined by

$${\Delta }_{f}u={\mathrm{div}}_f(\nabla u)=\Delta u-〈\nabla u,\nabla f〉.$$

(10)

The weighted mean curvature or f-mean curvature $H_f$ of ${\Sigma }^n$ is defined by (see [17])

$$n{H}_f=nH-〈N,\overline{\nabla }f〉.$$

(11)

From the splitting theorem (see [18], Theorem 1.2), if $-I{\times }_{\rho }{M}_f^n$ with a bounded weight function $f$ and ${\overline{\mathrm{Ric}}}_f(V,V)\ge 0$ for any timelike vector field $V$, then $f$ is constant on $I$. Consequently, throughout the paper, we assume that $f$ does not depend on $t\in I$, that is $〈\overline{\nabla }f,{\partial }_t〉=0$.

3. Uniqueness Results

The following lemmas play very important roles in this paper.

Lemma 1. 

Let $\psi :{\Sigma }^n-I{\times }_{\rho }{M}_f^n$ be a spacelike hypersurface in weighted generalized Robertson–Walker spacetimes $-I{\times }_{\rho }{M}_f^n$. Then, we have

$$\begin{array}{ll}\frac{1}{2}{\Delta }_f( |\nabla \tau {|}^2)& =n\theta 〈\nabla H_f,{\partial }_t〉+n\frac{{\rho }^{\prime }(\tau )}{\rho (\tau )}\theta H_f|\nabla \tau {|}^2+2\frac{{\rho }^{\prime }(\tau )}{\rho (\tau )}〈\nabla f,\nabla \tau 〉+|\nabla \theta {|}^2+\left|\mathrm{Hess}\right(\tau ){|}^2\\ & +\frac{{\rho }^{\prime }{(\tau )}^2}{\rho {(\tau )}^2}|\nabla \tau {|}^2(n+|\nabla \tau {|}^2)+{\theta }^2({\mathrm{Ric}}_f^M(N^{\ast },N^{\ast })-n(\log \rho {)}^{″}(\tau )|\nabla \tau {|}^2)\end{array}$$

(12)

where ${\mathrm{Ric}}_f^M$ is the Bakry–Émery Ricci curvature tensor on $M$;$N^{\ast }=N+〈N,{\partial }_t〉{\partial }_t$ is the projection of $N$ onto $M$.

Proof of Lemma 1. 

taking into account (10), it follows that

$${\Delta }_f(\rho (\tau )\theta )=\Delta (\rho (\tau )\theta )-〈\nabla f,\nabla (\rho (\tau )\theta )〉.$$

(13)

From (8) and (9), we obtain

$$\begin{array}{ll}\Delta (\rho (\tau )\theta )& =\theta \Delta \rho (\tau )+\rho (\tau )\Delta \theta +2〈\nabla \rho (\tau ),\nabla \theta 〉\\ & =-n\frac{{\rho }^{\prime }{(\tau )}^2}{\rho (\tau )}\theta +\rho (\tau )\theta |\nabla \tau {|}^2(\log \rho {)}^{″}(\tau )+2{\rho }^{\prime }(\tau )〈A\nabla \tau ,\nabla \tau 〉\\ & -2\frac{{\rho }^{\prime }{(\tau )}^2}{\rho (\tau )}\theta |\nabla \tau {|}^2-nH{\rho }^{\prime }(\tau ){\theta }^2+\rho (\tau )\Delta \theta .\end{array}$$

On the other hand,

$$\begin{array}{ll}〈\nabla f,\nabla (\rho (\tau )\theta )〉& =〈\nabla f,{\rho }^{\prime }(\tau )\theta \nabla \tau +\rho (\tau )\nabla \theta 〉\\ & ={\rho }^{\prime }(\tau )\theta 〈\nabla f,\nabla \tau 〉+\rho (\tau )〈\nabla f,\nabla \theta 〉\\ & =-{\rho }^{\prime }(\tau ){\theta }^2〈\overline{\nabla }f,N〉+\rho (\tau )〈\nabla f,\nabla \theta 〉\end{array}$$

Together with (10), (11), and (13), we have

$$\begin{array}{ll}{\Delta }_f(\rho (\tau )\theta )& =-n\frac{{\rho }^{\prime }{(\tau )}^2}{\rho (\tau )}\theta +\rho (\tau )\theta |\nabla \tau {|}^2(\log \rho {)}^{″}(\tau )+2{\rho }^{\prime }(\tau )〈A\nabla \tau ,\nabla \tau 〉\\ & -2\frac{{\rho }^{\prime }{(\tau )}^2}{\rho (\tau )}\theta |\nabla \tau {|}^2-n{H}_f{\rho }^{\prime }(\tau ){\theta }^2+\rho (\tau ){\Delta }_f\theta .\end{array}$$

(14)

From Lemma 1 in [3],

$$\begin{array}{ll}{\Delta }_f( \rho (\tau )\theta )& =n\rho (\tau )〈\nabla H_f,{\partial }_t〉+n{\rho }^{\prime }(\tau )H_f+\rho (\tau )\theta |A{|}^2+\rho (\tau )\theta \overline{\mathrm{Hess}}f(N,N)\\ & +\rho (\tau )\theta ({\mathrm{Ric}}^M(N^{\ast },N^{\ast })-(n-1)(\log \rho {)}^{″}(\tau )|\nabla \tau {|}^2) \end{array}$$

(15)

where ${\mathrm{Ric}}^M$ is the Ricci curvature tensor on $M$.

Combining $〈\overline{\nabla }f,{\partial }_t〉=0$, we have

$$\overline{\mathrm{Hess}}f(N,N)={\mathrm{Hess}}^{M}f(N^{\ast },N^{\ast }).$$

(16)

From (1), (14), (15), and (16), we obtain

$$\begin{array}{ll}{\Delta }_f\theta & =n〈\nabla H_f,{\partial }_t〉+n{H}_f\frac{{\rho }^{\prime }(\tau )}{\rho (\tau )}(1+{\theta }^2)-2\frac{{\rho }^{\prime }(\tau )}{\rho (\tau )}〈A\nabla \tau ,\nabla \tau 〉+\theta |A{|}^2\\ & +\frac{{\rho }^{\prime }{(\tau )}^2}{\rho {(\tau )}^2}\theta (n+2|\nabla \tau {|}^2)+\theta ({\mathrm{Ric}}_f^M(N^{\ast },N^{\ast })-n(\log \rho {)}^{″}(\tau )|\nabla \tau {|}^2) \end{array}$$

By taking the tangential component in (2) and using (3), we find that

$$|\mathrm{Hess}(\tau ){|}^2=2nH\frac{{\rho }^{\prime }(\tau )}{\rho (\tau )}\theta -2\frac{{\rho }^{\prime }(\tau )}{\rho (\tau )}\theta 〈A\nabla \tau ,\nabla \tau 〉+\frac{{\rho }^{\prime }{(\tau )}^2}{\rho {(\tau )}^2}(n-1+{\theta }^4)+{\theta }^2|A{|}^2.$$

Thus,

$$\begin{array}{ll}\theta {\Delta }_f\theta & =n\theta 〈\nabla H_f,{\partial }_t〉+n{H}_f\frac{{\rho }^{\prime }(\tau )}{\rho (\tau )}\theta |\nabla \tau {|}^2+2\frac{{\rho }^{\prime }(\tau )}{\rho (\tau )}〈\nabla f,\nabla \tau 〉+|\mathrm{Hess}(\tau ){|}^2\\ & +\frac{{\rho }^{\prime }{(\tau )}^2}{\rho {(\tau )}^2}(n+|\nabla \tau {|}^2)|\nabla \tau {|}^2+{\theta }^2({\mathrm{Ric}}_f^M(N^{\ast },N^{\ast })-n(\log \rho {)}^{″}(\tau )|\nabla \tau {|}^2) .\end{array}$$

(17)

Moreover,

$$\frac{1}{2}{\Delta }_f( |\nabla \tau {|}^2)=\frac{1}{2}{\Delta }_f({\theta }^2-1)=\theta {\Delta }_f\theta +|\nabla \theta {|}^2.$$

With the help of the above equality and (17), we complete the proof of (12). □

Lemma 2. 

Let $\psi :{\Sigma }^n-I{\times }_{\rho }{M}_f^n$ be a spacelike hypersurface with constant f-mean curvature $H_f$ in a weighted generalized Robertson–Walker spacetime $-I{\times }_{\rho }{M}_f^n$ with $(\log \rho {)}^{″}(\tau )\le 0$ and the Hessian of $f$ is bounded from below. If either

(i)

 $M$ has non-negative sectional curvature;

(ii)

 Or $\theta$ is bounded and the sectional curvature of $M$ is bounded from below.

then the Bakry–Émery Ricci curvature on ${\Sigma }^n$ is bounded from below.

Proof of Lemma 2. 

Choose a local orthonormal frame $E_1,E_2,\dots ,E_n$ in $\mathfrak{X}$$({\Sigma }^n)$. For any $X\in$$\mathfrak{X}$$({\Sigma }^n)$, from the Gauss equation, we obtain

$$\mathrm{Ric}(X,X)={\displaystyle \sum _{i=1}^n〈\overline{R}(X,E_i)X,E_i〉}+nH〈AX,X〉+|AX{|}^2,$$

(18)

From Proposition 7.42 in [9], we obtain

$$\begin{array}{ll}〈\overline{R}(X,E_i)X,E_i〉& =\rho {(\tau )}^2{K}_M(X^{\ast },E_i^{\ast })\left({〈X^{\ast },X^{\ast }〉}_M{〈E_i^{\ast },E_i^{\ast }〉}_M-{〈X^{\ast },E_i^{\ast }〉}_M^2\right)\\ & +{((\log \rho )\prime (\tau ))}^2\left(〈X,X〉-{〈X,E_i〉}^2\right)\\ & +(\log \rho {)}^{″}(\tau )〈X,\nabla \tau 〉\left(〈\nabla \tau ,E_i〉〈X,E_i〉-〈X,\nabla \tau 〉\right)\\ & -(\log \rho {)}^{″}(\tau )\left(〈E_i,\nabla \tau 〉〈X,X〉-〈X,\nabla \tau 〉〈X,E_i〉\right)〈E_i,\nabla \tau 〉,\end{array}$$

where $K_M$ is the sectional curvature of $M$, $X^{\ast }=X+〈X,{\partial }_t〉{\partial }_t$ and $E_i^{\ast }=E_i+〈E_i,{\partial }_t〉{\partial }_t$ are the projections of $X$ and $E_i$ on $M$, respectively. Moreover, through a direct computation, if (i) holds, we obtain

$$\begin{array}{ll}{\displaystyle \sum _{i=1}^n〈\overline{R}(X,E_i)X,E_i〉}\ge & -(\log \rho {)}^{″}(\tau )|X{|}^2|\nabla \tau {|}^2+(n-1)\frac{{\rho }^{\prime }{(\tau )}^2}{\rho {(\tau )}^2}|X{|}^2\\ & -(n-2)(\log \rho {)}^{″}(\tau ){〈X,\nabla \tau 〉}^2.\end{array}$$

(19)

Considering the hypothesis, $(\log \rho {)}^{″}(\tau )\le 0$. Thus, substituting (19) into (18), we have

$$\mathrm{Ric}(X,X)\ge nH〈AX,X〉+|AX{|}^2.$$

(20)

Furthermore, because the Hessian of weight function $f$ is bounded from below, there is a constant $\alpha >0$ satisfying $\overline{\mathrm{Hess}}f(X,X)\ge -\alpha |X{|}^2$ for any $X\in$$\mathfrak{X}$$(-I{\times }_{\rho }{M}_f^n)$. Thus,

$$\begin{array}{ll}\mathrm{Hess} f(X,X)=& \overline{\mathrm{Hess}}f(X,X)-〈\overline{\nabla }f,N〉〈AX,X〉\\ & \ge -\alpha |X{|}^2-〈\overline{\nabla }f,N〉〈AX,X〉.\end{array}$$

(21)

Therefore, from (1), (20), and (21), we obtain

$${\mathrm{Ric}}_f(X,X)\ge -\alpha |X{|}^2+n{H}_f〈AX,X〉+|AX{|}^2.$$

(22)

Moreover,

$$n{H}_f〈AX,X〉+|AX{|}^2=|AX+\frac{n{H}_f}{2}X{|}^2-\frac{n^2{H}_f^2}{4}|X{|}^2.$$

So, (22) becomes

$${\mathrm{Ric}}_f(X,X)\ge -\alpha |X{|}^2+|AX+\frac{n{H}_f}{2}X{|}^2-\frac{n^2{H}_f^2}{4}|X{|}^2.$$

Therefore, we can clearly see that the Bakry–Émery Ricci curvature on ${\Sigma }^n$ is bounded from below.

For (ii), the sectional curvature of $M$ is bounded from below, and there is a positive constant $k$, such that $\rho {(\tau )}^2{K}_M\ge -k$; we have

$$\begin{array}{ll}{\displaystyle \sum _{i=1}^n〈\overline{R}(X,E_i)X,E_i〉}\ge & -k\left((n-1)|X{|}^2+(n-2){〈X,\nabla \tau 〉}^2+|X{|}^2|\nabla \tau {|}^2\right)+(n-1)\frac{{\rho }^{\prime }{(\tau )}^2}{\rho {(\tau )}^2}|X{|}^2\\ & -(\log \rho {)}^{″}(\tau )|X{|}^2|\nabla \tau {|}^2-(n-2)(\log \rho {)}^{″}(\tau ){〈X,\nabla \tau 〉}^2.\end{array}$$

and

$$\mathrm{Ric}(X,X)\ge -k\left((n-1)|X{|}^2+(n-2){〈X,\nabla \tau 〉}^2+|X{|}^2|\nabla \tau {|}^2\right)+nH〈AX,X〉+|AX{|}^2.$$

So,

$$\begin{array}{ll}{\mathrm{Ric}}_f(X,X)\ge & -k\left((n-1)|X{|}^2+(n-2){〈X,\nabla \tau 〉}^2+|X{|}^2|\nabla \tau {|}^2\right)\\ & -\alpha |X{|}^2+|AX+\frac{n{H}_f}{2}X{|}^2-\frac{n^2{H}_f^2}{4}|X{|}^2.\end{array}$$

(23)

Moreover, the classical Schwarz inequality and (4) ensure that

$$-k\left((n-1)|X{|}^2+(n-2){〈X,\nabla \tau 〉}^2+|X{|}^2|\nabla \tau {|}^2\right)\ge -k(n-1)|X{|}^2{\theta }^2.$$

Therefore, we can conclude from (23) that if $\theta$ on ${\Sigma }^n$ is bounded, then the Bakry–Émery Ricci curvature of ${\Sigma }^n$ is bounded from below. □

Theorem 1. 

Let $\psi :{\Sigma }^n-I{\times }_{\rho }{M}_f^n$ be a complete spacelike hypersurface with constant f-mean curvature $H_f$ and bounded angle function $\theta$ in a weighted generalized Robertson–Walker spacetime $-I{\times }_{\rho }{M}_f^n$ whose fiber $M^n$ has non-negative sectional curvature. If $f$ is convex ${\sup }_{\Sigma }(\log \rho {)}^{″}(\tau )<0$,$H_f{\rho }^{\prime }(\tau )\ge 0$, and ${\rho }^{\prime }(\tau )〈\nabla f,\nabla \tau 〉\ge 0$, then ${\Sigma }^n$ is a spacelike slice.

Proof of Theorem 1. 

Considering the assumptions of sectional curvature on $M^n$ and the weight function $f$, then ${\mathrm{Ric}}^M(N^{\ast },N^{\ast })\ge 0$ and $\overline{\mathrm{Hess}}f(N,N)\ge 0$. From (1) and (16), we have

$${\mathrm{Ric}}_f^M(N^{\ast },N^{\ast })={\mathrm{Ric}}^M(N^{\ast },N^{\ast })+\overline{\mathrm{Hess}}f(N,N)\ge 0.$$

Proceeding as above, it follows from (4) and (12) that

$$\frac{1}{2}{\Delta }_f( |\nabla \tau {|}^2)\ge -n(\log \rho {)}^{″}(\tau )|\nabla \tau {|}^2(|\nabla \tau {|}^2+1).$$

(24)

Moreover, Lemma 2 holds when ${\sup }_{\Sigma }(\log \rho {)}^{″}(\tau )<0$; thus, we find that the Bakry–Émery Ricci curvature on ${\Sigma }^n$ is bounded from below. From Remark 2.18 of [19], we find that the weak Omori–Yau maximum principle for ${\Delta }_f$ holds. Therefore, there is a sequence of points $\left\{p_j\right\},j\ge 1$ on ${\Sigma }^n$ which satisfies

$$\underset{j}{\lim }|\nabla \tau {|}^2(p_j)={\sup }_{\Sigma }|\nabla \tau {|}^2,\hspace{1em}\underset{j}{\lim }\sup {\Delta }_f(|\nabla \tau {|}^2(p_j))\le 0.$$

So, from (24), we have

$$0\ge \underset{j}{\lim }\sup {\Delta }_f(|\nabla \tau {|}^2(p_j))\ge -n{\sup }_{\Sigma }(\log \rho {)}^{″}(\tau )(1+{\sup }_{\Sigma }|\nabla \tau {|}^2){\sup }_{\Sigma }|\nabla \tau {|}^2.$$

(25)

From ${\sup }_{\Sigma }(\log \rho {)}^{″}(\tau )<0$ and (25), we have ${\sup }_{\Sigma }|\nabla \tau {|}^2=0$. So, $\tau$ is constant and ${\Sigma }^n$ is a spacelike slice. □

Remark 1. 

Note that by weakening the assumptions of Theorem 1, the uniqueness result of weighted generalized Robertson–Walker does not hold. In fact, the complete f-maximal spacelike hypersurfaces in $-I\times M_f^n$ satisfy all the assumptions but ${\sup }_{\Sigma }(\log \rho {)}^{″}(\tau )<0$; thus, we cannot obtain an analogous uniqueness result.

Moreover, consider a weighted Lorentzian product space$-I\times$${\mathbb{G}}^{\mathfrak{n}}$, where${\mathbb{G}}^{\mathfrak{n}}$is Gaussian space given by the Euclidean space ${ℝ}^n$ with the following measure:

$$e^{-f}d{x}^2={(2\pi )}^{-\frac{n}{2}}{e}^{-\frac{|x{|}^2}{2}}d{x}^2.$$

If we omit the assumption ${\sup }_{\Sigma }(\log \rho {)}^{″}(\tau )<0$ in Theorem 1, then there is a complete nontrivial f-maximal graph given by the function$u(x_2,\dots ,x_{n+1})=x_1,(x_2,\dots ,x_{n+1})\in$${\mathbb{G}}^{\mathfrak{n}}$in$-I\times$${\mathbb{G}}^{\mathfrak{n}}$ (see [20]).

4. Calabi–Bernstein-Type Results

Consider a vertical graph ${\Sigma }^n(u)=\{(u(x),x):x\in \Omega \}\subset -I{\times }_{\rho }{M}^n$ determined by a function $u\in C^{\infty }(M), u(M)\subseteq I$, where $-I{\times }_{\rho }{M}^n$ is a generalized Robertson–Walker spacetime and $\Omega \subset M^n$ is a connected domain with the metric

$$〈,〉=-d{u}^2+{\rho }^2(u){〈,〉}_M.$$

The graph ${\Sigma }^n(u)$ is spacelike if and only if the metric induced on $\Omega$ is Riemannian, that is, $\left|Du\right|<\rho (u)$ on $\Omega$, where $Du$ is the gradient of $u$ and $|Du{|}^2={〈Du,Du〉}_M$. Furthermore, ${\Sigma }^n(u)$ is called entire if $\Omega =M^n$. In fact, if $p\in M^n$, then $\tau (u(p),p)=u(p)$. Thus, $\tau$ and $u$ are identified naturally on ${\Sigma }^n(u)$.

The unitary normal vector field on the spacelike graph ${\Sigma }^n(u)$ is defined by

$$N=-\frac{1}{\rho (u)\sqrt{\rho {(u)}^2-|Du{|}^2}}(\rho {(u)}^2{\partial }_t+Du).$$

Using Proposition 7.35 in [9] again, the shape operator $A$ relevant to $N$ is given by

$$\begin{array}{ll}A(X)& =\frac{1}{\sqrt{\rho {(u)}^2-|Du{|}^2}}\left({\rho }^{\prime }(u)X+\frac{1}{\rho (u)}{D}_{X}Du\right)\\ & +\left(\frac{{〈D_{X}Du,Du〉}_M}{\rho (u){(\rho {(u)}^2-|Du{|}^2)}^{3/2}}-\frac{{\rho }^{\prime }(u){〈Du,X〉}_M}{{(\rho {(u)}^2-|Du{|}^2)}^{3/2}}\right)Du,\end{array}$$

(26)

for all $X\in$$\mathfrak{X}$$(\Omega )$. From (11) and (26), we find the f-mean curvature connected to $N$ is given by,

$$H_f=-{\mathrm{div}}_f\left(\frac{Du}{n\rho (u)\sqrt{\rho {(u)}^2-|Du{|}^2}}\right)-\frac{{\rho }^{\prime }(u)}{n\sqrt{\rho {(u)}^2-|Du{|}^2}}\left(n+\frac{|Du{|}^2}{\rho {(u)}^2}\right).$$

Thus, we have the following constant f-mean curvature spacelike hypersurface equations:

$${\mathrm{div}}_f\left(\frac{Du}{\rho (u)\sqrt{\rho {(u)}^2-|Du{|}^2}}\right)+\frac{{\rho }^{\prime }(u)}{\sqrt{\rho {(u)}^2-|Du{|}^2}}\left(n+\frac{|Du{|}^2}{\rho {(u)}^2}\right)=-n{H}_f,$$

(27)

$$\left|Du\right|<\lambda \rho (u), 0<\lambda <1.$$

(28)

Theorem 2. 

Let $-I{\times }_{\rho }{M}_f^n$ be a weighted generalized Robertson–Walker spacetime with nonnegative sectional curvature on $M^n$. Assume the weighted function $f$ is convex, $\inf \rho (u)>0$ and $\sup (\log \rho {)}^{″}(u)<0$. If ${\rho }^{\prime }(u)〈\nabla f,\nabla u〉\ge 0$ and $H_f{\rho }^{\prime }(u)\ge 0$, then the only entire solutions to the Equations (27) and (28) are $u=t_0, t_0\in I$ with ${\rho }^{\prime }(t_0)=0$.

Proof of Theorem 2. 

It is noted that

$$|\nabla \tau {|}^2={\theta }^2-1=\frac{|Du{|}^2}{\rho {(u)}^2-|Du{|}^2}.$$

(29)

From (28) and (29), we have

$$1\le \theta <\frac{1}{\sqrt{1-{\lambda }^2}}.$$

(30)

So, $\theta$ is bounded on ${\Sigma }^n(u)$.

From the Schwarz inequality,

$$〈X,X〉=-{〈Du,X〉}_M^2+\rho {(u)}^2{〈X,X〉}_M\ge (\rho {(u)}^2-|Du{|}^2){〈X,X〉}_M,$$

(31)

where $X\in$$\mathfrak{X}$$({\Sigma }^n(u))$. Therefore, from (28) and (31),

$$〈X,X〉\ge \frac{\rho {(u)}^2}{{\theta }^2}{〈X,X〉}_M.$$

(32)

We denote by $L(\beta )$ and $L_M(\beta )$ the length of a smooth curve $\beta$ on ${\Sigma }^n(u)$ corresponding to the metrics $〈,〉$ and ${〈,〉}_M$, respectively. From (30) and (32), we have

$$L(\beta )\ge (1-{\lambda }^2)(\inf \rho {(u)}^2)L_M(\beta ).$$

when $M^n$ is complete and $\inf \rho (u)>0$, then ${\Sigma }^n(u)$ is complete. Combining Theorem 1, we complete the proof of Theorem 2. □