A Rigidity Theorem on Spacelike Hypersurfaces in Generalized Robertson–Walker Spacetimes

Abstract

We study the properties of complete parabolic constant mean curvature spacelike hypersurfaces in generalized Robertson–Walker (GRW) spacetimes $I{\times }_{\phi }{P}^n$ whose warping function $\phi$ fulfills a certain convexity criterion such that $-\phi$ is convex, and whose Ricci curvature of the fiber $P^n$ is non-negative. Our approach is based on calculating the Laplacian of an appropriate function. Under appropriate conditions on the constant mean curvature, by using the parabolicity, we obtain a rigidity theorem and some corollaries of spacelike hypersurfaces. As a consequence, we solve new corresponding Calabi–Bernstein-type problems.

1. Introduction

In recent years, constant mean curvature (CMC) spacelike hypersurfaces immersed in GRW spacetimes have been studied from both a mathematical and a physical perspective. Prior to furnishing a detailed account of the current work, we give a succinct summary of recent research developments that are closely associated with ours.

In [1,2] the authors investigated the CMC spacelike hypersurfaces in spatially closed GRW spacetimes and obtained some uniqueness results under the Timelike Convergence Condition (TCC) by some Minkowski-type integral formulas. Later, in [3] the authors studied compact CMC spacelike hypersurfaces in conformally stationary spacetimes by the same approach. On the other hand, ref. [4] proved that each compact CMC spacelike hypersurface in a GRW spacetime obeying TCC must be umbilical. Afterwards, in [5] by using Bochner–Lichnerowiz formula, the authors derived some uniqueness theorems relating to noncompact CMC spacelike hypersurfaces in GRW spacetimes under appropriate assumptions, which extend several aforementioned results in [1] through the existence of a local maximum for a distinguished function changing the compactness of CMC spacelike hypersurfaces. Recently, in [6] the authors investigated complete maximal spacelike hypersurfaces immersed in GRW spacetimes with parabolic Riemannian fiber and proved some uniqueness results. Moreover, in [7] the authors used the result in [6] and provided some non-existence and uniqueness results on parabolic complete CMC spacelike hypersurfaces in GRW spacetimes by establishing sufficient conditions. Moreover, as a consequence, in [7] the authors also studied the Calabi–Bernstein-type properties in such ambient space. Based on maximum principles, in [8,9] the authors obtained uniqueness results on noncompact complete CMC spacelike hypersurfaces immersed in a GRW spacetime. More recently, in [10] the authors proved several results related to uniqueness and nonexistence concerning spacelike hypersurfaces by maximum principle jointly with a sharp Bochner formula inequality.

The main purpose of this paper is to prove new rigidity theorems for complete parabolic CMC spacelike hypersurfaces in a GRW spacetime. The structure of this paper is outlined as follows. Section 2 is devoted to collecting some preliminary results and fixing some notations of spacelike hypersurfaces in GRW spacetimes. In Section 3.1, we will finish the proof of our main rigidity theorem. Moreover, we recast the main Theorem in some particular cases. The proof of the Calabi–Bernstein-type results is given in Section 3.2.

2. Materials and Methods

The GRW spacetime (see [1]) is the Lorentzian warped product manifold ${\overline{M}}^{n+1}=I{\times }_{\phi }{P}^n$ with the Lorentzian metric tensor

$$\langle , \rangle =-{\pi }_I^{\ast }(\mathrm{d}{t}^2)+\phi {({\pi }_I)}^2{\pi }_P^{\ast }(g_p),$$

where ${\pi }_I:{\overline{M}}^{n+1}\to I$ and ${\pi }_P:{\overline{M}}^{n+1}\to P^n$ are the projections of ${\overline{M}}^{n+1}=I{\times }_{\phi }{P}^n$ onto $I$ and $P^n$, respectively. The open interval $I\subset ℝ$ with metric $-\mathrm{d}{t}^2$ is called the base of ${\overline{M}}^{n+1}=I{\times }_{\phi }{P}^n$, and the Riemannian manifold $(P^n, g_p)$ is referred to as the fiber of ${\overline{M}}^{n+1}=I{\times }_{\phi }{P}^n$. Furthermore, we call the smooth positive function $\phi :I\to {ℝ}^{+}$ the warping function. In any GRW spacetime ${\overline{M}}^{n+1}$, there exists a unit timelike vector field ${\partial }_t:=\partial /\partial t\in T({\overline{M}}^{n+1})$ globally defined on ${\overline{M}}^{n+1}$, where $T({\overline{M}}^{n+1})$ is the tangent bundle on ${\overline{M}}^{n+1}$. Thus ${\overline{M}}^{n+1}$ is a time-orientable manifold.

An immersion $\varphi :M^n\to {\overline{M}}^{n+1}$ of a manifold $M^n$ is called spacelike whenever the induced metric via $\varphi$ on $M^n$ is Riemannian. In such a case, $M^n$ is said to be a spacelike hypersurface in ${\overline{M}}^{n+1}$. Since each GRW spacetime ${\overline{M}}^{n+1}$ is time-orientable, then the spacelike hypersurface $M^n$ in ${\overline{M}}^{n+1}$ is automatically orientable (see [11]). Thus, we orient $M^n$ by the choice of a unique globally defined unit timelike vector field $N\in T{(M^n)}^{\perp }$ normal to $M^n$ with the same time-orientation as $-{\partial }_t$ such that $\langle N,-{\partial }_t\rangle <0$, where $T{(M^n)}^{\perp }$ is a normal bundle of spacelike hypersurface $M^n$. Using the Cauchy–Schwarz inequality (see the Proposition 5.30 in [11]), we have $\langle N,{\partial }_t\rangle \ge 1$ and $\langle N_p,{\partial }_t(p)\rangle =1$ at any point $p\in M^n$ if and only if $N_p=-{\partial }_t(p)$. From the considerations above, at a point $p\in M^n$, the hyperbolic angle $\theta$ between the unitary timelike vector fields $-{\partial }_t$ and $N$ is defined by $\langle N_P,{\partial }_t(p)\rangle =\cosh \theta (p)$. For simplicity, we refer to $\theta$ as the hyperbolic angle function on the spacelike hypersurface $M^n$.

Particularly, for $t_0\in I$, there is a family of spacelike hypersurfaces $M_{t_0}^n=\left\{t_0\right\}\times P^n$ in ${\overline{M}}^{n+1}$. Our rigidity results will deal with this case, so we proceed to the following definition: A spacelike hypersurface $M^n$ is a spacelike slice of ${\overline{M}}^{n+1}=I{\times }_{\phi }{P}^n$ if and only if $\langle N,{\partial }_t\rangle =1$, namely, if the hyperbolic angle function $\theta$ vanishes identically.

From a physical perspective, Lorentzian geometry serves as the geometric language of general relativity. In a GRW spacetime ${\overline{M}}^{n+1}=I{\times }_{\phi }{P}^n$, the comoving observers are said to be the integral curves of a unit timelike vector field ${\partial }_t$ and ${\partial }_t(p), p\in {\overline{M}}^{n+1}$ is called an instantaneous comoving observer. For any point $q\in M^n$, ${\partial }_t(q)$ and $N_q$ are the natural instantaneous observers at point $q$. Let $N_q^{*}$ be the projection of $N_q$ on $M^n$. From the orthogonal decomposition $N_q^{*}=N_q+\langle N_q,{\partial }_t(q)\rangle {\partial }_t(q)$, it follows that $\cosh \theta (p)=\langle N_P,{\partial }_t(p)\rangle$ is identical to the energy $e(q)=:-\langle N_q,{\partial }_t(q)\rangle$ that ${\partial }_t(q)$ measures for $N_q$. In addition, the speed $\Vert v(q)\Vert$ of the velocity $v(q)=:\left(1/e(q)\right)N_q^{*}$ measured by ${\partial }_t(q)$ for $N_q$ satisfies ${\Vert v(q)\Vert }^2={\tanh }^2\theta (q)$, (see [5,12]).

Specially, when $M^n$ is a spacelike slice, namely, if $\theta \equiv 0$, we have $\Vert v\Vert =0$. In fact, every spacelike slice corresponds to the physical space at a single moment of cosmic time for the family of observers related to ${\partial }_t.$ If $M^n$ is a compact spacelike hypersurface in GRW spacetimes, the speed function $\Vert v\Vert$ on $M^n$ achieves a global maximum, so that $\Vert v\Vert$ does not tend to the light speed $1$ on a compact spacelike hypersurface $M^n$. Along this line, the natural generalization of the compactness of $M^n$ consists of assuming $\sup \Vert v\Vert <1$ holds everywhere on $M^n$. Nevertheless, this natural postulate is much too weak. One approach to controlling the relative speeds $\Vert v\Vert$ on $M^n$ would be to suppose that $\theta$ is bounded, which is equivalent to the relative speeds satisfying $\Vert v\Vert <1$. In this paper, this is the hypothesis we impose on the CMC spacelike hypersurfaces in GRW spacetimes (see [5,12]).

3. Results

3.1. Rigidity Results

For a spacelike hypersurface $M^n$, we denote its projection on $I$ as the height function $\tau ={\pi }_I\circ \varphi :M^n\to I$ of $M^n$ with respect to ${\partial }_t$. Let $\overline{\nabla }$ be the Levi-Civita connection of the metric $\langle , \rangle$ on GRW spacetimes ${\overline{M}}^{n+1}$. Then, we have

$$\overline{\nabla }{\pi }_I=-\langle \overline{\nabla }{\pi }_I,{\partial }_t\rangle {\partial }_t=-{\partial }_t.$$

Let ${\partial }_t^{{\rm T}}:={\partial }_t+\langle N,{\partial }_t\rangle N$ be the tangential component of the vector field ${\partial }_t$ on $M^n$. Then, the gradient of the height function $\tau$ on $M^n$ is

$$\nabla \tau ={(\overline{\nabla }{\pi }_I)}^{{\rm T}}={\left(-{\partial }_t\right)}^{{\rm T}}=-{\partial }_t-\langle N,{\partial }_t\rangle N,$$

(1)

where $\nabla$ is the gradient on $M^n$ and, ${( \cdot )}^{\mathrm{T}}$ is the tangential component of a vector field on $M^n$. Moreover,

$$|\nabla \tau {|}^2=-1+{\langle N,{\partial }_t\rangle }^2,$$

(2)

where $|\cdot |$ denotes the norm of a vector field on $M^n$. Therefore, from (2) and the definition of spacelike slice, we also have that a spacelike hypersurface $\varphi :M^n\to {\overline{M}}^{n+1}$ is a spacelike slice if the height function $\tau$ is constant on $M^n$.

Let $T(M^n)$ be the tangent bundle over $M^n$, and let $A:T(M^n)\to T(M^n)$ be the shape operator associated to $N$. Then the mean curvature $H$ of $M^n$ with respect to $N$ is given by $H=-\mathrm{trace}(A)/n$. In particular, the shape operator $A$ of a spacelike slice $M_{t_0}^n$ is defined by $A={\phi }^{\prime }(t_0)/\phi (t_0)E$, where $E$ stands for identity operator, thus the associated mean curvature $H=-{\phi }^{\prime }(t_0)/\phi (t_0)$. Consequently, a spacelike slice $M_{t_0}^n$ is maximal if and only if ${\phi }^{\prime }(t_0)=0$, that is, the maximal spacelike slice is totally geodesic.

To establish our main theorem, we next compute the Laplacian of $\langle N,{\partial }_t\rangle$. So, if we choose a local orthonormal frame $\{e_1,e_2,\cdots ,e_n\}$ on $T(M^n)$, we have

$$\Delta \langle N,{\partial }_t\rangle =\mathrm{div}\left(\nabla \langle N,{\partial }_t\rangle \right)=\mathrm{trace}\left({\nabla }^2\langle N,{\partial }_t\rangle \right)={\displaystyle \sum _{i=1}^n\langle {\nabla }_{e_i}\nabla \langle N,{\partial }_t\rangle ,e_i\rangle },$$

(3)

where $\Delta \langle N,{\partial }_t\rangle :=\mathrm{div}\left(\nabla \langle N,{\partial }_t\rangle \right)$ is the Laplacian of $\langle N,{\partial }_t\rangle$, div is the divergence on $M^n$. In the following, we calculate the gradient of $\langle N,{\partial }_t\rangle .$ First, for any $X\in T(M^n)$, there exists $U\in T({\overline{M}}^{n+1})$, and tangent to $P^n$, such that

$$X=U-\langle X,{\partial }_t\rangle {\partial }_t.$$

Using the basic Levi-Civita fact, we have

$$\begin{array}{ll}{\overline{\nabla }}_X{\partial }_t& ={\overline{\nabla }}_{U-\langle X,{\partial }_t\rangle {\partial }_t}{\partial }_t={\overline{\nabla }}_U{\partial }_t-{\overline{\nabla }}_{\langle X,{\partial }_t\rangle {\partial }_t}{\partial }_t\\ & ={\overline{\nabla }}_U{\partial }_t-\langle X,{\partial }_t\rangle {\overline{\nabla }}_{{\partial }_t}{\partial }_t\\ & ={\overline{\nabla }}_U{\partial }_t,\end{array}$$

and since ${\overline{\nabla }}_{{\partial }_t}{\partial }_t=0$, thus the last equality holds.

Therefore, by the relation between the Levi-Civita connection $\overline{\nabla }$ of ${\overline{M}}^{n+1}=I{\times }_{\phi }{P}^n$ and those of the fiber $P^n$ and the base $I$ (further details can be found in Proposition 7.35 of [11]), we get

$${\overline{\nabla }}_X{\partial }_t={\overline{\nabla }}_U{\partial }_t={\displaystyle \frac{{\phi }^{\prime }(\tau )}{\phi (\tau )}}U={\displaystyle \frac{{\phi }^{\prime }(\tau )}{\phi (\tau )}}\langle X,{\partial }_t\rangle {\partial }_t+{\displaystyle \frac{{\phi }^{\prime }(\tau )}{\phi (\tau )}}X.$$

(4)

Using the Gauss and Weingarten formulas, for all $X, Y\in T(M^n)$, we obtain

$${\overline{\nabla }}_{X}Y={\nabla }_{X}Y-\langle AX,Y\rangle N,$$

(5)

where $\nabla$ denotes the Levi-Civita connection on $M^n$, and the shape operator $A$ corresponding to $N$ is given by

$$AX=-{\overline{\nabla }}_{X}N.$$

(6)

By a direct computation, from (4) and (6), we have that

$$\begin{array}{ll}\langle X,\nabla \langle N,{\partial }_t\rangle \rangle & =X\left(\nabla \langle N,{\partial }_t\rangle \right)=\langle {\overline{\nabla }}_X{\partial }_t,N\rangle +\langle {\partial }_t,{\overline{\nabla }}_{X}N\rangle \\ & =\langle {\displaystyle \frac{{\phi }^{\prime }(\tau )}{\phi (\tau )}}X+{\displaystyle \frac{{\phi }^{\prime }(\tau )}{\phi (\tau )}}\langle X,{\partial }_t\rangle {\partial }_t,N\rangle -\langle A{\partial }_t^{\mathrm{T}},X\rangle \\ & =\langle {\displaystyle \frac{{\phi }^{\prime }(\tau )}{\phi (\tau )}}\langle N,{\partial }_t\rangle {\partial }_t^{\mathrm{T}},X\rangle +\langle X,A\nabla \tau \rangle ,\end{array}$$

(7)

for any $X\in T(M^n)$. Therefore, (7) shows that

$$\nabla \langle N,{\partial }_t\rangle =-{\displaystyle \frac{{\phi }^{\prime }(\tau )}{\phi (\tau )}}\langle N,{\partial }_t\rangle \nabla \tau +A\nabla \tau .$$

(8)

Therefore, from (8), we get

$${\left|\nabla \langle N,{\partial }_t\rangle \right|}^2=-{\displaystyle \frac{{\phi }^{\prime }{(\tau )}^2}{\phi {(\tau )}^2}}{\langle N,{\partial }_t\rangle }^2{\left|\nabla \tau \right|}^2+{\left|A\nabla \tau \right|}^2-2{\displaystyle \frac{{\phi }^{\prime }(\tau )}{\phi (\tau )}}\langle N,{\partial }_t\rangle \langle A\nabla \tau ,\nabla \tau \rangle .$$

(9)

Substituting (8) into (3), one gets

$$\begin{array}{ll}\Delta \langle N,{\partial }_t\rangle & ={\displaystyle \sum _{i=1}^n\langle {\nabla }_{e_i}\left(-\frac{{\phi }^{\prime }(\tau )}{\phi (\tau )}\langle N,{\partial }_t\rangle \nabla \tau +A\nabla \tau \right),e_i\rangle }\\ & =-{\displaystyle \sum _{i=1}^n\langle {\nabla }_{e_i}\left(\frac{{\phi }^{\prime }(\tau )}{\phi (\tau )}\langle N,{\partial }_t\rangle \nabla \tau \right),e_i\rangle }+{\displaystyle \sum _{i=1}^n\langle {\nabla }_{e_i}\left(A\nabla \tau \right),e_i\rangle .}\end{array}$$

(10)

In order to compute the first summand of (10), note that

$$\begin{array}{l}{\displaystyle \sum _{i=1}^n\langle {\nabla }_{e_i}\left(\frac{{\phi }^{\prime }(\tau )}{\phi (\tau )}\langle N,{\partial }_t\rangle \nabla \tau \right),e_i\rangle }\\ ={\left({\displaystyle \frac{{\phi }^{\prime }(\tau )}{\phi (\tau )}}\right)}^{\prime }\langle N,{\partial }_t\rangle {\left|\nabla \tau \right|}^2-{\displaystyle \frac{{\phi }^{\prime }{(\tau )}^2}{\phi {(\tau )}^2}}\langle N,{\partial }_t\rangle {\left|\nabla \tau \right|}^2+{\displaystyle \frac{\phi \prime (\tau )}{\phi (\tau )}}\langle A\nabla \tau ,\nabla \tau \rangle \\ +{\displaystyle \frac{\phi \prime (\tau )}{\phi (\tau )}}\langle N,{\partial }_t\rangle {\displaystyle \sum _{i=1}^n\langle {\nabla }_{e_i}\nabla \tau ,e_i\rangle },\end{array}$$

(11)

and, it follows from (1) and (4)–(6) that

$$\begin{array}{ll}{\nabla }_X\nabla \tau & ={\overline{\nabla }}_X\nabla \tau +\langle AX,\nabla \tau \rangle N\\ & =-{\overline{\nabla }}_X{\partial }_t-\langle N,{\overline{\nabla }}_X{\partial }_t\rangle N+\langle N,{\partial }_t\rangle AX\\ & =-{\displaystyle \frac{{\phi }^{\prime }(\tau )}{\phi (\tau )}}X-{\displaystyle \frac{{\phi }^{\prime }(\tau )}{\phi (\tau )}}\langle X,\nabla \tau \rangle \nabla \tau +\langle N,{\partial }_t\rangle AX. \end{array}$$

(12)

Now, from (12), we obtain

$$\begin{array}{ll}{\displaystyle \sum _{i=1}^n\langle {\nabla }_{e_i}\nabla \tau ,e_i\rangle }& ={\displaystyle \sum _{i=1}^n\langle -\frac{{\phi }^{\prime }(\tau )}{\phi (\tau )}{e}_i-\frac{{\phi }^{\prime }(\tau )}{\phi (\tau )}\langle e_i,\nabla \tau \rangle \nabla \tau +\langle N,{\partial }_t\rangle A{e}_i,e_i\rangle }\\ & =-n{\displaystyle \frac{{\phi }^{\prime }(\tau )}{\phi (\tau )}}-{\displaystyle \frac{{\phi }^{\prime }(\tau )}{\phi (\tau )}}{\left|\nabla \tau \right|}^2+\langle N,{\partial }_t\rangle \mathrm{trace}(A).\end{array}$$

(13)

Observe that the normal bundle of the spacelike hypersurface $M^n$ is negative definite. Therefore, the Codazzi equation is given by

$$\langle \overline{R}(X,Y)Z,N\rangle =\langle ({\nabla }_{Y}A)X,Z\rangle -\langle ({\nabla }_{X}A)Y,Z\rangle ,$$

(14)

where $\overline{R}$ is the curvature tensor of ${\overline{M}}^{n+1}$ and ${\nabla }_{X}A:T(M^n)\to T(M^n)$ is the covariant derivative of $A$, such that

$$({\nabla }_{X}A)Y={\nabla }_X(AY)-A({\nabla }_{X}Y).$$

(15)

It follows from (14) and (15) that

$$\begin{array}{ll}{\displaystyle \sum _{i=1}^n\langle {\nabla }_{e_i}(A\nabla \tau ),e_i\rangle }& ={\displaystyle \sum _{i=1}^n\langle ({\nabla }_{e_i}A)\nabla \tau ,e_i\rangle }+{\displaystyle \sum _{i=1}^n\langle A({\nabla }_{e_i}\nabla \tau ),e_i\rangle }\\ & ={\displaystyle \sum _{i=1}^n\langle ({\nabla }_{\nabla \tau }A)e_i,e_i\rangle }-{\displaystyle \sum _{i=1}^n\langle \overline{R}(e_i,\nabla \tau )e_i,N\rangle }+{\displaystyle \sum _{i=1}^n\langle A({\nabla }_{e_i}\nabla \tau ),e_i\rangle }.\end{array}$$

(16)

Using (12) again,

$$\begin{array}{ll}{\displaystyle \sum _{i=1}^n\langle A({\nabla }_{e_i}\nabla \tau ),e_i\rangle }& ={\displaystyle \sum _{i=1}^n\langle A\left(-\frac{{\phi }^{\prime }(\tau )}{\phi (\tau )}{e}_i-\frac{{\phi }^{\prime }(\tau )}{\phi (\tau )}\langle e_i,\nabla \tau \rangle \nabla \tau +\langle N,{\partial }_t\rangle A{e}_i\right),e_i\rangle }\\ & =-{\displaystyle \frac{{\phi }^{\prime }(\tau )}{\phi (\tau )}}\mathrm{trace}(A)-{\displaystyle \frac{{\phi }^{\prime }(\tau )}{\phi (\tau )}}\langle A\nabla \tau ,\nabla \tau \rangle +\langle N,{\partial }_t\rangle \mathrm{trace}(A^2)\end{array}$$

(17)

and

$${\displaystyle \sum _{i=1}^n\langle ({\nabla }_{\nabla \tau }A)e_i,e_i\rangle }=\mathrm{trace}({\nabla }_{\nabla \tau }A)=\nabla \tau \mathrm{trace}(A)=\nabla \tau (-nH)=-n\langle \nabla H,\nabla \tau \rangle .$$

(18)

Applying (1), we get

$$\begin{array}{ll}{\displaystyle \sum _{i=1}^n\langle \overline{R}(e_i,\nabla \tau )e_i,N\rangle }& ={\displaystyle \sum _{i=1}^n\langle \overline{R}\left(-{\partial }_t-\langle N,{\partial }_t\rangle N,e_i\right)N,e_i\rangle }\\ & =-{\displaystyle \sum _{i=1}^n\langle \overline{R}\left({\partial }_t,e_i\right)N,e_i\rangle -\langle N,{\partial }_t\rangle {\displaystyle \sum _{i=1}^n\langle \overline{R}\left(N,e_i\right)N,e_i\rangle }}.\end{array}$$

(19)

In addition, we recall that the Gauss equation

$$\langle R(X,Y)Z,W\rangle =\langle \overline{R}(X,Y)Z,W\rangle +\langle AY,W\rangle \langle AX,Z\rangle -\langle AY,Z\rangle \langle AX,W\rangle ,$$

and equivalently,

$$R(X,Y)Z={\left(\overline{R}(X,Y)Z\right)}^{\mathrm{T}}-\langle AX,Z\rangle AY-\langle AY,Z\rangle AX,$$

where $R$ is the curvature tensor of $M^n$.

Using the Gauss equation and ${\pi }_P^{*}({\partial }_t)=0$, we obtain

$$\overline{R}\left({\partial }_t,e_i\right)N=\left({\displaystyle \frac{{\phi }^{\prime }{(\tau )}^2}{\phi {(\tau )}^2}}+{\left({\displaystyle \frac{{\phi }^{\prime }(\tau )}{\phi (\tau )}}\right)}^{\prime }\right)\langle N,{\partial }_t\rangle e_i={\displaystyle \frac{{\phi }^{″}(\tau )}{\phi (\tau )}}\langle N,{\partial }_t\rangle e_i.$$

Thus,

$${\displaystyle \sum _{i=1}^n\langle \overline{R}\left({\partial }_t,e_i\right)N,e_i\rangle }=n{\displaystyle \frac{{\phi }^{″}(\tau )}{\phi (\tau )}}\langle N,{\partial }_t\rangle .$$

(20)

If $N^{*}=N+\langle N,{\partial }_t\rangle {\partial }_t$ is the projection of $N$ on $M^n$, then

$$\langle N^{*},N^{*}\rangle ={\langle N,{\partial }_t\rangle }^2-1.$$

From Corollary 7.43 in [11], we get

$$\begin{array}{ll}\overline{\mathrm{Ric}}(N,N)& =\overline{\mathrm{Ric}}(N^{*},N^{*})+{\langle N,{\partial }_t\rangle }^2\overline{\mathrm{Ric}}({\partial }_t,{\partial }_t)\\ & ={\mathrm{Ric}}_P(N^{*},N^{*})+\langle N^{*},N^{*}\rangle \left\{{\displaystyle \frac{{\phi }^{″}(\tau )}{\phi (\tau )}}+(n-1){\displaystyle \frac{{\phi }^{\prime }{(\tau )}^2}{\phi {(\tau )}^2}}\right\}-n{\displaystyle \frac{{\phi }^{″}(\tau )}{\phi (\tau )}}{\langle N,{\partial }_t\rangle }^2\\ & ={\mathrm{Ric}}_P(N^{*},N^{*})-\left\{{\displaystyle \frac{{\phi }^{″}(\tau )}{\phi (\tau )}}+(n-1){\displaystyle \frac{{\phi }^{\prime }{(\tau )}^2}{\phi {(\tau )}^2}}\right\}-n{\left({\displaystyle \frac{{\phi }^{\prime }(\tau )}{\phi (\tau )}}\right)}^{\prime }{\langle N,{\partial }_t\rangle }^2,\end{array}$$

(21)

where $\overline{\mathrm{Ric}}$ and ${\mathrm{Ric}}_P$ denote the Ricci tensor of ${\overline{M}}^{n+1}$ and $P^n$, respectively. Therefore, using (11), (13), (16)–(21), and with $H=-\mathrm{trace}(A)/n$, we can rewrite (10) as

$$\begin{array}{ll}\Delta \langle N,{\partial }_t\rangle & =-n\langle \nabla H,\nabla \tau \rangle -{\displaystyle \frac{{\phi }^{″}(\tau )}{\phi (\tau )}}\langle N,{\partial }_t\rangle {\left|\nabla \tau \right|}^2+{\displaystyle \frac{{\phi }^{\prime }{(\tau )}^2}{\phi {(\tau )}^2}}\langle N,{\partial }_t\rangle \left(n+3{\left|\nabla \tau \right|}^2\right)\\ & -2{\displaystyle \frac{{\phi }^{\prime }(\tau )}{\phi (\tau )}}\langle A\nabla \tau ,\nabla \tau \rangle +nH{\displaystyle \frac{{\phi }^{\prime }(\tau )}{\phi (\tau )}}\left({\langle N,{\partial }_t\rangle }^2+1\right)+\langle N,{\partial }_t\rangle \mathrm{trace}(A^2)\\ & +\langle N,{\partial }_t\rangle \left({\mathrm{Ric}}_P(N^{*},N^{*})-(n-1){\left(\log \phi \right)}^{″}{\left|\nabla \tau \right|}^2\right).\end{array}$$

(22)

In the following, we will prove a new rigidity theorem for complete parabolic CMC spacelike hypersurfaces in a GRW spacetime. From [13], we know that a noncompact Riemannian manifold is called parabolic if the only positive superharmonic functions it admits are constant ones. To establish our main result, we first state the following proven fact (see Corollary 6.4 in [14]).

Lemma 1.

If $M$ is parabolic, then  $M$  is stochastically complete.

Let $B_r$ and $B_R (0<r<R)$ be geodesic balls of radius $r$ and $R$ at point $p\in M$, respectively, where $r$ and $R$ are positive constant. We denote

$${\displaystyle \frac{1}{{\mu }_{r,R}}}={{\displaystyle \underset{B_R\backslash {\overline{B}}_r}{\int }\left|\nabla {\omega }_{r,R}\right|}}^2\mathrm{d}V$$

as the capacity of the annulus $B_R\backslash {\overline{B}}_r$, where ${\omega }_{r,R}$ denotes the harmonic measure on $\partial B_R$ concerning the annulus $B_R\backslash {\overline{B}}_r$ (see [15]).

Moreover, a complete Riemannian manifold is parabolic if and only if ${\displaystyle \frac{1}{{\mu }_{r,R}}}\to 0$ as $R\to \infty$ (see [15]). Here, we provide the following result (see Lemma 3 in [7]).

Lemma 2.

Let $M^n$ be an  $n(\ge 2)$-dimensional Riemannian manifold and $\nu \in C^2(M)$ satisfying $\nu \Delta \nu \ge 0$. If $B_R$ is a geodesic ball of radius $R$ at point $p\in M^n$, for any $r$ such that $0<r<R$, then

$${{\displaystyle \underset{B_r}{\int }\left|\nabla \nu \right|}}^2\mathrm{d}V\le {\displaystyle \frac{4{\sup }_{B_R}{\nu }^2}{{\mu }_{r,R}}},$$

where $B_r$ denotes a geodesic ball of radius  $r$ at $p\in M^n$ and ${\displaystyle \frac{1}{{\mu }_{r,R}}}$ is the capacity of the annulus  $B_R\backslash {\overline{B}}_r$.

Summing up, we are thus able to obtain the next technical result which is the extension of Proposition 7 in [16].

Lemma 3.

Let $M^n$ be an $n(\ge 2)$-dimensional parabolic Riemannian manifold and let $\rho \in C^2(M^n)$ be a positive function on $M^n$ such that ${\sup }_{M^n}\rho <+\infty$. If $\Delta \rho$ does not change the sign on $M^n$, then $\rho$ is constant on $M^n$.

Proof of Lemma 3.

From our assumptions, we have $\Delta \rho \ge 0$ or $\Delta \rho \le 0$. If $\Delta \rho \le 0$, then $u$ is a superharmonic function on $M^n$. Therefore, $\rho$ must be constant on $M^n$.

If $\Delta \rho \ge 0$, then $\rho \Delta \rho \ge 0$. Regarding $\rho$ is a positive function bounded from above, there exists a positive constant $\lambda$ such that ${\rho }^2\le \lambda$ on $M^n$. Using Lemma 2, when $B_R$ is a geodesic ball of radius $R$ at $p\in M^n$, for each $r (0<r<R)$, we have

$${{\displaystyle \underset{B_r}{\int }\left|\nabla \rho \right|}}^2\mathrm{d}V\le {\displaystyle \frac{4\lambda }{{\mu }_{r,R}}}.$$

In view of the fact that $M^n$ is parabolic, we obtain ${\displaystyle \frac{1}{{\mu }_{r,R}}}\to 0$ as $R\to \infty$, namely, ${\left|\nabla \rho \right|}^2=0$. Therefore, $\rho$ is constant on $M^n$. □

Theorem 1.

Let ${\overline{M}}^{n+1}=I{\times }_{\phi }{P}^n (n\ge 2)$ be GRW spacetimes, whose fiber $P^n$ has non-negative Ricci curvature. Let $\varphi :M^n\to {\overline{M}}^{n+1}$ be complete parabolic CMC spacelike hypersurfaces with bounded hyperbolic angle $\theta$ and non-zero mean curvature $H$ such that $H^2\le {\displaystyle \frac{{\phi }^{\prime }{(\tau )}^2}{\phi {(\tau )}^2}}$. If ${\phi }^{″}(\tau )\le 0$, then $M^n$ is a spacelike slice.

Proof of Theorem 1.

We consider the function

$$f=\log \langle N,{\partial }_t\rangle +\alpha ,$$

where $\alpha \in {ℝ}^{+}$ is to make sure that $f>0$.

The key idea of the proof is to have a bound on the Laplacian of the function $f$. To do so, we have

$$\Delta f={\displaystyle \frac{\Delta \langle N,{\partial }_t\rangle }{\langle N,{\partial }_t\rangle }}-{\displaystyle \frac{{\left|\nabla \langle N,{\partial }_t\rangle \right|}^2}{{\langle N,{\partial }_t\rangle }^2}}={\displaystyle \frac{\langle N,{\partial }_t\rangle \Delta \langle N,{\partial }_t\rangle -{\left|\nabla \langle N,{\partial }_t\rangle \right|}^2}{{\langle N,{\partial }_t\rangle }^2}}.$$

Since $M^n$ has constant mean curvature, using (9) and (22), we can get

$$\begin{array}{l}\langle N,{\partial }_t\rangle \Delta \langle N,{\partial }_t\rangle -{\left|\nabla \langle N,{\partial }_t\rangle \right|}^2\\ =nH\langle N,{\partial }_t\rangle {\displaystyle \frac{{\phi }^{\prime }(\tau )}{\phi (\tau )}}\left(1+{\langle N,{\partial }_t\rangle }^2\right)+{\displaystyle \frac{{\phi }^{\prime }{(\tau )}^2}{\phi {(\tau )}^2}}{\langle N,{\partial }_t\rangle }^2\left((n+1){\left|\nabla \tau \right|}^2+n\right)\\ -n{\displaystyle \frac{{\phi }^{″}(\tau )}{\phi (\tau )}}{\langle N,{\partial }_t\rangle }^2{\left|\nabla \tau \right|}^2-{\left|A(\nabla \tau )\right|}^2+{\langle N,{\partial }_t\rangle }^2\mathrm{trace}(A^2)+{\langle N,{\partial }_t\rangle }^2{\mathrm{Ric}}_P(N^{*},N^{*}){\left|\nabla \tau \right|}^2.\end{array}$$

(23)

Moreover, it follows from (2) that

$$\begin{array}{l}nH\langle N,{\partial }_t\rangle {\displaystyle \frac{{\phi }^{\prime }(\tau )}{\phi (\tau )}}\left(1+{\langle N,{\partial }_t\rangle }^2\right)\\ =n\left(1+{\displaystyle \frac{1}{2}}{\left|\nabla \tau \right|}^2\right)\left\{{\left(H+{\displaystyle \frac{{\phi }^{\prime }(\tau )}{\phi (\tau )}}\langle N,{\partial }_t\rangle \right)}^2-H^2-{\displaystyle \frac{{\phi }^{\prime }{(\tau )}^2}{\phi {(\tau )}^2}}{\langle N,{\partial }_t\rangle }^2\right\}.\end{array}$$

(24)

Therefore, from (24), we rewrite (23) as

$$\begin{array}{l}\langle N,{\partial }_t\rangle \Delta \langle N,{\partial }_t\rangle -{\left|\nabla \langle N,{\partial }_t\rangle \right|}^2\\ =n\left(1+{\displaystyle \frac{1}{2}}{\left|\nabla \tau \right|}^2\right){\left(H+{\displaystyle \frac{{\phi }^{\prime }(\tau )}{\phi (\tau )}}\langle N,{\partial }_t\rangle \right)}^2-n{\displaystyle \frac{{\phi }^{″}(\tau )}{\phi (\tau )}}{\langle N,{\partial }_t\rangle }^2{\left|\nabla \tau \right|}^2\\ +{\displaystyle \frac{n}{2}}\left({\displaystyle \frac{{\phi }^{\prime }{(\tau )}^2}{\phi {(\tau )}^2}}-H^2\right){\left|\nabla \tau \right|}^2+\left({\displaystyle \frac{n}{2}}+1\right){\displaystyle \frac{{\phi }^{\prime }{(\tau )}^2}{\phi {(\tau )}^2}}{\left|\nabla \tau \right|}^4+{\displaystyle \frac{{\phi }^{\prime }{(\tau )}^2}{\phi {(\tau )}^2}}{\left|\nabla \tau \right|}^2\\ +{\langle N,{\partial }_t\rangle }^2\mathrm{trace}(A^2)-{\left|A(\nabla \tau )\right|}^2-n{H}^2+{\langle N,{\partial }_t\rangle }^2{\mathrm{Ric}}_P(N^{*},N^{*}){\left|\nabla \tau \right|}^2.\end{array}$$

(25)

By Schwarz’s inequality, we obtain

$${\left|A(\nabla \tau )\right|}^2\le \mathrm{trace}(A^2){\left|\nabla \tau \right|}^2.$$

Considering (2), we have

$$\mathrm{trace}(A^2)\le \mathrm{trace}(A^2){\langle N,{\partial }_t\rangle }^2-{\left|A(\nabla \tau )\right|}^2.$$

(26)

Using Schwarz’s inequality again, we get

$$n{H}^2\le \mathrm{trace}(A^2),$$

and the equality holds precisely when $M^n$ is totally umbilical in ${\overline{M}}^{n+1}$ (see [5] (p. 7) for further details). Therefore,

$${\langle N,{\partial }_t\rangle }^2\mathrm{trace}(A^2)-{\left|A(\nabla \tau )\right|}^2-n{H}^2\ge 0.$$

According to assumptions of Theorem 1 and (25), we know

$$\langle N,{\partial }_t\rangle \Delta \langle N,{\partial }_t\rangle -{\left|\nabla \langle N,{\partial }_t\rangle \right|}^2\ge 0.$$

So, $\Delta f\ge 0$ and by Lemma 3, we obtain $f$ is constant. Thus, $\Delta f=0$. From (25) we get ${\left|\nabla \tau \right|}^2=0$, that is $\langle N,{\partial }_t\rangle =1$. Therefore, $M^n$ is a spacelike slice. □

Remark 1.

Notice that by weakening the hypotheses of Theorem 1, the rigidity result on GRW spacetimes does not hold. In fact the complete parabolic non-zero CMC spacelike hyper surfaces in the de Sitter spacetime, namely, the GRW spacetimes

$${\mathbb{S}}_1^{n+1}=-ℝ{\times }_{\cosh t}{\mathbb{S}}^n$$

satisfy all the hypotheses but ${\phi }^{″}(\tau )\le 0$, then we cannot get homologous rigidity result (see [17]).

Moreover, a GRW spacetime ${\overline{M}}^{n+1}=I{\times }_{\phi }{P}^n$ is an Einstein GRW spacetime if its Ricci tensor $\overline{\mathrm{Ric}}$ is proportional to the metric

$$\langle , \rangle =-{\pi }_I^{\ast }(\mathrm{d}{t}^2)+\phi {({\pi }_I)}^2{\pi }_P^{\ast }(g_p),$$

namely,

$$\overline{\mathrm{Ric}}=\overline{c}\langle ,\rangle , \overline{c}\in ℝ .$$

This is equivalent to the Ricci curvature ${\mathrm{Ric}}_P$ of the fiber $P^n$ being a constant $c$ and the warping function satisfying the following differential equations

$${\displaystyle \frac{{\phi }^{″}}{\phi }}={\displaystyle \frac{\overline{c}}{n}} \mathit{and}{\displaystyle \frac{\overline{c}(n-1)}{n}}={\displaystyle \frac{c+(n-1){({\phi }^{\prime })}^2}{{\phi }^2}}.$$

We consider the solution for $c>0$, and $\overline{c}>0$:

$$\phi (t)=a{e}^{bt}+{\displaystyle \frac{cn}{4a\overline{c}(n-1)}}{e}^{-bt}, a>0, b=\sqrt{\overline{c}/n},$$

by direct computation, we obtain

$${\phi }^{″}(t)=b^2\phi >0,$$

then a complete parabolic CMC spacelike hypersurface $\varphi :M^n\to {\overline{M}}^{n+1}$ in Einstein GRW spacetime ${\overline{M}}^{n+1}=I{\times }_{\phi (t)}{P}^n$ with ${\mathrm{Ric}}_P=c, (c>0)$ and whose hyperbolic angle is bounded, but not a spacelike slice (see [17]).

In [6], the authors obtained the result for the parabolicity of a spacelike hypersurface $\varphi :M^n\to I{\times }_{\phi }{P}^n$ induced from the universal Riemannian covering of the fiber $P^n$ of GRW spacetime ${\overline{M}}^{n+1}=I{\times }_{\phi }{P}^n$ (see [6], Theorem 4.4).

Lemma 4.

Let ${\overline{M}}^{n+1}=I{\times }_{\phi }{P}^n$ be a GRW spacetime whose fiber $P^n$ has parabolic universal Riemannian covering. Let  $\varphi :M^n\to {\overline{M}}^{n+1}$be a complete spacelike hypersurface whose hyperbolic angle is bounded. If the warping function satisfies:

(i)

$\inf \phi (\tau )>0$, and

(ii)

$\sup \phi (\tau )<\infty .$

then  $M^n$  is parabolic.

Using Theorem 1 and Lemma 4, we have the next results.

Corollary 1.

Let ${\overline{M}}^{n+1}=I{\times }_{\phi }{P}^n$ be a GRW spacetime whose fiber $P^n$ has parabolic Riemannian universal covering and non-negative Ricci curvature. Let $\varphi :M^n\to {\overline{M}}^{n+1} (n\ge 2)$ be a complete non-zero CMC spacelike hypersurface whose hyperbolic angle is bounded. Assume that the warping function satisfies  ${\phi }^{″}(\tau )\le 0$, $\sup \phi <\infty$, and  $\inf \phi >0$. If  $H^2\le {\displaystyle \frac{{\phi }^{\prime }{(\tau )}^2}{\phi {(\tau )}^2}}$, then  $M^n$  is a spacelike slice.

Recalling that a spacelike hypersurface $\varphi :M^n\to I{\times }_{\phi }{P}^n$ is lying between two spacelike slices if there are $t_1,t_2\in I$ and $t_1<t_2$ satisfy  $\varphi (M^n)\subset [t_1,t_2]\times P^n$. Similarly, a spacelike hypersurface $\varphi :M^n\to I{\times }_{\phi }{P}^n$ >lies in an open slab if there is $t_0\in I$ such that $\tau >t_0$ (resp. $\tau <t_0$) on $M^n$.

Corollary 2.

Let ${\overline{M}}^{n+1}=I{\times }_{\phi }{P}^n$ be a GRW spacetime whose fiber $P^n$ has parabolic Riemannian universal covering and non-negative Ricci curvature. Let  $\varphi :M^n\to {\overline{M}}^{n+1} (n\ge 2)$  be a complete non-zero CMC spacelike hypersurface with bounded hyperbolic angle. If  ${\phi }^{″}(\tau )\le 0$,  $H^2\le {\displaystyle \frac{{\phi }^{\prime }{(\tau )}^2}{\phi {(\tau )}^2}}$, and  $M^n$  lies between two slices, then $M^n$ is a spacelike slice.

If $H=0$, then we obtain the next result.

Corollary 3.

Let ${\overline{M}}^{n+1}=I{\times }_{\phi }{P}^n$ be a GRW spacetime whose fiber $P^n$ has parabolic Riemannian universal covering and non-negative Ricci curvature. Let  $\varphi :M^n\to {\overline{M}}^{n+1} (n\ge 2)$  be a complete maximal spacelike hypersurface with a bounded hyperbolic angle. If  ${\phi }^{″}(\tau )\le 0$, then  $M^n$  is totally geodesic. Moreover, if  $M^n$  lies between two slices, then  $M^n$  is a spacelike slice.

Proof of Corollary 3.

Combining (25) and the condition $H=0$, we have

$$\begin{array}{l}\langle N,{\partial }_t\rangle \Delta \langle N,{\partial }_t\rangle -{\left|\nabla \langle N,{\partial }_t\rangle \right|}^2\\ =n\left(1+{\displaystyle \frac{1}{2}}{\left|\nabla \tau \right|}^2\right){\displaystyle \frac{{\phi }^{\prime }{(\tau )}^2}{\phi {(\tau )}^2}}{\langle N,{\partial }_t\rangle }^2-n{\displaystyle \frac{{\phi }^{″}(\tau )}{\phi (\tau )}}{\langle N,{\partial }_t\rangle }^2{\left|\nabla \tau \right|}^2\\ +\left({\displaystyle \frac{n}{2}}+1\right){\displaystyle \frac{{\phi }^{\prime }{(\tau )}^2}{\phi {(\tau )}^2}}\left({\left|\nabla \tau \right|}^4+{\left|\nabla \tau \right|}^2\right)+{\langle N,{\partial }_t\rangle }^2\mathrm{trace}(A^2)-{\left|A(\nabla \tau )\right|}^2\\ +{\langle N,{\partial }_t\rangle }^2{\mathrm{Ric}}_P(N^{*},N^{*}){\left|\nabla \tau \right|}^2.\end{array}$$

(27)

Using (26) and (27), we have $\Delta f\ge 0$, and thus $\Delta f=0$. From (27) we get ${\phi }^{\prime }=0$ and $\mathrm{trace}(A^2)=0$.

Therefore, $M^n$ is totally geodesic. Moreover, considering ${\phi }^{\prime }=0$ and (13), we obtain $\Delta \tau ={\displaystyle \sum _{i=1}^n\langle {\nabla }_{e_i}\nabla \tau ,e_i\rangle }=0$. Since $M^n$ lies between two slices, then $\tau$ is bounded. We can deduce that $\tau$ is constant, that is, $M^n$ is a spacelike slice. □

Moreover, if the ambient space is a static GRW spacetime, then the boundedness assumption on the spacelike hypersurface can be removed.

Corollary 4.

Let $\varphi :M^n\to {\overline{M}}^{n+1} (n\ge 2)$ be a complete maximal spacelike hypersurface in a static GRW spacetime ${\overline{M}}^{n+1}=I\times P^n$ whose fiber $P^n$ has parabolic Riemannian universal covering and non-negative Ricci curvature. If  $M^n$  has bounded hyperbolic angle, then  $M^n$  is totally geodesic. In addition, if  $M^n$  lies in an open slab, then  $M^n$  is a spacelike slice.

Remark 2.

We point out that there is no relation between the conditions “being contained in a slab” for a spacelike hypersurface in a GRW spacetime and “bounded hyperbolic angle” (see [7], Remark 16, [18], and Remark 5.3 for details).

Another way of obtaining (22) was used in [7] in order to obtain uniqueness results for complete parabolic constant mean curvature spacelike hypersurfaces in Lorentzian warped products ${\overline{M}}^{n+1}=I{\times }_{\phi }{P}^n$with dimension$n+1\le 5$under appropriate geometric assumptions.

Notice that Theorem 1 extends the Theorem 5 in [7] to the dimension$n+1>5$case in GRW spacetimes with non-negative Ricci curvature of the fiber. Furthermore, in the particular instance$M^n$is a complete maximal spacelike hypersurface, we can reobtain Theorem 5.7 in [6] and Corollary 8 in [7].

3.2. Calabi–Bernstein-Type Problems

In this section, we apply the rigidity results in Section 3.1 to solve the related Calabi–Bernstein-type problems. We consider a vertical entire graph on the fiber $(P^n,g_P)$ of a GRW spacetime ${\overline{M}}^{n+1}=I{\times }_{\phi }{P}^n$ given by

$$M^n(u)=\left\{\left(u(x),x\right):x\in P^n\right\}\subset {\overline{M}}^{n+1},$$

where $u:P^n\to I$ is a smooth positive function on $P^n$. Moreover, we equip the graph $M^n(u)$ with the following metric

$$\langle ,\rangle =-\mathrm{d}{u}^2+{\phi }^2(u)g_P.$$

Obviously, the metric is Riemannian if and only if $\left|Du\right|<\phi (u)$, where $Du$ represents the gradient of $u$ on $P^n$. For each point $p\in P^n$, $\tau \left(u(p),p\right)=u(p)$. So, $u$ and $\tau$ are naturally identifiable on $M^n(u)$.

When $M^n(u)$ is spacelike, the unit normal vector field $N$ such that $\langle N,{\partial }_t\rangle >0$ on $M^n(u)$ is

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

Therefore, the hyperbolic angle $\theta$ between $M^n(u)$ and ${\partial }_t$ at every point on $M^n(u)$ is expressed as

$$\cosh \theta =\langle N,{\partial }_t\rangle ={\displaystyle \frac{\phi (u)}{\sqrt{{\phi }^2(u)-|Du{|}^2}}}.$$

The related shape operator $A$ is

$$\begin{array}{ll}AX& ={\displaystyle \frac{1}{\sqrt{{\phi }^2(u)-|Du{|}^2}}}\left({\phi }^{\prime }(u)X+{\displaystyle \frac{1}{\phi (u)}}{D}_{X}Du\right)\\ & +\left({\displaystyle \frac{g_P(D_{X}Du,Du)}{\phi (u){({\phi }^2(u)-|Du{|}^2)}^{3/2}}}-{\displaystyle \frac{{\phi }^{\prime }(u)g_P(Du,X)}{{({\phi }^2(u)-|Du{|}^2)}^{3/2}}}\right)Du,\end{array}$$

for any tangent vector field $X\in T(M^n(u))$.

Hence the correlated mean curvature $H(u)$ is

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

where $\mathrm{div}$ is the divergence operator in $P^n$.

Under the restriction $\left|Du\right|<\phi (u)$, and if $H(u)$ is a constant $H$, then the equation $H(u)=H$ is referred to as the CMC spacelike hypersurface equation:

$$\mathrm{div}\left({\displaystyle \frac{Du}{\phi (u)\sqrt{{\phi }^2(u)-|Du{|}^2}}}\right)=-nH-{\displaystyle \frac{{\phi }^{\prime }(u)}{\sqrt{{\phi }^2(u)-|Du{|}^2}}}\left(n+{\displaystyle \frac{|Du{|}^2}{{\phi }^2(u)}}\right),$$

(28)

with

$$\left|Du\right|<\beta \phi (u),0<\beta <1.$$

(29)

In fact, by direct calculation, we have

$$|\nabla \tau {|}^2={\displaystyle \frac{{\left|Du\right|}^2}{{\phi }^2(u)-{\left|Du\right|}^2}}.$$

(30)

Using (2), (30) and the constraint (29), we have

$$\langle N,{\partial }_t\rangle =\cosh \theta <{\displaystyle \frac{1}{\sqrt{1-{\beta }^2}}},$$

(31)

where $\theta$ is the hyperbolic angle of graph $M^n(u)$. So, $M^n(u)$ has bounded hyperbolic angle.

Next, we derive some results for bounded solutions to equation (28) with (29) under appropriate hypotheses.

Theorem 2.

Let $M^n(u) (n\ge 2)$ be a parabolic non-zero CMC entire graph in a GRW spacetime $I{\times }_{\phi }{P}^n$ whose fiber $P^n$ is complete, with non-negative Ricci curvature. If ${\phi }^{″}(u)\le 0$, $\inf \phi (u)>0$, and $H^2\le {\displaystyle \frac{{\phi }^{\prime }{(u)}^2}{\phi {(u)}^2}}$, then the only entire solutions of equation (28) with (29) are the constant functions $u(x)=u_0$, with $H=-{\displaystyle \frac{{\phi }^{\prime }(u_0)}{\phi (u_0)}}$.

Proof of Theorem 2.

Using the Cauchy–Schwarz inequality, for any $X\in T(M^n(u))$, we have

$$\begin{array}{ll}\langle X,X\rangle & =-g_P{\left(X^{*},Du\right)}^2+\phi {(u)}^2{g}_P\left(X^{*},X^{*}\right)\\ & \ge (\phi {(u)}^2-|Du{|}^2)g_P\left(X^{*},X^{*}\right),\end{array}$$

(32)

where $X^{\ast }=X+\langle X,{\partial }_t\rangle {\partial }_t$ is the projection of $X$ on $P^n$. From (29) jointly with (32), we obtain

$$\langle X,X\rangle \ge {\displaystyle \frac{\phi {(u)}^2}{{\langle N,{\partial }_t\rangle }^2}}{g}_P\left(X^{*},X^{*}\right).$$

(33)

Therefore, (31) and (32) imply

$$L\ge (\inf \phi {(u)}^2)(1-{\beta }^2)L_P,$$

where $L_P$ and $L$ denote the length of a smooth curve on $M^n(u)$ corresponding with the metrics $g_P$ and $\langle , \rangle$, respectively.

Consequently, since $P^n$ is complete, if $\inf \phi (u)>0,$ and $0<\beta <1$, then the metric $\langle , \rangle$ is complete, and $M^n(u)$ is complete. Use Theorem 1 to complete the proof. □

Adopting the argument used within the proofs of Theorem 2 and Corollary 1, we have

Corollary 5.

Let $I{\times }_{\phi }{P}^n (n\ge 2)$ be a GRW spacetime whose fiber $P^n$ is complete, with parabolic Riemannian universal covering and non-negative Ricci curvature. Let $M^n(u)$ be an entire graph in $I{\times }_{\phi }{P}^n$ with $H\in {ℝ}^{*}$, and $H^2\le {\displaystyle \frac{{\phi }^{\prime }{(u)}^2}{\phi {(u)}^2}}$. If ${\phi }^{″}(u)\le 0$, then the only bounded entire solutions to equation (28) with (29) are the constant functions $u(x)=u_0$, with $H=-{\displaystyle \frac{{\phi }^{\prime }(u_0)}{\phi (u_0)}}$.

Moreover, if $H=0$, then from Corollary 3 we know

Corollary 6.

Let $I{\times }_{\phi }{P}^n (n\ge 2)$ be a GRW spacetime whose fiber $P^n$ is complete, with parabolic Riemannian universal covering, and non-negative Ricci curvature. If ${\phi }^{″}(u)\le 0$, then the only bounded entire solutions of equation

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

$$\left|Du\right|<\beta \phi (u), 0<\beta <1,$$

are the constant functions $u(x)=u_0$, with ${\phi }^{\prime }(u_0)\le 0$.

Finally, from Corollary 4, we have the next Corollary.

Corollary 7.

Let $I\times P^n(n\ge 2)$ be a static GRW spacetime whose fiber $P^n$ is complete, with parabolic Riemannian universal covering, and non-negative Ricci curvature. Then the only bounded entire solutions of equation

$$\mathrm{div}\left({\displaystyle \frac{Du}{\sqrt{1-|Du{|}^2}}}\right)=0,$$

$$\left|Du\right|<\beta , 0<\beta <1,$$

are the constant functions $u(x)=u_0$.

Remark 3.

Finally, we return to the physical motivation presented in Section 2. Thus, we obtain that the only complete parabolic CMC spacelike hypersurfaces possessing an upper bound of the speed observed by comoving observers are the spacelike slices in GRW spacetimes whose warping function satisfies a certain convexity criterion and the Ricci curvature of the fiber is non-negative.

4. Conclusions

In this paper, we study the complete parabolic CMC spacelike hypersurfaces in GRW spacetimes ${\overline{M}}^{n+1}=I{\times }_{\phi }{P}^n$, where $P^n$ is a Riemannian manifold and $I\subset ℝ$ is an open interval. We prove the following main rigidity results (Theorem 1 and its non-parametric counterpart Theorem 2).

Let ${\overline{M}}^{n+1}=I{\times }_{\phi }{P}^n (n\ge 2)$ be GRW spacetimes, whose fiber $P^n$ has non-negative Ricci curvature. Let $\varphi :M^n\to {\overline{M}}^{n+1}$ be complete parabolic CMC spacelike hypersurfaces with bounded hyperbolic angle $\theta$ and non-zero mean curvature $H$ such that $H^2\le {\displaystyle \frac{{\phi }^{\prime }{(\tau )}^2}{\phi {(\tau )}^2}}$. If ${\phi }^{″}(\tau )\le 0$, then $M^n$is a spacelike slice.

Let $M^n(u) (n\ge 2)$ be a parabolic non-zero CMC entire graph in a GRW spacetime $I{\times }_{\phi }{P}^n$ whose fiber $P^n$ is complete, with non-negative Ricci curvature. If ${\phi }^{″}(u)\le 0$, $\inf \phi (u)>0$, and $H^2\le {\displaystyle \frac{{\phi }^{\prime }{(u)}^2}{\phi {(u)}^2}}$, then the only entire solutions to equation (28) with (29) are the constant functions $u(x)=u_0$, with $H=-{\displaystyle \frac{{\phi }^{\prime }(u_0)}{\phi (u_0)}}$.

Moreover, we also obtain Corollary 2, which states that the image of such a spacelike hypersurface lying between two spacelike slices is necessarily a spacelike slice $M_{t_0}^n=\left\{t_0\right\}\times P^n$ of $I\times P^n$, for some $t_0\in I$, provided that the Ricci curvature of $P^n$ is non-negative and $-\phi$ is convex on its domains. In particular, we study maximal spacelike hypersurfaces and establish Corollary 3 and Corollary 6. Furthermore, we also consider the special case when the ambient space is static and provide Corollary 4 and Corollary 7.

The findings presented in this work not only extend previous results of GRW spacetimes and semi-Riemannian warped products but also further elucidate the important research value and great significance of parabolic CMC spacelike hypersurfaces in general relativity. Moreover, the research and the relative results enrich the theory of submanifolds geometry.