2.1 Some preliminaries

Let $(M^n,\langle \,,\rangle _M)$ be a connected, n-dimensional, oriented Riemannian manifold, let $I\subset \mathbb {R}$ be an open interval and let $f:I\rightarrow \mathbb {R}$ be a positive smooth function. Also, in the product manifold $\overline {M}^{n+1}=I\times M^{n}$ let $\pi _I$ and $\pi _M$ denote the canonical projections onto the factors I and $M^{n}$ , respectively.

The class of Lorentzian manifolds of concern here is the one obtained by furnishing $\overline {M}^{n+1}$ with the Lorentzian metric $\langle \,,\rangle $ given by

$$ \begin{align*} \langle\,,\rangle=-dt^{2}+f(t)^{2}\langle\,,\rangle_{M}, \end{align*} $$

where $-dt^{2}$ stands for the standard metric of $I\subset \mathbb R$ . In this article, we simply write

(2-1) $$ \begin{align} \overline{M}^{n+1}=-I\times_{f} {M}^{n}. \end{align} $$

According to the nomenclature established in [Reference Alías, Romero and SÁnchez11], we say that $\overline {M}^{n+1}$ is a GRW spacetime with warping function f and Riemannian fiber $M^n$ . When $M^n$ has constant sectional curvature, (2-1) has been known in the mathematical literature as a Robertson–Walker (RW) spacetime, an allusion to the fact that, for $n=3$ , it is an exact solution of Einstein’s field equations (see, for example, [Reference O’Neill35, Ch. 12]).

In this setting, we consider the timelike conformal closed vector field

(2-2) $$ \begin{align} \quad\quad \mathcal{K}(t,y)=f(t)\partial_{\,t}|_{(t,y)},\quad (t,y)\in -I\times_{f}M^n, \end{align} $$

globally defined on $\overline {M}$ , where $\partial _t={\partial }/{\partial _t}$ stands for the coordinate timelike vector field tangential to I. From the relationship between the Levi–Civita connections of $\overline {M}^{n+1}$ and those of I and $M^n$ (see [Reference O’Neill35, Proposition $7.35$ ]), it follows that

$$ \begin{align*} \overline{\nabla}_V\mathcal K=f'(\pi_I)V, \end{align*} $$

for all $V\in \mathfrak {X}(\overline {M})$ , where $\overline {\nabla }$ is the Levi–Civita connection of $\overline {M}^{n+1}$ .

Let $\Sigma ^n$ be an n-dimensional connected manifold. A smooth immersion ${\psi :\Sigma ^n\looparrowright \overline {M}^{n+1}}$ is said to be a spacelike hypersurface if $\Sigma ^n$ , furnished with the metric induced from $\langle \,,\rangle $ via $\psi $ , is a Riemannian manifold. We denote by $\nabla $ the Levi–Civita connection of $\Sigma ^n$ endowed with its induced metric (which will be denoted by $\langle \,,\rangle $ ). Since $\overline {M}$ is time-orientable, it follows from the connectedness of $\Sigma ^n$ that one can uniquely choose a globally defined timelike unit vector field $N\in \mathfrak {X}^\perp (\Sigma )$ that has the same time-orientation as $\partial _t$ , that is, such that $\langle N,\partial _t\rangle <0$ . In this case, one says that N is the future-pointing Gauss map of $\Sigma ^n$ and we always assume such a timelike orientation for $\Sigma ^n$ . From the inverse Cauchy–Schwarz inequality (see [Reference O’Neill35, Proposition $5.30$ ]), we have that ${\langle N,\partial _t\rangle \leq -1}$ , with the equality holding at a point $p\in \Sigma ^n$ if and only if $N=\partial _t$ at p.

We denote by A and $H=-\mathrm {trace}(A)$ the shape operator and the mean curvature function of the spacelike hypersurface $\psi :\Sigma ^n\looparrowright \overline {M}^{n+1}$ with respect to its future-pointing Gauss map N. Throughout this paper, the mean curvature H, taken with respect to such a choice of orientation N, will be called the future mean curvature of $\Sigma ^n$ . In particular, for a fixed $t_\ast \in I$ , from [Reference Alías, Brasil and Colares6, Example $5.6$ ] we have that the slice $\{t_\ast \}\times M^n$ has constant future mean curvature

$$ \begin{align*}H=n\frac{f'(t_\ast)}{f(t_\ast)}\end{align*} $$

with respect to $N=\partial _t$ .

It follows from (2-2) that

(2-3) $$ \begin{align} \overline{\nabla}_{V}\partial_{\,t}=\overline{\nabla}_{V}\bigg(\frac{1}{f(t)}\,\mathcal{K}\bigg)=-\frac{1}{f(t)^{2}}\,\langle V,\overline{\nabla}\,\overline{f}\rangle \mathcal{K}+\frac{1}{f(t)}\,f'(t)V \end{align} $$

and a simple computation shows that

(2-4) $$ \begin{align} \overline{\nabla}\pi_I=-\langle\overline{\nabla}\pi_I,\partial_t\rangle\partial_t=-{\partial_t}. \end{align} $$

So, from (2-3),

(2-5) $$ \begin{align} \overline{\nabla}_V\partial_t=\frac{f'(\pi_I)}{f(\pi_I)}\{V+\langle V,\partial_t\rangle\partial_t\}. \end{align} $$

2.2 Spacelike mean curvature flow solitons in GRW spacetimes

We recall that the spacelike mean curvature flow $\Psi :[0,T)\times \Sigma ^n\looparrowright \overline {M}^{n+1}$ related to a spacelike hypersurface $\psi :\Sigma ^n\looparrowright \overline {M}^{n+1}$ in an $(n+1)$ -dimensional Lorentzian manifold $\overline {M}^{n+1}$ , satisfying $\Psi (0,\cdot )=\psi (\cdot )$ , looks for solutions of the equation

$$ \begin{align*}\dfrac{\partial\Psi}{\partial t}=\vec{H},\end{align*} $$

where $\vec {H}(t,\cdot )$ is the (nonnormalized) mean curvature vector of $\Sigma ^n_t=\Psi (t,\Sigma ^n)$ (see, for example, [Reference Lambert and Lotay32]). In our context, according to [Reference Alías, de Lira and Rigoli9, Definition $(1.1)$ ] and [Reference Colombo, Mari and Rigoli24, Definition $(1.1)$ ], a spacelike hypersurface $\psi :\Sigma ^n\looparrowright \overline {M}^{n+1}$ immersed in a GRW spacetime ${\overline {M}^{n+1}=-I\times _fM^n}$ is called a spacelike mean curvature flow soliton with respect to ${\mathcal K=f(t)\partial _t}$ and has soliton constant $c\in \mathbb R$ when its (nonnormalized) future-pointing mean curvature vector $\vec {H}=HN$ satisfies

(2-6) $$ \begin{align} \vec{H}=c\,\mathcal K^\perp, \end{align} $$

where $\mathcal K^\perp $ stands for the orthogonal projection of $\mathcal K$ in the direction of the future-pointing Gauss map N. Adopting the terminology introduced in [Reference Alías, de Lira and Rigoli9, Reference Colombo, Mari and Rigoli24], we also consider the soliton function

(2-7) $$ \begin{align} \zeta_c(t)=nf'(t)+cf(t)^2. \end{align} $$

So, each slice $M_{t_\ast }=\{t_\ast \}\times M^n$ is a spacelike mean curvature flow soliton with respect to $\mathcal K=f(t)\partial _t$ and with soliton constant c given by

(2-8) $$ \begin{align} c=-n\dfrac{f'(t_\ast)}{f(t_\ast)^2}. \end{align} $$

Moreover, $t_\ast $ is implicitly given by the condition $\zeta _c(t_\ast )=0$ .

2.3 Standard examples

In this subsection, we quote standard examples of spacelike mean curvature flow solitons in GRW spacetimes.

Example 2.2. As in [Reference Alías, Colares and de Lima8, Section 4], the future temporal cone $\Lambda ^{+}$ of the Minkowski space $\mathbb R_1^{n+1}$ is defined as the set

$$ \begin{align*}\Lambda^{+}=\{x\in\mathbb R_1^{n+1}:\langle x,x\rangle<0 \text{ and } \langle x,e_{1}\rangle<0\},\end{align*} $$

where $e_{1}=(1,0,\ldots ,0)$ . We observe that $\Lambda ^{+}$ can be regarded as the GRW spacetime

$$ \begin{align*}-\mathbb R^+\times_t\mathbb H^n,\end{align*} $$

where $\mathbb H^n=\{x\in \mathbb R_1^{n+1}:\langle x,x\rangle =-1,\,\, x_{1}>0\}$ denotes the n-dimensional hyperbolic space. Indeed, it is not difficult to verify that the map $\Phi :-\mathbb R^+\times _t\mathbb H^n\rightarrow \Lambda ^{+}$ , given by $\Phi (t,x)=tx$ , is an isometry. In this setting, we have that the slices $\{\!\sqrt {-{n}/{c}}\}\times \mathbb H^n$ are spacelike mean curvature flow solitons with soliton constant $c<0$ with respect to vector field $\mathcal {K}=t\partial _{\,t}$ .

3.1 Some previous computations

Let $\psi :\Sigma ^n\looparrowright -I\times _{f}M^n$ be a spacelike mean curvature flow soliton, as described in Section 2. The height function of $\psi :\Sigma ^n\looparrowright -I\times _{f}M^n$ , denoted by h, is the restriction of the projection $\pi _{I}(t,y)=t$ to $\Sigma ^n$ : that is, $h:\Sigma ^n\rightarrow I$ is given by

(3-1) $$ \begin{align} h=\pi_{I}|_{\Sigma^n}=\pi_{I}\circ\psi. \end{align} $$

Thus, the hyperbolic angle $\Theta $ of $\Sigma ^n$ verifies

(3-2) $$ \begin{align} \Theta=\langle N,\partial_{\,t}\rangle\leq -1, \end{align} $$

where N denotes the future-pointing Gauss map of $\Sigma ^n$ . From (2-4), we have that the gradient of $\pi _{I}$ on $-I\times _{f}M^n$ is given by $\overline {\nabla }\pi _{I}=-\partial _{\,t}$ . Then, the gradient of h on $\Sigma ^n$ is given by

(3-3) $$ \begin{align} \nabla h=(\overline{\nabla}\pi_I)^\top=-\partial_t^{\top}=-\partial_t-\Theta N, \end{align} $$

where $\partial _{\,t}=\partial _{\,t}^{\top }+\partial _{\,t}^{\perp }$ . Here, $\partial _{\,t}^{\top }\in \mathfrak {X}(\Sigma ^n)$ and $\partial _{\,t}^{\perp }\in \mathfrak {X}^{\perp }(\Sigma ^n)$ denote, respectively, the tangential and normal components of $\partial _{\,t}$ .

Thus, (3-3) gives the relationship

(3-4) $$ \begin{align} |\nabla h|^2=\Theta^2-1, \end{align} $$

where $|\,\cdot \,|$ stands for the norm of a tangential vector field on $\Sigma ^n$ considered with its induced metric.

Hence, from (3-3) and (2-5), we deduce that, for any $X\in \mathfrak {X}(\Sigma ^n)$ , the Hessian of h in the metric $\langle \,,\rangle $ is given by

(3-5) $$ \begin{align} \nabla^2h(X,X)=&\ \langle\nabla_X\nabla h,X\rangle\nonumber\\ =&-\frac{f'(h)}{f(h)}\{|X|^2+\langle X,\nabla h\rangle^2\}+\langle AX,X\rangle\Theta. \end{align} $$

In what follows, we also consider the function

(3-6) $$ \begin{align} u=g(h)\in C^\infty(\Sigma^n), \end{align} $$

where $g:I\rightarrow \mathbb {R}$ is an arbitrary primitive of f. Since $g'=f>0$ , $u=g(h)$ can be thought as a reparametrization of the height function. In particular, from (3-3), we have that the gradient of u on $\Sigma ^n$ is given by

(3-7) $$ \begin{align} \nabla u=f(h)\nabla h\,=-\,f(h)\partial^{\top}_{\,t}\,=-\,\mathcal{K}^{\top}, \end{align} $$

where $\mathcal {K}^{\top }$ denotes the tangential component of the closed conformal vector field $\mathcal {K}$ , defined in (2-2). Taking into account this previous digression, we obtain the following auxiliary result.

Lemma 3.1. Let $\psi :\Sigma ^n\looparrowright -I\times _{f}M^n$ be a spacelike mean curvature flow soliton with respect to $\mathcal {K}=f(t)\partial _{\,t}$ and with soliton constant $c\not =0.$ Then,

(3-8) $$ \begin{align} H\langle AX,Y\rangle-c\nabla^2u(X,Y)=cf'(h)\langle X,Y\rangle, \end{align} $$

for all $X,Y\in \mathfrak {X}(\Sigma )$ . Furthermore,

$$ \begin{align*}\nabla H=cA(\nabla u).\end{align*} $$

Proof. First, we note that

$$ \begin{align*} \nabla^2u(X,X)=&\ \langle\nabla_X\nabla u,X\rangle\nonumber\\ =&\ \langle\nabla_X(f(h)\nabla h),X\rangle\nonumber\\ =&\ f(h)\langle\nabla_X\nabla h,X\rangle+\langle\nabla_X f(h)\nabla h,X\rangle\nonumber\\ =&\ f(h)\nabla^2h(X,X)+f'(h)\langle X,\nabla h\rangle^2. \end{align*} $$

Thus, from (3-5), we get that

$$ \begin{align*} \nabla^2u(X,X)=&f(h)\bigg(-\frac{f'(h)}{f(h)}\{|X|^2+\langle X,\nabla h\rangle^2\}+\langle AX,X\rangle\Theta\bigg)+f'(h)\langle X,\nabla h\rangle^2\nonumber\\ =& -f'(h)|X|^2-f'(h)\langle X,\nabla h\rangle^2+f(h)\langle AX,X\rangle\Theta+f'(h)\langle X,\nabla h\rangle^2\nonumber\\ =& -f'(h)|X|^2+f(h)\langle AX,X\rangle\Theta. \end{align*} $$

On the other hand,

(3-9) $$ \begin{align} \frac{1}{c}\langle \nabla H, X\rangle =&\ \langle \overline{\nabla}_X \mathcal{K},N\rangle+\langle \mathcal{K},\overline{\nabla}_{X}N\rangle\nonumber \\ =&-\langle A(X),\mathcal{K}\rangle=\langle X,A(\nabla u)\rangle, \end{align} $$

for every vector field $X\in \mathfrak {X}(\Sigma ^n)$ , so that, from (3-7), we conclude the desired result.

Naturally attached to $\psi :\Sigma ^n\looparrowright -I\times _{f}M^n$ , we can consider the support function

(3-10) $$ \begin{align} \varphi_{\mathcal{K}}:\Sigma^n&\rightarrow\mathbb{R}\nonumber\\ q&\mapsto\varphi_{\mathcal{K}}(q)\langle \mathcal{K}(q),N(q)\rangle. \end{align} $$

Hence, from (3-2),

(3-11) $$ \begin{align} \varphi_{\mathcal{K}}=f(h)\langle N,\partial_{\,t}\rangle=f(h)\Theta\leq -f(h)<0. \end{align} $$

Furthermore, from [Reference Caminha and de Lima19, Proposition $2.1$ ] and (3-4),

(3-12) $$ \begin{align} \Delta(\varphi_{\mathcal{K}})=\{\overline{\mathrm{Ric}}(N,N)+|A|^2\}\varphi_{\mathcal{K}}-\{nN(f')-Hf'\}+\langle \mathcal{K},\nabla H\rangle, \end{align} $$

where $\nabla H$ is the gradient of H in the metric of $\Sigma ^n$ , $\overline {\mathrm {Ric}}$ is the Ricci tensor of $\overline {M}^{n+1}$ and $|A|$ is the Hilbert–Schmidt norm of A.

In addition, we get that

(3-13) $$ \begin{align} N(f')=-f"\Theta=-\frac{f"}{f}\varphi_{\mathcal{K}}. \end{align} $$

On the other hand, since $N=N^{\ast }-\Theta \partial _{\,t}$ , where $N^\ast =\pi _M(N)$ is the orthogonal projection of N onto $M^n$ , it follows from [Reference O’Neill35, Corollary $7.43$ ] that

(3-14) $$ \begin{align} \overline{\mathrm{Ric}}(N,N)=&\ \overline{\mathrm{Ric}}(N^\ast,N^\ast)+\Theta^2\overline{\mathrm{Ric}}(\partial_{\,t},\partial_{\,t})\nonumber\\ =&\ \mathrm{Ric}_M(N^\ast,N^\ast)+\langle N^\ast,N^\ast\rangle\bigg\{\dfrac{f"}{f}+(n-1)\dfrac{(f')^2}{f^2}\bigg\}-\dfrac{nf"}{f}\Theta^2\nonumber\\ =&\ \mathrm{Ric}_M(N^\ast,N^\ast)-\bigg\{\dfrac{f"}{f}+(n-1)\dfrac{(f')^2}{f^2}\bigg\}-(n-1)\bigg(\dfrac{f'}{f}\bigg)^{'}\Theta^2, \end{align} $$

where $\mathrm {Ric}_{M}$ denotes the Ricci tensor of $M^n$ . We note that the relationship ${\langle N^\ast ,N^\ast \rangle =\Theta ^2-1}$ is used in the last equality above.

Thus, inserting (3-13) and (3-14) into (3-12), we obtain

(3-15) $$ \begin{align} \Delta(\varphi_{\mathcal{K}})=&\ \bigg\{\mathrm{Ric}_M(N^\ast,N^\ast)+|A|^2-\bigg\{\dfrac{f"}{f}+(n-1)\dfrac{(f')^2}{f^2}\bigg\}-(n-1)\bigg(\dfrac{f'}{f}\bigg)^{'}\Theta^2\bigg\}\varphi_{\mathcal{K}}\nonumber\\ &+\bigg\{n\dfrac{f"}{f}\varphi_{\mathcal{K}}+Hf'\bigg\}+\langle\mathcal{K},\nabla H\rangle\nonumber\\ =&\ \bigg\{\mathrm{Ric}_M(N^\ast,N^\ast)+|A|^2+\dfrac{f"f-(f')^2}{f^2}-(n-1)\bigg(\dfrac{f'}{f}\bigg)'\Theta^2\bigg\}\varphi_{\mathcal{K}}+Hf'+\langle\mathcal{K},\nabla H\rangle \nonumber\\ =&\ \{\mathrm{Ric}_M(N^\ast,N^\ast)+(n-1)(\ln f)"(1-\Theta^2)+|A|^2\}\varphi_{\mathcal{K}}+Hf'+\langle\mathcal{K},\nabla H\rangle \nonumber\\ =&\ \{\mathrm{Ric}_M(N^\ast,N^\ast)-(n-1)(\ln f)"|\nabla h|^2+|A|^2\}\varphi_{\mathcal{K}}+Hf'+\langle\mathcal{K},\nabla H\rangle. \end{align} $$

From Equations (2-6) and (3-10), we have that $H=c\,\varphi _{\mathcal {K}}$ , and from (3-7) we get $\nabla u=-\mathcal {K}^\top $ , where u is the reparametrization of the height function h given in (3-6). Consequently, we can rewrite (3-15) as

(3-16) $$ \begin{align} \Delta(\varphi_{\mathcal{K}})=\{cf'(h)+\mathrm{Ric}_M(N^\ast,N^\ast)-(n-1)(\ln f)"(h)|\nabla h|^2+|A|^2\}\varphi_{\mathcal{K}} +\langle\nabla(cu),\nabla(\varphi_{\mathcal{K}})\rangle. \end{align} $$

We recall that the drift Laplacian on $\Sigma ^n$ is defined by

(3-17) $$ \begin{align} \Delta_{cu}(\varphi)=\Delta(\varphi)-\langle\,\nabla(cu),\nabla\varphi\,\rangle \end{align} $$

for all $\varphi \in C^{\infty }(\Sigma ^n)$ . So, from (3-16) and (3-17), we conclude that the drift Laplacian $\Delta _{cu}$ acting on $\varphi _{\mathcal {K}}$ is given by

(3-18) $$ \begin{align} \Delta_{cu}(\varphi_{\mathcal{K}})=\{\tilde{\zeta}_{c}+\mathrm{Ric}_M(N^*,N^*)-(n-1)(\ln f)"(h)|\nabla h|^2\}\varphi_{\mathcal{K}}, \end{align} $$

where $\tilde {\zeta }_{c}\in C^{\infty }(\Sigma ^n)$ is the function defined by

(3-19) $$ \begin{align} \tilde{\zeta}_{c}(q)=cf'(h(q))+|A(q)|^2 \end{align} $$

for every $q\in \Sigma ^n$ , which will be called the second soliton function associated to the spacelike mean curvature flow soliton $\psi :\Sigma ^n\looparrowright -I\times _{f}M^n$ . Such nomenclature for $\tilde {\zeta }_{c}$ is motivated by [Reference Alías, de Lira and Rigoli9, Equation (6.11)].

3.2 Rigidity and nonexistence results

In this subsection, beyond nonexistence results, we establish several theorems that guarantee that, under some necessary hypothesis (see Remark 3.25), a spacelike mean curvature flow soliton must coincide with a slice of the ambient GRW spacetime. We observe that these results are related to each other in the sense that they ensure the rigidity of a complete spacelike mean curvature flow soliton with nonnegative second soliton function that lies in a timelike bounded region of a GRW spacetime and obeys a suitable convergence condition.

In what follows, we assume that the GRW spacetime $-I\times _{f}M^n$ satisfies the null convergence condition (NCC)

(3-20) $$ \begin{align} \mathrm{Ric}_M\geq(n-1)(ff"-f^{\prime 2})\langle\,,\rangle_M, \end{align} $$

which was originally established by Montiel [Reference Montiel34], where $\mathrm {Ric}_{M}$ denotes the Ricci tensor of the Riemannian fiber $M^n$ . It is not difficult to verify that all the GRW spacetimes described in Section 2.2 satisfy the NCC. For this, in the case of a steady-state-type spacetime (see Example 2.4), it is necessary to assume that its Riemannian fiber has nonnegative Ricci curvature.

Before we prove the first rigidity result, we start by quoting an extension of Hopf’s theorem on a complete Riemannian manifold $\Sigma ^n$ due to Yau in [Reference Yau45]. For this, we adopt the notation

$$ \begin{align*}\mathcal{L}^1(\Sigma^n)=\bigg\{\,\varphi\in C^{\infty}(\Sigma^n) : \int_{\Sigma^n}|\varphi|\,d\Sigma\ll+\infty\,\bigg\}\end{align*} $$

for the space of Lebesgue integrable functions on $\Sigma ^n$ , where $d\Sigma $ stands for the volume element induced by the metric of $\Sigma ^n$ , and we denote by $\mathcal L_{cu}^1(\Sigma ^n)$ the set of Lebesgue integrable functions on $\Sigma ^n$ with respect to the modified volume element

(3-21) $$ \begin{align} d\mu=e^{cu}\,d\Sigma. \end{align} $$

We also recall that a smooth function $\varphi $ on $\Sigma ^n$ is said to be $(cu)$ -subharmonic (respectively, $(cu)$ -superharmonic) if $\Delta _{cu}(\varphi )\geq 0$ (respectively, $\Delta _{cu}(\varphi )\leq 0$ ) on $\Sigma ^n$ . So, it is not difficult to verify that, from [Reference Caminha18, Proposition $2.1$ ], we obtain the following auxiliary lemma.

Given a GRW spacetime $\overline {M}^{n+1}=-I\times _{f}M^n$ obeying the NCC (3-20), we require a suitable behavior of the second soliton function associated to a spacelike mean curvature flow soliton $\psi :\Sigma ^n\looparrowright \overline {M}^{n+1}$ , and of the norm of the gradient of its mean curvature function, to establish our first uniqueness result. For this, we consider a timelike bounded region of $\overline {M}^{n+1}$ defined by

$$ \begin{align*} \mathcal{B}_{t_1,t_2}:=\{(t,p)\in-I\times_{f}M^n:t_1\leq t\leq t_2\mbox{ and }p\in M^n\}. \end{align*} $$

Taking into account that all spacelike mean curvature solitons that appear in this paper are considered with respect to the closed conformal vector field $\mathcal {K}=f(t)\partial _{\,t}$ , we are in position to present our first main result.

Proof. From (3-20), we obtain that

(3-22) $$ \begin{align} &\mathrm{Ric}_M(N^*,N^*)-(n-1)(\ln f)"(h)|\nabla h|^2\nonumber\\ &\quad\geq (n-1)(f(h)f"(h)-f'(h)^2)|N^*|_M^2-(n-1)(\ln f)"(h)|\nabla h|^2 \nonumber \\ &\quad=(n-1)(f(h)f"(h)-f'(h)^2)|N+\Theta\partial_{\,t}|_M^2-(n-1)\bigg(\dfrac{f'}{f}\bigg)'(h)|\nabla h|^2 \nonumber\\ &\quad=(n-1)\bigg\{(f(h)f"(h)-f'(h)^2)\dfrac{|\nabla h|^2}{f(h)^2}-\bigg(\dfrac{f(h)f"(h)-f'(h)^2}{f(h)^2}\bigg)|\nabla h|^2\bigg\}=0. \end{align} $$

Thus, since $\tilde {\zeta }_{c}\geq 0$ on $\Sigma ^n$ , from (3-18) and (3-22) we get that the support function $\varphi _K$ defined in (3-10) satisfies

(3-23) $$ \begin{align} \Delta_{cu}(\varphi_{\mathcal{K}})\leq f(h)\tilde{\zeta}_{c}\Theta\leq 0. \end{align} $$

On the other hand, since $\psi :\Sigma ^n\looparrowright -I\times _{f}M^n$ is contained in a timelike bounded region $\mathcal {B}_{t_1,t_2}$ of $-I\times _{f}M^n$ , h is bounded on $\Sigma ^n$ and, consequently, the same happens with $u=g(h)$ and $e^{cu}$ . So, since $c\not =0$ , from (3-21), (2-6), (3-10) and $|\nabla H|\in \mathcal L^1(\Sigma ^n)$ we get $|\nabla (\varphi _{\mathcal {K}})|\in \mathcal L_{cu}^1(\Sigma ^n)$ . Next, from Lemma 3.3 we obtain that $\Delta _{cu}(\varphi _{\mathcal {K}})=0$ on $\Sigma ^n$ . Since $f(h)>0$ and $\Theta <0$ on $\Sigma ^n$ , from (3-22) and (3-23) we must have on $\Sigma ^n$ that

$$ \begin{align*}\tilde{\zeta}_{c}=0 \quad \mathrm{and} \quad \mathrm{Ric}_M(N^*,N^*)-(n-1)(\ln f)"(h)|\nabla h|^2=0.\end{align*} $$

But, taking into account that the equality in (3-20) occurs only in isolated points of I, we can conclude that $|\nabla h|=0$ on $\Sigma ^n$ and, consequently, h is constant on $\Sigma ^n$ . Therefore, $\psi (\Sigma ^n)$ is a slice.

Remark 3.5. From (2-5), we have that the slice $M_{t}^n$ is a spacelike hypersurface whose shape operator (with respect to the orientation $\partial _{\,t}$ ) $A_{t}$ is given by

(3-24) $$ \begin{align} \mathcal{A}_{t_\ast}:\mathfrak{X}(M_{t_\ast}^n)&\rightarrow\mathfrak{X}(M_{t_\ast}^n)\nonumber\\ V&\mapsto\mathcal{A}_{t_\ast}(V)=-\overline{\nabla}_V(\partial_{\,t_\ast})=-\dfrac{f'(t_\ast)}{f(t_\ast)}\,V. \end{align} $$

Thus, from (3-24) we obtain that the principal curvatures $\kappa _i^{t_\ast }$ of the shape operator $\mathcal {A}_{t_\ast }$ of a slice $M_{t_\ast }^n=\{t_\ast \}\times M^n$ , $t_\ast \in I$ , are given by $\kappa _i^{t_\ast }=-{f'(t_\ast )}/{f(t_\ast )}$ for all $i\in \{1,\ldots ,n\}$ . So, from (2-8) and (3-19),

$$ \begin{align*}\tilde{\zeta}_{c}=c\,f'(t_\ast)+|\mathcal{A}_{t_\ast}|^2=\sum_{i=1}^n(\kappa_i^{t_\ast})^2+\bigg(-\dfrac{n\,f'(t_\ast)}{f^2(t_\ast)}\bigg)f'(t_\ast)=0\end{align*} $$

on $M_{t_\ast }^n$ . Hence, our restriction on the values of the second soliton function $\tilde {\zeta }_{c}$ in Theorem 3.4 constitutes a mild hypothesis in the sense that it is natural to detect slices of $-I\times _{f}M^n$ .

From Theorem 3.4, we derive the following consequence.

When the ambient space is a steady-state-type spacetime, Theorem 3.4 gives the following rigidity result.

From Theorem 3.4 we also get the following nonexistence results.

According to the classical terminology in linear potential theory, a Riemannian manifold $\Sigma ^n$ is called $(cu)$ -parabolic if the constant functions are the only functions $\varphi \in C^2(\Sigma )$ that are bounded from below and satisfy $\Delta _{cu}(\varphi )\leq 0$ . Inspired by the ideas of Romero et al. [Reference Romero, Rubio and Salamanca37, Reference Romero, Rubio and Salamanca38], Albujer et al. established in [Reference Albujer, de Lima, Oliveira and VelÁsquez4, Theorem $1$ ] the following parabolicity criterion, which provides conditions for a complete spacelike hypersurface immersed in GRW spacetime $-I\times _{f}M^n$ to be $(cu)$ -parabolic. For this, we consider the function $\tilde {u}:=g(\pi _I)\circ \tilde {\pi }$ , where $\tilde {\pi }:\widetilde {M}^n\rightarrow M^n$ is the universal covering map of the Riemannian fiber $M^n$ .

We can state the following rigidity result for spacelike mean curvature flow solitons in GRW spacetimes.

From Theorem 3.11, we obtain the following applications.

Considering the strong null convergence condition (SNCC)

(3-25) $$ \begin{align} K_M\geq\sup_I(ff"-f^{\prime 2}), \end{align} $$

which was introduced by Alías and Colares [Reference Alías and Colares7], where $K_M$ denotes the sectional curvature of the Riemannian fiber $M^n$ , and adding a suitable control to the growing of the height function through the second soliton function of a spacelike mean curvature flow soliton, we get the following version of Omori–Yau’s maximum principle.

Proposition 3.15. Let $\overline {M}^{n+1}=-I\times _{f}M^n$ be a GRW spacetime obeying the SNCC (3-25), and let $\psi :\Sigma ^n\looparrowright -I\times _{f}M^n$ be a complete spacelike mean curvature flow soliton with soliton constant $c\not =0$ . If the function $({(n-1)f"(h)+cf(h)f'(h)})/{f(h)}$ is bounded from below on $\Sigma ^n$ , then Omori–Yau’s maximum principle holds for the drift Laplacian $\Delta _{cu}$ : that is, for $\varphi \in C^2(\Sigma ^n)$ with $\sup _{\Sigma }\varphi <+\infty $ , there exists a sequence of points $\{p_k\}_{k\geq 1}$ in $\Sigma ^n$ such that

$$ \begin{align*}\lim_{k}\varphi(p_k)=\sup_{\Sigma}\varphi,\quad\lim_{k}|\nabla\varphi(p_k)|=0\quad\mbox{and}\quad\lim_{k}\Delta_{cu}\varphi(p_k)\leq0.\end{align*} $$

Proof. We recall that the curvature tensor R of $\Sigma ^n$ can be described in terms of its Weingarten operator A and the curvature tensor $\overline {R}$ of the ambient $-I\times _{f} M^{n}$ by the so-called Gauss equation, which is given by

(3-26) $$ \begin{align} \langle R(X,Y)Z,W\rangle=\langle\overline{R}(X,Y)Z,W\rangle-\langle AX,Z\rangle\langle AY,W\rangle+\langle AX,W\rangle\langle AY,Z\rangle, \end{align} $$

for every tangential vector field $X,Y,Z\in \mathfrak X(\Sigma ^n)$ . Here, as in [Reference O’Neill35], the curvature tensor R is given by

$$ \begin{align*} R(X,Y)Z=\nabla_{[X,Y]}Z-[\nabla_X,\nabla_Y]Z, \end{align*} $$

where $[~,~]$ denotes the Lie bracket and $X,Y,Z\in \mathfrak X(\Sigma ^n)$ .

We consider $X\in \mathfrak {X}(\Sigma ^n)$ and take a (local) orthonormal frame $\{E_1,\ldots ,E_n\}$ . It follows from the Gauss equation (3-26) that the Ricci curvature $\mathrm {Ric}$ of $\Sigma ^n$ satisfies

(3-27) $$ \begin{align} \mathrm{Ric}(X,X)=\sum_{i}\langle\overline{R}(X,E_i)X,E_i\rangle+ |AX|^{2}+H\langle AX,X\rangle. \end{align} $$

Thus, from (3-8) and (3-27), we get

(3-28) $$ \begin{align} \mathrm{Ric}(X,X)-c\nabla^{2}u(X,X)\geq \sum_{i}\langle\overline{R}(X,E_i)X,E_i\rangle+cf'(h)|X|^2. \end{align} $$

To estimate the first summand on the right-hand side of inequality (3-28), we consider $X^\ast =(\pi _M)_{\ast }(X)$ and $E_i^\ast =(\pi _M)_{\ast }(E_i)$ . So, from [Reference O’Neill35, Proposition 7.42] and (3-3),

(3-29) $$ \begin{align} \sum_{i}\langle\overline{R}(X,E_i)X,E_i\rangle=&\sum_{i}\langle{R}_M(X^\ast,E_i^\ast)X^\ast,E_i^\ast\rangle+(n-1)((\ln f)'(h))^2|X|^2\nonumber\\ &-(n-2)(\ln f)"(h)\langle X,\nabla h\rangle^2-(\ln f)"(h)|\nabla h|^2|X|^2, \end{align} $$

where ${R}_M$ denotes the curvature tensor of the Riemannian fiber $M^n$ . By writing ${X^\ast =X+\langle X,\partial _t\rangle \partial _t}$ , we can estimate the first summand on the right-hand side of (3-29) to get

(3-30) $$ \begin{align} \sum_{i}\langle{R}_M(X^\ast,E_i^\ast)X^\ast,E_i^\ast\rangle&=f^2(h)(|X^\ast|_ M^2|E^\ast|_ M^2-\langle X^\ast,E^\ast\rangle_M^2) K_M(X^\ast,E^\ast)\nonumber\\ &\geq\dfrac{1}{f^2(h)}((n-1)|X|^2+|\nabla h^2||X|^2\nonumber\\ &\quad+(n-2)\langle X,\nabla h\rangle^2)\min_{i}K_M(X^\ast,E_i^\ast). \end{align} $$

Consequently, since our ambient space obeys (3-25), from (3-30),

(3-31) $$ \begin{align} &\sum_{i}\langle R_M(X^\ast,E_i^\ast)X^\ast,E_i^\ast\rangle\nonumber\\ &\quad \geq((n-1)|X|^2+|\nabla h|^2|X|^2+(n-2)\langle X,\nabla h\rangle^2)(\ln f)"(h). \end{align} $$

Substituting (3-31) into (3-29) gives

(3-32) $$ \begin{align} \sum_{i}\langle\overline{R}(X,E_i)X,E_i\rangle&\geq((n-1)|X|^2\!+\!|\nabla h|^2|X|^2\!+\!(n-2)\langle X,\nabla h\rangle^2)(\ln f)"(h)\!\nonumber\\[-9pt] &\quad+(n-1)((\ln f)'(h))^2|X|^2-(n-2)(\ln f)"(h)\langle X,\nabla h\rangle^2\nonumber\\ &\quad-(\ln f)"(h)|\nabla h|^2|X|^2\nonumber\\ &=(n-1)\dfrac{f"(h)}{f(h)}|X|^2. \end{align} $$

Hence, from (3-28) and (3-32), we obtain

$$ \begin{align*}\mathrm{Ric}-c\nabla^{2}u\geq ((n-1)\dfrac{f"(h)}{f(h)}+cf'(h))\langle\,,\rangle.\end{align*} $$

Therefore, since the right-hand side of the above inequality is bounded from below, we conclude our proof by applying [Reference Chen and Qiu22, Theorem 1].

To proceed, we use Proposition 3.15 to establish the following result.

Theorem 3.16. Let $\overline {M}^{n+1}=-I\times _{f}M^n$ be a GRW spacetime obeying the SNCC (3-25), and let $\psi :\Sigma ^n\looparrowright -I\times _{f}M^n$ be a complete spacelike mean curvature flow soliton with soliton constant $c\not =0$ such that $({(n-1)f"(h)+cf(h)f'(h)})/{f(h)}$ is bounded from below. If $\inf _{\Sigma }f(h)>0$ , the second soliton function $\tilde {\zeta }_{c}=|A|^2+cf'(h)$ is nonnegative and the height function h satisfies

(3-33) $$ \begin{align} |\nabla h|\leq \inf_{\Sigma^n}\tilde{\zeta}_{c} \quad \mathrm{on} \quad \displaystyle\Sigma^n, \end{align} $$

then $\Sigma ^n$ is a slice of $\overline {M}^{n+1}$ .

Proof. Since $\varphi _{\mathcal {K}}<0$ on $\Sigma ^n$ , Proposition 3.15 ensures the existence of a sequence of points $\{p_k\}_{k\in \mathbb {N}}\subset \Sigma ^n$ such that

$$ \begin{align*}\lim_{k\rightarrow+\infty}\varphi_{\mathcal{K}}(p_j)=\sup_{\Sigma^n}\varphi_{\mathcal{K}}\, \quad \mathrm{and} \quad \lim_{k\rightarrow+\infty}\Delta_{cu}\,\varphi_{\mathcal{K}}(p_k)\leq 0.\end{align*} $$

Hence, from (3-23), we get

(3-34) $$ \begin{align} 0\geq\lim_{k\rightarrow+\infty}\Delta_{cu}(\varphi_{\mathcal{K}})(p_j)=\sup_{\Sigma^n}\varphi_{\mathcal{K}}\lim_{k\rightarrow+\infty}\tilde{\zeta}_{c}(p_k)\geq0. \end{align} $$

But, since we are assuming that $\inf _{\Sigma }f(h)>0$ , we have that $\sup _{\Sigma ^n}\varphi _{\mathcal {K}}<0$ . Consequently, from (3-34) we must have $\lim _{j\rightarrow +\infty }\tilde {\zeta }_{c}(p_k)=0$ , and hence $\inf _{\Sigma ^n}\tilde {\zeta }_{c}=0$ . Therefore, the result follows from hypothesis (3-33).

From Theorem 3.16, we obtain the following applications.

Our next result can be regarded as a sort of extension of [Reference Chen and Qiu22, Theorem 3].

Proof. Let us suppose, by contradiction, the existence of such a complete spacelike mean curvature flow soliton $\Sigma ^n$ immersed in $\overline {M}^{n+1}$ . Since we are supposing that $cf'(h)\geq 0$ and that $\overline {M}^{n+1}$ satisfies the SNCC (3-25), we conclude from (3-18) and (3-22) that

$$ \begin{align*}\Delta_{cu}\varphi_{\mathcal{K}}\leq |A|^{2}\varphi_{\mathcal{K}}.\end{align*} $$

From the above equation, we get

$$ \begin{align*}\Delta_{cu}\varphi_{\mathcal{K}}^{2}\geq 2 \varphi_{\mathcal{K}}\Delta_{cu}\varphi_{\mathcal{K}}\geq 2|A|^{2}\varphi_{\mathcal{K}}^{2}.\end{align*} $$

Since $\varphi _{\mathcal {K}}={H}/{c}$ ,

(3-35) $$ \begin{align} \Delta_{cu}H^{2}\geq 2H^{2}|A|^{2}\geq 2\frac{H^{4}}{n}. \end{align} $$

With a straightforward computation, we can verify that

(3-36) $$ \begin{align} \Delta_{cu}\bigg(\frac{-1}{\sqrt{1+H^{2}}}\bigg)=\frac{\Delta_{cu}H^{2}}{2(1+H^{2})^{3/2}}-\frac{3}{4}\frac{|\nabla H^{2}|^{2}}{(1+H^{2})^{5/2}}. \end{align} $$

Hence, from (3-35) and (3-36), we obtain

$$ \begin{align*}\Delta_{cu}\bigg(\frac{-1}{\sqrt{1+H^{2}}}\bigg)\geq \frac{H^{4}}{n(1+H^{2})^{3/2}}-\frac{3}{4}\frac{|\nabla H^{2}|^{2}}{(1+H^{2})^{5/2}}.\end{align*} $$

Therefore, since $({(n-1)f"(h)+cf(h)f'(h)})/{f(h)}$ is bounded from below, from Proposition 3.15 we can apply Omori–Yau’s maximum principle and reason as in the proof of [Reference Chen and Qiu22, Theorem 3] to conclude that $H\equiv 0$ , which corresponds to an absurdity.

From Theorem 3.20, we get the following nonexistence results.

Remark 3.25. Fixing a constant $c\in \mathbb R$ with $0<|c|<1$ , from [Reference de Lima and Lima26, Example $4.4$ ] we have that

$$ \begin{align*}\Sigma^n=\{(c\ln x_n,x_1,\ldots,x_n):x_n>0\}\subset-\mathbb R\times\mathbb H^n\end{align*} $$

is a complete spacelike translating soliton of the mean curvature flow with respect to $\partial _t$ that has soliton constant c and constant future mean curvature

$$ \begin{align*}H=\frac{c}{\sqrt{1-c^2}}=c\,\Theta.\end{align*} $$

Moreover, we also get that

$$ \begin{align*}|\nabla h|=\frac{|c|}{\sqrt{1-c^2}}=|A|.\end{align*} $$

Hence, since the static GRW spacetime $-\mathbb R\times \mathbb H^n$ obeys neither the NCC (3-20) nor the SNCC (3-25), we can verify that it works as a counterexample related to our previous theorems. Consequently, we conclude that some hypothesis is needed.

Now, we deal with compact (without boundary) mean curvature flow solitons.

Theorem 3.26. Let $\overline {M}^{n+1}=-I\times _{f}M^n$ be a GRW spacetime and let $\psi :\Sigma ^n\looparrowright \overline {M}^{n+1}$ be a compact mean curvature flow soliton with soliton constant $c\neq 0$ . If $c>0$ , then

$$ \begin{align*}\min_{\Sigma}H^{2}\leq -cnf'(h_{*}) \quad \mathrm{and} \quad \max_{\Sigma}H^{2}\geq - cnf'(h^{*}),\end{align*} $$

where $h_{*}$ and $h^{*}$ are the minimum and maximum of the height function on $\Sigma ^n$ . Similarly, if $c<0$ , then

$$ \begin{align*}\min_{\Sigma}H^{2}\leq -cnf'(h^{*}) \quad \mathrm{and} \quad \max_{\Sigma}H^{2}\geq - cnf'(h_{*}).\end{align*} $$

Proof. From (3-8),

(3-37) $$ \begin{align} c\Delta u=-ncf'(h)-H^{2}. \end{align} $$

We consider $c>0$ and let $p_0$ be a minimum point of the height function h. Since a primitive g of f is an increasing function, we have that $h(p_0)=h_{*}$ is a minimum point of the function $u=g(h)$ , and hence $\Delta u(p_0)\geq 0$ . Thus, from (3-37), we get that

$$ \begin{align*}\min_{\Sigma}H^{2}\leq H^{2}(p_0)\leq -ncf'(h_{*}).\end{align*} $$

Analogously, taking a maximum point of h, we are able to conclude that

$$ \begin{align*}\max_{\Sigma}H^{2}\geq -cnf'(h^{*}).\end{align*} $$

The proof of the case $c<0$ follows the same steps as the case $c>0$ .

From the above result, we conclude directly the following nonexistence result.

We finish this subsection by establishing a rigidity result derived from Theorem 3.26.

Proof. Indeed, since $f"(t)\leq 0$ , we have that $f'$ is nondecreasing. In addition, since H is constant, from Theorem 3.26 we conclude that

$$ \begin{align*}-ncf'(t)=H^{2}\end{align*} $$

for $h_{*}\leq t\leq h^{*}$ . Thus, from the above equation jointly with (3-37), we have that $\Delta u=0$ . Therefore, since $\Sigma ^n$ is compact, we conclude that u is constant, which means that $\Sigma ^n$ is a slice of $\overline {M}^{n+1}$ .

3.3 The spacelike mean curvature flow soliton equation

Let $\Omega \subseteq M^n$ be a connected domain and let $z\in C^\infty (\Omega )$ be a smooth function such that $z(\Omega )\subseteq I$ . Then $\Sigma ^n(z)$ will denote the (vertical) graph over $\Omega $ determined by z, that is,

$$ \begin{align*}\Sigma^n(z)=\{(z(p),p):p\in\Omega\}\subset\overline{M}^{n+1}=-I\times_{f}M^n.\end{align*} $$

The graph is said to be entire if $\Omega =M^n$ . Observe that $h(z(p),p)=z(p),p\in \Omega $ . Hence, h and z can be identified in a natural way. The metric induced on $\Omega $ from the Lorentzian metric of the ambient GRW spacetime via $\Sigma ^n(z)$ is

(3-38) $$ \begin{align} g_{z}=-dz^2+f^2(z)g_M. \end{align} $$

It follows from (3-38) that a graph $\Sigma ^n(z)$ is a spacelike hypersurface if and only if $|Dz|_{M}<f(z)$ , where $Dz$ stands for the gradient of z in $M^n$ and $|Dz|_M$ denotes its norm, both with respect to the metric $g_M$ . On the other hand, in the case where $M^n$ is a simply connected manifold, from [Reference Alías, Romero and SÁnchez11, Lemma 3.1] we have that every complete spacelike hypersurface $\psi :\Sigma ^n\looparrowright -I\times _{f}M^n$ such that the warping function f is bounded on $\Sigma ^n$ is an entire spacelike graph over $M^n$ . In particular, this happens for complete spacelike hypersurfaces lying in a timelike bounded region of $-I\times _{f}M^n$ . It is also interesting to point out that, in contrast to the case of graphs in a Riemannian space, an entire spacelike graph $\Sigma ^n(z)$ in a GRW spacetime is not necessarily complete, in the sense that the induced Riemannian metric (3-38) is not necessarily complete on $M^n$ . For example, Albujer [Reference Albujer1, Section 3] constructed explicit examples of noncomplete entire maximal spacelike graphs (that is, whose mean curvature is identically zero) in the Lorentzian product space $-\mathbb R\times \mathbb H^2$ .

The future-pointing Gauss map of a spacelike graph $\Sigma ^n(z)$ over $\Omega $ is given by the vector field

(3-39) $$ \begin{align} N(p)=\frac{f(z(p))}{\sqrt{f^2(z(p))-|Dz(p)|_M^2}}\bigg(\partial_t|_{(z(p),p)}+\frac{Dz(p)}{f^2(z(p))}\bigg)\quad \forall p\in\Omega. \end{align} $$

From (3-39), we have that the shape operator related to the future-pointing Gauss map (3-39) is given by

(3-40) $$ \begin{align} AX&=-\frac{1}{f(u)\sqrt{f^2(z)-|Dz|_M^2}}D_XDz-\frac{f'(z)}{\sqrt{f^2(z)-|Dz|_M^2}}X\nonumber\\ &\qquad+\bigg(\frac{-g_M(D_XDz,Dz)}{f(z)(f^2(z)-|Dz|_M^2)^{3/2}}+\frac{f'(z)g_M(Dz,X)}{(f^2(z)-|Dz|_M^2)^{3/2}}\bigg)Dz, \end{align} $$

for any vector field X tangential to $\Omega $ , where D denotes the Levi–Civita connection of $(M^n,g_M)$ . Consequently, if $\Sigma ^n(z)$ is a spacelike graph defined over a domain $\Omega \subseteq M^n$ , it is not difficult to verify from (3-40) that the future mean curvature function $H(z)$ of $\Sigma ^n(z)$ is given by

(3-41) $$ \begin{align} H(z)=\mathrm{div}_M\bigg(\dfrac{Dz}{nf(z)\sqrt{f^2(z)-|Dz|_M^2}}\bigg)+\dfrac{f'(z)}{n\sqrt{f^2(z)-|Dz|_M^2}}\bigg(n+\dfrac{|Dz|_M^2}{f^2(z)}\bigg), \end{align} $$

where $\mathrm {div}_M$ stands for the divergence operator computed in the metric $g_M$ .

Hence, from (2-6) and (3-41), we have that $\Sigma ^n(z)$ is a spacelike mean curvature flow soliton with respect to $K=f(t)\partial _t$ and with soliton constant c if and only if $|Dz|_{M}<f(z)$ and z is a solution of the nonlinear differential equation

(3-42) $$ \begin{align} \mathrm{div}_M\bigg(\dfrac{Dz}{f(z)\sqrt{f(z)^{2}-|Dz|_M^2}}\bigg)=-\dfrac{1}{\sqrt{f(z)^{2}-|Dz|_M^2}}\bigg\{cf(z)^2+f'(z)\bigg(n+\dfrac{|Dz|_M^2}{f(z)^{2}}\bigg)\bigg\}. \end{align} $$

We say that $z\in C^\infty (M)$ has finite $C^2$ norm when

$$ \begin{align*}||z||_{\mathcal{C}^{2}(M)}:=\sup_{|k|\leq2}|D^{k}z|_{L^{\infty}(M)}<+\infty.\end{align*} $$

In this context, we obtain the following result.

Proof. Let $z\in C^{\infty }(M)$ be such a solution of Equation (3-42). It follows from (3-40) that the shape operator A of $\Sigma ^n(z)$ is bounded provided that z has finite $C^2$ . We note also that the finiteness of the $C^2$ norm of z implies, in particular, that z is bounded, which, in turn, guarantees that $\Sigma ^n(z)$ is contained in a bounded timelike region of $\overline {M}^{n+1}$ . Consequently, since we are also assuming that $|Dz|_M\leq \alpha f(z)$ , for some constant $0<\alpha <1$ , we get that

$$ \begin{align*}|Dz|^2_M\leq f^2(z)-\beta\end{align*} $$

for $\beta =(1-\alpha ^2)\inf _{\Sigma (z)}f^2(z)$ . Thus, we can apply [Reference Albujer, de Lima, Oliveira and VelÁsquez3, Proposition 1] to conclude that $\Sigma ^n(z)$ is complete.

We also have that $N=N^\ast -\Theta \partial _t$ , where $N^\ast $ denotes the projection of N onto the fiber $M^n$ . Consequently, from (3-39), we get

(3-43) $$ \begin{align} |\nabla z|^2=\langle N^\ast,N^\ast\rangle=f^2(z)\langle N^\ast,N^\ast\rangle_{M}. \end{align} $$

Thus, from (3-39) and (3-43), we obtain

(3-44) $$ \begin{align} |\nabla z|^2=\frac{|Dz|_M^2}{f^2(z)-|Dz|_M^2}. \end{align} $$

On the other hand, it follows from (3-38) that $d\Sigma =\sqrt {|G|}dM$ , where $dM$ and $d\Sigma ^n$ stand for the Riemannian volume elements of $(M^n,g_M)$ and $(\Sigma ^n(z),g_z)$ , respectively, and $G=det(g_{ij})$ with

$$ \begin{align*}g_{ij}=g_z(E_i,E_j)=f^2(z)\delta_{ij}-E_i(z)E_j(z).\end{align*} $$

Here, $\{E_1,\ldots ,E^n\}$ denotes a local orthonormal frame with respect to the metric $g_M$ . So, it is not difficult to verify that

$$ \begin{align*}|G|=f^{2(n-1)}(z)(f^2(z)-|Dz|_{M}^2).\end{align*} $$

Consequently,

(3-45) $$ \begin{align} d\Sigma=f^{n-1}(z)\sqrt{f^2(z)-|Dz|_{M}^2}dM. \end{align} $$

Thus, from (3-44) and (3-45), we get

(3-46) $$ \begin{align} |\nabla z|d\Sigma=f(z)^{n-1}|Dz|_{M}dM. \end{align} $$

Hence, since z is bounded and $|Dz|_{M}\in \mathcal {L}^1(M)$ , from Equation (3-46) we conclude that $|\nabla z|\in \mathcal {L}^1(\Sigma ^n(z))$ . Consequently, from (3-9) we get that $|\nabla (\varphi _{\mathcal {K}})|\in \mathcal L_{cu}^1(\Sigma ^n(z))$ . Therefore, we can reason as in the last part of the proof of Theorem 3.4 to conclude the result.

From Theorem 3.11, we obtain the following consequence.

Proof. Observing that h satisfies (3-2) and (3-43), from (3-40) we obtain

(3-47) $$ \begin{align} |\nabla h|^2=\frac{|Dz|_M^2}{f(z)^2-|Dz|_M^2}. \end{align} $$

Hence, since we are assuming that z has finite $C^1$ norm and taking into account once more that $\Theta ^2=|\nabla h|^2+1$ , with the aid of (3-47) we conclude that $\Theta $ is bounded. Therefore, the result follows by applying Theorem 3.11.

From Theorem 3.16, we also obtain the following result.

Theorem 3.31. Let $\overline {M}^{n+1}=-I\times _fM^n$ be a GRW spacetime obeying the SNCC (3-20) whose Riemannian fiber $M^n$ is complete. Let $z\in C^{\infty }(M)$ be a bounded entire solution of Equation (3-42) for some constant $c\neq 0$ such that $|Dz|_M\leq \alpha f(z)$ , for some constant $0<\alpha <1$ , and the second soliton function $\tilde {\zeta }_{c}(z)=|A|^2+cf'(z)$ is nonnegative. If

(3-48) $$ \begin{align} |Dz|_M\leq\inf_{M}\tilde{\zeta}_{c}, \end{align} $$

then $z\equiv t_\ast $ for some $t_\ast \in I$ , which is implicitly given by the condition $\zeta (t_\ast )=0$ .

We close this subsection with the following application of Theorem 3.20