2. Spacelike hypersurfaces in a standard static spacetime

In this section, we quote some basic concepts, properties, and examples concerning standard static spacetimes, which will be used and addressed in the next sections. For a thorough and modern discussion about spacelike hypersurfaces immersed in a standard static spacetime, see [Reference de Lima, Bisci and Velásquez13, Chapter 4].

Let $(\overline{M}^{n+1}, \langle \ , \ \rangle)$ be an $(n+1)$-dimensional Lorentz manifold endowed with a timelike Killing vector field K. Recall that a vector field $K \in \mathfrak{X}(\overline{M})$ is said to be Killing if it satisfies

\begin{equation*} \langle \overline{\nabla}_V K, W\rangle + \langle V, \overline{\nabla}_W K\rangle = 0, \end{equation*}

for all vector fields $V, W \in \mathfrak{X}(\overline{M})$, where $\overline{\nabla}$ denotes the Levi–Civita connection in $\overline{M}^{n + 1}$. Let us suppose that the distribution $\mathcal D$ orthogonal to K has constant rank and it is integrable. We denote by $\Phi:M^n\times\mathbb{I}\rightarrow\overline{M}^{n+1}$ the flow generated by K, where Mⁿ is an arbitrarily fixed spacelike integral leaf of $\mathcal D$ labelled as t = 0, which we will assume to be connected, and $\mathbb{I}$ is the maximal interval of definition. Here, the term spacelike means that the induced metric of Mⁿ via the inclusion $M^n\hookrightarrow\overline{M}^{n+1}$ is a Riemannian metric. Without loss of generality, in what follows we will also consider $\mathbb{I}=\mathbb{R}$.

In this setting, $\overline{M}^{n+1}$ can be regarded as the standard static spacetime $M^n\times_{\rho}\mathbb{R}_1$, that is, the product manifold $M^n\times\mathbb{R}$ endowed with the standard static metric

(2.1)\begin{equation} \langle \ , \ \rangle = \pi_{M}^*(\langle \ , \ \rangle_M)-(\rho\circ\pi_M)^2\pi_{\mathbb{R}}^*(dt^2), \end{equation}

where π_M and $\pi_{\mathbb{R}}$ denote the canonical projections from $M^n\times\mathbb{R}$ onto each factor, $\langle \ , \ \rangle_M$ is the induced Riemannian metric on the Riemannian base Mⁿ, $\mathbb{R}_1$ is the manifold $\mathbb{R}$ endowed with the metric $-dt^2$ and $\rho\in C^{\infty}(M)$ is the warping function, which is given by $\rho=|K|=\sqrt{-\langle K, K \rangle}$, where $| \cdot |$ denotes the norm of a vector field on $\overline{M}^{n+1}$. In particular, when $\rho\equiv1$, the resulting standard static spacetime $(\overline{M}^{n+1}, \langle \ , \ \rangle)$ is just a Lorentzian product space with factors $(M^n,\langle \ , \ \rangle_M)$ and $(\mathbb{R},-dt^2)$. Note also that $t \in \mathbb{R} \rightarrow M^n\times\{t\}$, is a foliation of $\overline{M}^{n+1}$ whose leaves $M^n\times\{t\}$, called slices, are totally geodesic spacelike hypersurfaces.

Now, let us consider a (connected) spacelike hypersurface $\psi:\Sigma^n\rightarrow\overline{M}^{n+1}$ immersed in $\overline{M}^{n+1}=M^n\times_{\rho}\mathbb{R}_1$, that is, the induced metric, also denoted by $\langle \ , \ \rangle$, on $\Sigma^n$ via ψ is a Riemannian metric. Since K is a globally defined timelike vector field on $\overline{M}^{n + 1}$, it is a standard fact that there exists a unique unit timelike normal vector field N globally defined on $\Sigma^n$ which is in the same time-orientation as K. In particular, by using the inverse Cauchy-Schwarz inequality, we get

(2.2)\begin{equation} \langle N, K \rangle \leq-\rho \lt 0 \textrm{\ \ on \ \ } \Sigma^n. \end{equation}

We will refer to the normal vector field N as the future-pointing Gauss map of $\Sigma^n$. Throughout this work, N will always denote the future-pointing Gauss map of such a spacelike hypersurface.

The (vertical) height function and the angle function of $\Sigma^n$ are defined by $h=\pi_{\mathbb{R}}\circ \psi$ and $\Theta = \langle N,K \rangle$, respectively. From (2.2), we note that Θ is always a negative function. Moreover, from the decomposition $K=K^{\top}-\Theta N$, where $( \ )^{\top}$ denotes the tangential component of a vector field in $\mathfrak{X}(\overline{M})$ along $\Sigma^n$, we obtain

(2.3)\begin{equation} \nabla h=-\frac{1}{\rho^2}K^{\top} \textrm{\ \ and \ \ } |\nabla h|^2=\frac{\Theta^2-\rho^2}{\rho^4}. \end{equation}

Here, for the sake of simplicity, we are considering $\rho=\rho\circ\pi_M\circ\psi$ along $\Sigma^n$.

Let $\overline{\nabla}$ and $\nabla$ denote the Levi–Civita connections in $\overline{M}^{n + 1}$ and $\Sigma^n$, respectively. Then the Gauss and Weingarten formulas for the spacelike hypersurface $\psi : \Sigma^n \rightarrow \overline{M}^{n + 1}$ are given, respectively, by

\begin{equation*} \overline{\nabla}_XY = \nabla_XY - \langle AX,Y\rangle N \end{equation*}

and

\begin{equation*} AX=-\overline{\nabla}_XN, \end{equation*}

for all tangent vector fields $X,Y\in\mathfrak{X}(\Sigma)$, where A stands for the second fundamental form (or Weingarten operator) of $\Sigma^n$ with respect to its future-pointing Gauss map N. For our purpose, it is convenient to define the (non-normalized) mean curvature function of $\Sigma^n$ by $H = -{\rm tr} A$. When H vanishes identically on $\Sigma^n$ we just say that $\Sigma^n$ is maximal.

3. Auxiliary results

For the sake of completeness, in this section, we briefly quote some auxiliary results in order to use them in the proof of the main theorems. Let $(\Sigma^n, \langle \ , \ \rangle)$ be a Riemannian manifold and let V be a smooth vector field on $\Sigma^n$. Denote by

\begin{equation*}\Delta_{V}:=\Delta - \langle V,\nabla .\rangle\end{equation*}

the drift Laplacian of $\Sigma^n$ and

\begin{equation*}{\rm Ric}_{V}:= {\rm Ric} - \dfrac{1}{2}\mathcal{L}_{V}\langle \ , \ \rangle\end{equation*}

the Bakry–Émery–Ricci tensor of $\Sigma^n$, both of them with respect to $V\in\mathfrak{X}(\Sigma)$, where $\mathcal{L}_V$ denotes the Lie derivative in the direction of V. Also, let r be the distance function on $\Sigma^n$ from a fixed point $x_0 \in \Sigma^n$. Then [Reference Chen and Qiu9, Theorem 1] establishes the following version of the Omori–Yau maximum principle:

Lemma 1. Let $\Sigma^n$ be a complete Riemannian manifold and let V be a smooth vector field on $\Sigma^n$. Suppose that ${\rm Ric}_{V}\geq -F(r)\langle \ , \ \rangle$, where $F: \mathbb{R} \to \mathbb{R}$ is a positive continuous function satisfying

(3.1)\begin{equation} \varphi(t)=\int_{\rho_{0}+1}^{t}\dfrac{dr}{\int_{\rho_{0}}^{r}F(s)ds+1}\to +\infty\quad {\rm as} \quad t\to +\infty, \end{equation}

for some positive constant $\rho_{0}.$ Let $f\in C^{2}(\Sigma)$ with $\displaystyle\lim_{x\to +\infty}\dfrac{f(x)}{\varphi(r(x))}=0$, then there exist points $\{p_j\}\subset \Sigma^n$ such that

\begin{eqnarray*} \lim_{j\to +\infty}f(p_j)=\sup_{\Sigma} f,\quad \lim_{j\to +\infty}|\nabla f|(p_j)=0,\quad \lim_{j\to +\infty}\Delta_{V}f(p_j)\leq 0. \end{eqnarray*}

The next result guarantees that the Ricci curvature of a spacelike hypersurface of a standard static space is bounded from below and can be found in [Reference de Lima, de Lima, Lima and Medeiros12, Proposition 1].

Taking into account that K is a timelike Killing vector field and our convention of the mean curvature function, from [Reference Albujer, Aledo and Alías6, 1 and equation (7)], we quote the following expression for the Laplacian of the angle function above defined:

Lemma 3. The angle function $\Theta = \langle N, K\rangle$ of a spacelike hypersurface $\psi: \Sigma^{n}\to \overline{M}^{n+1}$ in a standard static spacetime $\overline{M}^{n+1}=M^{n}\times_{\rho}\mathbb{R}_{1}$ satisfies

\begin{equation*}\Delta\Theta = \langle K, \nabla H\rangle + (\overline{{\rm Ric}}(N,N)+|A|^2)\Theta,\end{equation*}

where $\nabla H$ denotes the gradient of the mean curvature function on $\Sigma^{n}$ and $\overline{{\rm Ric}}$ is the Ricci curvature of $\overline{M}^{n + 1}$.

We will also need the concept of convergence to zero at infinity. Given a connected, complete noncompact Riemannian manifold $\Sigma^n$ and denoting by $d(\cdot,o): \Sigma^n \rightarrow [0,+\infty)$ the Riemannian distance of $\Sigma^n$ measured from a fixed point $o \in \Sigma^n$, a function $f \in C^{\infty}(\Sigma)$ converges to zero at infinity when

\begin{equation*}\lim_{d(x,o) \rightarrow \infty} f(x) = 0.\end{equation*}

In this setting, we quote the following lemma due to Alías, Caminha and do Nascimento (see [Reference Alías, Caminha and do Nascimento3, Theorem 2.2]):

Let $\Sigma^n$ be a connected, oriented, complete noncompact Riemannian manifold. Denote by $B(p,t)$ the geodesic ball centred at p and with radius t > 0. According to [Reference Alías, Caminha and do Nascimento4], given a polynomial function $\sigma:(0,+\infty)\rightarrow(0,+\infty)$, we say that $\Sigma^n$ has polynomial volume growth like $\sigma(t)$ if for some (and hence, any) $p \in \Sigma$,

\begin{equation*} {\rm vol}(B(p,t)) = \mathcal{O}(\sigma(t)) \quad {\rm as} \quad t\to +\infty, \end{equation*}

where ${\rm vol}$ denotes the Riemannian volume of $\Sigma^n$. For the sake of simplicity, we will just say that $\Sigma^n$ has polynomial volume growth. In this setting, from [Reference Alías, Caminha and do Nascimento4, Theorem 2.1(a)] we have the following maximum principle:

4. Spacelike mean curvature flow solitons in a standard static spacetime

We recall that the spacelike mean curvature flow $\Psi:\Sigma^n\times[0,T)\rightarrow\overline{M}^{n+1}$ of a spacelike hypersurface $\psi:\Sigma^n\rightarrow\overline{M}^{n+1}$ in a standard static spacetime $\overline{M}^{n+1}=M^n\times_\rho\mathbb{R}_1$, satisfying $\Psi(0,\cdot)=\psi(\cdot)$, deals with the solutions of the equation

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

where $\vec{H}(\cdot,t)$ is the (non-normalized) mean curvature vector of $\Sigma^n_t=\Psi(\Sigma^n,t)$ (see [Reference Lambert and Lotay18] and references therein). In a similar spirit of [Reference Alías, de Lira and Rigoli5, Definition 1.1], [Reference Colombo, Mari and Rigoli10, Definition 1.1] and [Reference de Lira and Martín15, Definition 2], a spacelike hypersurface $\psi:\Sigma^n\rightarrow\overline{M}^{n+1}$ is said to be a spacelike mean curvature flow soliton (SMCFS for short) with respect to K and with soliton constant $c\in\mathbb R$ if its (non-normalized) mean curvature vector $\vec{H}=HN$ satisfies

(4.1)\begin{equation} \vec{H}=c\,K^\perp, \end{equation}

where $K^\perp$ stands for the orthogonal projection of K in the direction of the future-pointing Gauss map N. Since $\Theta=\langle K,N\rangle$, (4.1) can be rewritten as

(4.2)\begin{equation} H=-c\,\Theta. \end{equation}

In particular, we observed that the slices $M^n\times\{t\}$, $t \in \mathbb{R}$, of $\overline{M}^{n+1}$ are SMCFS with soliton constant c = 0.

In what follows, we will provide results regarding the geometry of SMCFS of standard static spacetimes under a lot of different geometric hypotheses and, in particular, non-existence and rigidity results. We begin by proving that the operator $\Delta_{cK^{\top}}$ of such an SMCFS satisfies the following version of the Omori–Yau generalized maximum principle:

Theorem 1. Let $\overline{M}^{n+1}=M^n\times_{\rho}\mathbb{R}_{1}$ be a standard static spacetime with sectional curvature ${\rm Sec} \geq -F(r)$, where $F : \mathbb{R} \rightarrow \mathbb{R}$ is a positive continuous function satisfying (3.1). Let $\psi:\Sigma^{n}\to \overline{M}^{n+1}$ be a complete noncompact SMCFS (with respect to K) with soliton constant $c\in \mathbb{R}$. If Θ is bounded on $\Sigma^n$, then there exists a sequence of points $\{p_j\}\subset \Sigma^n$ such that

(4.3)\begin{equation} \lim_{j\to +\infty}\Theta(p_j)=\inf_{\Sigma} \Theta,\quad \lim_{j\to +\infty}|\nabla \Theta|(p_j)=0,\quad \lim_{j\to +\infty}\Delta_{cK^{\top}}\Theta(p_j)\geq 0. \end{equation}

Proof. We begin by observing that, for every $X \in \mathfrak{X}(\Sigma)$, it is not difficult to see that

\begin{eqnarray*} {\rm Ric}_{cK^{\top}}(X,X) &=& {\rm Ric}(X,X) - \dfrac{1}{2}\left(\mathcal{L}_{cK^{\top}}\langle \ , \ \rangle\right)(X,X) \\ & = & {\rm Ric}(X,X) - c\langle\overline{\nabla}_{X}K^{\top}, X\rangle. \end{eqnarray*}

So, from the fact that K is a Killing vector field jointly with the decomposition $K^{\top} = K + \Theta N$, we arrive at

\begin{equation*} {\rm Ric}_{cK^{\top}}(X,X) = {\rm Ric}(X,X) + c\Theta \langle AX,X\rangle. \end{equation*}

that is,

(4.4)\begin{equation} {\rm Ric}_{cK^{\top}}(X,X) = {\rm Ric}(X, X) - H\langle AX, X\rangle, \end{equation}

because of $H = -c\Theta$. Let $\overline{R}$ and R be the curvature tensors of $\overline{M}^{n+1}$ and $\Sigma^n$, respectively. The Gauss equation is

\begin{equation*} R(X, Y)Z = (\overline{R}(X, Y)Z)^{\top} - \langle AX, Z\rangle AY + \langle AY, Z\rangle AX, \end{equation*}

for every vector fields $X, Y, Z \in \mathfrak{X}(\Sigma)$. Here, as in [Reference O’Neill19], the curvature tensor R is given by

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

where $[ \ , \ ]$ denotes the Lie bracket and $X, Y, Z \in \mathfrak{X}(\Sigma)$. Then, taking a local orthonormal frame $\{E_1, \ldots, E_n\}$ tangent to $\Sigma^n$, we have that the Ricci curvature of $\Sigma^n$ is given by

(4.5)\begin{equation} \textrm{Ric}(X, X) = \sum_{i = 1}^{n}\langle \overline{R}(X, E_i)X, E_i\rangle + H\langle AX, X\rangle + |AX|^2, \quad \forall X \in \mathfrak{X}(\Sigma). \end{equation}

It follows from equations (4.4) and (4.5) and our assumptions ${\rm Sec} \geq -F(r)$ that

\begin{equation*} {\rm Ric}_{cK^{\top}}(X,X) \geq -(n - 1)F(r)\langle X, X\rangle, \quad \forall X \in \mathfrak{X}(\Sigma). \end{equation*}

Moreover, since Θ is bounded on $\Sigma^{n}$, we have $\displaystyle\lim_{x \to +\infty}\dfrac{\Theta(x)}{\varphi(r(x))}=0$. Therefore, applying lemma 1 to the function $-\Theta$ we reach the desired result.

Proceeding, we are able to prove that every complete noncompact SMCFS in such a standard static spacetime must be maximal.

Proof. We have that lemma 3 yields

\begin{equation*} \Delta\Theta = \langle K, \nabla H\rangle + (\overline{{\rm Ric}}(N,N)+|A|^2)\Theta, \end{equation*}

which jointly with $H = -c\Theta$ imply that

\begin{equation*} \Delta_{cK^{\top}}\Theta = -2c\langle K^{\top}, \nabla \Theta\rangle + (\overline{{\rm Ric}}(N,N)+|A|^2)\Theta. \end{equation*}

Then, since $|A|^2 \geq \frac{1}{n}H^2 = \frac{c^2}{n}\Theta^2$, we get

(4.6)\begin{equation} \Delta_{cK^{\top}}\Theta \leq -2c\langle K^{\top}, \nabla \Theta\rangle + \frac{c^2}{n}\Theta^3. \end{equation}

Now, let us observe that, since Θ is bounded, we have $|K^{\top}|^2 = \Theta^2 - \rho^2$ is also bounded. From theorem 1, there exists a sequence of points $\{p_j\} \subset \Sigma^n$ satisfying (4.3). In particular,

\begin{equation*} |\langle K^{\top}, \nabla \Theta\rangle(p_j)| \leq |K^{\top}(p_j)||\nabla\Theta(p_j)| \rightarrow 0 \quad {\rm as} \quad j \rightarrow +\infty. \end{equation*}

Hence, evaluating (4.6) along the sequence $\{p_j\}$ and taking limits we obtain

\begin{equation*} 0 \leq \lim_{j\to +\infty}\Delta_{cK^{\top}} \Theta(p_j)\leq \frac{c^2}{n}(\inf_{\Sigma} \Theta)^3 \leq 0, \end{equation*}

so that c = 0, because $\inf \Theta \lt 0$. Therefore, $\Sigma^n$ is maximal.

It is worth pointing out that we will see through example 1 at the end of our paper that theorem 2 does not hold when the hypothesis $\overline{{\rm Ric}}(N, N) \geq 0$ is removed. Moreover, as a corollary of the proof of theorem 2, we get the nonexistence of complete noncompact SMCFS in a standard static spacetime when c ≠ 0:

In order to prove our next result, we will consider assumptions on the base Mⁿ and the warping function instead of $\overline{M}^{n + 1}$. In this setting, we get the following:

Proof. Since the angle function is bounded, then the mean curvature $H = -c\Theta$ is also bounded and, from lemma 2, the Ricci curvature of $\Sigma^{n}$ is bounded from below. Hence, the Omori–Yau generalized maximum principle [Reference Omori20, Reference Yau21] holds on $\Sigma^n$. In particular, there exists a sequence of points $\{p_j\} \subset \Sigma^n$ satisfying

\begin{equation*} \lim_{j\to +\infty}\Theta(p_j)=\inf_{\Sigma} \Theta,\quad \lim_{j\to +\infty}|\nabla \Theta|(p_j)=0,\quad \liminf_{j\to +\infty}\Delta\Theta(p_j)\geq 0. \end{equation*}

On the other hand, lemma 3 gives

\begin{equation*} \Delta\Theta = \langle K, \nabla H\rangle + (\overline{{\rm Ric}}(N,N)+|A|^2)\Theta. \end{equation*}

So, denoting by $N^*$ the orthogonal projection of N onto TM, from our hypotheses, it follows from equation (4.3) of [Reference de Lima, de Lima, Lima and Medeiros12] that

\begin{equation*} \Delta\Theta \leq -c\langle K, \nabla\Theta\rangle + ({\rm Ric}_{M}(N^{\ast},N^{\ast}) + |A|^2)\Theta, \end{equation*}

where Ric_M denotes the Ricci curvature of Mⁿ. Then evaluating the previous equation at p_j and taking limits we deduce

\begin{equation*} 0 \leq \liminf_{j\to +\infty}\Delta\Theta(p_j) \leq \frac{c^2}{n}(\inf_{\Sigma} \Theta)^3 \leq 0, \end{equation*}

proving that c = 0 and $\Sigma^n$ is maximal.

From theorem 3, we also get the following nonexistence result:

Proceeding, we establish the following rigidity result:

Proof. Let us assume that $\Theta(p) \lt \Theta^*$, for some $p\in \Sigma^{n}$. Then $f = \Theta^* - \Theta$ is a nonnegative nonvanishing function which converges to zero at infinity. Since $H = -c\Theta$, lemma 3 assures that

\begin{equation*} \Delta(-\Theta) = c\langle K,\nabla\Theta\rangle - (\overline{{\rm Ric}}(N, N) + |A|^2)\Theta. \end{equation*}

Hence, our assumption and equation (4.3) of [Reference de Lima, de Lima, Lima and Medeiros12] imply

(4.7)\begin{equation} {\rm div}\nabla (-\Theta) = -\Delta\Theta \geq -({\rm Ric}_{M}(N^*,N^*) + |A|^2)\Theta \geq -|A|^2\Theta\geq 0. \end{equation}

So, since

\begin{eqnarray*} \langle \nabla(-\Theta),\nabla f\rangle = |\nabla \Theta|^2\geq 0, \end{eqnarray*}

we can apply lemma 4 to arrive at $|\nabla \Theta| = 0$. Hence, we conclude that $\Theta\equiv \Theta^*$. So, returning to (4.7) and observing that $\Theta^* \lt 0$, we get $|A| = 0$ on $\Sigma^n$ and $\Sigma^n$ is totally geodesic. In this case, we deduce

\begin{equation*} X(\Theta) = \langle \overline{\nabla}_X N, K\rangle + \langle N, \overline{\nabla}_X K\rangle = \langle N, \overline{\nabla}_{X}K\rangle. \end{equation*}

Thus, writing

\begin{equation*} X = X^* - \frac{\langle X, K\rangle}{\rho^2}K \in TM \oplus T\mathbb{R}_1, \end{equation*}

and using [Reference O’Neill19, Proposition 7.35] we arrive at

\begin{equation*} X(\Theta) = \frac{1}{\rho}\left(\langle X, \overline{\nabla} \rho\rangle\Theta - \langle X, K\rangle\langle N, \overline{\nabla}\rho\rangle\right), \end{equation*}

that is,

\begin{equation*} 0 = \nabla\Theta = \frac{1}{\rho}\left(\Theta\overline{\nabla}\rho - \langle N, \overline{\nabla}\rho\rangle K\right) \in TM \oplus T\mathbb{R}_1. \end{equation*}

It follows that $\overline{\nabla}\rho = 0$ and ρ is constant. In particular, $\overline{{\rm Ric}}(N, N) = {\rm Ric}_M(N^*, N^*)$, where $N^*$ denotes the orthogonal projection of N onto TM. So, if Ric_M is positive at some point $p_0 \in \Sigma^n$, by equation (4.7) we see that ${\rm Ric}_{M}(p_0)(N^*, N^*) = 0$ and, consequently, $N^*(p_0) = 0$. On the other hand, a simple computation and the equation (2.3) give

\begin{equation*} |\nabla h|^2 = \frac{1}{\rho^2}\left(\frac{\Theta^2}{\rho^2} - 1\right) = \frac{1}{\rho^2}|N^*|^2, \end{equation*}

which shows that $\Sigma^n$ must be a slice of $\overline{M}^{n+1}$.

Next, we consider complete noncompact SMCFS having polynomial volume growth to get the following result:

Proof. Let us suppose the existence of a positive constant α such that

\begin{equation*}|A|\geq \alpha \gt 0.\end{equation*}

Consequently, taking $X=-\nabla\Theta$ and $f=-\Theta \gt 0$, we get from (4.7) that ${\rm div} X \geq \alpha f$. But, lemma 5 leads us to conclude that $f\leq 0$, which cannot occur. Therefore, there is no such a positive constant α and, hence, $|A|$ cannot be bounded away from zero.

It is not difficult to see that we can derive from theorem 5 the following consequence:

5. Nonexistence of entire Killing graphs

In this last section, we will use the theorems of the previous section in order to establish nonexistence results concerning entire Killing graphs constructed over the Riemannian base of a standard static spacetime $M^n\times_{\rho}\mathbb{R}_1$ and which are SMCFS with respect to K. For this, we need to recall some basic facts related to these graphs.

According to [Reference Dajczer and de Lira11], we define the entire graph $\Sigma(u)$ associated to a smooth function $u\in C^{\infty}(M)$ as the hypersurface given by

\begin{equation*} \Sigma(u)=\{\Phi(x,u(x)):x\in M^n\}\subset M^n\times_{\rho}\mathbb{R}_1, \end{equation*}

where $\Phi:M^n\times\mathbb{I}\rightarrow\overline{M}^{n+1}$ is the flow generated by the timelike Killing vector field K. The metric induced on Mⁿ from the Lorentzian metric (2.1) via $\Sigma(u)$ is given by

(5.1)\begin{equation} g_u = \langle \ , \ \rangle_M -\rho^2du^2. \end{equation}

From (5.1) we conclude that $\Sigma(u)$ is spacelike if, and only if, $\rho|Du|_{M} \lt 1$, where Du denotes the gradient of the function u with respect to the metric $\langle \ , \ \rangle_M$ of Mⁿ. In this case, we have that

\begin{equation*} N_u=\frac{1}{\rho(1-\rho^2|Du|_M^2)^{1/2}}(K+\rho^2\Psi_*(Du)) \end{equation*}

defines the future-pointing Gauss map of $\Sigma(u)$ and its corresponding angle function is given by

\begin{equation*} \Theta_u = -\frac{\rho}{(1-\rho^2|Du|_M^2)^{1/2}}. \end{equation*}

Moreover, for all vector field X tangent to Mⁿ, the second fundamental form A_u of $\Sigma(u)$ with respect to N_u is given by

(5.2)\begin{eqnarray} A_uX&=&-\frac{\rho}{(1-\rho^2|Du|_M^2)^{1/2}}D_XDu - \frac{\rho^3 \langle D_XDu,Du \rangle}{(1 - \rho^2|Du|_M^2)^{3/2}}Du - \frac{\rho^2\langle D\rho,X \rangle|Du|_M^2}{(1 - \rho^2|Du|_M^2)^{3/2}}Du\nonumber\\ && - \frac{\langle D\rho,X\rangle}{(1-\rho^2|Du|_M^2)^{1/2}}Du - \frac{\langle Du, X\rangle}{(1-\rho^2|Du|_M^2)^{1/2}}D\rho.\end{eqnarray}

So, it follows from (5.2) that the mean curvature H_u of the spacelike entire graph $\Sigma(u)$ is given by

(5.3)\begin{equation} H_u ={\rm Div}_M\left(\frac{\rho Du}{(1-\rho^2|Du|^2_M)^{1/2}}\right)+\frac{\langle Du,D\rho \rangle}{(1-\rho^2|Du|^2_M)^{1/2}}, \end{equation}

where Div_M stands for the divergence operator on Mⁿ with respect to its metric $\langle \ , \ \rangle_M$.

Hence, from (4.2) and (5.3) we have that $\Sigma(u)$ is an SMCFS with respect to K with soliton constant c if, and only if, $\rho|Du|_M \lt 1$ and u is a solution of the following nonlinear differential equation:

\begin{equation*} {\rm Div}_M\left(\frac{\rho Du}{(1-\rho^2|Du|^2_M)^{1/2}}\right)=\dfrac{1}{(1-\rho^2|Du|^2_M)^{1/2}}\big(c\rho - \langle Du,D\rho\rangle\big). \end{equation*}

In this context, we get from theorem 2 the following nonexistence result:

As far as the analogue of theorem 4, we have this other nonexistence result:

We close our paper exhibiting an example to show that the hypothesis $\overline{{\rm Ric}}(N_u,N_u) \geq 0$ in theorem 2 and the condition that the base of the ambient space has negative sectional curvature in theorem 4 cannot be dropped.

Example 1. Let us consider the standard static spacetime $\mathbb{H}^2\times\mathbb{R}_1$ whose Riemannian base is the hyperbolic space of sectional curvature −1, $\mathbb{H}^2 = \{(x, y) \in \mathbb{R}^2 \ ; \ y \gt 0\}$ endowed with its standard complete metric $\langle\,,\rangle_{\mathbb H^2}=\frac{1}{y^2}(dx^2+dy^2)$, and the function $u : \mathbb{H}^2 \rightarrow \mathbb{R}$ given by $u(x, y) = c \ln y$, with $0 \lt |c| \lt 1$. Note that $Du(x, y) = (0, cy)$ and $|Du|_{\mathbb H^2}^2=c^2$, where D denotes the gradient on $\mathbb{H}^2$.

According to [Reference de Lima and Lima14, Example 4.4], we have that

\begin{equation*} \Sigma(u) = \{(x, y, c\ln y);y \gt 0\}\subset \mathbb{H}^2\times\mathbb{R}_1 \end{equation*}

is a complete spacelike surface in $\mathbb{H}^2\times\mathbb{R}_1$ with

\begin{equation*} H_u = -\dfrac{c}{\sqrt{1-c^2}} \quad {\rm and} \quad \Theta_u = \langle N_u, \partial_t\rangle = -\frac{1}{\sqrt{1 - c^2}}, \end{equation*}

where

\begin{equation*} N_u = \frac{1}{\sqrt{1 - c^2}}(Du + \partial_t). \end{equation*}

Hence $H_u = c\Theta_u$ and $\Sigma(u)$ is a noncompact complete SMCFS with respect to the Killing vector field $\partial_t$ and soliton constant c. Moreover,

\begin{equation*} \overline{{\rm Ric}}(N_u, N_u) = {\rm Ric}_{\mathbb{H}^2}\left(\frac{1}{\sqrt{1 - c^2}}Du, \frac{1}{\sqrt{1 - c^2}}Du\right) = - \frac{c^2}{\sqrt{1 - c^2}}. \end{equation*}

In particular, this shows that theorem 2 does not work when $\overline{{\rm Ric}}(N_u, N_u) \lt 0$ and theorem 4 does not hold when the Riemannian base of the ambient spacetime has negative sectional curvature.