---
abstract: 'Given a generalized $\overline M^{n+1}=I\times_{\phi}F^n$ Robertson-Walker spacetime we will classify strongly stable spacelike hypersurfaces with constant mean curvature whose warping function verifies a certain convexity condition. More precisely, we will show that given $x:M^n\rightarrow\overline M^{n+1}$ a closed spacelike hypersurfaces of $\overline M^{n+1}$ with constant mean curvature $H$ and the warping function $\phi$ satisfying $\phi''''\geq\max\{H\phi'',0\}$, then $M^{n}$ is either minimal or a spacelike slice $M_{t_0}=\{t_0\}\times F$, for some $t_0\in I$.'
author:
- 'A. Barros, A. Brasil and A. Caminha'
title: Stability of Spacelike Hypersurfaces in Foliated Spacetimes
---

Introduction
============

Spacelike hypersurfaces with constant mean curvature in Lorentz manifolds have been object of great interest in recent years, both from physical and mathematical points of view. In [@ABC:03], the authors studied the uniqueness of spacelike hypersurfaces with CMC in generalized Robertson-Walker (GRW) spacetimes, namely, Lorentz warped products with 1-dimensional negative definite base and Riemannian fiber. They proved that in a GRW spacetime obeying the timelike convergence condition (i.e, the Ricci curvature is non-negative on timelike directions), every compact spacelike hypersurface with CMC must be umbilical. Recently, Alías and Montiel obtained, in [@AM:01], a more general condition on the warping function $f$ that is sufficient in order to guarantee uniqueness. More precisely, they proved the following

Let $f:I\rightarrow\mathbb R$ be a positive smooth function defined on an open interval, such that $ff^{''}-(f^{'})^{2}\leq 0$, that is, such that $-\log f$ is convex. Then, the only compact spacelike hypersurfaces immersed into a generalized Robertson-Walker spacetime $I\times_fF^{n}$ and having constant mean curvature are the slices $\{t\}\times F$, for a [(]{}necessarily compact[)]{} Riemannian manifold $F$.

Stability questions concerning CMC, compact hypersurfaces in Riemannian space forms began with Barbosa and do Carmo in [@BdC:84], and Barbosa, Do Carmo and Eschenburg in [@BdCE:88]. In the former paper, they introduced the notion of stability and proved that spheres are the only stable critical points for the area functional, for volume-preserving variations. In the setting of spacelike hypersurfaces in Lorentz manifolds, Barbosa and Oliker proved in [@Barbosa:93] that CMC spacelike hypersurfaces are critical points of volume-preserving variations. Moreover, by computing the second variation formula they showed that CMC embedded spheres in the de Sitter space $S_1^{n+1}$ maximize the area functional for such variations. In this paper, we give a characterization of [*strongly stable*]{}, CMC spacelike hypersurfaces in GRW spacetimes, the essential tool for the proof being a formula for the Laplacian of a new support function. More precisely, it is our purpose to show the following

Let $\overline M^{n+1}=I\times_{\phi}F^n$ be a generalized Robertson-Walker spacetime, and $x:M^n\rightarrow\overline M^{n+1}$ be a closed spacelike hypersurface of $\overline M^{n+1}$, having constant mean curvature $H$. If the warping function $\phi$ satisfies $\phi''\geq\max\{H\phi',0\}$ and $M^n$ is strongly stable, then $M^{n}$ is either minimal or a spacelike slice $M_{t_0}=\{t_0\}\times F$, for some $t_0\in I$.

Stable spacelike hypersurfaces
==============================

In what follows, $\overline M^{n+1}$ denotes an orientable, time-oriented Lorentz manifold with Lorentz metric $\overline g=\langle\,\,,\,\,\rangle$ and semi-Riemannian connection $\overline\nabla$. If $x:M^n\rightarrow\overline M^{n+1}$ is a spacelike hypersurface of $\overline M^{n+1}$, then $M^n$ is automatically orientable ([@O'Neill:83], p. 189), and one can choose a globally defined unit normal vector field $N$ on $M^n$ having the same time-orientation of $V$, that is, such that $$\langle V,N\rangle<0$$ on $M$. One says that such an $N$ [*points to the future*]{}.

A [*variation*]{} of $x$ is a smooth map $$X:M^n\times(-\epsilon,\epsilon)\rightarrow\overline M^{n+1}$$ satisfying the following conditions:

1.  For $t\in(-\epsilon,\epsilon)$, the map $X_t:M^n\rightarrow\overline M^{n+1}$ given by $X_t(p)=X(t,p)$ is a spacelike immersion such that $X_0=x$.

2.  $X_t\big|_{\partial M}=x\big|_{\partial M}$, for all $t\in(-\epsilon,\epsilon)$.

The [*variational field*]{} associated to the variation $X$ is the vector field $\frac{\partial X}{\partial t}$. Letting $f=-\langle\frac{\partial X}{\partial t},N\rangle$, we get $$\frac{\partial X}{\partial t}\Big|_M=fN+\left(\frac{\partial X}{\partial t}\right)^T,$$ where $T$ stands for tangential components. The [*balance of volume*]{} of the variation $X$ is the function $\mathcal V:(-\epsilon,\epsilon)\rightarrow\mathbb R$ given by $$\mathcal V(t)=\int_{M\times[0,t]}X^*(d\overline M),$$ where $d\overline M$ denotes the volume element of $\overline M$.

The [*area functional*]{} $\mathcal A:(-\epsilon,\epsilon)\rightarrow\mathbb R$ associated to the variation $X$ is given by $$\mathcal A(t)=\int_MdM_t,$$ where $dM_t$ denotes the volume element of the metric induced in $M$ by $X_t$. Note that $dM_0=dM$ and $\mathcal A(0)=\mathcal A$, the volume of $M$. The following lemma is classical:

\[lemma:first variation\] Let $\overline M^{n+1}$ be a time-oriented Lorentz manifold and $x:M^n\rightarrow\overline M^{n+1}$ a spacelike closed hypersurface having mean curvature $H$. If $X:M^n\times(-\epsilon,\epsilon)\rightarrow\overline M^{n+1}$ is a variation of $x$, then $$\frac{d\mathcal V}{dt}\Big|_{t=0}=\int_MfdM,\ \   \ \ \frac{d\mathcal A}{dt}\Big|_{t=0}=\int_MnHfdM.$$

Set $H_0=\frac{1}{\mathcal A}\int_MdM$ and $\mathcal J:(-\epsilon,\epsilon)\rightarrow\mathbb R$ given by $$\mathcal J(t)=\mathcal A(t)-nH_0\mathcal V(t).$$ $\mathcal J$ is called the [*Jacobi functional*]{} associated to the variation, and it is a well known result [@BdCE:88] that $x$ has constant mean curvature $H_0$ if and only if $\mathcal J'(0)=0$ for all variations $X$ of $x$.

We wish to study here immersions $x:M^n\rightarrow\overline M^{n+1}$ that maximize $\mathcal J$ for all variations $X$. Since $x$ must be a critical point of $\mathcal J$, it thus follows from the above discussion that $x$ must have constant mean curvature. Therefore, in order to examine whether or not some critical immersion $x$ is actually a maximum for $\mathcal J$, one certainly needs to study the second variation $\mathcal J''(0)$. We start with the following

Let $x:M^n\rightarrow\overline M^{n+1}$ be a closed spacelike hypersurface of the time-oriented Lorentz manifold $\overline M^{n+1}$, and $X:M^n\times(-\epsilon,\epsilon)\rightarrow\overline M^{n+1}$ be a variation of $x$. Then, $$\label{eq:fundamental relation}
n\frac{\partial H}{\partial t}=\Delta f-\left\{\overline{Ric}(N,N)+|A|^2\right\}f-n\langle\left(\frac{\partial X}{\partial t}\right)^T,\nabla H\rangle.$$

Although the above proposition is known to be true, we believe there is a lack, in the literature, of a clear proof of it in this degree of generality, so we present a simple proof here.

Let $p\in M$ and $\{e_k\}$ be a moving frame on a neighborhood $U\subset M$ of $p$, geodesic at $p$ and diagonalizing $A$ at $p$, with $Ae_k=\lambda_ke_k$ for $1\leq k\leq n$. Extend $N$ and the $e_k's$ to a neighborhood of $p$ in $\overline M$, so that $\langle N,e_k\rangle=0$ and $(\overline\nabla_Ne_k)(p)=0$. Then $$\begin{aligned}
n\frac{\partial H}{\partial t}&=&-{\rm tr}\left(\frac{\partial A}{\partial t}\right)=-\sum_k\langle\frac{\partial A}{\partial t}e_k,e_k\rangle=-\sum_k\langle\left(\overline\nabla_{\frac{\partial X}{\partial t}}A\right)e_k,e_k\rangle\\
&=&-\sum_k\left\{\langle\overline\nabla_{\frac{\partial X}{\partial t}}Ae_k,e_k\rangle-\langle A\overline\nabla_{\frac{\partial X}{\partial t}}e_k,e_k\rangle\right\}\\
&=&\sum_k\langle\overline\nabla_{\frac{\partial X}{\partial t}}\overline\nabla_{e_k}N,e_k\rangle+\sum_k\langle A\overline\nabla_{e_k}\frac{\partial X}{\partial t},e_k\rangle,\end{aligned}$$ where in the last equality we used the fact that $[\frac{\partial X}{\partial t},e_k]=0$. Letting $$I=\sum_k\langle\overline\nabla_{\frac{\partial X}{\partial t}}\overline\nabla_{e_k}N,e_k\rangle\ \ \text{and}\ \ II=\sum_k\langle A\overline\nabla_{e_k}\frac{\partial X}{\partial t},e_k\rangle,$$ we have $$\begin{aligned}
I&=&\sum_k\left\{\langle\overline\nabla_{\frac{\partial X}{\partial t}}\overline\nabla_{e_k}N-\overline\nabla_{e_k}\overline\nabla_{\frac{\partial X}{\partial t}}N+\overline\nabla_{[e_k,\frac{\partial X}{\partial t}]}N,e_k\rangle+\overline\nabla_{e_k}\overline\nabla_{\frac{\partial X}{\partial t}}N,e_k\rangle\right\}\\
&=&\sum_k\left\{\langle\overline R\left(e_k,\frac{\partial X}{\partial t}\right)N,e_k\rangle+\langle\overline\nabla_{e_k}\overline\nabla_{\frac{\partial X}{\partial t}}N,e_k\rangle\right\}\\
&=&-\overline{Ric}\left(\frac{\partial X}{\partial t},N\right)+\sum_k\langle\overline\nabla_{e_k}\overline\nabla_{\frac{\partial X}{\partial t}}N,e_k\rangle.\end{aligned}$$

Since the frame $\{e_k\}$ is geodesic at $p$, it follows that $$\langle\overline\nabla_{\frac{\partial X}{\partial t}}N,\overline\nabla_{e_k}e_k\rangle=\langle\overline\nabla_{\frac{\partial X}{\partial t}}N,N\rangle\langle\overline\nabla_{e_k}e_k,N\rangle=0$$ at $p$, and hence $$\begin{aligned}
\langle\overline\nabla_{e_k}\overline\nabla_{\frac{\partial X}{\partial t}}N,e_k\rangle&=&e_k\langle\overline\nabla_{\frac{\partial X}{\partial t}}N,e_k\rangle=-e_k\langle N,\overline\nabla_{\frac{\partial X}{\partial t}}e_k\rangle=-e_k\langle N,\overline\nabla_{e_k}\frac{\partial X}{\partial t}\rangle\\
&=&-e_ke_k\langle N,\frac{\partial X}{\partial t}\rangle+e_k\langle\overline\nabla_{e_k}N,\frac{\partial X}{\partial t}\rangle\\
&=&e_ke_k(f)+e_k\langle\overline\nabla_{e_k}N,\left(\frac{\partial X}{\partial t}\right)^T\rangle\\
&=&e_ke_k(f)+\langle\overline\nabla_{e_k}\overline\nabla_{e_k}N,\left(\frac{\partial X}{\partial t}\right)^T\rangle-\langle Ae_k,\overline\nabla_{e_k}\left(\frac{\partial X}{\partial t}\right)^T\rangle.\end{aligned}$$

For $II$, we have $$\begin{aligned}
II&=&\sum_k\langle Ae_k,\overline\nabla_{e_k}\frac{\partial X}{\partial t}\rangle=\sum_k\langle Ae_k,\overline\nabla_{e_k}(fN+\left(\frac{\partial X}{\partial t}\right)^T)\rangle\\
&=&\sum_k\langle Ae_k,f\overline\nabla_{e_k}N\rangle+\sum_k\langle Ae_k,\overline\nabla_{e_k}\left(\frac{\partial X}{\partial t}\right)^T\rangle\\
&=&-f|A|^2+\sum_k\langle Ae_k,\overline\nabla_{e_k}\left(\frac{\partial X}{\partial t}\right)^T\rangle\\\end{aligned}$$

Therefore, $$\label{eq:aux_I}
n\frac{\partial H}{\partial t}=-\overline{Ric}\left(\frac{\partial X}{\partial t},N\right)+\Delta f-f|A|^2+\sum_k\langle\overline\nabla_{e_k}\overline\nabla_{e_k}N,\left(\frac{\partial X}{\partial t}\right)^T\rangle.$$

Now, letting $$\frac{\partial X}{\partial t}=\sum_l^n\alpha_le_l+fN$$ and $Ae_k=\sum_jh_{jk}e_j$, one successively gets $$\begin{aligned}
\overline{Ric}\left(\frac{\partial X}{\partial t},N\right)&=&\sum_l\alpha_l\overline{Ric}(N,e_l)+f\overline{Ric}(N,N)\\
&=&\sum_{k,l}\alpha_l\langle\overline R(e_k,e_l)e_k,N\rangle+f\overline{Ric}(N,N)\\\end{aligned}$$ and, since $(\overline\nabla_Ne_k)(p)=0$, $$\begin{aligned}
\langle\overline R(e_k,e_l)e_k,N\rangle_p&=&\langle\overline\nabla_{e_l}\overline\nabla_{e_k}e_k-\overline\nabla_{e_k}\overline\nabla_{e_l}e_k,N\rangle_p\\
&=&e_l\langle\overline\nabla_{e_k}e_k,N\rangle_p-\langle\overline\nabla_{e_k}e_k,\overline\nabla_{e_l}N\rangle_p-e_k\langle\overline\nabla_{e_l}e_k,N\rangle_p\\
&=&-e_l\langle e_k,\overline\nabla_{e_k}N\rangle_p+e_k\langle e_k,\overline\nabla_{e_l}N\rangle_p\\
&=&e_l(h_{kk})-e_k(h_{kl}),\end{aligned}$$ so that $$\label{eq:aux_II}
\overline{Ric}\left(\frac{\partial X}{\partial t},N\right)_p=\sum_{k,l}\alpha_le_l(h_{kk})-\sum_{k,l}\alpha_le_k(h_{kl})+f\overline{Ric}(N,N)_p.$$

Also, $$\begin{aligned}
\alpha_l\langle\overline\nabla_{e_k}\overline\nabla_{e_k}N,e_l\rangle&=&\alpha_l\langle\nabla_{e_k}\overline\nabla_{e_k}N,e_l\rangle=-\alpha_l\sum_j\langle\nabla_{e_k}h_{kj}e_j,e_l\rangle\\
&=&-\alpha_l\sum_j\left\{e_k(h_{kj})\delta_{lj}+h_{kj}\langle\nabla_{e_k}e_j,e_l\rangle\right\}\\
&=&-\alpha_le_k(h_{kl}),\end{aligned}$$ and hence $$\label{eq:aux_III}
\sum_k\langle\overline\nabla_{e_k}\overline\nabla_{e_k}N,\left(\frac{\partial X}{\partial t}\right)^T\rangle=-\sum_{k,l}\alpha_le_k(h_{kl}).$$

Substituting (\[eq:aux\_II\]) and (\[eq:aux\_III\]) into (\[eq:aux\_I\]), we finally arrive at $$\begin{aligned}
n\frac{\partial H}{\partial t}&=&-\sum_{k,l}\alpha_le_l(h_{kk})-f\overline{Ric}(N,N)_p+\Delta f-f|A|^2\\
&=&-\left(\frac{\partial X}{\partial t}\right)^T(nH)-f\overline{Ric}(N,N)_p+\Delta f-f|A|^2.\end{aligned}$$

Let $\overline M^{n+1}$ be a Lorentz manifold and $x:M^n\rightarrow\overline M^{n+1}$ be a closed spacelike hypersurface having constant mean curvature $H$. If $X:M^n\times(-\epsilon,\epsilon)\rightarrow\overline M^{n+1}$ is a variation of $x$, then $$\label{eq:second variation of J}
\mathcal J''(0)(f)=\int_Mf\left\{\Delta f-\left(\overline{Ric}(N,N)+|A|^2\right)f\right\}dM.$$

In the notations of the above discussion, set $f=f(0)$ and note that $H(0)=H$. It follows from lemma \[lemma:first variation\] that $$\mathcal J'(t)=\int_Mn\left\{H(t)-H\right\}f(t)dM_t.$$ Therefore, differentiating with respect to $t$ once more $$\begin{aligned}
\mathcal J''(0)&=&\int_MnH'(0)f(0)dM_0+\int_Mn\left\{H(0)-H\right\}\frac{d}{dt}f(t)dM_t\Big|_{t=0}\\
&=&\int_MnH'(0)fdM.\end{aligned}$$

Taking into account that $H$ is constant, relation (\[eq:fundamental relation\]) finally gives formula \[eq:second variation of J\]

It follows from the previous result that $\mathcal J''(0)=\mathcal J''(0)(f)$ depends only on $f\in C^{\infty}(M)$, for which there exists a variation $X$ of $M^n$ such that $\left(\frac{\partial X}{\partial t}\right)^{\bot}=fN$. Therefore, the following definition makes sense:

Let $\overline M^{n+1}$ be a Lorentz manifold and $x:M^n\rightarrow\overline M^{n+1}$ be a closed spacelike hypersurface having constant mean curvature $H$. We say that $x$ is strongly stable if, for every function $f\in C^{\infty}(M)$ for which there exists a variation $X$ of $M^n$ such that $\left(\frac{\partial X}{\partial t}\right)^{\bot}=fN$, one has $\mathcal J''(0)(f)\leq 0$.

Conformal vector fields
=======================

As in the previous section, let $\overline M^{n+1}$ be a Lorentz manifold. A vector field $V$ on $\overline M^{n+1}$ is said to be [*conformal*]{} if $$\mathcal L_V\langle\,\,,\,\,\rangle=2\psi\langle\,\,,\,\,\rangle$$ for some function $\psi\in C^{\infty}(\overline M)$, where $\mathcal L$ stands for the Lie derivative of the Lorentz metric of $\overline M$. The function $\psi$ is called the [*conformal factor*]{} of $V$.

Since $\mathcal L_V(X)=[V,X]$ for all $X\in\mathcal X(\overline M)$, it follows from the tensorial character of $\mathcal L_V$ that $V\in\mathcal X(\overline M)$ is conformal if and only if $$\label{eq:1.1}
\langle\overline\nabla_XV,Y \rangle+\langle X,\overline\nabla_YV\rangle=2\psi\langle X,Y\rangle,$$ for all $X,Y\in\mathcal X(\overline M)$. In particular, $V$ is a Killing vector field relatively to $\overline g$ if and only if $\psi\equiv 0$.

Any Lorentz manifold $\overline M^{n+1}$, possessing a globally defined, timelike conformal vector field is said to be a [*conformally stationary spacetime*]{}.

\[prop:Laplacian of conformal vector field\] Let $\overline M^{n+1}$ be a conformally stationary Lorentz manifold, with conformal vector field $V$ having conformal factor $\psi:\overline M^{n+1}\rightarrow\mathbb R$. Let also $x:M^n\rightarrow\overline M^{n+1}$ be a spacelike hypersurface of $\overline M^{n+1}$, and $N$ a future-pointing, unit normal vector field globally defined on $M^n$. If $f=\langle V,N\rangle$, then $$\label{eq:Laplacian formula_I}
\Delta f=n\langle V,\nabla H\rangle+f\left\{\overline{Ric}(N,N)+|A|^2\right\}+n\left\{H\psi-N(\psi)\right\},$$ where $\overline{Ric}$ denotes the Ricci tensor of $\overline M$, $A$ is the second fundamental form of $x$ with respect to $N$, $H=-\frac{1}{n}{\rm tr}(A)$ is the mean curvature of $x$ and $\nabla H$ denotes the gradient of $H$ in the metric of $M$.

Fix $p\in M$ and let $\{e_k\}$ be an orthonormal moving frame on $M$, geodesic at $p$. Extend the $e_k$ to a neighborhood of $p$ in $\overline M$, so that $(\overline\nabla_Ne_k)(p)=0$, and let $$V=\sum_l^n\alpha_le_l-fN.$$ Then $$\begin{aligned}
f=\langle N,V\rangle\Rightarrow e_k(f)&=&\langle\overline\nabla_{e_k}N,V\rangle+\langle N,\overline\nabla_{e_k}V\rangle\\
&=&-\langle Ae_k,V\rangle+\langle N,\overline\nabla_{e_k}V\rangle,\end{aligned}$$ so that $$\begin{aligned}
\label{eq:I}
\Delta f&=&\sum_ke_k(e_k(f))=-\sum_ke_k\langle Ae_k,V\rangle+\sum_ke_k\langle N,\overline\nabla_{e_k}V\rangle\nonumber\\
&=&-\sum_k\langle\overline\nabla_{e_k}Ae_k,V\rangle-2\sum_k\langle Ae_k,\overline\nabla_{e_k}V\rangle+\sum_k\langle N,\overline\nabla_{e_k}\overline\nabla_{e_k}V\rangle.\end{aligned}$$

Now, differentiating $Ae_k=\sum_lh_{kl}e_l$ with respect to $e_k$, one gets at $p$ $$\begin{aligned}
\label{eq:II}
\sum_k\langle\overline\nabla_{e_k}Ae_k,V\rangle&=&\sum_{k,l}e_k(h_{kl})\langle e_l,V\rangle+\sum_{k,l}h_{kl}\langle\overline\nabla_{e_k}e_l,V\rangle\nonumber\\
&=&\sum_{k,l}\alpha_le_k(h_{kl})-\sum_{k,l}h_{kl}\langle\overline\nabla_{e_k}e_l,N\rangle\langle V,N\rangle\nonumber\\
&=&\sum_{k,l}\alpha_le_k(h_{kl})-\sum_{k,l}h_{kl}^2f\nonumber\\
&=&\sum_{k,l}\alpha_le_k(h_{kl})-f|A|^2.\end{aligned}$$

Asking further that $Ae_k=\lambda_ke_k$ at $p$ (which is always possible), we have at $p$ $$\label{eq:III}
\sum_k\langle Ae_k,\overline\nabla_{e_k}V\rangle=\sum_k\lambda_k\langle e_k,\overline\nabla_{e_k}V\rangle=\sum_k\lambda_k\psi=-nH\psi.$$

In order to compute the last summand of (\[eq:I\]), note that the conformality of $V$ gives $$\langle\overline\nabla_NV,e_k\rangle+\langle N,\overline\nabla_{e_k}V\rangle=0$$ for all $k$. Hence, differentiating the above relation in the direction of $e_k$, we get $$\langle\overline\nabla_{e_k}\overline\nabla_NV,e_k\rangle+\langle\overline\nabla_NV,\overline\nabla_{e_k}e_k\rangle+\langle\overline\nabla_{e_k}N,\overline\nabla_{e_k}V\rangle+\langle N,\overline\nabla_{e_k}\overline\nabla_{e_k}V\rangle=0.$$ However, at $p$ one has $$\begin{aligned}
\langle\overline\nabla_NV,\overline\nabla_{e_k}e_k\rangle&=&-\langle\overline\nabla_NV,\langle\overline\nabla_{e_k}e_k,N\rangle N\rangle=-\langle\overline\nabla_NV,\lambda_kN\rangle\\
&=&-\lambda_k\psi\langle N,N\rangle=\lambda_k\psi\end{aligned}$$ and $$\langle\overline\nabla_{e_k}N,\overline\nabla_{e_k}V\rangle=-\lambda_k\langle e_k,\overline\nabla_{e_k}V\rangle=-\lambda_k\psi,$$ so that $$\label{eq:IV}
\langle\overline\nabla_{e_k}\overline\nabla_NV,e_k\rangle+\langle N,\overline\nabla_{e_k}\overline\nabla_{e_k}V\rangle=0$$ at $p$. On the other hand, since $$[N,e_k](p)=(\overline\nabla_Ne_k)(p)-(\overline\nabla_{e_k}N)(p)=\lambda_ke_k(p),$$ it follows from (\[eq:IV\]) that $$\begin{aligned}
\langle\overline R(N,e_k)V,e_k\rangle_p&=&\langle\overline\nabla_{e_k}\overline\nabla_NV-\overline\nabla_N\overline\nabla_{e_k}V+\overline\nabla_{[N,e_k]}V,e_k\rangle_p\\
&=&-\langle N,\overline\nabla_{e_k}\overline\nabla_{e_k}V\rangle_p-N\langle\overline\nabla_{e_k}V,e_k\rangle_p+\langle\overline\nabla_{\lambda_ke_k}V,e_k\rangle_p\\
&=&-\langle N,\overline\nabla_{e_k}\overline\nabla_{e_k}V\rangle_p-N(\psi)+\lambda_k\psi,\end{aligned}$$ and hence $$\label{eq:V}
\sum_k\langle N,\overline\nabla_{e_k}\overline\nabla_{e_k}V\rangle_p=-nN(\psi)-nH\psi-\overline{Ric}(N,V)_p$$ Finally, $$\begin{aligned}
\overline{Ric}(N,V)&=&\sum_l\alpha_l\overline{Ric}(N,e_l)-f\overline{Ric}(N,N)\\
&=&\sum_{k,l}\alpha_l\langle\overline R(e_k,e_l)e_k,N\rangle-f\overline{Ric}(N,N),\end{aligned}$$ and $$\begin{aligned}
\langle\overline R(e_k,e_l)e_k,N\rangle_p&=&\langle\overline\nabla_{e_l}\overline\nabla_{e_k}e_k-\overline\nabla_{e_k}\overline\nabla_{e_l}e_k,N\rangle_p\\
&=&e_l\langle\overline\nabla_{e_k}e_k,N\rangle_p-\langle\overline\nabla_{e_k}e_k,\overline\nabla_{e_l}N\rangle_p-e_k\langle\overline\nabla_{e_l}e_k,N\rangle_p\\
&&+\langle\overline\nabla_{e_l}e_k,\overline\nabla_{e_k}N\rangle_p\\
&=&-e_l\langle e_k,\overline\nabla_{e_k}N\rangle_p+e_k\langle e_k,\overline\nabla_{e_l}N\rangle_p\\
&=&e_l(h_{kk})-e_k(h_{kl}),\end{aligned}$$ so that $$\overline{Ric}(N,V)_p=\sum_{k,l}\alpha_le_l(h_{kk})-\sum_{k,l}\alpha_le_k(h_{kl})-f\overline{Ric}(N,N)_p,$$ and it follows from (\[eq:V\]) that $$\begin{aligned}
\label{eq:VI}
\sum_k\langle N,\overline\nabla_{e_k}\overline\nabla_{e_k}V\rangle_p&=&-nN(\psi)-nH\psi+V^T(nH)\nonumber\\
&&+\sum_{k,l}\alpha_le_k(h_{kl})+f\overline{Ric}(N,N).\end{aligned}$$

Substituting (\[eq:II\]), (\[eq:III\]) and (\[eq:VI\]) into (\[eq:I\]), one gets the desired formula (\[eq:Laplacian formula\_I\]).

Applications
============

A particular class of conformally stationary spacetimes is that of [*generalized Robertson-Walker*]{} spacetimes [@ABC:03], namely, warped products $\overline M^{n+1}=I\times_{\phi}F^n$, where $I\subseteq\mathbb R$ is an interval with the metric $-dt^2$, $F^n$ is an $n$-dimensional Riemannian manifold and $\phi:I\rightarrow\mathbb R$ is positive and smooth. For such a space, let $\pi_I:\overline M^{n+1}\rightarrow I$ denote the canonical projection onto the $I-$factor. Then the vector field $$V=(\phi\circ\pi_I)\frac{\partial}{\partial t}$$ is conformal, timelike and closed (in the sense that its dual $1-$form is closed), with conformal factor $\psi=\phi'$, where the prime denotes differentiation with respect to $t$. Moreover, according to [@Montiel:99], for $t_0\in I$, orienting the (spacelike) leaf $M_{t_0}^n=\{t_0\}\times F^n$ by using the future-pointing unit normal vector field $N$, it follows that $M_{t_0}$ has constant mean curvature $$H=\frac{\phi'(t_0)}{\phi(t_0)}.$$

If $\overline M^{n+1}=I\times_{\phi}F^n$ is a generalized Robertson-Walker spacetime and $x:M^n\rightarrow\overline M^{n+1}$ is a complete spacelike hypersurface of $\overline M^{n+1}$, such that $\phi\circ\pi_I$ is limited on $M$, then $\pi_F\big|_M:M^n\rightarrow F^n$ is necessarily a covering map ([@ABC:03]). In particular, if $M^n$ is closed, then $F^n$ is automatically closed.

One has the following corollary of proposition \[prop:Laplacian of conformal vector field\]:

Let $\overline M^{n+1}=I\times_{\phi}F^n$ be a generalized Robertson-Walker spacetime, and $x:M^n\rightarrow\overline M^{n+1}$ a spacelike hypersurface of $\overline M^{n+1}$, having constant mean curvature $H$. Let also $N$ be a future-pointing unit normal vector field globally defined on $M^n$. If $V=(\phi\circ\pi_I)\frac{\partial}{\partial t}$ and $f=\langle V,N\rangle$, then $$\label{eq:Laplacian formula_II}
\Delta f=\left\{\overline{Ric}(N,N)+|A|^2\right\}f+n\left\{H\phi'+\phi''\langle N,\frac{\partial}{\partial t}\rangle\right\}.$$ where $\overline{Ric}$ denotes the Ricci tensor of $\overline M$, $A$ is the second fundamental form of $x$ with respect to $N$, and $H=-\frac{1}{n}{\rm tr}(A)$ is the mean curvature of $x$.

First of all, $f=\langle V,N\rangle=\phi\langle N,\frac{\partial}{\partial t}\rangle$, and it thus follows from (\[eq:Laplacian formula\_I\]) that $$\Delta f=\left\{\overline{Ric}(N,N)+|A|^2\right\}f+n\left\{H\phi'-N(\phi')\right\}.$$ However, $$\overline\nabla\phi'=-\langle\overline\nabla\phi',\frac{\partial}{\partial t}\rangle\frac{\partial}{\partial t}=-\phi''\frac{\partial}{\partial t},$$ so that $$N(\phi')=\langle N,\overline\nabla\phi'\rangle=-\phi''\langle N,\frac{\partial}{\partial t}\rangle$$

We can now state and prove our main result:

Let $\overline M^{n+1}=I\times_{\phi}F^n$ be a generalized Robertson-Walker spacetime, and $x:M^n\rightarrow\overline M^{n+1}$ be a closed spacelike hypersurface of $\overline M^{n+1}$, having constant mean curvature $H$. If the warping function $\phi$ satisfies $\phi''\geq\max\{H\phi',0\}$ and $M^n$ is strongly stable, then $M^{n}$ is either minimal or a spacelike slice $M_{t_0}=\{t_0\}\times F$, for some $t_0\in I$.

Since $M^n$ is strongly stable, we have $$0\geq\mathcal J''(0)(g)=\int_Mg\left\{\Delta g-\left(\overline{Ric}(N,N)+|A|^2\right)g\right\}dM$$ for all $g\in C^{\infty}(M)$ for which $gN$ is the normal component of the variational field of some variation of $M^n$. In particular, if $f=\langle V,N\rangle=\phi\langle N,\frac{\partial}{\partial t}\rangle$, where $V=(\phi\circ\pi_I)\frac{\partial}{\partial t}$, and $g=-f=-\langle V,N\rangle$, then $$\Delta g=\left\{\overline{Ric}(N,N)+|A|^2\right\}g-n\left\{H\phi'+\phi''\langle N,\frac{\partial}{\partial t}\rangle\right\}.$$ Therefore, $M^n$ stable implies $$0\geq\int_M\phi\langle N,\frac{\partial}{\partial t}\rangle\left\{H\phi'+\phi''\langle N,\frac{\partial}{\partial t}\rangle\right\}dM$$ Letting $\theta$ be the hyperbolic angle between $N$ and $\frac{\partial}{\partial t}$, it follows from the reversed Cauchy-Schwarz inequality that $\cosh\theta=-\langle N,\frac{\partial}{\partial t}\rangle$, with $\cosh\theta\equiv 1$ if and only if $N$ and $\frac{\partial}{\partial t}$ are collinear at every point, that is, if and only if $M^n$ is a spacelike leaf $M_{t_0}$ for some $t_0\in I$. Hence, $$0\geq\int_M\phi\cosh\theta\left\{-H\phi'+\phi''\cosh\theta\right\}dM.$$

Now, notice that $-H\phi'+\phi''\cosh\theta\geq-\phi''+\phi''\cosh\theta$, which gives $$\phi\cosh\theta(-H\phi'+\phi''\cosh\theta)\geq\phi\phi''\cosh\theta(\cosh\theta-1).$$ Therefore, $$0\geq\int_M\phi\cosh\theta(-H\phi'+\phi''\cosh\theta)dM\geq\int_M\phi\phi''\cosh\theta(\cosh\theta-1)\geq 0,$$ and hence $$\phi''(\cosh\theta-1)=0\ \ \text{and}\ \ \phi''=H\phi'$$ on $M$. If, for some $p\in M$, one has $\phi''(p)=0$, then $\phi'H=0$ at $p$. If $H\neq 0$, then $\phi'(p)=0$. But if this is the case, then proposition 7.35 of [@O'Neill:83] gives that $$\overline\nabla_V\frac{\partial}{\partial t}=\frac{\phi'}{\phi}V=0$$ at $p$ for any $V$, and $M$ is totally geodesic at $p$. In particular, $H=0$, a contradiction. Therefore, either $\phi''(p)=0$ for some $p\in M$, and $M$ is minimal, or $\phi''\neq 0$ on all of $M$, whence $\cosh\theta=1$ always, and $M$ is an umbilical leaf such that $\phi''=H\phi'$.

Note that $\frac{\phi''}{\phi'}=H=\frac{\phi'}{\phi}$, i.e., $\phi''\phi-(\phi')^2=0$, which is a limit case of Alías and Montiel’s timelike convergent condition.

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