# Stable maximal hypersurfaces in Lorentzian spacetimes

Giulio Colombo  Dipartimento di Matematica, Università degli Studi di Milano,
20133 Milano, Italy<EMAIL_ADDRESS>,  José A. S. Pelegrín
Departamento de Geometría y Topología, Universidad de Granada, 18071 Granada,
Spain Departamento de Matemática Aplicada y Estadística, Universidad CEU San
Pablo, 28003 Madrid, Spain<EMAIL_ADDRESS><EMAIL_ADDRESS>and
Marco Rigoli  Dipartimento di Matematica, Università degli Studi di Milano,
20133 Milano, Italy<EMAIL_ADDRESS>
###### Abstract.

We study the geometry of stable maximal hypersurfaces in a variety of
spacetimes satisfying various physically relevant curvature assumptions, for
instance the Timelike Convergence Condition (TCC). We characterize stability
when the target space has constant sectional curvature as well as give
sufficient conditions on the geometry of the ambient spacetime (e.g., the
validity of TCC) to ensure stability. Some rigidity results and height
estimates are also proven in GRW spacetimes. In the last part of the paper we
consider $k$-stability of spacelike hypersurfaces, a concept related to mean
curvatures of higher orders.

MSC 2010 Primary: 53C24, 53C42, 35J20; Secondary: 35P15, 53C50, 53C80, 83C99

Keywords Stable maximal hypersurface $\cdot$ $k$-stability $\cdot$ Spacetime
of constant sectional curvature $\cdot$ Generalized Robertson-Walker spacetime

##  1\. Introduction

In the last decades, maximal hypersurfaces in spacetimes have attracted a
great deal of mathematical and physical interest. The importance of this
family of spacelike hypersurfaces in General Relativity is well-known and a
summary of several reasons justifying this opinion can be found, for instance,
in [30]. Among them, we emphasize the key role they play in the study of the
Cauchy problem [19, 28] as well as their importance in the proof of the
positivity of the gravitational mass [41]. Furthermore, maximal hypersurfaces
describe, in some relevant cases, the transition between the expanding and
contracting phases of a relativistic universe. Finally, the existence of
constant mean curvature (and in particular maximal) hypersurfaces is useful in
the study of the structure of singularities in the space of solutions of the
Einstein equations [8]. At last, we should also mention their use in numerical
relativity for integrating forward in time [27].

From a mathematical point of view, maximal hypersurfaces in a spacetime
$\overline{M}$ are (locally) critical points for a natural variational
problem, namely, that of the area functional (see, for instance, [12]) and
their study is helpful for understanding the structure of $\overline{M}$ [9].
In particular, for some asymptotically flat spacetimes, maximal hypersurfaces
produce a foliation of the spacetime, defining a time function [14]. Classical
papers dealing with uniqueness of maximal hypersurfaces are, for instance,
[14, 18], although a previous relevant result in this direction was the proof
given by Cheng and Yau [17] of the Bernstein-Calabi conjecture [15]: spacelike
affine hyperplanes are the only complete maximal hypersurfaces in the the
$(m+1)$-dimensional Lorentz-Minkowski spacetime. Nishikawa [32] extended their
result by proving that any complete maximal hypersurface immersed in a
spacetime $\overline{M}$ is totally geodesic when $\overline{M}$ belongs to a
family of locally symmetric Lorentzian manifolds that includes spacetimes of
nonnegative constant curvature. Ishihara [26] showed that this property is not
shared by spacetimes of negative constant curvature by exhibiting an example
of a complete maximal hypersurface with constant nonzero norm of the shape
operator in the $(m+1)$-dimensional anti-de Sitter spacetime of curvature
$-1$. In Theorem 7 below we prove a slight generalization of Nishikawa’s
result by proving an upper bound on the norm of the shape operator first
obtained by Ishihara in the case of ambient spacetimes of constant curvature.
More recently, new uniqueness results for maximal hypersurfaces have been
found in a large variety of spacetimes by means of different techniques [6,
35, 37].

In this paper we will focus on a particular family of maximal hypersurfaces,
namely, stable maximal hypersurfaces, that is, critical points of the volume
functional for compactly supported normal variations with non-positive second
variation. A mild condition on the curvature of the ambient spacetime is
enough to ensure stability of maximal hypersurfaces.

######  Theorem A.

Let $\overline{M}$ be a spacetime with nonnegative Ricci curvature on timelike
vectors. If $\psi:M\longrightarrow\overline{M}$ is a (not necessarily
complete) oriented maximal hypersurface, then $\psi$ is stable. If $M$ is also
compact, then $\psi$ is totally geodesic.

Note that in an oriented spacetime $\overline{M}$, the time orientation of
$\overline{M}$ ensures that every spacelike hypersurface is oriented. In
General Relativity, a spacetime with nonnegative Ricci curvature on timelike
vectors is said to obey the Timelike Convergent Condition (TCC). It is usually
argued that the TCC is the mathematical way to express that gravity, on
average, attracts (see [33]). Theorem A generalizes Corollary 5.6 of [6] and
Theorem 1 of [35], where the authors show that compact maximal hypersurfaces
in a spacetime $\overline{M}$ obeying the TCC are totally geodesic by also
assuming the existence of certain infinitesimal symmetries in $\overline{M}$.
As a corollary, we also have an alternative proof of Theorem 4.1 of [14], a
uniqueness result for vacuum spacetimes.

######  Corollary B.

Let $M$ be a compact maximal hypersurface in a spacetime that obeys the
Einstein vacuum equations without cosmological constant. Then, $M$ is totally
geodesic.

If a maximal hypersurface $\psi:M\to\overline{M}$ is unstable, then there
exist spacelike hypersurfaces of larger volume in $\overline{M}$ nearby
$\psi$. This happens, for instance, for the equator of de Sitter spacetime,
which is a saddle point for the volume functional. In fact, we have the
following

######  Theorem C.

Let $\overline{M}$ be an $(m+1)$-dimensional spacetime of constant curvature
$\overline{\kappa}$ and let $\psi:M\longrightarrow\overline{M}$ be a complete
oriented maximal hypersurface.

  * i)

If $\overline{\kappa}>0$ then $M$ is compact and the immersion $\psi$ is
totally geodesic and unstable.

  * ii)

If $\overline{\kappa}=0$ then $\psi$ is totally geodesic and stable.

  * iii)

If $\overline{\kappa}<0$ then $\psi$ is stable and the shape operator $A$ and
the scalar curvature $\mathrm{S}$ of $M$ satisfy

$\mathrm{trace}(A^{2})\leq-m\overline{\kappa},\qquad\mathrm{S}\leq(m-2)m\overline{\kappa}.$

If $M$ is also compact, then $\psi$ is totally geodesic.

In case the ambient manifold $\overline{M}$ is a $3$-dimensional spacetime of
constant curvature, we are able to provide more information on the topology of
orientable complete maximal surfaces.

######  Theorem D.

Let $\overline{M}$ be a $3$-dimensional spacetime of constant curvature
$\overline{\kappa}$, $M$ a complete maximal oriented surface in
$\overline{M}$.

  * i)

If $\overline{\kappa}>0$ then $M$ is a totally geodesic, unstable round sphere
of constant curvature $\overline{\kappa}$.

  * ii)

If $\overline{\kappa}=0$ then $M$ is totally geodesic, stable and it is either
a Euclidean plane, or a flat cylinder or a flat torus.

  * iii)

If $\overline{\kappa}<0$ then $M$ is stable and has non-positive Gaussian
curvature. If $M$ is compact then it is totally geodesic and its Euler
characteristic satisfies

$\chi(M)=\frac{\overline{\kappa}}{2\pi}\,\mathrm{Vol}(M).$

If $M$ is non-compact but its total curvature and its Euler characteristic are
finite, then

$\chi(M)\geq\frac{\overline{\kappa}}{2\pi}\,\mathrm{Vol}(M).$

Stable maximal hypersurfaces have been previously studied in [14] and [20],
where the authors introduced the relative variational formulas and some
characterizations in certain ambient spacetimes. More recently, sufficient
conditions to ensure stability, in some physically relevant spacetimes, have
been given in [21]. In fact, a maximal hypersurface
$\psi:M\longrightarrow\overline{M}$, with unit normal vector $N$ and shape
operator $A$, is stable if and only if the differential operator
$L=\Delta-\overline{\mathrm{Ric}}(N,N)-\mathrm{trace}(A^{2})$ has non-negative
first eigenvalue on $M$. More generally, a maximal hypersurface is said to
have finite index if the stability operator $L$ has finite index. In a
$3$-dimensional spacetime with nonnegative Ricci curvature on spacelike
directions, we prove that any complete maximal surface with finite index has
either finite or positive infinite total curvature, provided it is well
defined. We remark that when $M$ is an oriented surface with Gaussian
curvature $\mathrm{K}$, its total curvature is defined as

$\int_{M}\mathrm{K}=\int_{M}\mathrm{K}_{+}-\int_{M}\mathrm{K}_{-},$

where $\mathrm{K}_{+}$ and $\mathrm{K}_{-}$ are the positive and negative
parts of $\mathrm{K}$. Hence, when $M$ is noncompact, the total curvature is
well defined only if at least one of the integrals on the right side is
finite.

######  Theorem E.

Let $\overline{M}$ be a $3$-dimensional spacetime with nonnegative Ricci
curvature on spacelike vectors, $\psi:M\longrightarrow\overline{M}$ a complete
maximal oriented surface immersed in $\overline{M}$. If $M$ has finite index
and its total curvature is well defined, then

$\int_{M}\mathrm{K}>-\infty.$

A physically relevant family of Lorentzian manifolds is that of Generalized
Robertson-Walker (GRW) spacetimes. They can be defined as product manifolds
$\overline{M}=I\times F$, where $I\subseteq\mathbb{R}$ is an open interval
with the standard negative definite metric $-dt^{2}$ and $(F,g_{F})$ is a
Riemannian manifold. On $\overline{M}$ we put a Lorentzian warped product
metric of the form $\overline{g}=-dt^{2}+\rho(t)^{2}g_{F}$, with $\rho$ a
smooth positive function on $I$. In these ambient manifolds we give the
following generalization of the first part of Theorem D, suggested by the work
of Albujer and Alías, [1], where the authors consider maximal surfaces in
Lorentzian products, that is, the case $\rho\equiv 1$.

######  Theorem F.

Let $\overline{M}$ be a $3$-dimensional GRW spacetime
$\overline{M}=I\times_{\rho}F$ with nonnegative sectional curvatures on
spacelike $2$-planes and let $\psi:M\longrightarrow\overline{M}$ be a complete
maximal surface. Then $\psi$ is totally geodesic and one of the following
cases occurs:

  * i)

$\psi(M)$ is a spacelike slice $\\{t_{0}\\}\times F$ for some $t_{0}\in I$
such that $\rho^{\prime}(t_{0})=0$,

  * ii)

$F$ is a Riemann surface with a complete, flat metric $g_{F}$ and
$\overline{M}$ is the product manifold $\mathbb{R}\times F$ with the flat
metric $-dt^{2}+g_{F}$,

  * iii)

$F$ is a compact Riemann surface with a metric $g_{F}$ of constant positive
Gaussian curvature, $M$ is a round sphere and the spacetime $\overline{M}$ has
constant positive curvature in the smallest slab $I^{\prime}\times
F\subseteq\overline{M}$, $I^{\prime}\subseteq I$, such that $\psi(M)\subseteq
I^{\prime}\times F$.

In a GRW spacetime with sectional curvatures bounded below, the image of a
complete maximal hypersurface whose projection on the $I$ factor is relatively
compact in $I$ must always intersect at least one totally geodesic spacelike
slice of the ambient spacetime. More precisely, we have the following

######  Theorem G.

Let $\overline{M}=I\times_{\rho}F$ be a GRW spacetime whose sectional
curvatures on spacelike $2$-planes are bounded below. Let
$\psi:M\longrightarrow\overline{M}$ be a complete maximal hypersurface,
$I^{\prime}\subseteq I$ the smallest interval closed in $I$ such that
$\psi(M)\subseteq I^{\prime}\times F$. If $I^{\prime}=[t_{\ast},t^{\ast}]$ for
some $t_{\ast},t^{\ast}\in I$, then $\rho^{\prime}(t^{\ast})\geq 0$ and
$\rho^{\prime}(t_{\ast})\leq 0$. In particular, if $\psi(M)$ is contained in a
slab $[a,b]\times F$, $a,b\in I$, then there exists $t_{0}\in[a,b]$ such that
$\rho^{\prime}(t_{0})=0$ and

  * i)

if $\rho^{\prime}<0$ on $[a,t_{0})$ and $\rho^{\prime}>0$ on $(t_{0},b]$, then
$\psi(M)$ must intersect the spacelike slice $\\{t_{0}\\}\times F$,

  * ii)

if $\rho^{\prime}>0$ on $[a,t_{0})$ and $\rho^{\prime}<0$ on $(t_{0},b]$, then
$\psi(M)=\\{t_{0}\\}\times F$.

As an application of Theorems G and C, we give a simple proof of the following
Frankel type result.

######  Corollary H.

Let $\overline{M}=\mathbb{S}^{m+1}_{1}(\overline{\kappa})$ be the
$(m+1)$-dimensional de Sitter spacetime of constant curvature
$\overline{\kappa}>0$ and let $\psi_{0}:M_{0}\to\overline{M}$,
$\psi_{1}:M_{1}\to\overline{M}$ be two complete maximal hypersurfaces. Then
$\psi_{0}(M_{0})\cap\psi_{1}(M_{1})\neq\varnothing$.

If $\rho$ satisfies $\rho^{\prime\prime}\leq 0$ on $I$, then every maximal
hypersurface in $\overline{M}=I\times_{\rho}F$ is stable, as observed in
Theorem 9 of [21]. On the other hand, if a compact maximal hypersurface $M$ is
stable in $\overline{M}$ and $\rho^{\prime\prime}\geq 0$ on $I$, then
$\rho^{\prime\prime}\equiv 0$ on $M$. More precisely, we have the following

######  Theorem I.

Let $M$ be a complete oriented stable maximal hypersurface in a GRW spacetime
$\overline{M}=I\times_{\rho}F$ and let $I^{\prime}\subseteq I$ be the smallest
interval such that $\psi(M)\subseteq I^{\prime}\times F$.

  * i)

If $M$ is compact then either $\rho^{\prime\prime}\equiv 0$ or
$\rho^{\prime\prime}$ attains both positive and negative values on
$I^{\prime}$.

  * ii)

If $M$ is non-compact and, for some $o\in M$, the normal vector field $N$ of
$M$ satisfies

$\lim_{r\to+\infty}\int_{a}^{r}\left(\int_{\partial_{B_{r}}}\overline{g}(T,N)^{2}\right)^{-1}=+\infty,$

for some (hence any) $a>0$, where $B_{r}$ is the geodesic ball of $(M,g)$ with
radius $r$ centered at $o$ and $T=\rho(t)\partial_{t}$, then either
$\rho^{\prime\prime}\equiv 0$ on $I^{\prime}$ or there exists $t_{0}\in
I^{\prime}$ such that $\rho^{\prime\prime}(t_{0})<0$.

In this work we also obtain new results for higher order mean curvatures. In
particular, we study the $k$-stability of spacelike hypersurfaces with zero
$(k+1)$-th mean curvature in spacetimes of constant curvature. The notion of
$k$-stability has been previously studied in the Lorentzian setting in [13] as
well as in [16]. The following two results are somehow companions of Theorems
A, E and G in this context. Note that the requirements on the sign of the
$k$-th mean curvature function $H_{k}$ and on the rank of the shape operator
$A$ are minimal to guarantee the ellipticity of the $k$-stability operator of
$\psi$, as defined in Section 2.2.

######  Theorem J.

Let $\psi:M\longrightarrow\overline{M}$ be a complete oriented spacelike
hypersurface with zero $(k+1)$-th mean curvature, for some $1\leq k\leq m-2$,
in a spacetime $\overline{M}$ of constant curvature $\overline{\kappa}$.
Suppose that the $k$-th mean curvature function $H_{k}$ is positive and that
the shape operator $A$ has rank $>k$ on $M$.

  * i)

If $\overline{\kappa}\leq 0$, then $M$ is non-compact and $\psi$ is
$k$-stable.

  * ii)

If $\overline{\kappa}>0$ and $M$ is compact, then $\psi$ is not $k$-stable.

  * iii)

If $\overline{\kappa}>0$ and we assume that $M$ is non-compact and that, for
some $o\in M$ and for some (hence any) $a\in\mathbb{R}$,

$\displaystyle\lim_{r\to+\infty}\int_{a}^{r}\left(\int_{\partial
B_{r}}H_{k}\right)^{-1}=+\infty,$
$\displaystyle\lim_{r\longrightarrow+\infty}\int_{B_{r}}\left(\overline{\kappa}\
\mathrm{trace}(P_{k})-\mathrm{trace}(A^{2}P_{k})\right)=+\infty,$

where $B_{r}$ is the geodesic ball of $(M,g)$ with radius $r$ centered at $o$
and $P_{k}$ is the $k$-th Newton operator associated to $\psi$ (as defined in
Section 2.2), then for every compact subset $K\subseteq M$ the hypersurface
$\psi|_{M\setminus K}:M\setminus K\longrightarrow\overline{M}$ is not
$k$-stable.

We remark that the quantity $\overline{\kappa}\
\mathrm{trace}(P_{k})-\mathrm{trace}(A^{2}P_{k})$ appearing in the statement
of Theorem J can be expressed in terms of the higher order mean curvature
functions $H_{1},\dots,H_{m}$ of $\psi$ as

(1) $\begin{split}\overline{\kappa}\
\mathrm{trace}(P_{k})-\mathrm{trace}(A^{2}P_{k})&=\binom{m}{k}(m-k)\overline{\kappa}H_{k}\\\
&\phantom{=\;}+\binom{m}{k+1}\left((m-k-1)H_{k+2}-mH_{1}H_{k+1}\right).\end{split}$

######  Theorem K.

Let $\overline{M}=I\times_{\rho}F$ be a Robertson-Walker spacetime of constant
sectional curvature and let $\psi:M\longrightarrow\overline{M}$ be a complete
oriented spacelike hypersurface with zero $(k+1)$-th mean curvature. Suppose
that there exists $C>0$ such that $C^{-1}<H_{k}<C$, that $\mathrm{rank}(A)>k$
on $M$ and that for some $o\in M$ and for some (hence any) $a\in\mathbb{R}$,
one of the following conditions is satisfied:

$\displaystyle(i)$
$\displaystyle\qquad\lim_{r\to+\infty}\int_{a}^{r}\frac{dt}{\mathrm{Vol}(\partial
B_{r})}=+\infty$ $\displaystyle\qquad\text{and}\qquad\mathrm{Vol}(M)=+\infty,$
or $\displaystyle(ii)$
$\displaystyle\qquad\lim_{r\to+\infty}\int_{a}^{r}\frac{dt}{\mathrm{Vol}(\partial
B_{r})}<+\infty$
$\displaystyle\qquad\text{and}\qquad\lim_{r\to+\infty}\frac{\log(\mathrm{Vol}(\partial
B_{r}))}{r}=0,$

where $B_{r}$ is the geodesic ball of $(M,g)$ with radius $r$ centered at $o$.
If $\psi(M)$ is contained in a slab $[a,b]\times F$, then there exists
$t_{0}\in[a,b]$ such that $\rho^{\prime}(t_{0})=0$ and $\psi(M)$ must
intersect the spacelike slice $\\{t_{0}\\}\times F$.

Our paper is organized as follows. In Section 2 we recall the notion of
stability and $k$-stability for spacelike hypersurfaces with vanishing mean
curvature functions together with some general properties of Schrödinger
differential operators that we will need in the subsequent sections. In
Section 3 we mainly deal with maximal hypersurfaces in locally symmetric
spacetimes and we prove Theorems A, C, D and E above (see Theorems 9, 10, 11
and 12, respectively). In Section 4 we study maximal hypersurfaces in GRW
spacetimes and we prove Theorems F, G, I and Corollary H (see Theorems 22, 24
and Corollaries 20, 21), also giving a characterization of GRW spacetimes with
spacelike sectional curvatures bounded below (Lemma 16). In section 5 we
consider hypersurfaces with zero $(k+1)$-th order mean curvature in spacetimes
of constant curvature and we prove Theorems J and K (see Theorems 27 and 28).

##  2\. Preliminaries

We devote this section to introduce the basic concepts concerning the
stability of maximal hypersurfaces in general ambient spacetimes as well as
their natural generalization to the case of $k$-stable spacelike hypersurfaces
with zero $(k+1)$-th mean curvature.

###  2.1. Stability of maximal hypersurfaces

Let $\psi:M\longrightarrow\overline{M}$ be a spacelike hypersurface immersed
in a spacetime $(\overline{M},\overline{g})$ and let
$g=\psi^{\ast}\overline{g}$ be the Riemannian metric induced on $M$. We will
denote by $N$ a chosen unit normal vector field to $M$ and by $A$ the shape
operator in the direction of $N$, determined by the validity of the Weingarten
formula $AX=-\overline{\nabla}_{X}N$ for any $X\in TM$, with
$\overline{\nabla}$ the Levi-Civita connection of
$(\overline{M},\overline{g})$. The linear operator $A$ is self-adjoint at each
tangent space and its eigenvalues $\kappa_{1},\dots,\kappa_{m}$ are, by
definition, the principal curvatures of the hypersurface. The mean curvature
function $H$ of $\psi$ in the direction of $N$ is given by the normalized
trace of $-A$, that is,

(2)
${\color[rgb]{0,0,0}H=-\frac{1}{m}\mathrm{trace}(A)=-\frac{\kappa_{1}+\cdots+\kappa_{m}}{m}.}$

A spacelike hypersurface is called a maximal hypersurface if it is a critical
point of the volume functional for compactly supported normal variations of
the immersion. This condition is equivalent to the hypersurface having zero
mean curvature. If $\psi$ is a maximal hypersurface and we take a normal
variation driven by a variational vector field $\phi N$, with $\phi$ a smooth
function supported in a relatively compact domain $\Omega\subseteq M$, then
the second variation of the volume of $\Omega$ is (see Theorem 2.1 of [14] and
Theorem 1 of [20])

(3)
$\int_{\Omega}\left(\Delta\phi-\left(\overline{\mathrm{Ric}}(N,N)+\mathrm{trace}(A^{2})\right)\phi\right)\phi\
dV,$

where $\nabla$ is the Levi-Civita connection of $(M,g)$, $\Delta$ is the
corresponding Laplacian and $\overline{\mathrm{Ric}}$ is the Ricci tensor of
$(\overline{M},\overline{g})$. In view of the identity

(4)
$\mathrm{S}=\overline{\mathrm{S}}+2\overline{\mathrm{Ric}}(N,N)+\mathrm{trace}(A^{2})$

satisfied by the scalar curvatures $\mathrm{S}$, $\overline{\mathrm{S}}$ of
$M$ and $\overline{M}$ (see formula (28) in Section 2.4), we have

(5)
$\overline{\mathrm{Ric}}(N,N)+\mathrm{trace}(A^{2})=\mathrm{S}-\left(\overline{\mathrm{S}}+\overline{\mathrm{Ric}}(N,N)\right)=\sum_{i=1}^{m}\left(\mathrm{Ric}(E_{i},E_{i})-\overline{\mathrm{Ric}}(E_{i},E_{i})\right)$

for any local orthonormal frame $\\{E_{1},\dots,E_{m}\\}$ on $TM$, where
$\mathrm{Ric}$ is the Ricci tensor of $(M,g)$. The immersion $\psi$ is said to
be stable if the second variation of the volume of $M$ is non-positive for
every compactly supported normal variation. Stability is detected by the sign
of the first eigenvalue $\lambda_{1}^{L}(M)$ of the stability operator $L$
defined by

(6) $Lu=\Delta
u-\left(\overline{\mathrm{Ric}}(N,N)+\mathrm{trace}(A^{2})\right)u\qquad\text{for
every }u\in C^{2}(M),$

analogously to what happens in the Riemannian case for minimal hypersurfaces
(see for instance [23, 42]). In fact, since $\lambda_{1}^{L}(M)$ is
variationally characterized by

(7) $\lambda_{1}^{L}(M)=\inf_{\begin{subarray}{c}\phi\in C_{c}^{\infty}(M)\\\
\ \phi\neq
0\end{subarray}}\frac{\int_{M}|\nabla\phi|^{2}+\left(\overline{\mathrm{Ric}}(N,N)+\mathrm{trace}(A^{2})\right)\phi^{2}}{\int_{M}\phi^{2}},$

by applying the divergence theorem to (3) we see that $\psi$ is stable if and
only if $\lambda_{1}^{L}(M)\geq 0$. More generally, $M$ is said to have finite
index if the operator $L$ has finite index. When $M$ is compact, this is
always the case, while for a complete, non-compact hypersurface this happens
if and only if there exists a relatively compact open set $\Omega\subseteq M$
such that the second variation of the volume of $M$ is non-positive for every
normal variation compactly supported in $M\setminus\overline{\Omega}$. In this
case, we also say that $M$ is stable at infinity.

###  2.2. $k$-stability of spacelike hypersurfaces with zero $(k+1)$-th mean
curvature

We can generalize the concepts above to study the $k$-stability of spacelike
hypersurfaces with zero $(k+1)$-th mean curvature in spacetimes of constant
curvature. In order to do so, let $\psi:M\longrightarrow\overline{M}$ be a
spacelike hypersurface in a spacetime $\overline{M}$ with constant curvature
$\overline{\kappa}$ and let $A$ be the shape operator of the immersion with
respect to a unit normal timelike vector $N$. We associate to $A$ the
algebraic invariants $S_{1},\dots,S_{m}$ and the mean curvature functions
$H_{1},\dots,H_{m}$ of orders $1,\dots,m$ in the direction of $N$ by setting

(8) $S_{k}=\sum_{1\leq i_{1}<\cdots<i_{k}\leq
m}\kappa_{i_{1}}\cdots\kappa_{i_{m}},\qquad
H_{k}=\frac{(-1)^{k}}{\binom{m}{k}}S_{k}\qquad\text{for }1\leq k\leq m.$

The Newton tensors $P_{k}:TM\longrightarrow TM$, $0\leq k\leq m$, are
inductively defined by

(9) $P_{0}=\mathbb{I}\qquad\text{and}\qquad\
P_{k}=\binom{m}{k}H_{k}\mathbb{I}+A\circ P_{k-1}\qquad\text{for }1\leq k\leq
m,$

where $\mathbb{I}$ denotes the identity on $TM$, and they satisfy the
following identities, proved in [2].

######  Lemma 1.

For $0\leq k\leq m-1$, let $c_{k}=(m-k)\binom{m}{k}$. Then, for every
$X\in\mathfrak{X}(M)$

(10)
$\begin{split}&\mathrm{trace}(P_{k})=c_{k}H_{k},\quad\mathrm{trace}(P_{k}\circ
A)=-c_{k}H_{k+1},\\\
&\mathrm{trace}(P_{k}\circ\nabla_{X}A)=-\binom{m}{k+1}\nabla_{X}H_{k+1}.\end{split}$

Note that the identity (1) stated in the Introduction follows by applying
relations (9) and (10). For $0\leq k\leq m$, let $L_{k}$ be the second order
linear differential operator given by

(11)
$L_{k}u=\mathrm{trace}\left(P_{k}\circ\mathrm{Hess}(u)\right)\equiv\mathrm{div}(P_{k}(\nabla
u))\qquad\text{for every }u\in C^{2}(M),$

where the second equality holds because $P_{k}$ is divergence-free as long as
$\overline{M}$ has constant curvature. Note that $H_{1}=H$ and that
$L_{0}=\Delta$. The operator $L_{k}$ is elliptic if and only if $P_{k}$ is
positive definite. For $1\leq k\leq m-2$, if $H_{k+1}=0$ at some point then
$P_{k}$ is positive definite there if and only if $H_{k}>0$ and
$\mathrm{rank}(A)>k$, see Corollary 2.3 of [25].

Now, for $0\leq k\leq m-1$ we can define the $k$-volume functional for a
relatively compact domain $\Omega\subset M$ by setting

$\mathrm{Vol}_{k}(\Omega)=\int_{\Omega}F_{k}(S_{1},S_{2},\dots,S_{k})dV,$

where $S_{1},\dots,S_{m}$ are the invariants defined above and the functions
$F_{0},\dots,F_{m-1}$ are recursively defined by

$F_{0}=1,\qquad F_{1}=-S_{1},\qquad
F_{k}=(-1)^{k}S_{k}-\frac{\overline{\kappa}(m-k+1)}{k-1}F_{k-2}\qquad\text{for
}2\leq k\leq m-1.$

According to [16], if $M$ has zero $(k+1)$-th mean curvature for some $1\leq
k\leq m-1$ and we take a normal variation of $M$ given by $\phi N$, with
$\phi$ supported in a relatively compact domain $\Omega\subseteq M$, then the
second variation of the $k$-volume functional of $M$ is

(12) $(k+1)\int_{\Omega}\left(L_{k}\phi+\overline{\kappa}\
\mathrm{trace}(P_{k})\phi-\mathrm{trace}(A^{2}P_{k})\phi\right)\phi\ dV.$

As a generalization of (5), for every local orthonormal frame
$\\{E_{1},\dots,E_{m}\\}$ on $TM$ we have

$-\overline{\kappa}\
\mathrm{trace}(P_{k})+\mathrm{trace}(A^{2}P_{k})=\sum_{i=1}^{m}\left(\mathrm{Ric}(P_{k}E_{i},E_{i})-\overline{\mathrm{Ric}}(P_{k}E_{i},E_{i})\right),$

as a consequence of formula (29) of Section 2.4 and of Lemma 1. The immersion
$\psi$ is said to be $k$-stable if the second variation of the $k$-volume of
$M$ is non-positive for every compactly supported normal variation.
Analogously to the maximal case, $k$-stability is detected by the sign of the
first eigenvalue $\lambda_{1}^{\widetilde{L}_{k}}(M)$ of the $k$-stability
operator $\widetilde{L}_{k}$ defined by

(13)
$\widetilde{L}_{k}u=L_{k}u-\left(\mathrm{trace}(A^{2}P_{k})-\overline{\kappa}\
\mathrm{trace}(P_{k})\right)u\qquad\text{for every }u\in C^{2}(M).$

Since $\lambda_{1}^{\widetilde{L}_{k}}(M)$ is variationally characterized by

(14) $\lambda_{1}^{\widetilde{L}_{k}}(M)=\inf_{\begin{subarray}{c}\phi\in
C_{c}^{\infty}(M)\\\ \ \phi\neq
0\end{subarray}}\frac{\int_{M}g(P_{k}(\nabla\phi),\nabla\phi)+\left(\mathrm{trace}(A^{2}P_{k})-\overline{\kappa}\
\mathrm{trace}(P_{k})\right)\phi^{2}}{\int_{M}\phi^{2}},$

we see that a spacelike hypersurface $M$ with zero $(k+1)$-th mean curvature
is $k$-stable if and only if $\lambda_{1}^{\widetilde{L}_{k}}(M)\geq 0$.
Similarly to what happens with the usual stability operator, a complete, non-
compact, spacelike hypersurface with zero $(k+1)$-th mean curvature is said to
be $k$-stable at infinity if the $k$-stability operator has finite index.

###  2.3. General facts on Schrödinger operators

Consider a Riemannian manifold $(M,g)$, a function $F(x)\in
L^{\infty}_{\mathrm{loc}}(M)$ and let $P:TM\longrightarrow TM$ be a positive
definite, self-adjoint, endomorphism of class $C^{1}$. Define the second order
linear elliptic operator $L$ by setting

(15) $Lu=\mathrm{div}(P(\nabla u))-F(x)u\qquad\text{for every }u\in C^{2}(M).$

For every open set $\Omega\subseteq M$, let $\lambda_{1}^{L}(\Omega)$ be the
first eigenvalue of $L$ on $\Omega$, given by

(16) $\lambda_{1}^{L}(\Omega)=\inf_{\begin{subarray}{c}\phi\in
C_{c}^{\infty}(\Omega)\\\ \ \phi\neq
0\end{subarray}}\frac{\int_{\Omega}g(P(\nabla\phi),\nabla\phi)+F(x)\phi^{2}}{\int_{\Omega}\phi^{2}}=\inf_{\begin{subarray}{c}\phi\in
C_{c}^{\infty}(\Omega)\\\ \ \phi\neq 0\end{subarray}}\frac{\int_{\Omega}-\phi
L\phi}{\int_{\Omega}\phi^{2}}$

If $\Omega$ is a relatively compact domain with sufficiently regular boundary,
say of class $C^{2}$, then the infimum in the RHS of (16) is achieved by the
non-zero solutions of the Dirichlet problem

$\displaystyle Lu+\lambda_{1}^{L}(\Omega)u$ $\displaystyle=0$
$\displaystyle\quad\text{on }\Omega,$ $\displaystyle u$ $\displaystyle=0$
$\displaystyle\quad\text{on }\partial\Omega,$

which belong to $C^{0,\alpha}(\overline{\Omega})\cap H^{2}(\Omega)\cap
C^{1}(\Omega)$ for some $0<\alpha<1$ (see Theorems 8.6, 8.12, 8.29 of [24] and
Theorem 1 of [43]). For operators of the form (15), we have the next
monotonicity property of the eigenvalues with respect to the domain.

######  Proposition 2.

Let $(M,g)$ be a Riemannian manifold and for $F(x)\in
L^{\infty}_{\mathrm{loc}}(M)$, $P:TM\longrightarrow TM$ let $L$ be the
operator defined in (15). Let $\Omega_{1},\Omega_{2}$ be two relatively
compact domains in $M$ such that $\Omega_{1}\subseteq\Omega_{2}$. Then

(17) $\lambda_{1}^{L}(\Omega_{1})\geq\lambda_{1}^{L}(\Omega_{2}).$

If $\Omega_{1}$ and $\Omega_{2}$ have $C^{2}$ boundaries and the interior of
$\Omega_{2}\setminus\Omega_{1}$ is nonempty, then (17) holds with strict
inequality sign.

  * Proof.

Observe that (17) is a trivial consequence of the definition (16). To prove
the last statement we will proceed as follows. Consider an open subset
$\Omega\subseteq M$ and two functions $u,v\in C^{1}(\Omega)$, with $v\neq 0$
on $\Omega$: since $P$ is positive definite and self-adjoint, a direct
computation yields the following extension of the classic Picone’s identity

(18) $0\leq g\left(P\left(\nabla u-\frac{u}{v}\nabla v\right),\nabla
u-\frac{u}{v}\nabla v\right)=g(P(\nabla u),\nabla
u)-g\left(\nabla\left(\frac{u^{2}}{v}\right),P(\nabla v)\right).$

In particular,

$g\left(P\left(\nabla u-\frac{u}{v}\nabla v\right),\nabla u-\frac{u}{v}\nabla
v\right)\equiv 0\qquad\text{on }\Omega$

if and only if $u=Cv$ for some constant $C\in\mathbb{R}$. Now, let us suppose
that $\Omega_{1}$ and $\Omega_{2}$ have $C^{2}$ boundaries and let $u$ and $v$
be non-zero solutions of the Dirichlet problems

$\displaystyle Lu+\lambda_{1}^{L}(\Omega_{1})u$ $\displaystyle=0$
$\displaystyle\quad\text{in }\Omega_{1},$ $\displaystyle
Lv+\lambda_{1}^{L}(\Omega_{2})v$ $\displaystyle=0$ $\displaystyle\quad\text{in
}\Omega_{2},$ $\displaystyle u$ $\displaystyle=0$ $\displaystyle\quad\text{on
}\partial\Omega_{1},$ $\displaystyle v$ $\displaystyle=0$
$\displaystyle\quad\text{on }\partial\Omega_{2}.$

Note that we can suppose $v>0$ on $\Omega_{2}$. Taking (18) into account,
since $u=0$ on $\partial\Omega_{1}$ we get

$\displaystyle 0$ $\displaystyle\leq\int_{\Omega_{1}}g(P(\nabla u),\nabla
u)-g\left(\nabla\left(\frac{u^{2}}{v}\right),P(\nabla
v)\right)=\left(\lambda_{1}^{L}(\Omega_{1})-\lambda_{1}^{L}(\Omega_{2})\right)\int_{\Omega_{1}}u^{2}.$

We now reason by contradiction assuming that
$\overset{\circ}{\Omega_{2}\setminus\Omega_{1}}\neq\varnothing$ and
$\lambda_{1}^{L}(\Omega_{1})=\lambda_{1}^{L}(\Omega_{2})$. From the above
inequalities it follows that on the connected components of $\Omega_{1}$,
$u=Cv$ for some $C\in\mathbb{R}$. Choose one of the component, say
$\widetilde{\Omega}_{1}$, with
$\partial\widetilde{\Omega}_{1}\cap\Omega_{2}\neq\varnothing$ (this is
possible since $\Omega_{2}\setminus\Omega_{1}$ has non-empty interior). Since
$u\equiv 0$ on $\partial\Omega_{1}$, we have $v\equiv 0$ on
$\partial\widetilde{\Omega}_{1}\cap\Omega_{2}\neq\varnothing$, reaching a
contradiction. $\square$

The following generalization of Barta’s theorem also holds.

######  Proposition 3.

Let $(M,g)$ be a Riemannian manifold and for $F(x)\in
L^{\infty}_{\mathrm{loc}}(M)$, $P:TM\longrightarrow TM$ let $L$ be the
operator defined in (15). If $u\in C^{2}(M)$ is a positive function, we have

(19) $\lambda_{1}^{L}(M)\geq\inf_{M}\left(-\frac{Lu}{u}\right).$

  * Proof.

Let $\phi\in C_{c}^{\infty}(M)$ and consider the vector field

$Z=-\frac{\phi^{2}}{u}P(\nabla u).$

Taking the divergence and integrating we get

(20) $2\int_{M}\frac{\phi}{v}g\left(P(\nabla
u),\nabla\phi\right)=-\int_{M}\phi^{2}\mathrm{div}\left(\frac{P(\nabla
u)}{u}\right).$

Since $P$ is positive definite and self-adjoint, by Cauchy-Schwarz and Young’s
inequalities

(21) $\begin{split}2\frac{\phi}{u}g\left(P(\nabla u),\nabla\phi\right)&\leq
2\frac{|\phi|}{u}\sqrt{g(P(\nabla u),\nabla
u)}\sqrt{g(P(\nabla\phi),\nabla\phi)}\\\ &\leq\frac{\phi^{2}}{u^{2}}g(P(\nabla
u),\nabla u)+g(P(\nabla\phi),\nabla\phi).\end{split}$

Using (21), (20) and the divergence theorem again we obtain

(22) $\begin{split}\int_{M}-\phi
L\phi&=\int_{M}g(P(\nabla\phi),\nabla\phi)+\int_{M}F(x)\phi^{2}\\\
&\geq-\int_{M}\frac{\phi^{2}}{u^{2}}g(P(\nabla u),\nabla
u)+2\int_{M}\frac{\phi}{v}g\left(P(\nabla
u),\nabla\phi\right)+\int_{M}F(x)\phi^{2}\\\
&=-\int_{M}\left(g\left(\frac{\nabla u}{u^{2}},P(\nabla
u)\right)+\mathrm{div}\left(\frac{P(\nabla
u)}{u}\right)-F(x)\right)\phi^{2}\\\
&=-\int_{M}\left(\frac{\mathrm{div}(P(\nabla
u))}{u}-F(x)\right)\phi^{2}=\int_{M}-\frac{Lv}{v}\phi^{2}\geq\inf_{M}\left(-\frac{Lv}{v}\right)\int_{M}\varphi^{2}\end{split}$

and by definition (16) of $\lambda_{1}^{L}(M)$ we obtain inequality (19).
$\square$

We conclude this paragraph with the following characterization of non-
negativity of $\lambda_{1}^{L}(\Omega)$ for an open subset $\Omega\subseteq
M$. For $P=\mathbb{I}:TM\longrightarrow TM$ and $F(x)\in C^{\infty}(M)$, it is
given as Theorem 1 in [23]. For $P=\mathbb{I}$ and $F(x)\in
L^{\infty}_{\mathrm{loc}}(M)$ it is proved as Lemma 3.10 of [39]. The proof
for a general self-adjoint, positive definite $C^{1}$ endomorphism $P$ is a
straightforward extension of the proof given in [39] for $P=\mathbb{I}$.

######  Lemma 4.

Let $(M,g)$ be a Riemannian manifold, $\Omega\subseteq M$ an open set with
possibly non-compact closure. For $F(x)\in L^{\infty}_{\mathrm{loc}}(M)$,
$P:TM\longrightarrow TM$ as above, let $\lambda_{1}^{L}(\Omega)$ be the first
eigenvalue on $\Omega$ of the operator $L$ defined in (15). Then, the
following conditions are equivalent:

  1. (1)

$\lambda_{1}^{L}(\Omega)\geq 0$.

  2. (2)

There exists $u\in C^{1}(\Omega)$, $u>0$, weak solution of $Lu=0$ on $\Omega$.

  3. (3)

There exists $u\in H^{1}_{\mathrm{loc}}(\Omega)$, $u>0$, weak solution of
$Lu\leq 0$ on $\Omega$.

We remark that when $P$ and $F(x)$ are smooth, which will always be the case
in the following, standard elliptic regularity ensures that solutions of
$Lu=0$ are also smooth.

###  2.4. Gauss equations

Our main reference is O’Neill’s book [33]. However, we remark that we adopt
the convention of defining the Riemann curvature operator $\mathrm{R}$ of a
semi-Riemannian manifold $(M,g)$ by setting

$R(X,Y)Z=\nabla_{X}(\nabla_{Y}Z)-\nabla_{Y}(\nabla_{X}Z)-\nabla_{[X,Y]}Z\qquad\text{for
every }X,Y,Z\in\mathfrak{X}(M),$

so we have $R(X,Y)Z=-R_{XY}Z=R_{YX}Z$ for every $X,Y,Z\in\mathfrak{X}(M)$,
where $R_{(\;\cdot\;,\;\cdot\;)}(\;\cdot\;)$ is the notation used in [33]. The
$(0,4)$-form Riemann curvature tensor $\mathrm{Riem}$ is then given by

$\mathrm{Riem}(W,Z,X,Y)=g(R(X,Y)Z,W)\qquad\text{for every
}X,Y,Z,W\in\mathfrak{X}(M),$

the sectional curvature of any non-degenerate $2$-plane $X\wedge Y\subseteq
TM$ spanned by a couple of vectors $X,Y\in TM$ is

(23) $\mathrm{Sect}(X\wedge
Y)=\frac{\mathrm{Riem}(X,Y,X,Y)}{g(X,X)g(Y,Y)-g(X,Y)^{2}}.$

For every $p\in M$ and for every choice of a $g$-orthonormal basis
$\\{E_{i},\dots,E_{\dim M}\\}$ of $T_{p}M$, the values of the Ricci tensor
$\mathrm{Ric}$ and of the scalar curvature $\mathrm{S}$ are given by

(24) $\displaystyle\mathrm{Ric}(X,Y)$ $\displaystyle=\sum_{i=1}^{\dim
M}g(E_{i},E_{i})\mathrm{Riem}(X,E_{i},Y,E_{i})\qquad\text{for every }X,Y\in
T_{p}M,$ (25) $\displaystyle\mathrm{S}(p)$ $\displaystyle=\sum_{i=1}^{\dim
M}g(E_{i},E_{i})\mathrm{Ric}(E_{i},E_{i}).$

Let $\psi:M\longrightarrow\overline{M}$ be a spacelike hypersurface immersed
in a spacetime $\overline{M}$, with unit timelike vector field $N$ and shape
operator $A$ in the direction of $N$. For every $X,Y,Z,W\in\mathfrak{X}(M)$ we
have the validity of Gauss equations (see Theorem 4.5 and Lemma 4.19 of [33])

(26)
$\mathrm{Riem}(X,Z,Y,W)=\overline{\mathrm{Riem}}(X,Z,Y,W)-g(AX,Y)g(AZ,W)+g(AX,W)g(AZ,Y),$

for any $X,Y,Z,W\in TM$, where $\mathrm{Riem}$ and $\overline{\mathrm{Riem}}$
are the Riemann curvature tensors of $(M,g)=(M,\psi^{\ast}\overline{g})$ and
$(\overline{M},\overline{g})$, respectively. For every $p\in M$, $X,Z\in
T_{p}M$ and for any orthonormal basis $\\{E_{1},\dots,E_{m}\\}$ of $T_{p}M$ we
have, by (24) and (26),

(27)
$\begin{split}\mathrm{Ric}(X,Y)&=\sum_{i=1}^{m}\overline{\mathrm{Riem}}(X,E_{i},Y,E_{i})-\sum_{i=1}^{m}g(AX,Y)g(AE_{i},E_{i})\\\
&\phantom{=\;}+\sum_{i=1}^{m}g(AX,E_{i})g(AY,E_{i})\\\
&=\overline{\mathrm{Ric}}(X,Y)+\overline{\mathrm{Riem}}(X,N,Y,N)+mHg(AX,Y)+g(AX,AY),\end{split}$

recalling that $mH=-\mathrm{trace}(A)$. Since $\\{E_{1},\dots,E_{m},N\\}$ is a
$\overline{g}$-orthonormal basis of $T_{\psi(p)}\overline{M}$, by (25), (24)
and (27) we get

(28)
$\begin{split}\mathrm{S}&=\sum_{i=1}^{m}\left(\overline{\mathrm{Ric}}(E_{i},E_{i})+\overline{\mathrm{Riem}}(N,E_{i},N,E_{i})+mHg(AE_{i},E_{i})+g(A^{2}E_{i},E_{i})\right)\\\
&=\sum_{i=1}^{m}\overline{\mathrm{Ric}}(E_{i},E_{i})+\overline{\mathrm{Ric}}(N,N)-m^{2}H^{2}+\mathrm{trace}(A^{2})\\\
&=\overline{\mathrm{S}}+2\overline{\mathrm{Ric}}(N,N)-m^{2}H^{2}+\mathrm{trace}(A^{2}).\end{split}$

If $\overline{M}$ has constant sectional curvature $\overline{\kappa}$, then,
for every $X,Y,Z,W\in\mathfrak{X}(\overline{M})$,

$\displaystyle\overline{\mathrm{Riem}}(X,Z,Y,W)$
$\displaystyle=\overline{\kappa}(g(X,Y)g(Z,W)-g(X,W)g(Y,Z)),$
$\displaystyle\overline{\mathrm{Ric}}(X,Y)$
$\displaystyle=m\overline{\kappa}\,\overline{g}(X,Y),$
$\displaystyle\overline{\mathrm{S}}$ $\displaystyle=m(m+1)\overline{\kappa}$

and for every self-adjoint endomorphism $P:TM\longrightarrow TM$ it follows
that

(29)
$\begin{split}\sum_{i=1}^{m}\mathrm{Ric}(PE_{i},E_{i})&=\sum_{i=1}^{m}\overline{\mathrm{Ric}}(PE_{i},E_{i})+\sum_{i=1}^{m}\overline{\kappa}\,\overline{g}(N,N)g(PE_{i},E_{i})\\\
&\phantom{=\;}+\sum_{i=1}^{m}mHg(APE_{i},E_{i})+\sum_{i=1}^{m}g(APE_{i},AE_{i})\\\
&=\sum_{i=1}^{m}\overline{\mathrm{Ric}}(PE_{i},E_{i})-\overline{\kappa}\,\mathrm{trace}(P)+mH\,\mathrm{trace}(AP)+\mathrm{trace}(A^{2}P).\end{split}$

##  3\. Maximal hypersurfaces in locally symmetric spacetimes

In this section we first prove Theorems A and C of the Introduction, then we
focus our attention to the case of maximal surfaces and we give proofs of
Theorems D and E. We start with a slight generalization of results obtained by
Nishikawa (Theorem B of [32]) and Ishihara (Theorems 1.1 and 1.2 of [26]),
whose proof relies on Theorem 6 below, a consequence of a more general result
which is proved as Theorem 3.6 in [4].

######  Definition 5 (Definitions 2.1 and 2.3 of [4]).

Let $M$ be a Riemannian manifold. The Omori-Yau maximum principle for the
Laplacian is said to hold on $M$ if, for any function $u\in C^{2}(M)$ with
$u^{\ast}=\sup_{M}u<+\infty$, there exists a sequence of points
$\\{x_{k}\\}_{k\in\mathbb{N}}\subseteq M$ satisfying

$\text{(i) }u(x_{k})>u^{\ast}-\frac{1}{k},\quad\text{(ii) }|\nabla
u(x_{k})|<\frac{1}{k}\quad\text{and}\quad\text{(iii) }\Delta
u(x_{k})<\frac{1}{k}\qquad\forall k\in\mathbb{N}.$

The weak maximum principle for the Laplacian is said to hold on $M$ if, for
any function $u$ as above, there exists a sequence of points
$\\{x_{k}\\}_{k\in\mathbb{N}}\subseteq M$ such that (i) and (iii) hold.

######  Theorem 6.

Let $(M,g)$ be a Riemannian manifold on which the Omori-Yau maximum principle
for the Laplacian holds, $a\in\mathbb{R}$ and $F$ a positive continuous
function on $[a,+\infty)$ satisfying

$\int_{a+\varepsilon}^{+\infty}\left(\int_{a}^{t}F(s)ds\right)^{-1/2}dt<+\infty\qquad\text{and}\qquad\limsup_{t\to+\infty}\frac{1}{tF(t)}\int_{a}^{t}F(s)ds<+\infty$

for some (hence, any) $\varepsilon>0$. If $u\in C^{2}(M)$, $f\in
C^{0}(\mathbb{R})$ are such that

$\Delta u\geq f(u)\quad\text{on
}M\qquad\text{and}\qquad\liminf_{t\to+\infty}\frac{f(t)}{F(t)}>0,$

then $u^{\ast}:=\sup_{M}u$ is finite and $f(u^{\ast})\leq 0$.

When $M$ is a complete Riemannian manifold, a sufficient condition for the
validity of the Omori-Yau maximum principle for the Laplacian on $M$ is the
existence of a constant $C\in\mathbb{R}$ such that $\mathrm{Ric}(X,X)\geq
C|X|^{2}$ for every $X\in TM$, see the book [4]. We are now ready to state and
prove the following result. Recall that a semi-Riemannian manifold is said to
be locally symmetric if its Riemannian curvature tensor is parallel. Semi-
Riemannian manifolds of constant curvature provide the simplest examples of
such manifolds.

######  Theorem 7.

Let $\overline{M}$ be a locally symmetric spacetime of dimension $m+1$ whose
Ricci and sectional curvatures satisfy

(30) $\displaystyle\overline{\mathrm{Ric}}(Z,Z)\geq c_{1}$
$\displaystyle\qquad\text{for all unit timelike vectors }Z\in T\overline{M}$
(31) $\displaystyle\overline{\mathrm{Sect}}(\Pi)\geq c_{2}$
$\displaystyle\qquad\text{for all non-degenerate tangent $2$-planes
}\Pi\subseteq T\overline{M}$

for some constants $c_{1},c_{2}\in\mathbb{R}$ and let
$\psi:M\longrightarrow\overline{M}$ be a complete maximal hypersurface. Then
the shape operator $A$ satisfies

(32)
$\mathrm{trace}(A^{2})\leq\max\left\\{0,-(c_{1}+2mc_{2})\right\\}\qquad\text{on
}M.$

In particular, if $c_{1}\geq-2mc_{2}$ then $\psi$ is totally geodesic.

  * Proof.

Let $q\in M$ be a given point and let $\\{E_{1},\dots,E_{m}\\}$ be an
orthonormal basis of $T_{q}M$ given by the principal directions of curvature,
that is, eigenvectors of $A(q)$ corresponding to the principal curvatures
$\kappa_{1},\dots,\kappa_{m}$. Since $\overline{M}$ is locally symmetric and
$\psi$ is maximal, following Nishikawa [32] we have Simons’ formula

$\displaystyle\frac{1}{2}\Delta\mathrm{trace}(A^{2})$ $\displaystyle=|\nabla
A|^{2}+\overline{\mathrm{Ric}}(N,N)\mathrm{trace}(A^{2})$
$\displaystyle\phantom{=\;}+2\sum_{1\leq i<j\leq
m}(\kappa_{i}-\kappa_{j})^{2}\,\overline{\mathrm{Sect}}(E_{i}\wedge
E_{j})+\left(\mathrm{trace}(A^{2})\right)^{2}$

at $q$ and we can estimate

$\displaystyle 2\sum_{1\leq i<j\leq
m}(\kappa_{i}-\kappa_{j})^{2}\,\overline{\mathrm{Sect}}(E_{i}\wedge E_{j})$
$\displaystyle\geq 2c_{2}\sum_{1\leq i<j\leq m}(\kappa_{i}-\kappa_{j})^{2}$
$\displaystyle=2mc_{2}\sum_{i=1}^{m}\kappa_{i}^{2}=2mc_{2}\mathrm{trace}(A^{2}),$
$\displaystyle\overline{\mathrm{Ric}}(N,N)\mathrm{trace}(A^{2})$
$\displaystyle\geq c_{1}\mathrm{trace}(A^{2}).$

So, the function $u=\mathrm{trace}(A^{2})$ satisfies $\Delta u\geq
2(c_{1}+2mc_{2}+u)u$. For any given unit vector $X\in T_{q}M$, by choosing an
orthonormal basis $\\{e_{1},\dots,e_{m-1}\\}$ of $X^{\bot}\subseteq T_{q}M$ we
deduce from Gauss equations (see formulas (26) and (27) in Section 2.4) that

$\mathrm{Ric}(X,X)=\sum_{i=1}^{m-1}\overline{\mathrm{Sect}}(X\wedge
e_{i})+|AX|^{2}\geq(m-1)c_{2}.$

By bilinearity, it follows that $\mathrm{Ric}(X,X)\geq(m-1)c_{2}|X|^{2}$ for
every $X\in TM$. Since $M$ is complete, the Omori-Yau maximum principle for
the Laplacian on $M$. Applying Theorem 6 with the choice $F(t)=t^{2}$, we
deduce that $u$ is bounded above and that $u^{\ast}=\sup_{M}u$ satisfies

$(c_{1}+2mc_{2}+u^{\ast})u^{\ast}\leq 0,$

that is, $u^{\ast}\leq\max\left\\{0,-(c_{1}+2mc_{2})\right\\}$. $\square$

######  Remark 8.

We remark that in Theorem B of [32] the general estimate (32) is not stated
and it is only proved that $\psi$ is totally geodesic when $c_{1}+2mc_{2}\geq
0$.

######  Theorem 9.

Let $\overline{M}$ be a spacetime such that $\overline{\mathrm{Ric}}(Z,Z)\geq
0$ for every timelike vector $Z\in T\overline{M}$. If
$\psi:M\longrightarrow\overline{M}$ is a maximal hypersurface, then $\psi$ is
stable. If $M$ is also compact, then $\psi$ is totally geodesic.

  * Proof.

The unit normal vector $N$ on $M$ is timelike so
$\overline{\mathrm{Ric}}(N,N)\geq 0$ and we have

$\int_{M}|\nabla\phi|^{2}+\left(\overline{\mathrm{Ric}}(N,N)+\mathrm{trace}(A^{2})\right)\phi^{2}\geq
0\qquad\text{for every }\phi\in C_{c}^{\infty}(M).$

Therefore $\psi$ is stable by definition and from Lemma 4 we deduce the
existence of a positive function $u$ satisfying $\Delta
u-\left(\overline{\mathrm{Ric}}(N,N)+\mathrm{trace}(A^{2})\right)u=0$. If $M$
is compact we have

$\int_{M}\left(\overline{\mathrm{Ric}}(N,N)+\mathrm{trace}(A^{2})\right)u=\int_{M}\Delta
u=0$

by the divergence theorem. But $\overline{\mathrm{Ric}}(N,N)\geq 0$,
$\mathrm{trace}(A^{2})\geq 0$ and $u>0$, so $\mathrm{trace}(A^{2})\equiv 0$,
that is, $\psi$ is totally geodesic. $\square$

From Theorems 7 and 9 we easily deduce the next

######  Theorem 10.

Let $\overline{M}$ be a spacetime of dimension $m+1$ and constant curvature
$\overline{\kappa}$ and let $\psi:M\longrightarrow\overline{M}$ be a complete
maximal hypersurface.

  * i)

If $\overline{\kappa}>0$ then $M$ is compact and the immersion $\psi$ is
totally geodesic and unstable.

  * ii)

If $\overline{\kappa}=0$ then $\psi$ is totally geodesic and stable.

  * iii)

If $\overline{\kappa}<0$ then $\psi$ is stable and the shape operator $A$ and
the scalar curvature $\mathrm{S}$ of $M$ satisfy

(33)
$\mathrm{trace}(A^{2})\leq-m\overline{\kappa},\qquad\mathrm{S}\leq(m-2)m\overline{\kappa}.$

If $M$ is also compact, then $\psi$ is totally geodesic.

  * Proof.

We have $\overline{\mathrm{Ric}}(Z,Z)=-m\overline{\kappa}$ for all unit
timelike vectors $Z\in T\overline{M}$, so conditions (30) and (31) are
satisfied with $c_{1}=-m\overline{\kappa}$, $c_{2}=\overline{\kappa}$. Hence,
$c_{1}+2mc_{2}=m\overline{\kappa}$ and by Theorem 7 we have
$\mathrm{trace}(A^{2})\leq\max\\{0,-m\overline{\kappa}\\}$. If
$\overline{\kappa}\geq 0$ it follows that $\psi$ is totally geodesic, while
for $\overline{\kappa}<0$ we obtain
$\mathrm{trace}(A^{2})\leq-m\overline{\kappa}$, that by (4) is equivalent to
saying that the scalar curvature $\mathrm{S}$ of $M$ satisfies

$\mathrm{S}=(m-1)m\overline{\kappa}+\mathrm{trace}(A^{2})\leq(m-2)m\overline{\kappa}.$

If $\overline{\kappa}>0$ then $M$ has constant positive sectional curvature
$\overline{\kappa}$ and therefore it must be compact by the Bonnet-Myers
theorem. Since $\psi$ is totally geodesic,
$\overline{\mathrm{Ric}}(N,N)+\mathrm{trace}(A^{2})=-m\overline{\kappa}<0$ on
$M$ and the constant, compactly supported function $\phi\equiv 1$ verifies

$\frac{\int_{M}|\nabla\phi|^{2}+\left(\overline{\mathrm{Ric}}(N,N)+\mathrm{trace}(A^{2})\right)\phi^{2}}{\int_{M}\phi^{2}}=-m\overline{\kappa}<0,$

implying $\lambda_{1}^{L}(M)<0$ by (7). When $\overline{\kappa}\leq 0$ we have
$\overline{\mathrm{Ric}}(Z,Z)=-m\overline{\kappa}\geq 0$ for every unit
timelike vector $Z\in T\overline{M}$, so the other statements are direct
consequences of Theorem 9. $\square$

The following Theorem 11 is a refinement of Theorem 10 for maximal surfaces in
$3$-dimensional spacetimes of constant sectional curvature. Let us recall from
(4) that the Gaussian curvature $\mathrm{K}$ of such a surface satisfies

(34) ${\color[rgb]{0,0,0}2\mathrm{K}}=\mathrm{S}=\overline{\mathrm{S}}+2\
\overline{\mathrm{Ric}}(N,N)+\mathrm{trace}(A^{2}),$

where $\overline{\mathrm{S}}$ denotes the scalar curvature of the ambient
spacetime $\overline{M}$.

######  Theorem 11.

Let $\psi:M\to\overline{M}$ be a complete maximal oriented surface in a
$3$-dimensional spacetime $\overline{M}$ of constant sectional curvature
$\overline{\kappa}$.

  * i)

If $\overline{\kappa}>0$ then $M$ is a totally geodesic, unstable round sphere
of constant curvature $\overline{\kappa}$.

  * ii)

If $\overline{\kappa}=0$ then $M$ is totally geodesic, stable and it is either
a Euclidean plane, a flat cylinder, or a flat torus.

  * iii)

If $\overline{\kappa}<0$ then $M$ is stable and has non-positive Gaussian
curvature. If $M$ is compact then it is totally geodesic and its Euler
characteristic satisfies

(35) $\chi(M)=\frac{\overline{\kappa}}{2\pi}\,\mathrm{Vol}(M).$

If $M$ is non-compact but its total curvature and its Euler characteristic are
finite, then

(36) $\chi(M)\geq\frac{\overline{\kappa}}{2\pi}\,\mathrm{Vol}(M).$

  * Proof.

By formula (34), the Gaussian curvature $\mathrm{K}$ of $M$ always satisfies

$\mathrm{K}=\frac{1}{2}\,\overline{\mathrm{S}}+\overline{\mathrm{Ric}}(N,N)+\frac{1}{2}\,\mathrm{trace}(A^{2})=\overline{\kappa}+\frac{1}{2}\,\mathrm{trace}(A^{2})\geq\overline{\kappa}.$

If $\overline{\kappa}>0$ then, in view of Theorem 10, we only have to show
that $M$ is a topological sphere. Since $M$ is a compact surface of constant
Gaussian curvature $\mathrm{K}=\overline{\kappa}$, by the Gauss-Bonnet theorem
the Euler characteristic $\chi(M)$ satisfies

$\chi(M)=\frac{1}{2\pi}\int_{M}\mathrm{K}=\frac{\overline{\kappa}}{2\pi}\,\mathrm{Vol}(M)>0.$

Since $\chi(M)=2-2g$ with $g$ the topological genus of $M$, we conclude that
$g=0$ and $M$ is a topological sphere.

If $\overline{\kappa}=0$ then $\psi$ is totally geodesic by Theorem 10 and $M$
is a flat surface. Note that all of the three cases described in point $ii)$
of the statement of the theorem can occur, for example when $M$ is a spacelike
slice of a Lorentzian product $\overline{M}=\mathbb{R}\times F$ with metric
$\overline{g}=-dt^{2}+g_{F}$ and $(F,g_{F})$ is a flat surface of one of the
three above types.

If $\overline{\kappa}<0$ and $M$ is compact, we obtain (35) by applying again
the Gauss-Bonnet theorem. If $M$ is non-compact but has finite total curvature
and finite Euler characteristic, inequality (36) follows by using the Cohn-
Vossen’s inequality (see page 86 in [34])

$\chi(M)\geq\frac{1}{2\pi}\int_{M}\mathrm{K},$

which is valid under our assumptions. $\square$

We conclude this section by restating and proving Theorem E from the
Introduction under slightly more general hypotheses, see Remark 14 below.

######  Theorem 12.

Let $\overline{M}$ be a $3$-dimensional spacetime satisfying
$\overline{\mathrm{S}}+\overline{\mathrm{Ric}}(Z,Z)\geq 0$ for every unit
timelike vector $Z\in T\overline{M}$ and let
$\psi:M\longrightarrow\overline{M}$ be a complete stable oriented maximal
surface. If $\psi$ is stable at infinity and $\mathrm{K}_{+}\in L^{1}(M)$,
then also $\mathrm{K}_{-}\in L^{1}(M)$, where $\mathrm{K}_{+}$ and
$\mathrm{K}_{-}$ are the positive and negative parts of the Gaussian curvature
$\mathrm{K}$ of $M$.

To prove Theorem 12 we need the following result due to Fischer-Colbrie, see
Theorem 1 of [22].

######  Lemma 13.

Let $(M,g)$ be a complete Riemann surface with Gaussian curvature
$\mathrm{K}$. If $u\in C^{\infty}(M)$ is a positive function such that $\Delta
u\leq\mathrm{K}u$ on $M\setminus\overline{\Omega}$ for some relatively compact
open set $\Omega\subseteq M$, then $(M,u^{2}g)$ is complete.

###### Proof of Theorem 12.

If $M$ is compact, then it has finite total curvature and we are done. Hence,
suppose that $M$ is complete and non-compact. By (34) and since
$\overline{\mathrm{S}}+\overline{\mathrm{Ric}}(N,N)\geq 0$, the Gaussian
curvature of $M$ satisfies
$2\mathrm{K}\geq\overline{\mathrm{Ric}}(N,N)+\mathrm{trace}(A^{2})$. Let
$\Omega_{0}\subseteq M$ be a relatively compact open set such that the
stability operator
$L=\Delta-\left(\overline{\mathrm{Ric}}(N,N)+\mathrm{trace}(A^{2})\right)$
satisfies $\lambda_{1}^{L}(M\setminus\overline{\Omega_{0}})\geq 0$. By the
variational characterization (7), for every $\phi\in
C^{\infty}(M\setminus\overline{\Omega_{0}})$ we have

$\displaystyle\int_{M\setminus\overline{\Omega_{0}}}|\nabla\phi|^{2}+\mathrm{K}\phi^{2}$
$\displaystyle\geq\frac{1}{2}\int_{M\setminus\overline{\Omega_{0}}}|\nabla\phi|^{2}+2\mathrm{K}\phi^{2}$
$\displaystyle\geq\frac{1}{2}\int_{M\setminus\overline{\Omega_{0}}}|\nabla\phi|^{2}+\left(\overline{\mathrm{Ric}}(N,N)+\mathrm{trace}(A^{2})\right)\phi^{2}\geq
0$

so the operator $\Delta-\mathrm{K}$ satisfies
$\lambda_{1}^{\Delta-\mathrm{K}}(M\setminus\overline{\Omega_{0}})\geq 0$ and
by Lemma 4 there exists a positive solution $u_{0}$ of $\Delta
u_{0}=\mathrm{K}u_{0}$ on $M\setminus\overline{\Omega_{0}}$. By standard
elliptic regularity results, $u_{0}$ is smooth. Let $\Omega\subseteq M$ be a
relatively compact open set such that $\overline{\Omega_{0}}\subseteq\Omega$
and let $u\in C^{\infty}(M)$ be a positive function such that $u=u_{0}$ on
$M\setminus\overline{\Omega}$, so that

(37) $\Delta u=\mathrm{K}u\qquad\text{on }M\setminus\overline{\Omega}.$

By Lemma 13, $M$ is complete in the conformally deformed metric
$\tilde{g}=u^{2}g$ and the Gaussian curvature $\tilde{\mathrm{K}}$ of
$(M,\tilde{g})$ is nonnegative on $M\setminus\Omega$ since

(38) $\tilde{\mathrm{K}}=\frac{1}{u^{2}}(\mathrm{K}-\Delta\log
u)=\frac{\mathrm{K}u-\Delta u}{u^{3}}+\frac{|\nabla u|^{2}}{u^{4}}.$

Let $o\in M$ be a given point and let $B_{r}$ denote the geodesic ball of
$(M,\tilde{g})$ centered at $o$ with radius $r$. Completeness of
$(M,\tilde{g})$ by the Hopf-Rinow theorem enables us to choose $R>0$
sufficiently large so that $\Omega\subseteq B_{R}$. As $\tilde{K}\geq 0$ on
$M\setminus B_{R}$, by the volume comparison theorem there exists $C>0$ such
that $\mathrm{Vol}(\partial B_{r})\leq Cr$ for a.e. $r\geq R$, with
$\mathrm{Vol}(\partial B_{r})$ the $1$-dimensional Hausdorff measure of
$\partial B_{r}$ induced by $\tilde{g}$. Note that this is well defined for
a.e. $r>0$. Thus

(39) $\frac{1}{\mathrm{Vol}(\partial B_{r})}\notin L^{1}(+\infty).$

We let $w(r)=\mathrm{Vol}(\partial B_{r})$ and we set

(40) $A(r)=\frac{1}{w(r)}\int_{\partial
B_{r}}-\frac{\mathrm{K}}{u^{2}}\,d\mathrm{Vol}_{\tilde{g}}.$

We then consider the Cauchy problem

(41) $\begin{cases}(w(r)z^{\prime})^{\prime}+A(r)w(r)z=0&\text{on
}\mathbb{R}^{+}\\\ z(0^{+})=1,\quad(wz^{\prime})(0^{+})=0.\end{cases}$

Applying Proposition 4.2 of [11] we deduce the existence of a weak solution
$z$ of (41). We now reason by contradiction and we suppose that
$\mathrm{K}_{+}\in L^{1}(M,d\mathrm{Vol}_{g})$ while $\mathrm{K}_{-}\notin
L^{1}(M,d\mathrm{Vol}_{g})$. Then

(42) $\lim_{r\to+\infty}\int_{B_{r}}\mathrm{K}\,d\mathrm{Vol}_{g}=-\infty.$

By the coarea formula,

(43)
$\int_{0}^{r}A(s)w(s)ds=\int_{B_{r}}-\frac{\mathrm{K}}{u^{2}}\,d\mathrm{Vol}_{\tilde{g}}=\int_{B_{r}}-\mathrm{K}\,d\mathrm{Vol}_{g}$

and therefore

(44) $\lim_{r\to+\infty}\int_{0}^{r}A(s)w(s)ds=+\infty.$

This, together with (39) implies, by Corollary 2.9 of [29], that $z$ is
oscillatory. Let $R\leq R_{1}<R_{2}$ be two consecutive zeros of $z$ such that
$z>0$ on $(R_{1},R_{2})$. Define a function
$\varphi\in\mathrm{Lip}_{0}(\overline{B_{R_{2}}}\setminus B_{R_{1}})$ by
setting $\varphi(x):=z(r(x))$ for each $x\in
B_{R_{2}}\setminus\overline{B_{R_{1}}}$, with $r(x)$ the distance from $x$ to
$o$ in the metric $\tilde{g}$. By the coarea formula and (40)

(45)
$\begin{split}\int_{B_{R_{2}}\setminus\overline{B_{R_{1}}}}|\tilde{\nabla}\varphi|_{\tilde{g}}^{2}\,d\mathrm{Vol}_{\tilde{g}}&+\int_{B_{R_{2}}\setminus\overline{B_{R_{1}}}}\frac{\mathrm{K}}{u^{2}}\varphi^{2}\,d\mathrm{Vol}_{\tilde{g}}=\\\
&=\int_{R_{1}}^{R_{2}}w(s)z^{\prime}(s)^{2}ds-\int_{R_{1}}^{R_{2}}A(s)w(s)z(s)^{2}ds.\end{split}$

Since $z$ is a weak solution of (41), using
$z\,\chi_{[R_{1},R_{2}]}\in\mathrm{Lip}_{0}([R_{1},R_{2}])$ as a test function
we get

(46)
${\color[rgb]{0,0,0}\int_{R_{1}}^{R_{2}}w(s)z^{\prime}(s)^{2}ds=\int_{R_{1}}^{R_{2}}A(s)w(s)z(s)^{2}ds.}$

Collecting (45) and (46), by the monotonicity property of eigenvalues we
obtain

${\color[rgb]{0,0,0}\lambda_{1}^{\tilde{L}}(M\setminus\overline{\Omega})<\lambda_{1}^{\tilde{L}}(B_{R_{2}}\setminus\overline{B_{R_{1}}})\leq\frac{\int_{B_{R_{2}}\setminus\overline{B_{R_{1}}}}|\tilde{\nabla}\varphi|_{\tilde{g}}^{2}\,d\mathrm{Vol}_{\tilde{g}}+\int_{B_{R_{2}}\setminus\overline{B_{R_{1}}}}\frac{\mathrm{K}}{u^{2}}\varphi^{2}\,d\mathrm{Vol}_{\tilde{g}}}{\int_{B_{R_{2}}\setminus\overline{B_{R_{1}}}}\varphi^{2}}=0,}$

where $\tilde{L}$ is the operator defined by

${\color[rgb]{0,0,0}\tilde{L}=\tilde{\Delta}-\frac{\mathrm{K}}{u^{2}}},$

with $\tilde{\Delta}$ the Laplace-Beltrami operator of $(M,\tilde{g})$. Since
$u^{2}\tilde{\Delta}f=\Delta f$ on $M$ for every $f\in C^{2}(M)$, we see from
(37) that $u$ is a positive solution of $\tilde{L}u=0$ on
$M\setminus\overline{\Omega}$, so
$\lambda_{1}^{\tilde{L}}(M\setminus\overline{\Omega})\geq 0$ by Lemma 4 and we
have reached the desired contradiction. ∎

######  Remark 14.

If $\\{E_{1},E_{2},Z\\}$ is a local Lorentz orthonormal frame on
$T\overline{M}$ and $Z$ is timelike then
$\overline{\mathrm{S}}+\overline{\mathrm{Ric}}(Z,Z)=\overline{\mathrm{Ric}}(E_{1},E_{1})+\overline{\mathrm{Ric}}(E_{2},E_{2})$,
so Theorem E is indeed a consequence of Theorem 12.

##  4\. Maximal hypersurfaces in GRW spacetimes

Let $(F,g_{F})$ be an $m$-dimensional (connected) Riemannian manifold, $m\geq
2$, $I$ an open interval in $\mathbb{R}$ endowed with the metric $-dt^{2}$ and
$\rho$ a positive smooth function defined on $I$. The Generalized Robertson-
Walker (GRW) spacetime $\overline{M}=I\times_{\rho}F$, with fiber $(F,g_{F})$,
base $(I,-dt^{2})$ and warping function $\rho$, is the product manifold
$\overline{M}=I\times F$ endowed with the Lorentzian metric

(47)
${\color[rgb]{0,0,0}\overline{g}=-\pi^{*}_{I}(dt^{2})+\rho(\pi_{I})^{2}\,\pi_{F}^{*}(g_{F}),}$

where, respectively, $\pi_{I}$ and $\pi_{F}$ denote the projections from
$\overline{M}$ onto $I$ and $F$. If the fiber has constant sectional
curvature, $\overline{M}$ is simply called a Robertson-Walker spacetime.

In any GRW spacetime $\overline{M}=I\times_{\rho}F$, the coordinate vector
field $\partial_{t}:=\partial/\partial t$ is a unit timelike vector field and
hence $\overline{M}$ is time-orientable. With a slight abuse of notation, we
write $\rho(t)$, $\rho^{\prime}(t)$, $\rho^{\prime\prime}(t)$ to denote
$\rho\circ\pi_{I}$, $\rho^{\prime}\circ\pi_{I}$,
$\rho^{\prime\prime}\circ\pi_{I}$. If we consider the timelike vector field

${\color[rgb]{0,0,0}T}:=\rho({\color[rgb]{0,0,0}t})\,\partial_{t},$

from the relation between the Levi-Civita connection of $\overline{M}$ and
those of the base and the fiber (see Corollary 7.35 of [33]) it follows that

(48)
$\overline{\nabla}_{X}{\color[rgb]{0,0,0}T}=\rho^{\prime}({\color[rgb]{0,0,0}t})X,$

for any $X\in\mathfrak{X}(\overline{M})$, where $\overline{\nabla}$ is the
Levi-Civita connection of the Lorentzian metric (47). Thus,
${\color[rgb]{0,0,0}T}$ is conformal and its metrically equivalent $1$-form is
closed, that is, ${\color[rgb]{0,0,0}T}$ is a closed conformal vector field.
The curvature tensors of $\overline{M}$ are given by the following formulas.

######  Lemma 15.

The GRW spacetime $\overline{M}=I\times_{\rho}F$ has Riemann and Ricci
curvature tensors given by

(49) $\displaystyle\overline{\mathrm{Riem}}$
$\displaystyle=\rho(t)^{2}{\pi_{F}}^{\ast}\mathrm{Riem}^{F}+\rho(t)^{2}\left(\rho^{\prime}(t)^{2}-\rho(t)\rho^{\prime\prime}(t)\right){\pi_{F}}^{\ast}(g_{F}\mathbin{\bigcirc\mspace{-15.0mu}\wedge\mspace{3.0mu}}g_{F})+\frac{\rho^{\prime\prime}(t)}{\rho(t)}(\overline{g}\mathbin{\bigcirc\mspace{-15.0mu}\wedge\mspace{3.0mu}}\overline{g}),$
(50) $\displaystyle\overline{\mathrm{Ric}}$
$\displaystyle={\pi_{F}}^{\ast}\mathrm{Ric}^{F}-(m-1)\left(\rho(t)\rho^{\prime\prime}(t)-\rho^{\prime}(t)^{2}\right){\pi_{F}}^{\ast}g_{F}+m\frac{\rho^{\prime\prime}(t)}{\rho(t)}\overline{g},$

where $\mathrm{Riem}^{F}$ and $\mathrm{Ric}^{F}$ are the Riemann and Ricci
tensors of $(F,g_{F})$ and
$\mathbin{\bigcirc\mspace{-15.0mu}\wedge\mspace{3.0mu}}$ denotes Kulkarni-
Nomizu product.

###### Proof.

The gradient and the Hessian of the warping function $\rho$ in the base
$(I,-dt^{2})$ are given by $\nabla^{I}\rho=-\rho^{\prime}\partial_{t}$ and
$\mathrm{Hess}(\rho)=\rho^{\prime\prime}dt\otimes dt$. Since
$\overline{\nabla}_{\partial_{t}}\partial_{t}=0$ on $\overline{M}$, the lift
$-\rho^{\prime}(t)\partial_{t}\in\mathfrak{X}(\overline{M})$ of
$\nabla^{I}\rho$ satisfies
$\overline{\nabla}_{\partial_{t}}(-\rho^{\prime}(t)\partial_{t})=-\rho^{\prime\prime}(t)\partial_{t}$
on $\overline{M}$. Let $p=(t,x)\in\overline{M}$ be given and let $X,Y,U,V,W\in
T_{p}\overline{M}$ be such that

${\pi_{F}}_{\ast}(X)={\pi_{F}}_{\ast}(Y)=0\in
T_{x}F,\qquad{\pi_{I}}_{\ast}(U)={\pi_{I}}_{\ast}(V)={\pi_{I}}_{\ast}(W)=0\in
T_{t}I.$

From formulas (2)-(5) of Proposition 7.42 of [33] we get

$\displaystyle\overline{\mathrm{Riem}}(V,W,U,\;\cdot\;)$
$\displaystyle=\rho(t)^{2}\pi_{F}^{\ast}\mathrm{Riem}^{F}(V,W,U,\;\cdot\;)$
$\displaystyle\phantom{=\;}+\frac{\rho^{\prime}(t)^{2}}{\rho(t)^{2}}\left(\overline{g}(V,U)\overline{g}(W,\;\cdot\;)-\overline{g}(W,U)\overline{g}(V,\;\cdot\;)\right),$
$\displaystyle\overline{\mathrm{Riem}}(V,X,W,\;\cdot\;)$
$\displaystyle=\frac{\rho^{\prime\prime}(t)}{\rho(t)}\overline{g}(V,W)\overline{g}(X,\;\cdot\;),$
$\displaystyle\overline{\mathrm{Riem}}(V,X,Y,\;\cdot\;)$
$\displaystyle=-\frac{\rho^{\prime\prime}(t)}{\rho(t)}\overline{g}(X,Y)\overline{g}(V,\;\cdot\;)$

and from formulas (1)-(3) of Corollary 7.43 of [33] we also have

$\displaystyle\overline{\mathrm{Ric}}(V,\;\cdot\;)$
$\displaystyle=\pi_{F}^{\ast}\mathrm{Ric}^{F}(V,\;\cdot\;)+\left(\frac{\rho^{\prime\prime}(t)}{\rho(t)}+(m-1)\frac{\rho^{\prime}(t)^{2}}{\rho(t)^{2}}\right)\overline{g}(V,\;\cdot\;),$
$\displaystyle\overline{\mathrm{Ric}}(X,\;\cdot\;)$
$\displaystyle=m\frac{\rho^{\prime\prime}(t)}{\rho(t)}\overline{g}(X,\;\cdot\;)$

A direct computation shows that the RHS’s of (49) and (50) also satisfy the
identities above. By the symmetry properties of $\overline{\mathrm{Riem}}$,
these identities uniquely determine its action on $T_{p}\overline{M}$. ∎

As we see from (49), $\overline{M}$ has constant curvature $\overline{\kappa}$
if and only if the fiber $F$ has constant curvature $\kappa_{F}$ and the
warping function $\rho$ satisfies

(51)
$\overline{\kappa}=\frac{\rho^{\prime\prime}}{\rho}\qquad\text{and}\qquad\kappa_{F}=\rho\rho^{\prime\prime}-(\rho^{\prime})^{2}\qquad\text{on
}I.$

These equations are not independent. In fact, if there exists $C\in\mathbb{R}$
such that $\rho\rho^{\prime\prime}-(\rho^{\prime})^{2}=C$ on an interval
$I_{0}\subseteq I$, then

$\left(\frac{\rho^{\prime\prime}}{\rho}\right)^{\prime}=\left(\frac{C}{\rho^{2}}+\left(\frac{\rho^{\prime}}{\rho}\right)^{2}\right)^{\prime}=-\frac{2C\rho^{\prime}}{\rho^{3}}+2\frac{\rho^{\prime}}{\rho}\left(\frac{\rho^{\prime}}{\rho}\right)^{\prime}=-\frac{2C\rho^{\prime}}{\rho^{3}}+\frac{2\rho^{\prime}}{\rho}\frac{\rho\rho^{\prime\prime}-(\rho^{\prime})^{2}}{\rho^{2}}=0,$

that is, $\rho^{\prime\prime}/\rho$ is constant on $I_{0}$. We also
characterize GRW spacetimes with spacelike sectional curvatures bounded below.

######  Lemma 16.

Let $\overline{M}=I\times_{\rho}F$ be a GRW spacetime. For every
$C_{1}\in\mathbb{R}$, the following are equivalent:

  * i)

$\overline{\mathrm{Sect}}(\Pi)\geq C_{1}$ for every spacelike $2$-plane
$\Pi\subseteq T\overline{M}$,

  * ii)

there exists $C_{2}\in\mathbb{R}$ such that

$\displaystyle\mathrm{Sect}^{F}(\Pi_{0})$ $\displaystyle\geq C_{2}$
$\displaystyle\qquad\text{for every $2$-plane }\Pi_{0}\subseteq TF$
$\displaystyle\frac{C_{2}-\rho\rho^{\prime\prime}+(\rho^{\prime})^{2}}{\rho^{2}}$
$\displaystyle\geq\max\left\\{C_{1}-\frac{\rho^{\prime\prime}}{\rho},0\right\\}$
$\displaystyle\qquad\text{on }I.$

  * Proof.

For every $t\in I$, define the $(0,4)$-tensor $V_{t}:TF\otimes TF\otimes
TF\otimes TF\to C^{\infty}(F)$ by setting

(52) $\begin{split}V_{t}(X,Y,Z,W)&=\rho(t)^{2}\mathrm{Riem}^{F}(X,Y,Z,W)\\\
&\phantom{=\;}+\rho(t)^{2}\left(\rho^{\prime}(t)^{2}-\rho(t)\rho^{\prime\prime}(t)\right)(g_{F}\mathbin{\bigcirc\mspace{-15.0mu}\wedge\mspace{3.0mu}}g_{F})(X,Y,Z,W)\end{split}$

for every $X,Y,Z,W\in T_{x}F$, $x\in F$.

Assume that i) holds. Let $(t,x)\in\overline{M}$, $\alpha\in\mathbb{R}$, $X,Y$
be given, with

(53) $X,Y\in T_{x}F\qquad\text{such that}\qquad
g_{F}(X,X)=g_{F}(Y,Y)=1,g_{F}(X,Y)=0.$

The vectors

(54) $E_{1}=\frac{1}{\rho(t)}\,X,\qquad
E_{2}=\frac{\cosh\alpha}{\rho(t)}\,Y+\sinh\alpha\,\partial_{t}$

belong to $T_{(t,x)}\overline{M}$ and satisfy
$\overline{g}(E_{1},E_{1})=\overline{g}(E_{2},E_{2})=1$,
$\overline{g}(E_{1},E_{2})=0$, so, by (49), (52) and (54),

(55) $C_{1}\leq\overline{\mathrm{Sect}}(E_{1}\wedge
E_{2})=\overline{\mathrm{Riem}}(E_{1},E_{2},E_{1},E_{2})=\frac{\cosh^{2}\alpha}{\rho(t)^{4}}V_{t}(X,Y,X,Y)+\frac{\rho^{\prime\prime}(t)}{\rho(t)},$

that is,

(56) $\frac{\mathrm{Sect}^{F}(X\wedge
Y)-\rho(t)\rho^{\prime\prime}(t)+\rho^{\prime}(t)^{2}}{\rho(t)^{2}}=\frac{V_{t}(X,Y,X,Y)}{\rho(t)^{4}}\geq\frac{1}{\cosh^{2}\alpha}\left(C_{1}-\frac{\rho^{\prime\prime}(t)}{\rho(t)}\right).$

For $\alpha=0$ and $\alpha\to+\infty$ we respectively get

(57) $\begin{split}\frac{\mathrm{Sect}^{F}(X\wedge
Y)-\rho(t)\rho^{\prime\prime}(t)+\rho^{\prime}(t)^{2}}{\rho(t)^{2}}&\geq
C_{1}-\frac{\rho^{\prime\prime}(t)}{\rho(t)},\\\
\frac{\mathrm{Sect}^{F}(X\wedge
Y)-\rho(t)\rho^{\prime\prime}(t)+\rho^{\prime}(t)^{2}}{\rho(t)^{2}}&\geq
0.\end{split}$

For any fixed $t$, these inequalities must hold for every $x\in F$ and for
every $X,Y$ as in (53), so
$C=\inf\\{\mathrm{Sect}^{F}(\Pi_{0}):\Pi_{0}\subseteq TF\text{ is a
$2$-plane}\\}$ is finite and ii) follows with $C_{2}=C$.

Vice versa, assume that ii) holds. Let $(t,x)\in\overline{M}$ and
$\Pi\subseteq T_{(t,x)}\overline{M}$ a spacelike $2$-plane be given. We can
find a $\overline{g}$-orthonormal basis $\\{E_{1},E_{2}\\}$ for $\Pi$ of the
form (54), with $\alpha\in\mathbb{R}$ and $X,Y$ as in (53). Since inequalities
(57) hold by assumption, we have (56) and therefore (55), that is,
$\overline{\mathrm{Sect}}(\Pi)\geq C_{1}$. As $\Pi\subseteq T\overline{M}$ is
arbitrarily given, we obtain i). $\square$

Let $\psi:M\longrightarrow\overline{M}$ be a spacelike hypersurface immersed
in the GRW spacetime $\overline{M}=I\times_{\rho}F$. Consider the unit
timelike vector $N$ normal to $M$ with the same time orientation as
$\partial_{t}$ and let $A$ and $H$ be the shape operator and the mean
curvature of $\psi$ in the direction of $N$ as described in Section 2. The
height function $\tau$ of the immersion $\psi$ onto the factor $I$ and the
amplitude $\theta$ of the hyperbolic angle between $N$ and $\partial_{t}$ are
given by

(58)
${\color[rgb]{0,0,0}\tau=\pi_{I}\circ\psi,\qquad\cosh\theta=-\overline{g}(N,\partial_{t}).}$

Note that $\theta$ is well defined (up to a sign) by the wrong-way Cauchy
Schwarz inequality, since $N$ and $\partial_{t}$ are unit timelike vectors
with the same time-orientation. As above, we write $\rho(\tau)$,
$\rho^{\prime}(\tau)$, $\rho^{\prime\prime}(\tau)$ to denote $\rho\circ\tau$,
$\rho^{\prime}\circ\tau$, $\rho^{\prime\prime}\circ\tau$. For a fixed
$t_{0}\in I$, we set

(59) $\overline{\eta}(t)=\int_{t_{0}}^{t}\rho(s)ds\quad\forall t\in
I,\qquad\eta=\overline{\eta}\circ\tau.$

Since $\rho>0$, $\overline{\eta}$ is strictly increasing on $I$. We also
consider the positive function

(60) $v=-\overline{g}(T,N)=\rho(\tau)\cosh\theta.$

In the sequel we will make extensive use of these auxiliary functions.
Denoting by $\nabla$ and $\Delta$ the Levi-Civita connection and the Laplace-
Beltrami operator of $(M,g)$, we have the following computational result.

######  Lemma 17.

Let $\psi:M\longrightarrow\overline{M}$ be a spacelike hypersurface immersed
in a GRW spacetime $\overline{M}=I\times_{\rho}F$ and let $T$, $N$, $A$, $H$,
$\theta$, $\tau$, $\eta$ and $v$ be as above. Then

(61) $\displaystyle{\color[rgb]{0,0,0}|\nabla\tau|^{2}}$
$\displaystyle{\color[rgb]{0,0,0}\,=\sinh^{2}\theta,}$ (62)
$\displaystyle{\color[rgb]{0,0,0}|\nabla\eta|^{2}}$
$\displaystyle{\color[rgb]{0,0,0}\,=\rho(\tau)^{2}\sinh^{2}\theta=v^{2}-\rho(\tau)^{2},}$
(63) $\displaystyle{\color[rgb]{0,0,0}|\nabla v|^{2}}$
$\displaystyle{\color[rgb]{0,0,0}\,\leq\mathrm{trace}(A^{2})|\nabla\eta|^{2}=\mathrm{trace}(A^{2})(v^{2}-\rho(\tau)^{2}),}$
(64) $\displaystyle{\color[rgb]{0,0,0}\Delta\eta}$
$\displaystyle{\color[rgb]{0,0,0}\,=-m\rho^{\prime}(\tau)+mHv,}$ (65)
$\displaystyle{\color[rgb]{0,0,0}\Delta v}$
$\displaystyle{\color[rgb]{0,0,0}\,=\left(\overline{\mathrm{Ric}}(N,N)+\mathrm{trace}(A^{2})+m\frac{\rho^{\prime\prime}(\tau)}{\rho(\tau)}\right)v-mg(T^{\top},\nabla
H)-mH\rho^{\prime}(\tau),}$

where $T^{\top}$ is the tangential part of $T$ along $\psi$. If $\psi$ is
maximal, then

(66) $\displaystyle{\color[rgb]{0,0,0}|\nabla v|^{2}}$
$\displaystyle{\color[rgb]{0,0,0}\,\leq\frac{m-1}{m}\mathrm{trace}(A^{2})(v^{2}-\rho(\tau)^{2}),}$
(67) $\displaystyle{\color[rgb]{0,0,0}\Delta\eta}$
$\displaystyle{\color[rgb]{0,0,0}\,=-m\rho^{\prime}(\tau),}$ (68)
$\displaystyle{\color[rgb]{0,0,0}\Delta v}$
$\displaystyle{\color[rgb]{0,0,0}\,=\left(\overline{\mathrm{Ric}}(N,N)+\mathrm{trace}(A^{2})+m\frac{\rho^{\prime\prime}(\tau)}{\rho(\tau)}\right)v.}$

  * Proof.

As $dt=\overline{g}(\;\cdot\;,\partial_{t})$ in $\overline{M}$, we have that
$\nabla\tau=-\partial_{t}^{\top}$ on $M$, where $\partial_{t}^{\top}$ is the
tangential part of $\partial_{t}$ along $\psi$. From (59) it follows that

(69) $\nabla\eta=\rho(\tau)\nabla\tau=-T^{\top}.$

Using the orthogonal decomposition
$\partial_{t}=\partial_{t}^{\top}+\cosh\theta N$ and
$\overline{g}(\partial_{t},\partial_{t})=\overline{g}(N,N)=-1$ we get

${\color[rgb]{0,0,0}|\nabla\tau|^{2}=g(\partial_{t}^{\top},\partial_{t}^{\top})=\overline{g}(\partial_{t},\partial_{t})-\cosh^{2}\theta\,\overline{g}(N,N)=\sinh^{2}\theta,\qquad|\nabla\eta|^{2}=\rho(\tau)^{2}\sinh^{2}\theta,}$

proving (61) and (62) in view of (60). Since the tangential component of
${\color[rgb]{0,0,0}T}$ along $\psi$ is
${\color[rgb]{0,0,0}T}^{\top}={\color[rgb]{0,0,0}T}+\overline{g}({\color[rgb]{0,0,0}T},N)N{\color[rgb]{0,0,0}\,=T-vN}$,
a direct computation using (48) gives

(70) $\nabla
v=A{\color[rgb]{0,0,0}T}^{\top}{\color[rgb]{0,0,0}=-A\nabla\eta.}$

Denoting by $\lambda_{1},\dots,\lambda_{m}$ the eigenvalues of $A:T_{p}M\to
T_{p}M$ at a given point $p\in M$, we have

(71) ${\color[rgb]{0,0,0}|AX|^{2}=g(A^{2}X,X)\leq|X|^{2}\max_{1\leq i\leq
m}\lambda_{i}^{2}\leq\mathrm{trace}(A^{2})|X|^{2}}$

for each vector $X\in T_{p}M$. So, (63) follows from (70) and (62). If $\psi$
is maximal, then $\mathrm{trace}(A)\equiv 0$ and we obtain a “refined
Kato”-type inequality: for each $1\leq i\leq m$ we have
$\lambda_{i}=-\sum_{1\leq j\leq m,j\neq i}\lambda_{j}$ and using Cauchy
inequality we get

${\color[rgb]{0,0,0}\lambda_{i}^{2}+\frac{1}{m-1}\lambda_{i}^{2}=\lambda_{i}^{2}+\frac{1}{m-1}\left(\sum_{\begin{subarray}{c}1\leq
j\leq m\\\ j\neq
i\end{subarray}}\lambda_{j}\right)^{2}\leq\lambda_{i}^{2}+\frac{m-1}{m-1}\sum_{\begin{subarray}{c}1\leq
j\leq m\\\ j\neq i\end{subarray}}\lambda_{j}^{2}=\mathrm{trace}(A^{2}).}$

Hence, $\lambda_{i}^{2}\leq\frac{m-1}{m}\mathrm{trace}(A^{2})$ for each $1\leq
i\leq m$, thus $|AX|^{2}\leq\frac{m-1}{m}\mathrm{trace}(A^{2})|X|^{2}$ for
every $X\in T_{p}M$, proving the refined version (66).

In order to prove (64) and (65), we recall that Gauss and Weingarten formulas
for the immersion $\psi$ are respectively given by

(72) $\overline{\nabla}_{X}Y=\nabla_{X}Y-g(AX,Y)N\qquad\text{and}\qquad
AX=-\overline{\nabla}_{X}N$

for any $X,Y\in\mathfrak{X}(M)$ and that the covariant derivative $\nabla A$
of $A$, defined by $(\nabla_{X}A)Y=\nabla_{X}(AY)-A(\nabla_{X}Y)$ for every
$X,Y\in\mathfrak{X}(M)$, satisfies Codazzi equation

(73)
$g((\nabla_{X}A)Y,Z)=g((\nabla_{Y}A)X,Z)-\overline{g}(\overline{\mathrm{R}}(X,Y)N,Z)$

for every $X,Y,Z\in\mathfrak{X}(M)$. Taking the tangential component in (48)
and using (72) and (73) together with the definition of $\nabla A$, we get

(74) $\displaystyle\nabla_{X}{\color[rgb]{0,0,0}T}^{\top}$
$\displaystyle=-\rho(\tau)\,\overline{g}(N,\partial_{t})AX+\rho^{\prime}(\tau)X{\color[rgb]{0,0,0}\,=vAX+\rho^{\prime}(\tau)X},$
(75) $\displaystyle\nabla_{X}(AT^{\top})$
$\displaystyle=(\nabla_{X}A)T^{\top}+A(\nabla_{X}T^{\top})$ (76)
$\displaystyle=(\nabla_{T^{\top}}A)X-(\overline{\mathrm{R}}(X,T^{\top})N)^{\top}+vA^{2}X+\rho^{\prime}(\tau)AX$

for any $X\in\mathfrak{X}(M)$. By definition, for every function $u\in
C^{\infty}(M)$,

$\Delta u=\mathrm{trace}(\nabla(\nabla
u))=\sum_{i=1}^{m}g(\nabla_{E_{i}}\nabla u,E_{i})$

for any choice of a local orthonormal frame $\\{E_{1},\dots,E_{m}\\}$ on $TM$.
By (69), (70), (74), (75) and since $\mathrm{trace}(A)=-mH$ and
$\mathrm{trace}(\nabla_{T^{\top}}A)=\nabla_{T^{\top}}(\mathrm{trace}(A))=-m\nabla_{T^{\top}}H$,
we obtain

$\Delta\eta=-m\rho^{\prime}(\tau)-mHv,\quad\Delta
v=-\overline{\mathrm{Ric}}(T^{\top},N)+\mathrm{trace}(A^{2})v-m\nabla_{T^{\top}}H-mH\rho^{\prime}(\tau),$

and (64) is proved. Writing
${\color[rgb]{0,0,0}T}^{\top}={\color[rgb]{0,0,0}T}-vN$, by (50) we have

(77)
${\color[rgb]{0,0,0}-\overline{\mathrm{Ric}}(T^{\top},N)=-\overline{\mathrm{Ric}}(T,N)+\mbox{{\rm
Ric}}(N,N)v=\left(m\frac{\rho^{\prime\prime}(\tau)}{\rho(\tau)}+\mbox{{\rm
Ric}}(N,N)\right)v,}$

as ${\pi_{F}}^{\ast}\mathrm{Ric}^{F}(\partial_{t},\;\cdot\;)=0$ and
$(dt\otimes dt)(T,N)=-\overline{g}(T,N)=\rho(\tau)\cosh\theta=v$. This
concludes the proof of (65). If $\psi$ is maximal, then $H\equiv 0$ and (67)
and (68) follow at once. $\square$

######  Remark 18.

The spacelike slices $\\{t\\}\times F$, $t\in I$ of $\overline{M}$ are totally
umbilical hypersurfaces, in other words they satisfy
$\mathrm{trace}(A^{2})=mH^{2}$, and they have mean curvature
$H=\rho^{\prime}(t)/\rho(t)$ in the direction of the future-pointing normal.
This is a consequence of (64) and (65), as the image $\psi(M)$ of an immersed
hypersurface $\psi:M\longrightarrow\overline{M}$ is contained in a spacelike
slice if and only if $\tau$ is constant on $M$, in which case $N=\partial_{t}$
and $v\equiv\rho(t)$.

As a first application of equation (67), we prove Theorem G of the
Introduction as a corollary of the following result, which generalizes Theorem
3.7 of [7].

######  Theorem 19.

Let $\psi:M\longrightarrow\overline{M}$ be a maximal hypersurface in a GRW
spacetime $\overline{M}=I\times_{\rho}F$. If the weak maximum principle for
the Laplacian holds on $M$ and $\psi(M)$ is contained in a slab $[a,b]\times
F\subseteq I\times F$, then $\rho^{\prime}(\tau^{\ast})\geq 0$ and
$\rho^{\prime}(\tau_{\ast})\leq 0$, where $\tau^{\ast}=\sup_{M}\tau$,
$\tau_{\ast}=\inf_{M}\tau$.

  * Proof.

Let $\eta$ be defined as in (59). Since $\tau(M)\subseteq[a,b]$, we have

(78)
$-\infty<\int_{t_{0}}^{a}\rho(s)ds=\overline{\eta}(a)\leq\eta\leq\overline{\eta}(b)=\int_{t_{0}}^{b}\rho(s)ds<+\infty\qquad\text{on
}M.$

By the weak maximum principle applied to $\eta$ and $-\eta$, see Definition 5,
we can find two sequences of points
$\\{x_{k}\\}_{k\in\mathbb{N}},\\{y_{k}\\}_{k\in\mathbb{N}}\subseteq M$ such
that

$\displaystyle\lim_{k\to+\infty}\eta(x_{k})=\sup_{M}\eta,$
$\displaystyle\qquad\limsup_{k\to+\infty}\Delta\eta(x_{k})\leq 0,$
$\displaystyle\lim_{k\to+\infty}\eta(y_{k})=\inf_{M}\eta,$
$\displaystyle\qquad\liminf_{k\to+\infty}\Delta\eta(y_{k})\geq 0.$

We recall that $\eta$ is a strictly monotonic function of $\tau$ and that
$\Delta\eta=-m\rho^{\prime}(\tau)$ on $M$. Hence,

$-m\rho^{\prime}(\tau^{\ast})=\lim_{k\to+\infty}-m\rho^{\prime}(\tau(x_{k})))=\lim_{k\to+\infty}\Delta\eta(x_{k})\leq
0$

and similarly we have $-m\rho^{\prime}(\tau_{\ast})\geq 0$. $\square$

######  Corollary 20.

Let $\overline{M}=I\times_{\rho}F$ be a GRW spacetime whose sectional
curvatures on spacelike $2$-planes are bounded below and let
$\psi:M\longrightarrow\overline{M}$ be a complete maximal hypersurface. If
$\psi(M)$ is contained in a slab $[a,b]\times F$, then
$\rho^{\prime}(\tau^{\ast})\geq 0$, $\rho^{\prime}(\tau_{\ast})\leq 0$, where
$\tau^{\ast}=\sup_{M}\tau$, $\tau_{\ast}=\inf_{M}\tau$. In particular, if
there exist $t_{0},t_{1}\in[a,b]$ such that $\rho^{\prime}=0$ on
$[t_{0},t_{1}]$, then the following implications hold:

  * i)

if $\rho^{\prime}<0$ on $[a,t_{0})$ and $\rho^{\prime}>0$ on $(t_{1},b]$, then
$\psi(M)$ must intersect every spacelike slice $\\{t\\}\times F$ with
$t_{0}\leq t\leq t_{1}$;

  * ii)

if $\rho^{\prime}>0$ on $[a,t_{0})$ and $\rho^{\prime}<0$ on $(t_{1},b]$, then
$\psi(M)\subseteq[t_{0},t_{1}]\times F$.

  * Proof.

As in the proof of Theorem 7, since $M$ is maximal and there exists
$C\in\mathbb{R}$ such that $\overline{\mathrm{Sect}}(\Pi)\geq C$ for every
spacelike $2$-plane $\Pi\subseteq T\overline{M}$, the Ricci curvature of $M$
is bounded below by $(\dim M-1)C$. Since $(M,g)$ is complete, the weak maximum
principle for the Laplacian holds on $M$ and the main statement is a direct
consequence of Theorem 19. If $t_{0}\in[a,b]$ is such that $\rho^{\prime}<0$
(respectively, $\rho^{\prime}>0$) on $[a,t_{0})$, then $\tau^{\ast}\geq t_{0}$
(resp., $\tau_{\ast}\geq t_{0}$). Similarly, if $t_{1}\in[a,b]$ is such that
$\rho^{\prime}>0$ (resp., $\rho^{\prime}<0$) on $(t_{1},b]$, then
$\tau_{\ast}\leq t_{1}$ (resp., $\tau^{\ast}\leq t_{1}$). This concludes the
proof. $\square$

The following consequence is a Frankel type result.

######  Corollary 21.

Let $\mathbb{S}^{m+1}_{1}(\overline{\kappa})$ be the $(m+1)$-dimensional de
Sitter spacetime of constant curvature $\overline{\kappa}>0$ and let
$\psi_{0}:M_{0}\longrightarrow\mathbb{S}^{m+1}_{1}(\overline{\kappa})$,
$\psi_{1}:M_{1}\longrightarrow\mathbb{S}^{m+1}_{1}(\overline{\kappa})$ be two
complete maximal hypersurfaces. Then
$\psi_{0}(M_{0})\cap\psi_{1}(M_{1})\neq\varnothing$.

  * Proof.

Let $F=\mathbb{S}^{m}(\overline{\kappa})$ be the standard $m$-sphere of
constant curvature $\overline{\kappa}>0$, set
$\rho(t)=\cosh(\sqrt{\overline{\kappa}}t)$ for every $t\in\mathbb{R}$ and let
$\overline{M}=\mathbb{R}\times_{\rho}F$. The GRW spacetime $\overline{M}$ is
isometric to $\mathbb{S}^{m+1}_{1}(\overline{\kappa})$ (see, for instance,
page 339 of [44]). By Theorem 10, $M_{0}$ is compact and $\psi_{0}$ is totally
geodesic, so there exists an isometry
$\Psi:\mathbb{S}^{m+1}_{1}(\overline{\kappa})\longrightarrow\overline{M}$ such
that $\Psi\circ\psi_{0}:M_{0}\to\overline{M}$ sends $M_{0}$ into the totally
geodesic spacelike slice $\\{0\\}\times F$. Since $M_{0}$ is compact and $F$
is connected, $(\Psi\circ\psi_{0})(M_{0})=\\{0\\}\times F$. By Theorem 10
again, $M_{1}$ is compact. Let $\psi:=\Psi\circ\psi_{1}:M_{1}\to\overline{M}$.
The projection of $\psi(M_{1})$ on the $\mathbb{R}$-factor of $\overline{M}$
is compact. Moreover, $\rho^{\prime}<0$ on $(-\infty,0)$ and $\rho^{\prime}>0$
on $(0,+\infty)$. So, we apply point i) of Corollary 20 to obtain that
$(\\{0\\}\times F)\cap\psi(M_{1})\neq\varnothing$. Since
$\psi_{0}(M_{0})=\Psi^{-1}(\\{0\\}\times F)$,
$\psi_{1}(M_{1})=\Psi^{-1}(\psi(M_{1}))$, we obtain
$\psi_{0}(M_{0})\cap\psi_{1}(M_{1})\neq\varnothing$. $\square$

We are now ready to prove Theorem F of the Introduction.

######  Theorem 22.

Let $\psi:M\longrightarrow\overline{M}$ be a complete maximal surface in a
$3$-dimensional GRW spacetime $\overline{M}=I\times_{\rho}F$. Suppose that
$\overline{M}$ has nonnegative sectional curvatures on spacelike tangent
$2$-planes. Then $\psi$ is totally geodesic and one of the following cases
occurs:

  * i)

$\psi(M)$ is a spacelike slice $\\{t\\}\times F$ for some $t\in I$ such that
$\rho^{\prime}(t)=0$,

  * ii)

$(F,g_{F})$ is a flat, complete Riemann surface and $\overline{M}$ is the
product manifold $\mathbb{R}\times F$ endowed with the flat metric
$-dt^{2}+g_{F}$,

  * iii)

$(F,g_{F})$ is a compact Riemann surface of constant positive Gaussian
curvature, $M$ is a round sphere and the spacetime $\overline{M}$ has constant
positive sectional curvature in the slab $\tau(M)\times F$.

  * Proof.

As already remarked, the assumption that $\overline{M}$ has nonnegative
sectional curvatures on spacelike $2$-planes implies that $\mathrm{Ric}\geq 0$
on $M$ in the sense of quadratic forms, that is, the Gaussian curvature
$\mathrm{K}$ of $M$ is nonnegative. Moreover, denoting by $\mathrm{K}^{F}$ the
Gaussian curvature of $F$, by Lemma 16 there exists $C_{2}\in\mathbb{R}$ such
that, for each $(t,x)\in\overline{M}$,

(79) $\begin{split}\mathrm{K}^{F}(x)\geq
C_{2},\qquad\frac{C_{2}-\rho(t)\rho^{\prime\prime}(t)+\rho^{\prime}(t)^{2}}{\rho(t)^{2}}&\geq
0,\\\
\frac{C_{2}-\rho(t)\rho^{\prime\prime}(t)+\rho^{\prime}(t)^{2}}{\rho(t)^{2}}&\geq-\frac{\rho^{\prime\prime}(t)}{\rho(t)}.\end{split}$

Let $v=-\overline{g}(T,N)$ on $M$. By (68), for any $\alpha\in\mathbb{R}$ the
function $v^{-\alpha}$ satisfies

(80) $\begin{split}\Delta v^{-\alpha}&=-\alpha v^{-\alpha-1}\Delta
v+\alpha(\alpha+1)v^{-\alpha-2}|\nabla v|^{2}\\\ &=-\alpha
v^{-\alpha}\left(\overline{\mathrm{Ric}}(N,N)+\mathrm{trace}(A^{2})+2\frac{\rho^{\prime\prime}(\tau)}{\rho(\tau)}-(\alpha+1)\frac{|\nabla
v|^{2}}{v^{2}}\right)\end{split}$

on $M$. Since $\pi_{F}^{\ast}g_{F}(N,N)=\rho(t)^{-2}(\overline{g}+dt\otimes
dt)(N,N)=\rho(t)^{-2}(-1+\cosh^{2}\theta)=\rho(t)^{-2}\sinh^{2}\theta$, from
(50) and the first two inequalities in (79) we have

(81)
$\overline{\mathrm{Ric}}(N,N)+2\frac{\rho^{\prime\prime}(t)}{\rho(t)}=\sinh^{2}\theta\left(\frac{\mathrm{K}^{F}(x)}{\rho(t)^{2}}-\frac{\rho^{\prime\prime}(t)}{\rho(t)}+\frac{\rho^{\prime}(t)^{2}}{\rho(t)^{2}}\right)\geq
0$

and by (66) we deduce, for every $\alpha\in[-1,1]$,

(82) $\mathrm{trace}(A^{2})-(\alpha+1)\frac{|\nabla
v|^{2}}{v^{2}}\geq\frac{1-\alpha}{2}\mathrm{trace}(A^{2})\geq 0.$

Inserting these inequalities into (80) we find that for every $\alpha\in[0,1]$
the positive function $v^{-\alpha}$ is superharmonic on $M$. In particular,
$1/v$ is a positive superharmonic function on $M$. If $(M,g)$ is complete,
then it is parabolic because of its nonnegative Gaussian curvature, so $1/v$
is constant on $M$. Therefore $v$ is also constant and from (68) we obtain

(83)
$\overline{\mathrm{Ric}}(N,N)+2\frac{\rho^{\prime\prime}(\tau)}{\rho(\tau)}\equiv
0,\qquad\mathrm{trace}(A^{2})\equiv 0\qquad\text{on }M$

in view of (81). Hence, $\psi$ is totally geodesic.

Suppose that $\psi(M)$ is not a spacelike slice: then $\tau$ is not constant
on $M$ and by (61) the hyperbolic angle $\theta$ is not identically null, so
$M_{0}:=\\{q\in M:\sinh^{2}\theta(q)\neq 0\\}$ is a nonempty open subset of
$M$. Since $\overline{\eta}$ is a strictly increasing function on $I$, the
function $\eta$ defined in (59) is nonconstant. Moreover, equation (67) reads

(84) $\Delta\eta=-2(\rho^{\prime}\circ\overline{\eta}^{-1})(\eta)$

and the function $-2(\rho^{\prime}\circ\overline{\eta}^{-1})$ is of class
$C^{1}$ on its domain; hence the unique continuation property holds for
equation (84), that is, $\eta$ is constant on some nonempty open subset of $M$
if and only if it is constant on $M$ (see Theorem A.5 of [39]). Therefore,
$M_{0}$ is a dense open subset of $M$. Finally,
$v=\rho(\tau)\cosh\theta\geq\rho(\tau)$ implies that $\rho$ is bounded on
$\tau(M)\subseteq I$, as $v$ is constant. We set

(85) $I_{0}=\\{t\in
I:\rho(t)\rho^{\prime\prime}(t)-\rho^{\prime}(t)^{2}=C_{2}\\},\qquad
F_{0}=\\{x\in F:\mathrm{K}^{F}(x)=C_{2}\\}.$

Let $q\in M_{0}$ be given and set $(t,x)=\psi(q)\in\overline{M}$. From (83) it
follows that (81) holds with the equality sign. Since $\sinh^{2}\theta(q)\neq
0$, the same is true for the first two inequalities in (79). Therefore,
$\tau(M_{0})\subseteq I_{0}$ and $(\pi_{F}\circ\psi)(M_{0})\subseteq F_{0}$.
Note that $I_{0}$ and $F_{0}$ are closed in $I$ and $F$, respectively, and
that $\tau$ is constant on each connected component of $M\setminus M_{0}$.
Hence, $\tau(M)=\tau(\overline{M_{0}})\subseteq I_{0}$,
$(\pi_{F}\circ\psi)(M)=(\pi_{F}\circ\psi)(\overline{M_{0}})\subseteq F_{0}$.
As observed at the beginning of this section, $\rho^{\prime\prime}/\rho$ is
constant on every interval contained in $I_{0}$. Since $M$ is connected,
$\tau(M)$ is an interval and there exists $\overline{\kappa}\in\mathbb{R}$
such that

(86) $\frac{\rho^{\prime\prime}(t)}{\rho(t)}=\overline{\kappa}\qquad\text{for
each }t\in\tau(M).$

By the third inequality in (79), $\overline{\kappa}\geq 0$. So far, we have
proved that $\overline{M}$ has constant curvature $\overline{\kappa}\geq 0$ in
the cylinder

(87) $\Sigma=\tau(M)\times(\pi_{F}\circ\psi)(M)\subseteq I_{0}\times
F_{0}\subseteq I\times F.$

We conclude the proof by showing that $(\pi_{F}\circ\psi)(M)=F$ and that in
case where $\overline{\kappa}=0$ it must be $\tau(M)=\mathbb{R}$ and
$\rho^{\prime}\equiv 0$ on $\mathbb{R}$.

First suppose that $\overline{\kappa}=0$. Then, $\rho^{\prime}$ is constant on
$\tau(M)$. If $\rho^{\prime}\equiv C_{1}$ on $\tau(M)$ for a positive constant
$C_{1}>0$, then $\eta$ is a nonconstant superharmonic function on $M$ because
it satisfies equation (84), so it cannot be bounded above on the parabolic
surface $M$. This implies that $\tau(M)$ is not bounded above, otherwise we
would have

$\sup_{M}\eta\leq\int_{t_{0}}^{\sup\tau(M)}\rho(s)ds=(\sup\tau(M)-t_{0})\,\rho(t_{0})+\frac{C_{1}}{2}\,(\sup\tau(M)-t_{0})^{2}<+\infty.$

Since $\rho^{\prime}\equiv C_{1}>0$ on $\tau(M)$ and $\tau(M)$ is not bounded
above, we obtain that $\rho$ is not bounded on $\tau(M)$ and we reach a
contradiction. Similarly, we conclude that $\rho^{\prime}$ cannot be
identically equal to a negative constant on $\tau(M)$, so we are left with the
case where $\rho^{\prime}\equiv 0$ on $\tau(M)$. In this case, by (84) we have
that $\eta$ is a nonconstant harmonic function on $M$ and therefore it is not
bounded above nor below. Arguing as above we can show that $\tau(M)$ is not
bounded above nor below, so $\mathbb{R}=\tau(M)\subseteq I\subseteq\mathbb{R}$
and we conclude that $I=\mathbb{R}$. By (85) and (87), since
$\rho^{\prime}\equiv 0$ on $\tau(M)$ we deduce $C_{2}=0$ and
$\mathrm{K}^{F}=0$ on $F_{0}$. If we set $F^{\prime}=(\pi_{F}\circ\psi)(M)$
and we endow the surface $M$ with the metric
$\sigma:=(\pi_{F}\circ\psi)^{\ast}g_{F}$, then $(M,\sigma)$ is complete
because $\sigma=g+d\tau^{2}\geq g$ and $(\pi_{F}\circ\psi):M\to F^{\prime}$ is
a local Riemannian isometry, so $(F^{\prime},(g_{F})|_{F^{\prime}})$ is also
complete and we conclude that $F=F^{\prime}\subseteq F_{0}\subseteq F$. In
particular, $F_{0}=F$ and $\mathrm{K}^{F}\equiv 0$ on $F$.

Now, suppose that (86) holds with $\overline{\kappa}>0$. $\psi(M)$ is
contained in the cylinder $\Sigma$ and $\psi$ is totally geodesic, so $M$ has
constant positive Gaussian curvature $\overline{\kappa}$ and therefore it is
compact by Bonnet theorem. In particular, from Theorem 11 it follows that
$(M,g)$ is a round sphere. The map $\pi_{F}\circ\psi$ is continuous and open
(being a local diffeomorphism), so $(\pi_{F}\circ\psi)(M)$ is compact and open
in $F$. Since $F$ is connected, we conclude $(\pi_{F}\circ\psi)(M)=F$.
Moreover, since the second of (79) holds with the equality sign for every
$t\in\tau(M)$, we have $(\log\rho)^{\prime\prime}=C_{2}/\rho^{2}$ on
$\tau(M)$. Suppose, by contradiction, that $C_{2}\leq 0$. Then $\log\rho$ is
concave and the sign of $\rho^{\prime}$ is nonincreasing on $\tau(M)$. As
$\tau(M)$ is compact, we can apply the last statement of Corollary 20 to
deduce that $\rho^{\prime}(\tau)\equiv 0$ on $M$ and by (84) we get that
$\eta$ is a nonconstant harmonic function on the compact surface $M$,
contradiction. Therefore, $C_{2}>0$ and by (85) we conclude that $F=F_{0}$ has
constant positive Gaussian curvature. $\square$

We conclude this section with the following two results.

######  Theorem 23.

Let $\overline{M}=I\times_{\rho}F$ be a GRW spacetime with
$\rho^{\prime\prime}\leq 0$. Let $\psi:M\longrightarrow\overline{M}$ be a
complete, non-compact, maximal hypersurface in $\overline{M}$ such that, for
some $o\in M$ and for some (hence, any) $a>0$,

(88) $\lim_{r\to+\infty}\int_{a}^{r}\frac{dt}{\mathrm{Vol}(\partial
B_{t})}=+\infty,$

where $B_{t}$ is the geodesic ball of $(M,g)$ centered at $o$ with radius $t$.
Then

(89)
$\limsup_{r\to+\infty}\int_{B_{r}}\left(\overline{\mathrm{Ric}}(N,N)+\mathrm{trace}(A^{2})\right)\geq
0.$

  * Proof.

From (68) we know that the positive function
$v=-\overline{g}({\color[rgb]{0,0,0}T},N)$ satisfies

(90) $Lv=\Delta
v-\left(\overline{\mathrm{Ric}}(N,N)+\mathrm{trace}(A^{2})\right)v=m\frac{\rho^{\prime\prime}(\tau)}{\rho(\tau)}v\leq
0.$

Therefore, by Lemma 4, the operator $L$ satisfies $\lambda_{1}^{L}(M)\geq 0$.
Set

(91) $w(r)=\mathrm{Vol}(\partial B_{r}),\qquad
A(r)=-\frac{1}{w(r)}\int_{\partial
B_{r}}\left(\overline{\mathrm{Ric}}(N,N)+\mathrm{trace}(A^{2})\right)$

and consider the weak solution
$z\in\mathrm{Lip}_{\mathrm{loc}}(\mathbb{R}_{0}^{+})$ of the Cauchy problem

(92) $\begin{cases}(w(r)z^{\prime})^{\prime}+A(r)w(r)z=0&\text{on
}\mathbb{R}^{+}\\\ z(0^{+})=1,\quad(wz^{\prime})(0^{+})=0.\end{cases}$

We have $z>0$ on $\mathbb{R}^{+}_{0}$. If not, let $R_{0}$ be the first
positive zero of $z$ and set $\varphi(x)=z(r(x))$ for $x\in B_{R_{0}}$, with
$r(x)$ the distance from $x$ to $o$ in $(M,g)$. Since $z$ solves problem (92),
by Proposition 2 and using the coarea formula as in the proof of Theorem 12 we
get

$0\leq\lambda_{1}^{L}(M)<\lambda_{1}^{L}(B_{R_{0}})\leq\frac{\int_{0}^{R_{0}}w(r)z^{\prime}(r)^{2}dr+\int_{0}^{R_{0}}A(r)w(r)z(r)^{2}dr}{\int_{0}^{R_{0}}w(r)z(r)^{2}}=0,$

contradiction. So, problem (92) has a positive solution and (88) holds. By
Theorem 2.8 of [29],

(93)
$0\geq\liminf_{R\longrightarrow+\infty}\int_{0}^{R}A(s)w(s)ds=-\limsup_{R\longrightarrow+\infty}\int_{B_{R}}\left(\overline{\mathrm{Ric}}(N,N)+\mathrm{trace}(A^{2})\right).$

$\square$

######  Theorem 24.

Let $\psi:M\longrightarrow\overline{M}$ be a complete stable maximal
hypersurface in a GRW spacetime $\overline{M}=I\times_{\rho}F$ and let $\tau$,
$v$ be as above.

  * i)

If $M$ is compact, then either $\rho^{\prime\prime}(\tau)\equiv 0$ on $M$ or
$\rho^{\prime\prime}(\tau)$ attains both positive and negative values on $M$.

  * ii)

If $M$ is non-compact and for some $o\in M$ the function $v$ satisfies

(94) $\lim_{r\to+\infty}\int_{a}^{r}\left(\int_{\partial
B_{s}}v^{2}\right)^{-1}ds=+\infty$

for some (hence any) $a>0$, where $B_{s}$ is the geodesic ball of $(M,g)$
centered at $o$ with radius $s$, then either $\rho^{\prime\prime}(\tau)\equiv
0$ on $M$ or $\rho^{\prime\prime}(\tau)$ attains negative values at some
points of $M$.

  * Proof.

Since $M$ is stable, from Lemma 4 there exists a positive function $u$
satisfying $\Delta
u=\left(\overline{\mathrm{Ric}}(N,N)+\mathrm{trace}(A^{2})\right)u$ on $M$.
Set $\varphi=u/v$. By (68), a direct computation shows that

(95)
$\frac{\mathrm{div}(v^{2}\nabla\varphi)}{v^{2}}=\Delta\varphi+2g\left(\frac{\nabla
v}{v},\nabla\varphi\right)=\frac{\Delta u}{v}-\frac{u\Delta
v}{v^{2}}=-m\frac{\rho^{\prime\prime}(\tau)}{\rho(\tau)}\varphi\qquad\text{on
}M.$

i) Suppose that $M$ is compact. We apply the divergence theorem to obtain

$0=\int_{M}\mathrm{div}(v^{2}\nabla\varphi)=-m\int_{M}\frac{\rho^{\prime\prime}(\tau)}{\rho(\tau)}v^{2}\varphi.$

Since $v^{2}\varphi/\rho(\tau)>0$ on $M$, if $\rho^{\prime\prime}(\tau)\geq 0$
or $\rho^{\prime\prime}(\tau)\leq 0$ on $M$ then it must be
$\rho^{\prime\prime}(\tau)\equiv 0$.

ii) Suppose that $M$ is non-compact. (94) ensures that any positive function
$w\in C^{2}(M)$ such that $\mathrm{div}(v^{2}\nabla w)\leq 0$ on $M$ must be
constant, see Theorem 4.14 of [4]. Since $\varphi>0$, if
$\rho^{\prime\prime}(\tau)\geq 0$ on $M$ then from (95) we deduce that
$\varphi$ is constant and therefore $\rho^{\prime\prime}(\tau)\equiv 0$.
$\square$

######  Remark 25.

Let $\psi:M\longrightarrow\overline{M}=I\times_{\rho}F$ be a maximal
hypersurface such that $\tau(M)\subseteq T_{0}=\\{t\in
I:\rho^{\prime\prime}(t)\leq 0\\}$. Since $v>0$ on $M$, by condition 3 of
Lemma 4 and (68) we immediately deduce that $M$ is stable. Similarly, if
$\tau(M\setminus\overline{P})\subseteq T_{0}$ for some relatively compact open
set $P$, then $\psi$ is stable at infinity. For $T_{0}=I$, stability of $\psi$
is observed in Theorem 7 of [21].

##  5\. Higher order mean curvatures in Robertson-Walker spacetimes

In this section we will consider spacelike hypersurfaces in spacetimes of
constant sectional curvature. Let $\psi:M\longrightarrow\overline{M}$ be such
a hypersurface and suppose that the $(k+1)$-th mean curvature function
$H_{k+1}$ vanishes on $M$, for some $1\leq k\leq m-2$. Then, the $k$-th Newton
tensor $P_{k}$ corresponding to the shape operator $A$ of $\psi$ is positive
definite if and only if $H_{k}>0$ and $\mathrm{rank}(A)>k$ on $M$, see
Proposition 6.27 of [11]. Hence, the differential operator $L_{k}$ defined in
(11) is elliptic if and only if these conditions are satisfied. Furthermore,
since $\overline{M}$ has constant curvature, $L_{k}$ can be written in
divergence form, that is,

(96) $L_{k}u=\mathrm{div}(P_{k}(\nabla u))\qquad\text{for every }u\in
C^{2}(M).$

If the ambient spacetime has a Robertson-Walker spacetime structure
$\overline{M}=I\times_{\rho}F$, then we can consider the functions $\eta$ and
$v$ on $M$ as in the previous section. The action of $L_{k}$ on $\eta$ and $v$
is given by identities (97) and (98) below, also proved in Lemma 4.1 of [3]
and Lemma 3.1 of [16].

######  Lemma 26.

Let $\psi:M\longrightarrow\overline{M}$ be a spacelike hypersurface in a
Robertson-Walker spacetime $\overline{M}=I\times_{\rho}F$ of constant
curvature and let $\eta$ and $v$ be defined in (59) and (60). For $0\leq k\leq
m-1$,

(97) $\displaystyle L_{k}\eta$
$\displaystyle=-c_{k}\rho^{\prime}(\tau)H_{k}+c_{k}H_{k+1}v,$ (98)
$\displaystyle L_{k}v$
$\displaystyle=\mathrm{trace}(A^{2}P_{k})v-\binom{m}{k+1}g(T^{\top},\nabla
H_{k+1})-c_{k}H_{k+1}\rho^{\prime}(\tau).$

  * Proof.

In the proof of Lemma 17 we have already calculated the second covariant
derivatives of $\eta$ and $v$. More precisely, from (74) we can write

(99) $\displaystyle\nabla_{X}\nabla\eta$
$\displaystyle=vAX+\rho^{\prime}(\tau)X,$ (100) $\displaystyle\nabla_{X}\nabla
v$
$\displaystyle=(\nabla_{T^{\top}}A)X-\left(\overline{\mathrm{R}}\left(X,T^{\top}\right)N\right)^{\top}+vA^{2}X+\rho^{\prime}(\tau)AX$

for every $X\in\mathfrak{X}(M)$. Recalling the definition (11) of $L_{k}$,
formulas (97) and (98) follow from Lemma 1 and from the fact that
$g(\overline{\mathrm{R}}\left(X,Y\right)N,Z)=\overline{\mathrm{Riem}}(Z,N,X,Y)=0$
for every $X,Y,Z\in\mathfrak{X}(M)$ since $\overline{M}$ has constant
curvature. $\square$

The next theorem collects some observations about the $k$-stability of
hypersurfaces with zero $(k+1)$-th mean curvature and positive definite $k$-th
Newton tensor.

######  Theorem 27.

Let $\psi:M\longrightarrow\overline{M}$ be a complete spacelike hypersurface
with zero $(k+1)$-th mean curvature, for some $1\leq k\leq m-2$, in a
spacetime $\overline{M}$ of dimension $m+1$ and constant curvature
$\overline{\kappa}$. Suppose that $H_{k}>0$ on $M$ and that
$\mathrm{rank}(A)>k$ on $M$.

  * i)

If $\overline{\kappa}\leq 0$, then $M$ is non-compact and $\psi$ is
$k$-stable.

  * ii)

If $\overline{\kappa}>0$ and $M$ is compact and simply connected, then $\psi$
is not $k$-stable.

  * iii)

If $\overline{\kappa}>0$ and we assume that $M$ is non-compact and that, for
some $o\in M$ and for some (hence any) $a>0$,

(101) $\begin{split}&\lim_{r\to+\infty}\int_{a}^{r}\left(\int_{\partial
B_{r}}H_{k}\right)^{-1}=+\infty,\\\
&\lim_{r\longrightarrow+\infty}\int_{B_{r}}\left(\overline{\kappa}\
\mathrm{trace}(P_{k})-\mathrm{trace}(A^{2}P_{k})\right)=+\infty,\end{split}$

where $B_{r}$ is the geodesic ball of $(M,g)$ with radius $r$ centered at $o$,
then $\widetilde{L}_{k}$ has infinite index.

  * Proof.

Firstly, note that $H_{k+1}\equiv 0$ and $\mathrm{rank}(A)>k$ on $M$ guarantee
that the self-adjoint operator $P_{k}$ is positive definite (see Proposition
6.27 of [11]). Since $A^{2}$ and $P_{k}$ are simultaneously diagonalizable,
$A^{2}P_{k}$ is also self-adjoint and positive definite and therefore

(102) $\mathrm{trace}(P_{k})>0,\qquad\mathrm{trace}(A^{2}P_{k})>0.$

Moreover, the operator $L_{k}$ is elliptic and it can be put in divergence
form due to the fact that $\overline{M}$ has constant sectional curvature
$\overline{\kappa}$, that is, we have

(103) $L_{k}u=\mathrm{div}(P_{k}(\nabla u))\qquad\text{for every }u\in
C^{2}(M).$

i) Suppose that $\overline{\kappa}\leq 0$. By (102) it follows that
$\left(\overline{\kappa}\
\mathrm{trace}(P_{k})-\mathrm{trace}(A^{2}P_{k})\right)\leq 0$, so

$\int_{M}-\phi\widetilde{L}_{k}\phi=\int_{M}g(P_{k}(\nabla\phi),\nabla\phi)-\left(\overline{\kappa}\
\mathrm{trace}(P_{k})-\mathrm{trace}(A^{2}P_{k})\right)\phi^{2}\geq 0$

for every $\phi\in C^{\infty}_{c}(M)$, and $\psi$ is $k$-stable by definition.
By Lemma 4, there exists a positive function $u\in C^{\infty}(M)$ such that
$L_{k}u=(\mathrm{trace}(A^{2}P_{k})-\overline{\kappa}\
\mathrm{trace}(P_{k}))u$ on $M$. Suppose, by contradiction, that $M$ is
compact. By (103) and the divergence theorem,

$0=\int_{M}L_{k}u=\int_{M}(\mathrm{trace}(A^{2}P_{k})-\overline{\kappa}\
\mathrm{trace}(P_{k}))u.$

From (102) and $\overline{\kappa}\leq 0$ it then follows that $u\equiv 0$ on
$M$, contradiction.

ii) Suppose, by contradiction, that $\overline{\kappa}>0$, $M$ is compact and
simply connected and $\psi$ is $k$-stable. Let
$\pi:\widetilde{\overline{M}}\longrightarrow\overline{M}$ be the Lorentzian
universal covering of $\overline{M}$. $\widetilde{\overline{M}}$ is isometric
to de Sitter spacetime of dimension $m+1$ and curvature $\overline{\kappa}$,
which in turn is isometric to the GRW spacetime
$\mathbb{R}\times_{\cosh(\sqrt{\overline{\kappa}}t)}\mathbb{S}^{m}(\overline{\kappa})$.
$M$ is simply connected, so for every $p\in M$, $\tilde{p}\in\pi^{-1}(p)$
there exists a unique immersion
$\widetilde{\psi}:M\longrightarrow\widetilde{\overline{M}}$ such that
$\widetilde{\psi}(p)=\tilde{p}$ and $\pi\circ\widetilde{\psi}=\psi$. The shape
operator induced by $\widetilde{\psi}$ is equal to $A$, up to a change of
sign, because $\pi$ is a local isometry. Then, by Lemma 26, $M$ supports a
positive smooth functions $v$ satisfying $L_{k}v=\mathrm{trace}(A^{2}P_{k})v$.
Since $\psi$ is $k$-stable, by Lemma 4, $M$ also supports a positive smooth
function $u$ satisfying $L_{k}u=(\mathrm{trace}(A^{2}P_{k})-\overline{\kappa}\
\mathrm{trace}(P_{k}))u$. A direct computation shows that the positive
function $\varphi=u/v$ satisfies

$\frac{\mathrm{div}(v^{2}P_{k}(\nabla\varphi))}{v^{2}}=L_{k}\varphi+2g\left(\frac{\nabla
v}{v},P_{k}(\nabla\varphi)\right)=-\overline{\kappa}\
\mathrm{trace}(P_{k})\varphi.$

Since $M$ is compact, we have

$0=\int_{M}\mathrm{div}(v^{2}P_{k}(\nabla\varphi))=-\int_{M}\overline{\kappa}\
\mathrm{trace}(P_{k})v^{2}\varphi.$

From (102) and $\overline{\kappa}>0$ it then follows that $v^{2}\varphi\equiv
0$ on $M$, contradiction.

iii) Suppose, by contradiction, that $\overline{\kappa}>0$, that condition
(101) holds for some $o\in M$ and that $\widetilde{L}_{k}$ has finite index.
Then there exists a relatively compact open set $\Omega\subseteq M$ such that

(104) $\lambda_{1}^{\widetilde{L}_{k}}(M\setminus\overline{\Omega})\geq 0.$

Since $(M,g)$ is complete, there exists $R>0$ such that
$\overline{\Omega}\subseteq B_{R}$. Next we define

(105) $v_{k}(r)=\int_{\partial B_{r}}c_{k}H_{k},\qquad
A(r)=\frac{1}{v_{k}(r)}\int_{\partial B_{r}}\left(\overline{\kappa}\
\mathrm{trace}(P_{k})-\mathrm{trace}(A^{2}P_{k})\right).$

We consider the Cauchy problem

(106) $\begin{cases}(v_{k}(r)z^{\prime})^{\prime}+A(r)v_{k}(r)z=0\ \text{on}\
\mathbb{R}^{+}\\\ z(0^{+})=Z_{0}>0,\
w(0^{+})z^{\prime}(0^{+})=0\in\mathbb{R}.\end{cases}$

Since $v_{k}\in L_{\mathrm{loc}}^{\infty}(\mathbb{R}_{0}^{+})$, there exists a
solution $z$ of (106) with
$z\in\mathrm{Lip}_{\mathrm{loc}}(\mathbb{R}_{0}^{+})$ due to Proposition 4.2
of [11]. Moreover, from the coarea formula and (101) we obtain

(107)
$\liminf_{r\longrightarrow+\infty}\int_{0}^{r}A(s)v_{k}(s)ds=\liminf_{r\longrightarrow+\infty}\int_{B_{r}}\left(\overline{\kappa}\
\mathrm{trace}(P_{k})-\mathrm{trace}(A^{2}P_{k})\right)=+\infty.$

This condition and the fact that $v_{k}^{-1}\in
L_{\mathrm{loc}}^{\infty}(\mathbb{R^{+}})$ and $v_{k}^{-1}\notin
L^{1}(+\infty)$ enable us to use Corollary 2.9 of [29] to obtain that any
solution $z$ of (106) is oscillatory. Taking now $R\leq R_{1}<R_{2}$ two
consecutive zeros of $z$ such that $z>0$ on $(R_{1},R_{2})$ we define the
function $\varphi(x):=z(r(x))$, where $r(x)$ is the distance from $x$ to $o$
in $(M,g)$, and compute

(108)
$\begin{split}\int_{B_{R_{2}}\setminus\overline{B_{R_{1}}}}-\varphi\widetilde{L}_{k}\varphi&=\int_{B_{R_{2}}\setminus\overline{B_{R_{1}}}}g\left(P_{k}(\nabla\varphi),\nabla\varphi\right)-\left(\overline{\kappa}\
\mathrm{trace}(P_{k})-\mathrm{trace}(A^{2}P_{k})\right)\varphi^{2}\\\
&\leq\int_{B_{R_{2}}\setminus\overline{B_{R_{1}}}}\mathrm{trace}(P_{k})|\nabla\varphi|^{2}-\left(\overline{\kappa}\
\mathrm{trace}(P_{k})-\mathrm{trace}(A^{2}P_{k})\right)\varphi^{2}.\end{split}$

With the aid of the coarea formula, integrating by parts and using (106) we
obtain

(109)
$\int_{B_{R_{2}}\setminus\overline{B_{R_{1}}}}-\varphi\widetilde{L}_{k}\varphi\leq-\int_{R_{1}}^{R_{2}}\left[(v_{k}z^{\prime})^{\prime}(s)+A(s)v_{k}(s)z(s)\right]z(s)ds=0.$

Therefore, from (109) and Proposition 2 we deduce

(110)
$\lambda_{1}^{\widetilde{L}_{k}}(M\setminus\overline{\Omega})<\lambda_{1}^{\widetilde{L}_{k}}(B_{R_{2}}\setminus\overline{B_{R_{1}}})\leq\frac{\int_{B_{R_{2}}\setminus\overline{B_{R_{1}}}}-\varphi\widetilde{L}_{k}\varphi}{\int_{B_{R_{2}}\setminus\overline{B_{R_{1}}}}\varphi^{2}}\leq
0,$

contradicting (104). $\square$

######  Theorem 28.

Let $\overline{M}=I\times_{\rho}F$ be a Robertson-Walker spacetime of constant
sectional curvature and let $\psi:M\longrightarrow\overline{M}$ be a complete
non-compact spacelike hypersurface with zero $(k+1)$-th mean curvature.
Suppose that $H_{k}>0$, $\sup_{M}H_{k}<+\infty$, $\mathrm{rank}(A)>k$ on $M$
and that, for some $o\in M$ and for some (hence any) $a\in\mathbb{R}$, one of
the following conditions is satisfied:

$\displaystyle(i)$
$\displaystyle\qquad\lim_{r\to+\infty}\int_{a}^{r}\left(\int_{\partial
B_{r}}H_{k}\right)^{-1}dt=+\infty$
$\displaystyle\qquad\text{and}\qquad\lim_{r\to+\infty}\int_{B_{r}}H_{k}=+\infty,\qquad\text{or}$
$\displaystyle(ii)$
$\displaystyle\qquad\lim_{r\to+\infty}\int_{a}^{r}\left(\int_{\partial
B_{r}}H_{k}\right)^{-1}dt<+\infty$
$\displaystyle\qquad\text{and}\qquad\lim_{r\to+\infty}\frac{1}{r}\log\left(\int_{\partial
B_{r}}H_{k}\right)=0,$

where $B_{r}$ is the geodesic ball of $(M,g)$ with radius $r$ centered at $o$.
If $\psi(M)$ is contained in a slab $[a,b]\times F$, then
$\inf_{M}|H_{k}\rho^{\prime}(\tau)|=0$. In particular, if $\inf_{M}H_{k}>0$,
then $\inf_{M}|\rho^{\prime}(\tau)|=0$ and therefore there exists $t_{0}$ in
the closure of $\tau(M)\subseteq[a,b]$ such that $\rho^{\prime}(t_{0})=0$.

  * Proof.

First observe that $H_{k+1}\equiv 0$, $H_{k}>0$ and $\mathrm{rank}(A)>k$
guarantee that $P_{k}$ is positive definite and therefore $L_{k}$ is elliptic
(see Proposition 6.27 of [11]). Set $w(r)=\int_{\partial
B_{r}}\mathrm{trace}(P_{k})$ for every $r>0$ and $\Lambda=\sup_{M}H_{k}$.

Let $R>0$ be given and suppose that condition (i) is satisfied. Then

$\lim_{r\to+\infty}\int_{a}^{r}\frac{dt}{w(t)}=\lim_{r\to+\infty}\int_{a}^{r}w(t)dt=+\infty,$

since $\mathrm{trace}(P_{k})=c_{k}H_{k}$. Let $\varepsilon>0$ be given. By
Theorem 6.6 of [11], the solution
$z\in\mathrm{Lip}_{\mathrm{loc}}([R,+\infty))$ of the Cauchy problem

(111) $\begin{cases}(w(r)z^{\prime})^{\prime}+\varepsilon w(r)z=0&\text{on
}[R,+\infty)\\\ z(R)=1,\quad(wz^{\prime})(R^{+})=0\end{cases}$

is oscillatory. Let $R<R_{1}<R_{2}$ be two consecutive zeros of $z$ such that
$z>0$ on $(R_{1},R_{2})$ and let $\varphi(x)=z(r(x))$, where $r(x)$ is the
distance from $x$ to $o$ in $(M,g)$. By the coarea formula, the inequality
$\mathrm{Vol}(\partial B_{r})\geq w(r)/\Lambda$ and the fact that $z$ solves
(111), we have

$\displaystyle\lambda_{1}^{L_{k}}(B_{R_{2}}\setminus\overline{B_{R_{1}}})$
$\displaystyle\leq\frac{\int_{B_{R_{2}}\setminus\overline{B_{R_{1}}}}g(P_{k}(\nabla\varphi),\nabla\varphi)}{\int_{B_{R_{2}}\setminus\overline{B_{R_{1}}}}\varphi^{2}}=\frac{\int_{R_{1}}^{R_{2}}w(r)z^{\prime}(r)^{2}}{\int_{R_{1}}^{R_{2}}\mathrm{Vol}(\partial
B_{r})z(r)^{2}}$ $\displaystyle\leq\frac{\int_{R_{1}}^{R_{2}}\varepsilon
w(r)z(r)^{2}}{\int_{R_{1}}^{R_{2}}\frac{w(r)}{\Lambda}z(r)^{2}}=\varepsilon\Lambda,$

and by Proposition 2 we get

(112)
$0\leq\lambda_{1}^{L_{k}}(M\setminus\overline{B_{R}})<\varepsilon\Lambda.$

Suppose now that (ii) is satisfied. Then
$\lim_{r\to+\infty}\int_{a}^{r}\frac{dt}{w(t)}=0$ and for every
$\varepsilon>0$ there exists $C_{\varepsilon}>0$ such that $w(r)\leq
C_{\varepsilon}e^{\varepsilon r}$ for every $r>0$. Let $0<\varepsilon<4$ be
given. Then, for every $r>0$,

$\displaystyle\varepsilon$
$\displaystyle>\frac{\varepsilon^{2}}{4}=\left(\lim_{s\to+\infty}\left(-\frac{1}{2(s-r)}\log\int_{s}^{+\infty}\frac{dt}{C_{\varepsilon}e^{\varepsilon
t}}\right)\right)^{2}$
$\displaystyle\geq\left(\inf_{s>r}\left(-\frac{1}{2(s-r)}\log\int_{s}^{+\infty}\frac{dt}{C_{\varepsilon}e^{\varepsilon
t}}\right)\right)^{2}$

and by Proposition 6.9 of [11] this condition is sufficient to deduce that for
every $R>0$ the solution $z$ of the Cauchy problem (111) is oscillatory, so
inequality (112) follows again.

Letting $\varepsilon\to 0^{+}$ and $R\to 0^{+}$, we deduce that
$\lambda_{1}^{L_{k}}(M)=0$. We have $\psi(M)\subseteq[a,b]\times F$, with
$a,b\in I$. Choose $t_{0}=a$ and let $\overline{\eta}$, $\eta$ be as in (59).
Then $0=\overline{\eta}(a)\leq\eta\leq\overline{\eta}(b)<+\infty$ on $M$. Fix
$\varepsilon>0$. By Proposition 3 applied to the positive functions
$\eta+\varepsilon$, $\overline{\eta}(b)+\varepsilon-\eta$ and by (97) we have

$0\geq\inf_{M}\left(-\frac{L_{k}\eta}{\eta+\varepsilon}\right)=\inf_{M}\frac{c_{k}H_{k}\rho^{\prime}(\tau)}{\eta+\varepsilon},\quad
0\geq\inf_{M}\left(\frac{L_{k}\eta}{\overline{\eta}(b)+\varepsilon-\eta}\right)=\inf_{M}\frac{-c_{k}H_{k}\rho^{\prime}(\tau)}{\overline{\eta}(b)+\varepsilon-\eta}.$

Since $\eta+\varepsilon>\varepsilon$,
$\overline{\eta}(b)+\varepsilon-\eta>\varepsilon$ and $c_{k}>0$, we deduce
$\inf_{M}|H_{k}\rho^{\prime}(\tau)|=0$. $\square$

######  Remark 29.

Under the hypothesis $C^{-1}<H_{k}<C$ we have
$c_{k}C^{-1}\mathrm{Vol}(\partial B_{r})\leq w(r)\leq
c_{k}C\mathrm{Vol}(\partial B_{r})$ for every $r>0$ and therefore conditions
(i) and (ii) in the statement of Theorem K are equivalent to the corresponding
conditions in the statement of Theorem 28.

## Acknowledgements

The second author is supported by Spanish MINECO and ERDF project
MTM2016-78807-C2-1-P.

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