STABILITY OF SPACELIKE HYPERSURFACES IN FOLIATED SPACETIMES
ABDˆENAGO BARROS, ALDIR BRASIL, AND ANT ˆONIO CAMINHA
Abstract. Given a generalizedMn+1 =I×φFnRobertson-Walker space-time we will classify strongly stable spacelike hypersurfaces with constant mean curvature whose warping function verifies a certain convexity condition. More precisely, we will show that givenx:Mn→Mn+1a closed spacelike hyper-surfaces ofMn+1with constant mean curvatureH and the warping function
φsatisfyingφ′′≥max{Hφ′,0}, thenMnis either minimal or a spacelike slice
Mt0={t0} ×F, for somet0∈I.
1. Introduction
Spacelike hypersurfaces with constant mean curvature in Lorentz manifolds have been object of great interest in recent years, both from physical and mathematical points of view. In [1], the authors studied the uniqueness of spacelike hypersurfaces with CMC in generalized Robertson-Walker (GRW) spacetimes, namely, Lorentz warped products with 1-dimensional negative definite base and Riemannian fiber. They proved that in a GRW spacetime obeying the timelike convergence condition (i.e, the Ricci curvature is non-negative on timelike directions), every compact spacelike hypersurface with CMC must be umbilical. Recently, Al´ıas and Montiel obtained, in [2], a more general condition on the warping functionfthat is sufficient in order to guarantee uniqueness. More precisely, they proved the following
Theorem 1.1. Let f : I → R be a positive smooth function defined on an open interval, such thatf f′′−(f′)2≤0, that is, such that−logf is convex. Then, the
only compact spacelike hypersurfaces immersed into a generalized Robertson-Walker spacetime I×f Fn and having constant mean curvature are the slices{t} ×F, for
a(necessarily compact)Riemannian manifoldF.
Stability questions concerning CMC, compact hypersurfaces in Riemannian space forms began with Barbosa and do Carmo in [4], and Barbosa, Do Carmo and Es-chenburg in [5]. In the former paper, they introduced the notion of stability and proved that spheres are the only stable critical points for the area functional, for volume-preserving variations. In the setting of spacelike hypersurfaces in Lorentz manifolds, Barbosa and Oliker proved in [6] that CMC spacelike hypersurfaces are critical points of volume-preserving variations. Moreover, by computing the second variation formula they showed that CMC embedded spheres in the de Sitter space
S1n+1 maximize the area functional for such variations. In this paper, we give a characterization of strongly stable, CMC spacelike hypersurfaces in GRW space-times, the essential tool for the proof being a formula for the Laplacian of a new support function. More precisely, it is our purpose to show the following
Theorem 1.2. LetMn+1=I×φFnbe a generalized Robertson-Walker spacetime,
andx:Mn→Mn+1 be a closed spacelike hypersurface ofMn+1, having constant mean curvatureH. If the warping functionφ satisfiesφ′′≥max{Hφ′,0} andMn
is strongly stable, then Mn is either minimal or a spacelike sliceMt0 ={t0} ×F,
for somet0∈I.
2. Stable spacelike hypersurfaces
In what follows, Mn+1 denotes an orientable, time-oriented Lorentz manifold with Lorentz metric g = h, i and semi-Riemannian connection ∇. If x: Mn → Mn+1 is a spacelike hypersurface of Mn+1, then Mn is automatically orientable
([8], p. 189), and one can choose a globally defined unit normal vector fieldN on
Mn having the same time-orientation of V, that is, such that
hV, Ni<0
onM. One says that such anN points to the future. Avariationofxis a smooth map
X :Mn×(−ǫ, ǫ)→Mn+1
satisfying the following conditions:
(1) For t ∈(−ǫ, ǫ), the map Xt: Mn →M n+1
given by Xt(p) =X(t, p) is a
spacelike immersion such thatX0=x.
(2) Xt
∂M =x
∂M, for allt∈(−ǫ, ǫ).
Thevariational fieldassociated to the variationX is the vector field ∂X
∂t. Letting f =−h∂X∂t, Ni, we get
∂X ∂t
M =f N+
∂X ∂t
T
,
whereT stands for tangential components. Thebalance of volumeof the variation
X is the functionV: (−ǫ, ǫ)→Rgiven by
V(t) = Z
M×[0,t]
X∗(dM),
wheredM denotes the volume element of M.
Thearea functionalA: (−ǫ, ǫ)→Rassociated to the variationX is given by
A(t) = Z
M dMt,
where dMt denotes the volume element of the metric induced in M byXt. Note
thatdM0=dM andA(0) =A, the volume ofM. The following lemma is classical:
Lemma 2.1. Let Mn+1 be a time-oriented Lorentz manifold andx:Mn →Mn+1
a spacelike closed hypersurface having mean curvature H. IfX :Mn×(−ǫ, ǫ)→
Mn+1 is a variation ofx, then
dV
dt
t=0=
Z
M
f dM, dA
dt
t=0=
Z
M
SetH0= A1
R
MdM andJ : (−ǫ, ǫ)→Rgiven by
J(t) =A(t)−nH0V(t).
J is called theJacobi functionalassociated to the variation, and it is a well known result [5] that x has constant mean curvature H0 if and only ifJ′(0) = 0 for all
variationsX ofx.
We wish to study here immersions x : Mn → Mn+1 that maximize J for all variations X. Since x must be a critical point of J, it thus follows from the above discussion thatx must have constant mean curvature. Therefore, in order to examine whether or not some critical immersion x is actually a maximum for J, one certainly needs to study the second variation J′′
(0). We start with the following
Proposition 2.2. Let x: Mn →Mn+1 be a closed spacelike hypersurface of the
time-oriented Lorentz manifoldMn+1, andX :Mn×(−ǫ, ǫ)→Mn+1be a variation
of x. Then,
(2.1) n∂H
∂t = ∆f−
Ric(N, N) +|A|2 f−nh
∂X ∂t
T
,∇Hi.
Although the above proposition is known to be true, we believe there is a lack, in the literature, of a clear proof of it in this degree of generality, so we present a simple proof here.
Proof. Let p∈M and {ek} be a moving frame on a neighborhoodU ⊂M of p,
geodesic atpand diagonalizingAatp, withAek=λkek for 1≤k≤n. ExtendN
and thee′ksto a neighborhood ofpin M, so thathN, eki= 0 and (∇Nek)(p) = 0.
Then
n∂H
∂t = −tr
∂A ∂t
=−X
k
h∂A
∂tek, eki=−
X
k
h∇∂X
∂tA
ek, eki
= −X
k
n h∇∂X
∂tAek, eki − hA∇ ∂X
∂tek, eki
o
= X
k
h∇∂X
∂t∇ekN, eki+
X
k
hA∇ek
∂X ∂t , eki,
where in the last equality we used the fact that [∂X∂t, ek] = 0. Letting
I=X
k
h∇∂X
∂t∇ekN, eki and II=
X
k
hA∇ek
∂X ∂t , eki,
we have
I = X
k
n h∇∂X
∂t ∇ekN− ∇ek∇
∂X
∂tN+∇[ek,∂X∂t]N, eki+∇ek∇ ∂X
∂tN, eki
o = X k hR
ek, ∂X
∂t
N, eki+h∇ek∇∂X ∂tN, eki
= −Ric
∂X ∂t , N
+X
k
h∇ek∇∂X ∂tN, eki.
Since the frame{ek} is geodesic atp, it follows that
h∇∂X
∂t N,∇ekeki=h∇
∂X
atp, and hence
h∇ek∇∂X
∂tN, eki = ekh∇
∂X
∂tN, eki=−ekhN,∇
∂X
∂teki=−ekhN,∇ek
∂X ∂t i
= −ekekhN, ∂X
∂t i+ekh∇ekN,
∂X ∂t i
= ekek(f) +ekh∇ekN,
∂X ∂t
T i
= ekek(f) +h∇ek∇ekN,
∂X ∂t
T
i − hAek,∇ek
∂X ∂t
T i.
ForII, we have
II = X
k
hAek,∇ek
∂X
∂t i=
X
k
hAek,∇ek(f N+
∂X ∂t T )i = X k
hAek, f∇ekNi+
X
k
hAek,∇ek
∂X ∂t
T i
= −f|A|2+X
k
hAek,∇ek
∂X ∂t T i Therefore,
(2.2) n∂H
∂t =−Ric
∂X
∂t , N
+ ∆f−f|A|2+X
k
h∇ek∇ekN,
∂X ∂t T i. Now, letting ∂X ∂t = n X l
αlel+f N
andAek =Pjhjkej, one successively gets
Ric
∂X ∂t , N
= X
l
αlRic(N, el) +f Ric(N, N)
= X
k,l
αlhR(ek, el)ek, Ni+f Ric(N, N)
and, since (∇Nek)(p) = 0,
hR(ek, el)ek, Nip = h∇el∇ekek− ∇ek∇elek, Nip
= elh∇ekek, Nip− h∇ekek,∇elNip−ekh∇elek, Nip
= −elhek,∇ekNip+ekhek,∇elNip
= el(hkk)−ek(hkl),
so that
(2.3) Ric
∂X
∂t , N
p
=X
k,l
αlel(hkk)−
X
k,l
Also,
αlh∇ek∇ekN, eli = αlh∇ek∇ekN, eli=−αl
X
j
h∇ekhkjej, eli
= −αl
X
j
{ek(hkj)δlj+hkjh∇ekej, eli}
= −αlek(hkl),
and hence
(2.4) X
k
h∇ek∇ekN,
∂X ∂t
T
i=−X
k,l
αlek(hkl).
Substituting (2.3) and (2.4) into (2.2), we finally arrive at
n∂H
∂t = −
X
k,l
αlel(hkk)−f Ric(N, N)p+ ∆f −f|A|2
= −
∂X ∂t
T
(nH)−f Ric(N, N)p+ ∆f −f|A|2.
Proposition 2.3. LetMn+1be a Lorentz manifold andx:Mn→Mn+1be a closed
spacelike hypersurface having constant mean curvature H. If X :Mn×(−ǫ, ǫ)→ Mn+1 is a variation ofx, then
(2.5) J′′(0)(f) = Z
M
f
∆f− Ric(N, N) +|A|2
f dM.
Proof. In the notations of the above discussion, setf =f(0) and note thatH(0) =
H. It follows from lemma 2.1 that
J′(t) = Z
M
n{H(t)−H}f(t)dMt.
Therefore, differentiating with respect totonce more
J′′
(0) = Z
M
nH′(0)f(0)dM0+
Z
M
n{H(0)−H} d
dtf(t)dMt
t=0
= Z
M
nH′(0)f dM.
Taking into account thatHis constant, relation (2.1) finally gives formula 2.5
It follows from the previous result that J′′
(0) = J′′
(0)(f) depends only on
f ∈C∞
(M), for which there exists a variationX of Mn such that ∂X ∂t
⊥ =f N. Therefore, the following definition makes sense:
Definition 2.4. Let Mn+1 be a Lorentz manifold and x : Mn → Mn+1 be a
closed spacelike hypersurface having constant mean curvatureH. We say thatxis strongly stable if, for every functionf ∈C∞(
M) for which there exists a variation
X ofMn such that ∂X ∂t
⊥
3. Conformal vector fields
As in the previous section, letMn+1 be a Lorentz manifold. A vector fieldV on
Mn+1is said to be conformalif
(3.1) LVh, i= 2ψh, i
for some functionψ∈C∞
(M), whereLstands for the Lie derivative of the Lorentz metric ofM. The functionψis called theconformal factorofV.
SinceLV(X) = [V, X] for allX ∈ X(M), it follows from the tensorial character
ofLV thatV ∈ X(M) is conformal if and only if
(3.2) h∇XV, Yi+hX,∇YVi= 2ψhX, Yi,
for allX, Y ∈ X(M). In particular,V is a Killing vector field relatively togif and only ifψ≡0.
Any Lorentz manifold Mn+1, possessing a globally defined, timelike conformal vector field is said to be aconformally stationary spacetime.
Proposition 3.1. Let Mn+1 be a conformally stationary Lorentz manifold, with conformal vector field V having conformal factor ψ : Mn+1 → R. Let also x :
Mn →Mn+1 be a spacelike hypersurface ofMn+1, and N a future-pointing, unit
normal vector field globally defined onMn. Iff =hV, Ni, then
(3.3) ∆f =nhV,∇Hi+f
Ric(N, N) +|A|2 +n{Hψ−N(ψ)},
where Ric denotes the Ricci tensor of M,A is the second fundamental form ofx
with respect toN,H =−1ntr(A)is the mean curvature of xand∇H denotes the gradient ofH in the metric ofM.
Proof. Fixp∈M and let{ek}be an orthonormal moving frame onM, geodesic at p. Extend theek to a neighborhood ofpinM, so that (∇Nek)(p) = 0, and let
V =
n
X
l
αlel−f N.
Then
f =hN, Vi ⇒ek(f) = h∇ekN, Vi+hN,∇ekVi
= −hAek, Vi+hN,∇ekVi,
so that
∆f = X
k
ek(ek(f)) =−
X
k
ekhAek, Vi+
X
k
ekhN,∇ekVi
= −X
k
h∇ekAek, Vi −2
X
k
hAek,∇ekVi+
X
k
hN,∇ek∇ekVi.
Now, differentiatingAek=Plhklelwith respect to ek, one gets atp
X
k
h∇ekAek, Vi =
X
k,l
ek(hkl)hel, Vi+
X
k,l
hklh∇ekel, Vi
= X
k,l
αlek(hkl)−
X
k,l
hklh∇ekel, NihV, Ni
= X
k,l
αlek(hkl)−
X
k,l h2klf
= X
k,l
αlek(hkl)−f|A|2.
(3.5)
Asking further thatAek=λkek at p(which is always possible), we have atp
(3.6) X
k
hAek,∇ekVi=
X
k
λkhek,∇ekVi=
X
k
λkψ=−nHψ.
In order to compute the last summand of (3.4), note that the conformality ofV
gives
h∇NV, eki+hN,∇ekVi= 0
for allk. Hence, differentiating the above relation in the direction ofek, we get
h∇ek∇NV, eki+h∇NV,∇ekeki+h∇ekN,∇ekVi+hN,∇ek∇ekVi= 0.
However, atpone has
h∇NV,∇ekeki = −h∇NV,h∇ekek, NiNi=−h∇NV, λkNi
= −λkψhN, Ni=λkψ
and
h∇ekN,∇ekVi=−λkhek,∇ekVi=−λkψ,
so that
(3.7) h∇ek∇NV, eki+hN,∇ek∇ekVi= 0
atp. On the other hand, since
[N, ek](p) = (∇Nek)(p)−(∇ekN)(p) =λkek(p),
it follows from (3.7) that
hR(N, ek)V, ekip = h∇ek∇NV − ∇N∇ekV +∇[N,ek]V, ekip
= −hN,∇ek∇ekVip−Nh∇ekV, ekip+h∇λkekV, ekip
= −hN,∇ek∇ekVip−N(ψ) +λkψ,
and hence
(3.8) X
k
hN,∇ek∇ekVip=−nN(ψ)−nHψ−Ric(N, V)p
Finally,
Ric(N, V) = X
l
αlRic(N, el)−f Ric(N, N)
= X
k,l
and
hR(ek, el)ek, Nip = h∇el∇ekek− ∇ek∇elek, Nip
= elh∇ekek, Nip− h∇ekek,∇elNip−ekh∇elek, Nip
+h∇elek,∇ekNip
= −elhek,∇ekNip+ekhek,∇elNip
= el(hkk)−ek(hkl),
so that
Ric(N, V)p=
X
k,l
αlel(hkk)−
X
k,l
αlek(hkl)−f Ric(N, N)p,
and it follows from (3.8) that X
k
hN,∇ek∇ekVip = −nN(ψ)−nHψ+V T(nH)
+X
k,l
αlek(hkl) +f Ric(N, N).
(3.9)
Substituting (3.5), (3.6) and (3.9) into (3.4), one gets the desired formula (3.3).
4. Applications
A particular class of conformally stationary spacetimes is that of generalized Robertson-Walkerspacetimes [1], namely, warped productsMn+1=I×φFn, where I ⊆ R is an interval with the metric −dt2, Fn is an n-dimensional Riemannian
manifold andφ:I→Ris positive and smooth. For such a space, letπI :M n+1
→I
denote the canonical projection onto theI−factor. Then the vector field
V = (φ◦πI) ∂ ∂t
is conformal, timelike and closed (in the sense that its dual 1−form is closed), with conformal factorψ=φ′
, where the prime denotes differentiation with respect tot. Moreover, according to [7], fort0∈I, orienting the (spacelike) leafMtn0 ={t0}×F
n
by using the future-pointing unit normal vector field N, it follows that Mt0 has
constant mean curvature
H =φ
′
(t0) φ(t0) .
IfMn+1=I×φFnis a generalized Robertson-Walker spacetime and x:Mn→ Mn+1is a complete spacelike hypersurface ofMn+1, such thatφ◦πI is limited on M, thenπF
M :M
n→Fn is necessarily a covering map ([1]). In particular, ifMn
is closed, thenFn is automatically closed.
One has the following corollary of proposition 3.1:
Corollary 4.1. LetMn+1=I×φFn be a generalized Robertson-Walker spacetime,
and x : Mn → Mn+1 a spacelike hypersurface of Mn+1, having constant mean
curvatureH. Let alsoNbe a future-pointing unit normal vector field globally defined onMn. IfV = (φ◦π
I)∂t∂ andf =hV, Ni, then
(4.1) ∆f =
Ric(N, N) +|A|2 f+n
Hφ′+φ′′hN, ∂ ∂ti
where Ric denotes the Ricci tensor of M,A is the second fundamental form ofx
with respect toN, andH=−1ntr(A)is the mean curvature ofx.
Proof. First of all,f =hV, Ni=φhN,∂t∂i, and it thus follows from (3.3) that
∆f =
Ric(N, N) +|A|2 f+n{Hφ′−N(φ′)}.
However,
∇φ′ =−h∇φ′, ∂ ∂ti
∂
∂t =−φ
′′∂
∂t,
so that
N(φ′) =hN,∇φ′i=−φ′′hN, ∂ ∂ti
We can now state and prove our main result:
Theorem 4.2. LetMn+1=I×φFnbe a generalized Robertson-Walker spacetime,
andx:Mn→Mn+1 be a closed spacelike hypersurface ofMn+1, having constant
mean curvatureH. If the warping functionφ satisfiesφ′′≥max{Hφ′,0} andMn
is strongly stable, then Mn is either minimal or a spacelike sliceM
t0 ={t0} ×F,
for somet0∈I.
Proof. SinceMn is strongly stable, we have
0≥ J′′(0)(g) = Z
M
g
∆g− Ric(N, N) +|A|2
g dM
for allg∈C∞(M) for whichgNis the normal component of the variational field of some variation ofMn. In particular, iff =hV, Ni=φhN, ∂
∂ti, whereV = (φ◦πI) ∂ ∂t,
andg=−f =−hV, Ni, then
∆g=
Ric(N, N) +|A|2 g−n
Hφ′+φ′′hN, ∂ ∂ti
.
Therefore,Mn stable implies
0≥ Z
M
φhN, ∂ ∂ti
Hφ′+φ′′hN, ∂ ∂ti
dM
Lettingθ be the hyperbolic angle betweenN and ∂
∂t, it follows from the reversed
Cauchy-Schwarz inequality that coshθ =−hN, ∂
∂ti, with coshθ ≡1 if and only if N and ∂t∂ are collinear at every point, that is, if and only ifMn is a spacelike leaf Mt0 for somet0∈I. Hence,
0≥ Z
M
φcoshθ{−Hφ′+φ′′coshθ}dM.
Now, notice that−Hφ′
+φ′′
coshθ≥ −φ′′
+φ′′
coshθ, which gives
φcoshθ(−Hφ′+φ′′coshθ)≥φφ′′coshθ(coshθ−1).
Therefore,
0≥ Z
M
φcoshθ(−Hφ′+φ′′coshθ)dM ≥ Z
M
φφ′′coshθ(coshθ−1)≥0,
and hence
onM. If, for somep∈M, one hasφ′′
(p) = 0, thenφ′
H = 0 at p. IfH 6= 0, then
φ′
(p) = 0. But if this is the case, then proposition 7.35 of [8] gives that
∇V ∂
∂t =
φ′
φV = 0
atpfor anyV, andM is totally geodesic atp. In particular,H = 0, a contradiction. Therefore, eitherφ′′
(p) = 0 for somep∈M, andM is minimal, orφ′′
6= 0 on all of
M, whence coshθ= 1 always, andM is an umbilical leaf such thatφ′′
=Hφ′
.
Remark 4.3. Note that φ′′
φ′ =H =
φ′
φ, i.e.,φ ′′
φ−(φ′)2= 0, which is a limit case of
Al´ıas and Montiel’s timelike convergent condition.
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Abdˆenago Barros, Departamento de Matem´atica-UFC, 60455-760-Fortaleza-CE-Br
E-mail address:abbarros@mat.ufc.br
Aldir Brasil, Departamento de Matem´atica-UFC, 60455-760-Fortaleza-CE-Br
E-mail address:aldir@mat.ufc.br
Antˆonio Caminha, Departamento de Matem´atica-UFC, 60455-760-Fortaleza-CE-Br
