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Stable space-like hypersurfaces with constant scalar
curvature in generalized Roberston-Walker
spacetimes
Ximin Liu and Biaogui Yang
Abstract. In this paper we study stable spacelike hyersurfaces with conn+1
=
stant scalar curvature in generalized Roberston-Walker spacetime M
−I ×φ F n .
M.S.C. 2000: 53B30, 53C42, 53C50.
Key words: stability, spacelike hypersurface, constant scalar curvature, generalized
Roberston-Walker spacetime.
1
Introduction
Hyersurfaces M n with constant r-mean curvature in Riemannian manifolds or Lorentz
n+1
manifolds M
(c) with constant sectional curvature c are critical points of some area
functional variations which keep constant a certain volume function. Stable hyersurfaces with constant mean curvature(CMC) (or constant r-mean curvature) in real
space form are very interesting geometrical objects that were investigated by many
geometricians. Barbosa and do Carmo [2] gave definition of stability of hyersurfaces
with constant mean curvature in the Eucildean space Rn+1 and proved the round
spheres are the only compact stable hyersurfaces with CMC in Rn+1 . Later, Barbosa,
do Carmo and Eschenburg [3] extended ambient spaces to Riemannian manifolds
and obtained the corresponding results. In [5] Barbosa and Oliker discussed stable
spacelile hyersurfaces with CMC in Lorentz manifolds. At the same time, Alencar, do
Carmo and Colares [1] investigated stable hyersurfaces with constant scalar curvature
in Riemannian manifolds and obtained geodesic sphere is the only stable compact orientable hyersurface in Riemannain spaces. On the other hand, Barbosa and Colares
[4] studied compact hyersurfaces without boundary immersed in space forms with
constant r-mean curvature. Recently, Liu and Deng [9] also discussed stable spacelike hyersurfaces with constant scalar curvature in de Siter space S1n+1 . Barros, Brasil
and Caminha [6] classified strongly stable spacelike hypersurfaces with constant mean
curvature whose warping function satisfied a certain convexity condition.
∗
Balkan
Journal of Geometry and Its Applications, Vol.13, No.1, 2008, pp. 66-76.
c Balkan Society of Geometers, Geometry Balkan Press 2008.
°
67
Stable space-like hypersurfaces
In this paper we will study stable spacelike hypersurfaces with constant scalar
n+1
= −I ×φ F n .
curvature in generalized Roberston-Walker spacetime M
2
Preliminaries
Consider F n an n-dimensional manifold, let I be a 1-dimensional manifold (either
n+1
a circle or an open interval of R). We denote by M
= −I ×φ F n the (n + 1)dimensional product manifold I × F endowed with the Lorentzian metric
(2.1)
g = h, i = −dt2 + f 2 (t)h, iM ,
where f > 0 is positive function on I, and h, iM stands for the Riemannian metric
on F n . We refer to −I ×φ F n as a generalized Robertson-Walker (GRW) spacetime.
In particular, when the Riemannian factor F n has constant sectional curvature, then
−I ×φ F n is classically called a Robertson-Walker (RW) spacetime.
n+1
A vector field V on a Lorentz manifold M
is said to be conformal if
(2.2)
LV g = 2ψg,
n+1
→ R, where L stands for the Lie derivative of
for some smooth function ψ : M
n+1
. The function ψ is called the conformal factor of V . V ∈ T M
Lorentz metric of M
is conformal if and only if
(2.3)
h∇X V, Y i + h∇Y V, Xi = 2ψhX, Y i,
for all X, Y ∈ T (M ).
n+1
Any Lorentz manifold M
, possessing a globally defined, timelike conformal
vector field is said to be a conformally stationary (CS) spacetime.
n+1
denote an orientable spacelike hyersurface in the timeLet x : M n → M
n+1
and N be a globally defined unit normal vector
oriented Lorentz manifold M
field on M n . ∇ and ∇ denote the Levi-Civita connection of M n and ambient space
n+1
M
respectively. R and Ric denote the curvature tensor and Ricci curvature tensor
n+1
on M
respectively, which are defined by
(2.4)
R(X, Y )Z = ∇X ∇Y Z − ∇Y ∇X Z − ∇[X,Y ] Z,
and
(2.5)
R(W, Z, X, Y ) = h∇X ∇Y Z, W i − h∇Y ∇X Y, W i − h∇[X,Y ] Z, W i,
then
(2.6)
Ric(X, Y ) =
n+1
X
R(ek , X, ek , Y ),
k=1
where X, Y, Z, W ∈ T M , and {ek }nk=1 is a basis of Tp M , en+1 = N . In particular we
have
68
Ximin Liu and Biaogui Yang
(2.7)
Ric(N, N ) =
n
X
R(ek , N, ek , N ).
k=1
The shape operator A associated to N of M n , defined by
A = −∇N (i.e Aek = −∇ek N )
(2.8)
is a self-adjoint linear operator in each tangent space Tp M . Its eigenvalues are the
principal curvatures of immersion and are represented by λ1 , λ2 , · · · , λn .The elementary symmetic functions Sr associated to A can be defined, using the characteristic
polynomial of A, by
n
X
det(tI − A) =
(−1)k Sk tn−k ,
k=0
where S0 = 1. If p ∈ M , and {ek } is a basis of Tp M formed by eigenvector of Ap ,
with corresponding eigenvalues λk , one immediately sees that
Sr = σr (λ1 , · · · , λn ),
where σr is the r-th elementary symmetric polynomial. In particular
X
(2.9)
kAk2 =
λ2k = S12 − 2S2 ,
k
and
X
(2.10)
λ3k = S13 − 3S1 S2 + 3S3 .
k
The r-th classical Newton transformation Pr on M is defined as following
P0 = I,
Pr = Sr I − APr−1 ,
1 ≤ r ≤ n.
Associated to each Newton transformation Pr of immersion x : M n → M
have a second order differential operator defined by
(2.11)
When M
n+1
, we
Lr (f ) = trace(Pr ◦ Hessf ).
n+1
has constant sectional curvature, then
(2.12)
Lr (f ) = div(Pr ∇f ),
where div stands for the divergence of a vector field on M , it was proved by H.
Rosenberg in [12].
Remark 1.1. According (2.11) or (2.12), when r = 0,
L0 f = div(P0 ∇f ) = △f
n
is Laplace operator on M , and if r = 1, then
L1 f
(2.13)
= div[P1 ◦ hessf ] = div[(S1 I − AP0 ) ◦ hessf ]
X
=
(S1 δij − hij )fij
i,j
become Cheng-Yau’s operator ✷ on M n , where hij and fij denote the component of
A and hessf respectively.
69
Stable space-like hypersurfaces
3
The variational problem in Lorentz manifolds
n+1
Let x : M n → M
denotes an orientable spacelike hyersurface in the time-oriented
n+1
and N be a globally defined unit normal vector field on M n .
Lorentz manifold M
n+1
satisfying the following
A variation of x is a smooth map X : M n × (−ε, ε) → M
conditions:
n+1
(1) For t ∈ (−ε, ε), the map Xt : M n → M
given by Xt (p) = X(t, p) is a
spaelike immersion such that X0 = x.
(2) Xt |∂M = x|∂M , for all t ∈ (−ε, ε).
∂
) = ∂X
The variational field vector associated the variation X is vector field X∗ ( ∂t
∂t .
∂X
Let f = h ∂t , N i, we have
(3.1)
∂X ⊤
∂X
=(
) − f N,
∂t
∂t
where ⊤ denotes tangential components. The balance of volume of the variation X is
the function V : (−ε, ε) → R given by
Z
X ∗ (dM ),
(3.2)
V (t) =
M ×[0,t]
where dM denotes the volume element of M .
The area functional A : (−ε, ε) → R is given by
Z
S1 dMt ,
(3.3)
A(t) =
M
where dMt denotes the volume element of the metric induced in M by Xt . Then we
have the following classical result.
n+1
n+1
Lemma 3.1. Let M
be a time-oriented Lorentz manifold and x : M n → M
n+1
is a variation of x, then
a spacelike hyersurface. If X : M n × (−ε, ε) → M
(i)
Z
dV (t)
f dM ;
|t=0 =
(3.4)
dt
M
(ii)
(3.5)
∂(dMt )
∂X ⊤
= (S1 + div(
) )dMt .
∂t
∂t
Proof. For (i) see [3, 9], and for (ii) see [4, 11]. ✷
Barros, Brasil and Caminha [6] proved the following proposition:
n+1
Proposition 3.2. Let x : M n → M
be a spacelike hypersurface of the timen+1
, and N be a globally defined unit normal vector field
oriented Lorentz manifold M
n+1
on M n . If X : M n × (−ε, ε) → M
is a variation of x, then
(3.6)
∂X ⊤
dS1
= △f − (Ric(N, N ) + kAk2 )f + h(
) , ∇S1 i.
dt
∂t
70
Ximin Liu and Biaogui Yang
Suppose λ is a constant, and J : (−ε, ε) → R is given by
(3.7)
J(t) = A(t) + λV (t),
J is called the Jacobi functional associated to the variation X. Then we have the
following proposition:
n+1
be a spacelike hypersurface in the timeProposition 3.3. Let x : M n → M
n+1
oriented Lorentz manifold M
, and N be a globally defined unit normal vector field
n+1
is a variation of x, then
on M n . If X : M n × (−ε, ε) → M
Z
dJ(t)
∂X ⊤
(3.8)
[div(S1 (
=
) ) + ∆f − (Ric(N, N ) + kAk2 − S12 − λ)f ]dMt .
dt
∂t
M
n+1
has constant sectional curvature c, then
In particular, when M n is closed and M
Z
dJ(t)
(3.9)
(2S2 − cn + λ)f dMt .
=
dt
M
Proof. We can get this result from Lemma 3.1 and Proposition 3.2. In fact,
Z
Z
Z
dS1
dJ(t)
∂X ⊤
S1 (S1 f + div(
λf dMt
=
dMt +
) )dMt +
dt
dt
∂t
M
M
ZM
∂X ⊤
∂X ⊤
) , ∇S1 i + S1 div(
) + △f
[h(
=
∂t
∂t
M
−(Ric(N, N ) + kAk2 )f + S12 f + λf ]dMt
Z
∂X ⊤
[div(S1 (
=
) ) + △f − (Ric(N, N ) + kAk2 f − S12 f − λ)f ]dMt .
∂t
M
When M n is closed and M
n+1
Z
has constant sectional curvature c, then we have
div(S1 (
M
Z
∂X ⊤
) )dMt = 0,
∂t
△f dMt = 0,
M
and Ric(N, N ) = nc, then using (2.9), we have (3.9).
n+1
Proposition 3.4. Let x : M n → M
is a spacelike hypersurface in Lorentz
n+1
(c) with constant sectional curvature c, and X : M n × (−ε, ε) →
space form M
n+1
M
is a variation of x, then
(3.10)
∂X ⊤
dS2
= L1 (f ) − (S1 S2 − 3S3 )f − f (n − 1)cS1 + h(
) , ∇S2 i.
dt
∂t
In particular, if S2 is a constant, then one has
(3.11)
dS2
= L1 (f ) − (S1 S2 − 3S3 )f − f (n − 1)cS1 .
dt
71
Stable space-like hypersurfaces
Proof. According to the proof of proposition 3.2 in [6], we can get
dhkk
∂X ⊤
= fkk − cf − h2kk f + h∇hkk , (
) i.
dt
∂t
(3.12)
Using (2.9), we can get
dS1 X
dhkk
dS2
= S1
−
hkk
.
dt
dt
dt
(3.13)
k
Substituting (3.6) and (3.12) into (3.13), using (2.9) and (2.10), then we have
dS2
dt
=
∂X ⊤
) , ∇S1 i]
∂t
X
∂X ⊤
−
hkk [fkk − cf − h2kk f + h∇hkk , (
) i]
∂t
S1 [△f − (Ric(N, N ) + kAk2 )f + h(
k
=
S1 △f − S1 (nc + S12 − 2S2 )f + S1 h(
−S1
X
k
=
fkk + cS1 f + f
X
k
X
∂X ⊤
) , ∇S1 i +
(S1 fkk − hkk fkk )
∂t
k
1
∂X ⊤
λ3k − h∇(S12 − 2S2 ), (
) i
2
∂t
L1 (f ) − (S1 S2 − 3S3 )f − f (n − 1)cS1 + h(
∂X ⊤
) , ∇S2 i.
∂t
If S2 is constant, then the last term in the above is equal to zero, so we have (3.11).
If M has constant normalized scalar curvature R, and we choose
(3.14)
λ = 2S2 − nc = n(n − 1)(c − R) − nc,
then λ is a constant too, so we have
n+1
Proposition 3.5. Let x : M n → M
(c) is a spacelike hypersurface in the timen+1
n+1
(c), and X : M n × (−ε, ε) → M
is a variation of
oriented Lorentz manifold M
x, and S2 is constant, then
Z
d2 J(0)
(f
)
=
2
[L1 (f ) − (S1 S2 − 3S3 )f − f (n − 1)cS1 ]f dM.
(3.15)
dt2
M
Proof. Since λ = 2S2 − nc = n(n − 1)(c − R) − nc, using (3.9) and (3.11), we can get
Z
Z
d2 J(0)
dS2 (0)
[L1 (f ) − (S1 S2 − 3S3 )f − f (n − 1)cS1 ]f dM.
(f
)
=
2
f
dM
=
2
dt2
dt
M
M
n+1
Definition 3.6. Suppose x : M n → M
(c) has constant scalar curvature. The
immersion x is stable if
Z
d2 J(0)
(3.16)
[L1 (f ) − (S1 S2 − 3S3 )f − f (n − 1)cS1 ]f dM ≤ 0,
(f ) = 2
dt2
M
72
Ximin Liu and Biaogui Yang
for all volume-presering variations of x. If M n is noncompact, x is stable if for every
conpact submanifolds M ′ ⊂ M n with boundary, the restriction x |M ′ is stable.
For conformally stationary spacetimes, we have the following proposition.
n+1
be a conformally stationary Lorentz manifold, with
Proposition 3.7. Let M
n+1
n+1
→ R. Suppose x : M n → M
conformal vector V having conformal factor ψ : M
n+1
is a spacelike hypersurface in M
= I ×φ F n with constant sectional curvature c,
and N a future-pointing, unit normal vector field globally defined on M n , f = hV, N i,
then
(3.17) L1 (f ) = (S1 S2 − 3S3 )f + f (n − 1)cS1 − (n − 1)S1 N (ψ) − 2S2 ψ − hV ⊤ , ∇S2 i.
In particular, if R is constant, then S2 is a constant too, so
(3.18)
✷f = L1 (f ) = (S1 S2 − 3S3 )f + f (n − 1)cS1 − (n − 1)S1 N (ψ) − 2S2 ψ.
Proof. We can choose {ek } as a moving frame on neighborhood U ⊂ M of p, geodesic
at p, and diagonalizing the shape operator A of M at p, with Aek = λk ek , for 1 ≤
k ≤ n. Extend N and ek (1 ≤ k ≤ n) to a neighborhood of p in M , such that
hN, ek i = 0 and (∇N ek )(p) = 0.
Let
V =
X
αl el − f N,
l
so we have
ek (f ) = h∇ek N, V i + hN, ∇ek V i = −hAek , V i + hN, ∇ek V i.
Then
ek ek (f )
(3.19)
= −ek hAek , V i + ek hN, ∇ek V i
= −h∇ek (Aek ), V i − 2hAek , ∇ek V i + hN, ∇ek ∇ek V i.
For the first term in (3.19), we have
h∇ek (Aek ), V i
= h∇ek (Aek ),
X
=
j
(3.20)
X
=
l
X
l
ek (hkj )hej ,
αl el − f N i
X
αl el i +
X
hkj h∇ek ej , −f N i
j
l
αl ek (hkl ) −
X
h2kl f.
l
For the second term in (3.19), we have
X
(3.21)
hkj hej , ∇ek V i = λk hek , ∇ek V i = λk ψ,
hAek , ∇ek V i =
j
where in the last equality we use the fact that V is conformal vector having conformal
factor ψ, and we have
73
Stable space-like hypersurfaces
hN, ∇ek V i + hek , ∇N V i = 2ψhek , N i = 0,
then we can get
(3.22)
h∇ek N, ∇ek V i + hN, ∇ek ∇ek V i + h∇ek ek , ∇N V i + hek , ∇ek ∇N V i = 0.
Since
h∇ek N, ∇ek V i = −hAek , ∇ek V i = −λk ψ,
and
h∇N V, ∇ek ek i
= −h∇N V, h∇ek ek , N iN i
= −h∇N V, hkk N i = −hkk ψhN, N i = λk ψ,
then
hN, ∇ek ∇ek V i = −hek , ∇ek ∇N V i.
(3.23)
On the other hand, noting that
[N, ek ](p) = −∇N ek (p) − ∇ek N (p) = −λk ek (p),
so we have
hR(N, ek )V, ek i
= h∇N ∇ek V − ∇ek ∇N V − ∇[N,ek ] V, ek ip
= N h∇ek V, ek i + hN, ∇ek ∇ek V i − h∇λk ek V, ek i
= hN, ∇ek ∇ek V i + N (ψ) − λk ψ.
For the third term in (3.19), we have
(3.24)
hN, ∇ek ∇ek V i = hR(N, ek )V, ek i − N (ψ) + λk ψ.
Also we have
hR(N, ek )V, ek i =
=
hR(N, ek )(
X
X
αl el − f N ), ek i
l
αl R(N, ek )el , ek i − f hR(N, ek )N, ek i,
l
and
hR(N, ek )el , ek i = R(ek , el , N, ek ) = ek (hkl ) − el (hkk ),
so (3.24) become
hN, ∇ek ∇ek V i =
(3.25)
X
[αl ek (hkl ) − el (hkk )] + f R(N, ek , N, ek ) − N (ψ) + λk ψ.
l
Substituting (3.20), (3.21) and (3.25) into (3.19) we can get
74
Ximin Liu and Biaogui Yang
=
fkk
ek ek (f ) = −
X
αl ek (hkl ) +
l
X
h2kl f − 2λk ψ +
l
X
(αl ek (hkl ) − αl el (hkk )) + f R(N, ek , N, ek ) − N (ψ) + λk ψ
l
= (
(3.26)
X
h2kl + R(N, ek , N, ek ))f − hV ⊤ , ∇hkk i − N (ψ) − λk ψ.
l
So we have
(3.27)
△f = (kAk2 + Ric(N, N ))f − hV ⊤ , ∇S1 i − nN (ψ) − S1 ψ,
and
X
λk fkk
=
−2
k
X
λ2k ψ −
k
(3.28)
=
−
k
λk ek (hkl ) +
k,l
−el (hkk )) +
X
X
X
+
X
λk h2kl f +
k
f λk R(N, ek , N, ek ) −
k
λ2k ψ
X
λ3k f
−
k
X
X
X
λk αl (ek (hkl )
k
λk N (ψ) +
k
X
λ2k ψ
k
αl λk el (λk ) + f cS1 − S1 N (ψ).
k,l
Note that (2.9) and (2.10)
X
λ2k = S12 − 2S2 ,
X
k
λ3k = S13 − 3S1 S2 + 3S3 ,
k
substituting (3.27) and (3.28) into (2.13), so we can get
X
X
X
L1 (f ) =
(S1 δij − hij )fij =
S1 fii −
hii fii
i,j
i
i
= (S1 S2 − 3S3 )f + f (n − 1)cS1 − (n − 1)S1 N (ψ) − 2S2 ψ − hV ⊤ , ∇S2 i.
4
Stable hyersurfaces with constant scalar curvature in GRW
In the following, we will consider the generalized Roberston-Walker spaces M
−I ×φ F n , let
πI : M
n+1
n+1
=
→I
denote the canonical projection onto the I. Then the vector field
V = (φ ◦ πI )
∂
∂t
is conformal, timelike and closed (in the sense that its metrically equavalent 1-form
is closed), with conformal factor ψ = φ′ . Now we have the follow corollary
75
Stable space-like hypersurfaces
Corollary 4.1. If M n is a closed spacelike hypersurface having constant normaln+1
= −I ×φ F n
ized scalar curvature R in generalized Roberston-Walker spaces M
with constant sectional curvature c. Let N be a future-pointing unit normal vector
∂
field globally defined on M n . If V = (φ ◦ πI ) ∂t
and f = hV, N i, then
(4.1)
L1 (f ) = (S1 S2 − 3S3 )f + f (n − 1)cS1 + (n − 1)S1 φ′′ hN,
∂
i − 2S2 ψ.
∂t
Proof. Since we have
∇φ′ = −h∇φ′ ,
(4.2)
∂ ∂
∂
i = −φ′′ ,
∂t ∂t
∂t
then
N (φ′ ) = hN, ∇φ′ i = −φ′′ h
(4.3)
∂
, N i.
∂t
Substituting (4.3) into (3.18), we can get (4.1).
Now we can state and prove our main result:
Theorem 4.2. If M n a is closed hypersurface, having constant normalized scalar
n+1
curvature R in generalized Roberston-Walker spaces M
= −I ×φ F n with constant
sectional curvature c. If the warping function φ is not constant and satisfies Hφ′′ ≥
max{(R − c)φ′ , 0}, and M n is stable, then
(I) R = c on M , or
(II) M is spacelike slice Mt0 = t0 × F n , for some t0 ∈ I, satisfying
Hφ′′ = (R − c)φ′ .
Proof. Using Proposition 3.5 and Corollary 4.1, we can get
Z
∂
[(n − 1)S1 φ′′ hN, i − 2S2 φ′ ]f dM
J ′′ (0)(f ) = 2
∂t
M
Z
∂
∂
[(n − 1)S1 φ′′ hN, i − n(n − 1)(c − R)φ′ ]φhN, idM.
= 2
∂t
∂t
M
∂
i = − cosh θ, where θ denotes the hyperbolic angle between the timelike
Let hN, ∂t
∂
verctor fields N and ∂t
. Since M n is stable, so
Z
∂
∂
[(n − 1)S1 φ′′ hN, i − n(n − 1)(c − R)φ′ ]φhN, idM
0 ≥2
∂t
∂t
ZM
′′
(n − 1)S1 φφ cosh θ(cosh θ − 1)dM ≥ 0,
≥2
M
and hence
(4.4)
Hφ′′ (cosh θ − 1) = 0 and Hφ′′ = (R − c)φ′
holds on M n . If R 6= c and φ′ 6= 0 then Hφ′′ 6= 0, and cosh θ = 1, so M is an umbilical
leaf satisfying Hφ′′ = (R − c)φ′ .
✷
Acknowledgements. The first author is supported by RFDP((20050141011),
NCET(06-0276) and MXDUT073011.
76
Ximin Liu and Biaogui Yang
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[5] J. L. Barbosa and V. Oliker, Stable hyersurface with constant mean curvature in
Lorentz space, Geom. Global Analysis, Tohoku Universtiy, Sendai(1993), 161-164.
[6] A. Barros, A. Brasil and A. Caminha, Stability of spacelike hypersurfaces in
foliated spacetimes, preprint.
[7] A. Brasil and A. G. Colares, Stability of spacelike hypersurface with constant
r-mean curvature in de Sitter space, Proceedings of the XII Fall Workshop on
Geometry and Physics, 139–145, Publ. R. Soc. Mat. Esp., 7, R. Soc. Mat. Esp.,
Madrid, 2004.
[8] A. Caminha, On the spacelke hyersurfaces of constant sectional curvature Lorentz
manifolds, J. Geom. Phy. 56(2006), 1144-1174.
[9] X. Liu and J. Deng, Stable space-like hyersurfaces in the De Sitter space, Arch.
Math. 40(2004), 111-117.
[10] B. O’Nelill, Semi-Riemannian Geometry with Appications to Relativity, Academic Press, New York, 1983.
[11] R. C. Reilly, Variational properties of functions of constant mean curvature hyersurface in space forms, J. Diff. Geom. 8(1973), 465-477.
[12] H. Rosenberg, Hpyersurfaces of constant curvature in space forms, Bull. Sc.
Math., 2e Sérié 117(1993), 211-239.
Authors’ address:
Ximin Liu, Biaogui Wang
Department of Applied Mathematics
Dalian University of Technology
Dalian 116024, P.R. China.
E-mail: [email protected], [email protected]
curvature in generalized Roberston-Walker
spacetimes
Ximin Liu and Biaogui Yang
Abstract. In this paper we study stable spacelike hyersurfaces with conn+1
=
stant scalar curvature in generalized Roberston-Walker spacetime M
−I ×φ F n .
M.S.C. 2000: 53B30, 53C42, 53C50.
Key words: stability, spacelike hypersurface, constant scalar curvature, generalized
Roberston-Walker spacetime.
1
Introduction
Hyersurfaces M n with constant r-mean curvature in Riemannian manifolds or Lorentz
n+1
manifolds M
(c) with constant sectional curvature c are critical points of some area
functional variations which keep constant a certain volume function. Stable hyersurfaces with constant mean curvature(CMC) (or constant r-mean curvature) in real
space form are very interesting geometrical objects that were investigated by many
geometricians. Barbosa and do Carmo [2] gave definition of stability of hyersurfaces
with constant mean curvature in the Eucildean space Rn+1 and proved the round
spheres are the only compact stable hyersurfaces with CMC in Rn+1 . Later, Barbosa,
do Carmo and Eschenburg [3] extended ambient spaces to Riemannian manifolds
and obtained the corresponding results. In [5] Barbosa and Oliker discussed stable
spacelile hyersurfaces with CMC in Lorentz manifolds. At the same time, Alencar, do
Carmo and Colares [1] investigated stable hyersurfaces with constant scalar curvature
in Riemannian manifolds and obtained geodesic sphere is the only stable compact orientable hyersurface in Riemannain spaces. On the other hand, Barbosa and Colares
[4] studied compact hyersurfaces without boundary immersed in space forms with
constant r-mean curvature. Recently, Liu and Deng [9] also discussed stable spacelike hyersurfaces with constant scalar curvature in de Siter space S1n+1 . Barros, Brasil
and Caminha [6] classified strongly stable spacelike hypersurfaces with constant mean
curvature whose warping function satisfied a certain convexity condition.
∗
Balkan
Journal of Geometry and Its Applications, Vol.13, No.1, 2008, pp. 66-76.
c Balkan Society of Geometers, Geometry Balkan Press 2008.
°
67
Stable space-like hypersurfaces
In this paper we will study stable spacelike hypersurfaces with constant scalar
n+1
= −I ×φ F n .
curvature in generalized Roberston-Walker spacetime M
2
Preliminaries
Consider F n an n-dimensional manifold, let I be a 1-dimensional manifold (either
n+1
a circle or an open interval of R). We denote by M
= −I ×φ F n the (n + 1)dimensional product manifold I × F endowed with the Lorentzian metric
(2.1)
g = h, i = −dt2 + f 2 (t)h, iM ,
where f > 0 is positive function on I, and h, iM stands for the Riemannian metric
on F n . We refer to −I ×φ F n as a generalized Robertson-Walker (GRW) spacetime.
In particular, when the Riemannian factor F n has constant sectional curvature, then
−I ×φ F n is classically called a Robertson-Walker (RW) spacetime.
n+1
A vector field V on a Lorentz manifold M
is said to be conformal if
(2.2)
LV g = 2ψg,
n+1
→ R, where L stands for the Lie derivative of
for some smooth function ψ : M
n+1
. The function ψ is called the conformal factor of V . V ∈ T M
Lorentz metric of M
is conformal if and only if
(2.3)
h∇X V, Y i + h∇Y V, Xi = 2ψhX, Y i,
for all X, Y ∈ T (M ).
n+1
Any Lorentz manifold M
, possessing a globally defined, timelike conformal
vector field is said to be a conformally stationary (CS) spacetime.
n+1
denote an orientable spacelike hyersurface in the timeLet x : M n → M
n+1
and N be a globally defined unit normal vector
oriented Lorentz manifold M
field on M n . ∇ and ∇ denote the Levi-Civita connection of M n and ambient space
n+1
M
respectively. R and Ric denote the curvature tensor and Ricci curvature tensor
n+1
on M
respectively, which are defined by
(2.4)
R(X, Y )Z = ∇X ∇Y Z − ∇Y ∇X Z − ∇[X,Y ] Z,
and
(2.5)
R(W, Z, X, Y ) = h∇X ∇Y Z, W i − h∇Y ∇X Y, W i − h∇[X,Y ] Z, W i,
then
(2.6)
Ric(X, Y ) =
n+1
X
R(ek , X, ek , Y ),
k=1
where X, Y, Z, W ∈ T M , and {ek }nk=1 is a basis of Tp M , en+1 = N . In particular we
have
68
Ximin Liu and Biaogui Yang
(2.7)
Ric(N, N ) =
n
X
R(ek , N, ek , N ).
k=1
The shape operator A associated to N of M n , defined by
A = −∇N (i.e Aek = −∇ek N )
(2.8)
is a self-adjoint linear operator in each tangent space Tp M . Its eigenvalues are the
principal curvatures of immersion and are represented by λ1 , λ2 , · · · , λn .The elementary symmetic functions Sr associated to A can be defined, using the characteristic
polynomial of A, by
n
X
det(tI − A) =
(−1)k Sk tn−k ,
k=0
where S0 = 1. If p ∈ M , and {ek } is a basis of Tp M formed by eigenvector of Ap ,
with corresponding eigenvalues λk , one immediately sees that
Sr = σr (λ1 , · · · , λn ),
where σr is the r-th elementary symmetric polynomial. In particular
X
(2.9)
kAk2 =
λ2k = S12 − 2S2 ,
k
and
X
(2.10)
λ3k = S13 − 3S1 S2 + 3S3 .
k
The r-th classical Newton transformation Pr on M is defined as following
P0 = I,
Pr = Sr I − APr−1 ,
1 ≤ r ≤ n.
Associated to each Newton transformation Pr of immersion x : M n → M
have a second order differential operator defined by
(2.11)
When M
n+1
, we
Lr (f ) = trace(Pr ◦ Hessf ).
n+1
has constant sectional curvature, then
(2.12)
Lr (f ) = div(Pr ∇f ),
where div stands for the divergence of a vector field on M , it was proved by H.
Rosenberg in [12].
Remark 1.1. According (2.11) or (2.12), when r = 0,
L0 f = div(P0 ∇f ) = △f
n
is Laplace operator on M , and if r = 1, then
L1 f
(2.13)
= div[P1 ◦ hessf ] = div[(S1 I − AP0 ) ◦ hessf ]
X
=
(S1 δij − hij )fij
i,j
become Cheng-Yau’s operator ✷ on M n , where hij and fij denote the component of
A and hessf respectively.
69
Stable space-like hypersurfaces
3
The variational problem in Lorentz manifolds
n+1
Let x : M n → M
denotes an orientable spacelike hyersurface in the time-oriented
n+1
and N be a globally defined unit normal vector field on M n .
Lorentz manifold M
n+1
satisfying the following
A variation of x is a smooth map X : M n × (−ε, ε) → M
conditions:
n+1
(1) For t ∈ (−ε, ε), the map Xt : M n → M
given by Xt (p) = X(t, p) is a
spaelike immersion such that X0 = x.
(2) Xt |∂M = x|∂M , for all t ∈ (−ε, ε).
∂
) = ∂X
The variational field vector associated the variation X is vector field X∗ ( ∂t
∂t .
∂X
Let f = h ∂t , N i, we have
(3.1)
∂X ⊤
∂X
=(
) − f N,
∂t
∂t
where ⊤ denotes tangential components. The balance of volume of the variation X is
the function V : (−ε, ε) → R given by
Z
X ∗ (dM ),
(3.2)
V (t) =
M ×[0,t]
where dM denotes the volume element of M .
The area functional A : (−ε, ε) → R is given by
Z
S1 dMt ,
(3.3)
A(t) =
M
where dMt denotes the volume element of the metric induced in M by Xt . Then we
have the following classical result.
n+1
n+1
Lemma 3.1. Let M
be a time-oriented Lorentz manifold and x : M n → M
n+1
is a variation of x, then
a spacelike hyersurface. If X : M n × (−ε, ε) → M
(i)
Z
dV (t)
f dM ;
|t=0 =
(3.4)
dt
M
(ii)
(3.5)
∂(dMt )
∂X ⊤
= (S1 + div(
) )dMt .
∂t
∂t
Proof. For (i) see [3, 9], and for (ii) see [4, 11]. ✷
Barros, Brasil and Caminha [6] proved the following proposition:
n+1
Proposition 3.2. Let x : M n → M
be a spacelike hypersurface of the timen+1
, and N be a globally defined unit normal vector field
oriented Lorentz manifold M
n+1
on M n . If X : M n × (−ε, ε) → M
is a variation of x, then
(3.6)
∂X ⊤
dS1
= △f − (Ric(N, N ) + kAk2 )f + h(
) , ∇S1 i.
dt
∂t
70
Ximin Liu and Biaogui Yang
Suppose λ is a constant, and J : (−ε, ε) → R is given by
(3.7)
J(t) = A(t) + λV (t),
J is called the Jacobi functional associated to the variation X. Then we have the
following proposition:
n+1
be a spacelike hypersurface in the timeProposition 3.3. Let x : M n → M
n+1
oriented Lorentz manifold M
, and N be a globally defined unit normal vector field
n+1
is a variation of x, then
on M n . If X : M n × (−ε, ε) → M
Z
dJ(t)
∂X ⊤
(3.8)
[div(S1 (
=
) ) + ∆f − (Ric(N, N ) + kAk2 − S12 − λ)f ]dMt .
dt
∂t
M
n+1
has constant sectional curvature c, then
In particular, when M n is closed and M
Z
dJ(t)
(3.9)
(2S2 − cn + λ)f dMt .
=
dt
M
Proof. We can get this result from Lemma 3.1 and Proposition 3.2. In fact,
Z
Z
Z
dS1
dJ(t)
∂X ⊤
S1 (S1 f + div(
λf dMt
=
dMt +
) )dMt +
dt
dt
∂t
M
M
ZM
∂X ⊤
∂X ⊤
) , ∇S1 i + S1 div(
) + △f
[h(
=
∂t
∂t
M
−(Ric(N, N ) + kAk2 )f + S12 f + λf ]dMt
Z
∂X ⊤
[div(S1 (
=
) ) + △f − (Ric(N, N ) + kAk2 f − S12 f − λ)f ]dMt .
∂t
M
When M n is closed and M
n+1
Z
has constant sectional curvature c, then we have
div(S1 (
M
Z
∂X ⊤
) )dMt = 0,
∂t
△f dMt = 0,
M
and Ric(N, N ) = nc, then using (2.9), we have (3.9).
n+1
Proposition 3.4. Let x : M n → M
is a spacelike hypersurface in Lorentz
n+1
(c) with constant sectional curvature c, and X : M n × (−ε, ε) →
space form M
n+1
M
is a variation of x, then
(3.10)
∂X ⊤
dS2
= L1 (f ) − (S1 S2 − 3S3 )f − f (n − 1)cS1 + h(
) , ∇S2 i.
dt
∂t
In particular, if S2 is a constant, then one has
(3.11)
dS2
= L1 (f ) − (S1 S2 − 3S3 )f − f (n − 1)cS1 .
dt
71
Stable space-like hypersurfaces
Proof. According to the proof of proposition 3.2 in [6], we can get
dhkk
∂X ⊤
= fkk − cf − h2kk f + h∇hkk , (
) i.
dt
∂t
(3.12)
Using (2.9), we can get
dS1 X
dhkk
dS2
= S1
−
hkk
.
dt
dt
dt
(3.13)
k
Substituting (3.6) and (3.12) into (3.13), using (2.9) and (2.10), then we have
dS2
dt
=
∂X ⊤
) , ∇S1 i]
∂t
X
∂X ⊤
−
hkk [fkk − cf − h2kk f + h∇hkk , (
) i]
∂t
S1 [△f − (Ric(N, N ) + kAk2 )f + h(
k
=
S1 △f − S1 (nc + S12 − 2S2 )f + S1 h(
−S1
X
k
=
fkk + cS1 f + f
X
k
X
∂X ⊤
) , ∇S1 i +
(S1 fkk − hkk fkk )
∂t
k
1
∂X ⊤
λ3k − h∇(S12 − 2S2 ), (
) i
2
∂t
L1 (f ) − (S1 S2 − 3S3 )f − f (n − 1)cS1 + h(
∂X ⊤
) , ∇S2 i.
∂t
If S2 is constant, then the last term in the above is equal to zero, so we have (3.11).
If M has constant normalized scalar curvature R, and we choose
(3.14)
λ = 2S2 − nc = n(n − 1)(c − R) − nc,
then λ is a constant too, so we have
n+1
Proposition 3.5. Let x : M n → M
(c) is a spacelike hypersurface in the timen+1
n+1
(c), and X : M n × (−ε, ε) → M
is a variation of
oriented Lorentz manifold M
x, and S2 is constant, then
Z
d2 J(0)
(f
)
=
2
[L1 (f ) − (S1 S2 − 3S3 )f − f (n − 1)cS1 ]f dM.
(3.15)
dt2
M
Proof. Since λ = 2S2 − nc = n(n − 1)(c − R) − nc, using (3.9) and (3.11), we can get
Z
Z
d2 J(0)
dS2 (0)
[L1 (f ) − (S1 S2 − 3S3 )f − f (n − 1)cS1 ]f dM.
(f
)
=
2
f
dM
=
2
dt2
dt
M
M
n+1
Definition 3.6. Suppose x : M n → M
(c) has constant scalar curvature. The
immersion x is stable if
Z
d2 J(0)
(3.16)
[L1 (f ) − (S1 S2 − 3S3 )f − f (n − 1)cS1 ]f dM ≤ 0,
(f ) = 2
dt2
M
72
Ximin Liu and Biaogui Yang
for all volume-presering variations of x. If M n is noncompact, x is stable if for every
conpact submanifolds M ′ ⊂ M n with boundary, the restriction x |M ′ is stable.
For conformally stationary spacetimes, we have the following proposition.
n+1
be a conformally stationary Lorentz manifold, with
Proposition 3.7. Let M
n+1
n+1
→ R. Suppose x : M n → M
conformal vector V having conformal factor ψ : M
n+1
is a spacelike hypersurface in M
= I ×φ F n with constant sectional curvature c,
and N a future-pointing, unit normal vector field globally defined on M n , f = hV, N i,
then
(3.17) L1 (f ) = (S1 S2 − 3S3 )f + f (n − 1)cS1 − (n − 1)S1 N (ψ) − 2S2 ψ − hV ⊤ , ∇S2 i.
In particular, if R is constant, then S2 is a constant too, so
(3.18)
✷f = L1 (f ) = (S1 S2 − 3S3 )f + f (n − 1)cS1 − (n − 1)S1 N (ψ) − 2S2 ψ.
Proof. We can choose {ek } as a moving frame on neighborhood U ⊂ M of p, geodesic
at p, and diagonalizing the shape operator A of M at p, with Aek = λk ek , for 1 ≤
k ≤ n. Extend N and ek (1 ≤ k ≤ n) to a neighborhood of p in M , such that
hN, ek i = 0 and (∇N ek )(p) = 0.
Let
V =
X
αl el − f N,
l
so we have
ek (f ) = h∇ek N, V i + hN, ∇ek V i = −hAek , V i + hN, ∇ek V i.
Then
ek ek (f )
(3.19)
= −ek hAek , V i + ek hN, ∇ek V i
= −h∇ek (Aek ), V i − 2hAek , ∇ek V i + hN, ∇ek ∇ek V i.
For the first term in (3.19), we have
h∇ek (Aek ), V i
= h∇ek (Aek ),
X
=
j
(3.20)
X
=
l
X
l
ek (hkj )hej ,
αl el − f N i
X
αl el i +
X
hkj h∇ek ej , −f N i
j
l
αl ek (hkl ) −
X
h2kl f.
l
For the second term in (3.19), we have
X
(3.21)
hkj hej , ∇ek V i = λk hek , ∇ek V i = λk ψ,
hAek , ∇ek V i =
j
where in the last equality we use the fact that V is conformal vector having conformal
factor ψ, and we have
73
Stable space-like hypersurfaces
hN, ∇ek V i + hek , ∇N V i = 2ψhek , N i = 0,
then we can get
(3.22)
h∇ek N, ∇ek V i + hN, ∇ek ∇ek V i + h∇ek ek , ∇N V i + hek , ∇ek ∇N V i = 0.
Since
h∇ek N, ∇ek V i = −hAek , ∇ek V i = −λk ψ,
and
h∇N V, ∇ek ek i
= −h∇N V, h∇ek ek , N iN i
= −h∇N V, hkk N i = −hkk ψhN, N i = λk ψ,
then
hN, ∇ek ∇ek V i = −hek , ∇ek ∇N V i.
(3.23)
On the other hand, noting that
[N, ek ](p) = −∇N ek (p) − ∇ek N (p) = −λk ek (p),
so we have
hR(N, ek )V, ek i
= h∇N ∇ek V − ∇ek ∇N V − ∇[N,ek ] V, ek ip
= N h∇ek V, ek i + hN, ∇ek ∇ek V i − h∇λk ek V, ek i
= hN, ∇ek ∇ek V i + N (ψ) − λk ψ.
For the third term in (3.19), we have
(3.24)
hN, ∇ek ∇ek V i = hR(N, ek )V, ek i − N (ψ) + λk ψ.
Also we have
hR(N, ek )V, ek i =
=
hR(N, ek )(
X
X
αl el − f N ), ek i
l
αl R(N, ek )el , ek i − f hR(N, ek )N, ek i,
l
and
hR(N, ek )el , ek i = R(ek , el , N, ek ) = ek (hkl ) − el (hkk ),
so (3.24) become
hN, ∇ek ∇ek V i =
(3.25)
X
[αl ek (hkl ) − el (hkk )] + f R(N, ek , N, ek ) − N (ψ) + λk ψ.
l
Substituting (3.20), (3.21) and (3.25) into (3.19) we can get
74
Ximin Liu and Biaogui Yang
=
fkk
ek ek (f ) = −
X
αl ek (hkl ) +
l
X
h2kl f − 2λk ψ +
l
X
(αl ek (hkl ) − αl el (hkk )) + f R(N, ek , N, ek ) − N (ψ) + λk ψ
l
= (
(3.26)
X
h2kl + R(N, ek , N, ek ))f − hV ⊤ , ∇hkk i − N (ψ) − λk ψ.
l
So we have
(3.27)
△f = (kAk2 + Ric(N, N ))f − hV ⊤ , ∇S1 i − nN (ψ) − S1 ψ,
and
X
λk fkk
=
−2
k
X
λ2k ψ −
k
(3.28)
=
−
k
λk ek (hkl ) +
k,l
−el (hkk )) +
X
X
X
+
X
λk h2kl f +
k
f λk R(N, ek , N, ek ) −
k
λ2k ψ
X
λ3k f
−
k
X
X
X
λk αl (ek (hkl )
k
λk N (ψ) +
k
X
λ2k ψ
k
αl λk el (λk ) + f cS1 − S1 N (ψ).
k,l
Note that (2.9) and (2.10)
X
λ2k = S12 − 2S2 ,
X
k
λ3k = S13 − 3S1 S2 + 3S3 ,
k
substituting (3.27) and (3.28) into (2.13), so we can get
X
X
X
L1 (f ) =
(S1 δij − hij )fij =
S1 fii −
hii fii
i,j
i
i
= (S1 S2 − 3S3 )f + f (n − 1)cS1 − (n − 1)S1 N (ψ) − 2S2 ψ − hV ⊤ , ∇S2 i.
4
Stable hyersurfaces with constant scalar curvature in GRW
In the following, we will consider the generalized Roberston-Walker spaces M
−I ×φ F n , let
πI : M
n+1
n+1
=
→I
denote the canonical projection onto the I. Then the vector field
V = (φ ◦ πI )
∂
∂t
is conformal, timelike and closed (in the sense that its metrically equavalent 1-form
is closed), with conformal factor ψ = φ′ . Now we have the follow corollary
75
Stable space-like hypersurfaces
Corollary 4.1. If M n is a closed spacelike hypersurface having constant normaln+1
= −I ×φ F n
ized scalar curvature R in generalized Roberston-Walker spaces M
with constant sectional curvature c. Let N be a future-pointing unit normal vector
∂
field globally defined on M n . If V = (φ ◦ πI ) ∂t
and f = hV, N i, then
(4.1)
L1 (f ) = (S1 S2 − 3S3 )f + f (n − 1)cS1 + (n − 1)S1 φ′′ hN,
∂
i − 2S2 ψ.
∂t
Proof. Since we have
∇φ′ = −h∇φ′ ,
(4.2)
∂ ∂
∂
i = −φ′′ ,
∂t ∂t
∂t
then
N (φ′ ) = hN, ∇φ′ i = −φ′′ h
(4.3)
∂
, N i.
∂t
Substituting (4.3) into (3.18), we can get (4.1).
Now we can state and prove our main result:
Theorem 4.2. If M n a is closed hypersurface, having constant normalized scalar
n+1
curvature R in generalized Roberston-Walker spaces M
= −I ×φ F n with constant
sectional curvature c. If the warping function φ is not constant and satisfies Hφ′′ ≥
max{(R − c)φ′ , 0}, and M n is stable, then
(I) R = c on M , or
(II) M is spacelike slice Mt0 = t0 × F n , for some t0 ∈ I, satisfying
Hφ′′ = (R − c)φ′ .
Proof. Using Proposition 3.5 and Corollary 4.1, we can get
Z
∂
[(n − 1)S1 φ′′ hN, i − 2S2 φ′ ]f dM
J ′′ (0)(f ) = 2
∂t
M
Z
∂
∂
[(n − 1)S1 φ′′ hN, i − n(n − 1)(c − R)φ′ ]φhN, idM.
= 2
∂t
∂t
M
∂
i = − cosh θ, where θ denotes the hyperbolic angle between the timelike
Let hN, ∂t
∂
verctor fields N and ∂t
. Since M n is stable, so
Z
∂
∂
[(n − 1)S1 φ′′ hN, i − n(n − 1)(c − R)φ′ ]φhN, idM
0 ≥2
∂t
∂t
ZM
′′
(n − 1)S1 φφ cosh θ(cosh θ − 1)dM ≥ 0,
≥2
M
and hence
(4.4)
Hφ′′ (cosh θ − 1) = 0 and Hφ′′ = (R − c)φ′
holds on M n . If R 6= c and φ′ 6= 0 then Hφ′′ 6= 0, and cosh θ = 1, so M is an umbilical
leaf satisfying Hφ′′ = (R − c)φ′ .
✷
Acknowledgements. The first author is supported by RFDP((20050141011),
NCET(06-0276) and MXDUT073011.
76
Ximin Liu and Biaogui Yang
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Authors’ address:
Ximin Liu, Biaogui Wang
Department of Applied Mathematics
Dalian University of Technology
Dalian 116024, P.R. China.
E-mail: [email protected], [email protected]