curvature in generalized Roberston-Walker
spacetimes
Ximin Liu and Biaogui Yang
Abstract.In this paper we study stable spacelike hyersurfaces with con-stant scalar curvature in generalized Roberston-Walker spacetimeMn+1= −I×φFn.
M.S.C. 2000: 53B30, 53C42, 53C50.
Key words: stability, spacelike hypersurface, constant scalar curvature, generalized Roberston-Walker spacetime.
1
Introduction
HyersurfacesMnwith constantr-mean curvature in Riemannian manifolds or Lorentz manifoldsMn+1(c) with constant sectional curvaturecare critical points of some area functional variations which keep constant a certain volume function. Stable hyersur-faces 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 inRn+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 ori-entable 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 space-like hyersurfaces with constant scalar curvature in de Siter spaceS1n+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
In this paper we will study stable spacelike hypersurfaces with constant scalar curvature in generalized Roberston-Walker spacetimeMn+1 =−I×φFn.
2
Preliminaries
Consider Fn an n-dimensional manifold, let I be a 1-dimensional manifold (either
a circle or an open interval of R). We denote by Mn+1 = −I×φ Fn the (n+
1)-dimensional product manifoldI×F endowed with the Lorentzian metric
(2.1) g=h,i=−dt2+f2(t)h,iM,
where f > 0 is positive function on I, and h,iM stands for the Riemannian metric
onFn. We refer to −I×
φFn as a generalized Robertson-Walker (GRW) spacetime.
In particular, when the Riemannian factorFn has constant sectional curvature, then
−I×φFn is classically called a Robertson-Walker (RW) spacetime.
A vector fieldV on a Lorentz manifoldMn+1is said to be conformal if
(2.2) LVg= 2ψg,
for some smooth function ψ: Mn+1 → R, where L stands for the Lie derivative of Lorentz metric ofMn+1. The functionψis called the conformal factor ofV.V ∈T M
is conformal if and only if
h∇XV, Yi+h∇YV, Xi= 2ψhX, Yi,
(2.3)
for allX, Y ∈T(M).
Any Lorentz manifold Mn+1, possessing a globally defined, timelike conformal vector field is said to be a conformally stationary (CS) spacetime.
Let x : Mn → Mn+1 denote an orientable spacelike hyersurface in the
time-oriented Lorentz manifold Mn+1 and N be a globally defined unit normal vector field onMn. ∇ and∇ denote the Levi-Civita connection of Mn and ambient space
Mn+1respectively. R and Ric denote the curvature tensor and Ricci curvature tensor onMn+1 respectively, which are defined by
(2.4) R(X, Y)Z =∇X∇YZ− ∇Y∇XZ− ∇[X,Y]Z,
and
(2.5) R(W, Z, X, Y) =h∇X∇YZ, Wi − h∇Y∇XY, Wi − h∇[X,Y]Z, Wi,
then
(2.6) Ric(X, Y) =
n+1
X
k=1
R(ek, X, ek, Y),
Ric(N, N) =
n
X
k=1
R(ek, N, ek, N).
(2.7)
The shape operator A associated toN ofMn, defined by
(2.8) A =−∇N (i.e Aek =−∇ekN)
is a self-adjoint linear operator in each tangent space TpM. Its eigenvalues are the principal curvatures of immersion and are represented byλ1, λ2, · · · , λn.The
elemen-tary symmetic functionsSr associated to A can be defined, using the characteristic
polynomial of A, by
det(tI−A) =
n
X
k=0
(−1)kSktn
−k,
where S0 = 1. If p∈ M, and {ek} is a basis of TpM formed by eigenvector of Ap,
with corresponding eigenvaluesλk, one immediately sees that
Sr=σr(λ1,· · · , λn),
whereσr is ther-th elementary symmetric polynomial. In particular
(2.9) kAk2=X
k
λ2k=S12−2S2,
and
(2.10) X
k
λ3k=S13−3S1S2+ 3S3.
Ther-th classical Newton transformation Pr onM is defined as following
P0= I,
Pr=SrI−APr−1, 1≤r≤n.
Associated to each Newton transformation Pr of immersion x : Mn → M n+1
, we have a second order differential operator defined by
(2.11) Lr(f) = trace(Pr◦Hessf).
WhenMn+1has 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), whenr= 0,
L0f = div(P0∇f) =△f
is Laplace operator onMn, and ifr= 1, then
L1f = div[P1◦hessf] = div[(S1I−AP0)◦hessf]
= X
i,j
(S1δij−hij)fij (2.13)
become Cheng-Yau’s operator✷onMn, whereh
ij andfij denote the component of
3
The variational problem in Lorentz manifolds
Letx:Mn →Mn+1 denotes an orientable spacelike hyersurface in the time-oriented
Lorentz manifoldMn+1andN be a globally defined unit normal vector field onMn. A variation ofxis a smooth mapX :Mn×(−ε, ε)→Mn+1 satisfying the following
conditions:
(1) For t ∈ (−ε, ε), the map Xt : Mn → M n+1
given by Xt(p) = X(t, p) is a
spaelike immersion such thatX0=x. (2)Xt|∂M =x|∂M, for allt∈(−ε, ε).
The variational field vector associated the variationXis vector fieldX∗(∂
∂t) = ∂X
∂t.
Letf =h∂X
∂t, Ni, we have
(3.1) ∂X
∂t = ( ∂X
∂t )
⊤ −f N,
where⊤denotes tangential components. The balance of volume of the variationX is the functionV : (−ε, ε)→R given by
(3.2) V(t) =
Z
M×[0,t]
X∗(dM),
wheredM denotes the volume element ofM. The area functionalA: (−ε, ε)→Ris given by
(3.3) A(t) =
Z
M
S1dMt,
wheredMt denotes the volume element of the metric induced inM byXt. Then we
have the following classical result.
Lemma 3.1.LetMn+1be a time-oriented Lorentz manifold andx:Mn→Mn+1
a spacelike hyersurface. IfX:Mn×(−ε, ε)→Mn+1 is a variation ofx, then
(i)
(3.4) dV(t)
dt |t=0=
Z
M f dM;
(ii)
(3.5) ∂(dMt)
∂t = (S1+ div( ∂X
∂t )
⊤ )dMt.
Proof.For (i) see [3, 9], and for (ii) see [4, 11].✷
Barros, Brasil and Caminha [6] proved the following proposition:
Proposition 3.2.Let x:Mn →Mn+1 be a spacelike hypersurface of the
time-oriented Lorentz manifoldMn+1, andNbe a globally defined unit normal vector field onMn. IfX:Mn×(−ε, ε)→Mn+1 is a variation ofx, then
(3.6) dS1
dt =△f−(Ric(N, N) +kAk
2)
f+h(∂X
∂t )
⊤
Supposeλis a constant, andJ : (−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:
Proposition 3.3.Let x:Mn →Mn+1 be a spacelike hypersurface in the
time-oriented Lorentz manifoldMn+1, andNbe a globally defined unit normal vector field onMn. IfX:Mn×(−ε, ε)→Mn+1 is a variation ofx, then
In particular, whenMn is closed andMn+1 has constant sectional curvature c, then
(3.9) dJ(t)
dt =
Z
M
(2S2−cn+λ)f dMt.
Proof. We can get this result from Lemma 3.1 and Proposition 3.2. In fact,
dJ(t)
WhenMn is closed and Mn+1 has constant sectional curvaturec, then we have
Z
Proposition 3.4. Let x : Mn → Mn+1 is a spacelike hypersurface in Lorentz space form Mn+1(c) with constant sectional curvaturec, and X : Mn×(−ε, ε)→
In particular, ifS2 is a constant, then one has
(3.11) dS2
Proof. According to the proof of proposition 3.2 in [6], we can get
Using (2.9), we can get
dS2
Substituting (3.6) and (3.12) into (3.13), using (2.9) and (2.10), then we have
dS2
IfS2 is constant, then the last term in the above is equal to zero, so we have (3.11).
IfM has constant normalized scalar curvatureR, and we choose
λ= 2S2−nc=n(n−1)(c−R)−nc,
immersionxis stable if
for all volume-presering variations ofx. IfMn is noncompact,xis stable if for every
conpact submanifoldsM′
⊂Mn with boundary, the restrictionx|
M′ is stable.
For conformally stationary spacetimes, we have the following proposition. Proposition 3.7.LetMn+1 be a conformally stationary Lorentz manifold, with conformal vectorV having conformal factorψ:Mn+1→R. Supposex:Mn→Mn+1
is a spacelike hypersurface inMn+1 =I×φFn with constant sectional curvature c,
andNa future-pointing, unit normal vector field globally defined onMn,f =hV, Ni,
then
(3.17) L1(f) = (S1S2−3S3)f+f(n−1)cS1−(n−1)S1N(ψ)−2S2ψ− hV ⊤
,∇S2i.
In particular, ifR is constant, thenS2 is a constant too, so
(3.18) ✷f = L1(f) = (S1S2−3S3)f +f(n−1)cS1−(n−1)S1N(ψ)−2S2ψ.
Proof. We can choose{ek}as a moving frame on neighborhoodU ⊂M ofp, geodesic at p, and diagonalizing the shape operator A ofM at p, with Aek =λkek, for 1 ≤
k≤n. ExtendN andek (1≤k≤n) to a neighborhood of pin M, such that
hN, eki= 0 and (∇Nek)(p) = 0.
Let
V =X
l
αlel−f N,
so we have
ek(f) =h∇ekN, Vi+hN,∇ekVi=−hAek, Vi+hN,∇ekVi.
Then
ekek(f) = −ekhAek, Vi+ekhN,∇ekVi
= −h∇ek(Aek), Vi −2hAek,∇ekVi+hN,∇ek∇ekVi. (3.19)
For the first term in (3.19), we have
h∇ek(Aek), Vi = h∇ek(Aek),
X
l
αlel−f Ni
= X
j
ek(hkj)hej,
X
l
αleli+
X
j
hkjh∇ekej,−f Ni
= X
l
αlek(hkl)−
X
l h2klf.
(3.20)
For the second term in (3.19), we have
hAek,∇ekVi=
X
j
hkjhej,∇ekVi=λkhek,∇ekVi=λkψ, (3.21)
hN,∇ekVi+hek,∇NVi= 2ψhek, Ni= 0,
then we can get
h∇ekN,∇ekVi+hN,∇ek∇ekVi+h∇ekek,∇NVi+hek,∇ek∇NVi= 0. (3.22)
Since
h∇ekN,∇ekVi=−hAek,∇ekVi=−λkψ,
and
h∇NV,∇ekeki = −h∇NV,h∇ekek, NiNi
= −h∇NV, hkkNi=−hkkψhN, Ni=λkψ,
then
hN,∇ek∇ekVi=−hek,∇ek∇NVi. (3.23)
On the other hand, noting that
[N, ek](p) =−∇Nek(p)− ∇ekN(p) =−λkek(p),
so we have
hR(N, ek)V, eki = h∇N∇ekV − ∇ek∇NV − ∇[N,ek]V, ekip = Nh∇ekV, eki+hN,∇ek∇ekVi − h∇λkekV, eki = hN,∇ek∇ekVi+N(ψ)−λkψ.
For the third term in (3.19), we have
hN,∇ek∇ekVi=hR(N, ek)V, eki −N(ψ) +λkψ. (3.24)
Also we have
hR(N, ek)V, eki = hR(N, ek)(
X
l
αlel−f N), eki
= X
l
αlR(N, ek)el, eki −fhR(N, ek)N, eki,
and
hR(N, ek)el, eki= R(ek, el, N, ek) =ek(hkl)−el(hkk),
so (3.24) become
hN,∇ek∇ekVi=
X
l
[αlek(hkl)−el(hkk)] +fR(N, ek, N, ek)−N(ψ) +λkψ.
(3.25)
fkk = ekek(f) =−X
Note that (2.9) and (2.10)
X
substituting (3.27) and (3.28) into (2.13), so we can get
L1(f) = X
4
Stable hyersurfaces with constant scalar
curva-ture in GRW
In the following, we will consider the generalized Roberston-Walker spacesMn+1 = −I×φFn, let
πI :M n+1
→I
denote the canonical projection onto theI. 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 factorψ=φ′
Corollary 4.1.IfMnis a closed spacelike hypersurface having constant
normal-ized scalar curvature R in generalized Roberston-Walker spaces Mn+1 =−I×φFn
with constant sectional curvature c. Let N be a future-pointing unit normal vector field globally defined onMn. If V = (φ◦π
Proof. Since we have
(4.2) ∇φ′
Substituting (4.3) into (3.18), we can get (4.1).
Now we can state and prove our main result:
Theorem 4.2.IfMna is closed hypersurface, having constant normalized scalar curvatureRin generalized Roberston-Walker spacesMn+1=−I×φFnwith constant
sectional curvaturec. If the warping function φis not constant and satisfies Hφ′′ ≥
Proof.Using Proposition 3.5 and Corollary 4.1, we can get
J′′ verctor fieldsN and ∂
∂t. SinceMn is stable, so leaf satisfyingHφ′′
= (R−c)φ′
. ✷
References
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Authors’ address:
Ximin Liu, Biaogui Wang
Department of Applied Mathematics Dalian University of Technology Dalian 116024, P.R. China.
