curvature in generalized Roberston-Walker spacetimes

(1)

curvature in generalized Roberston-Walker spacetimes

Ximin Liu and Biaogui Yang

Abstract.In this paper we study stable spacelike hyersurfaces with constant 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 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 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 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 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

°Balkan Society of Geometers, Geometry Balkan Press 2008.

(2)

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+1X

k=1

R(ek, X, ek, Y),

whereX, Y, Z, W ∈T M, and{e_k}ⁿ_k=1 is a basis ofT_pM,e_n+1=N. In particular we have

(3)

Ric(N, N) = Xn

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, · · · , λ_n.The elementary symmetic functionsSr associated to A can be defined, using the characteristic polynomial of A, by

det(tI−A) = Xn k=0

(−1)^kSkt^n−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

λ²_k=S₁²−2S2, and

(2.10) X

k

λ³_k=S₁³−3S1S2+ 3S3.

Ther-th classical Newton transformation Pr onM is defined as following P0= I,

P_r=S_rI−AP_r−1, 1≤r≤n.

Associated to each Newton transformation Pr of immersion x : Mⁿ → Mⁿ⁺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) =4f 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 operator2onMⁿ, whereh_ij andf_ij denote the component of A and hessf respectively.

(4)

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 → Mⁿ⁺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,

wheredM_t denotes the volume element of the metric induced inM byX_t. 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 )^>)dM_t. Proof.For (i) see [3, 9], and for (ii) see [4, 11].2

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 =4f−(Ric(N, N) +kAk²)f+h(∂X

∂t )^>,∇S1i.

(5)

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

(3.8) dJ(t) dt =

Z

M

[div(S1(∂X

∂t )^>) + ∆f−(Ric(N, N) +kAk²−S₁²−λ)f]dMt. 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)

dt = Z

M

dS1

dt dMt+ Z

M

S1(S1f+ div(∂X

∂t )^>)dMt+ Z

M

λf dMt

= Z

M

[h(∂X

∂t )^>,∇S1i+S1div(∂X

∂t )^>+4f

−(Ric(N, N) +kAk²)f+S₁²f+λf]dMt

= Z

M

[div(S1(∂X

∂t )^>) +4f−(Ric(N, N) +kAk²f−S²₁f−λ)f]dMt. WhenMⁿ is closed and Mⁿ⁺¹ has constant sectional curvaturec, then we have

Z

M

div(S1(∂X

∂t )^>)dMt= 0, Z

M

4f dMt= 0, and Ric(N, N) =nc, then using (2.9), we have (3.9).

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)→ Mⁿ⁺¹ is a variation ofx, then

(3.10) dS2

dt = L1(f)−(S1S2−3S3)f−f(n−1)cS1+h(∂X

∂t )^>,∇S2i.

In particular, ifS2 is a constant, then one has

(3.11) dS2

dt = L1(f)−(S1S2−3S3)f−f(n−1)cS1.

(6)

Proof. According to the proof of proposition 3.2 in [6], we can get dhkk

dt =fkk−cf−h²_kkf+h∇hkk,(∂X

∂t )^>i.

(3.12)

Using (2.9), we can get dS2

dt =S1dS1

dt −X

k

hkkdhkk

dt . (3.13)

Substituting (3.6) and (3.12) into (3.13), using (2.9) and (2.10), then we have dS2

dt = S₁[4f−(Ric(N, N) +kAk²)f+h(∂X

∂t )^>,∇S₁i]

−X

k

hkk[fkk−cf−h²_kkf+h∇hkk,(∂X

∂t )^>i]

= S14f−S1(nc+S₁²−2S2)f+S1h(∂X

∂t )^>,∇S1i+X

k

(S1fkk−hkkfkk)

−S1

X

k

fkk+cS1f +fX

k

λ³_k−1

2h∇(S²₁−2S2),(∂X

∂t )^>i

= L1(f)−(S1S2−3S3)f−f(n−1)cS1+h(∂X

∂t )^>,∇S2i.

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,

(3.14)

thenλis a constant too, so we have

Proposition 3.5.Letx:Mⁿ →Mⁿ⁺¹(c) is a spacelike hypersurface in the time- oriented Lorentz manifoldMⁿ⁺¹(c), andX:Mⁿ×(−ε, ε)→Mⁿ⁺¹ is a variation of x, andS2 is constant, then

(3.15) d²J(0) dt² (f) = 2

Z

M

[L1(f)−(S1S2−3S3)f−f(n−1)cS1]f dM.

Proof. Sinceλ= 2S2−nc=n(n−1)(c−R)−nc, using (3.9) and (3.11), we can get d²J(0)

dt² (f) = 2 Z

M

dS2(0)

dt f dM = 2 Z

M

[L₁(f)−(S₁S₂−3S₃)f −f(n−1)cS₁]f dM.

Definition 3.6.Supposex:Mⁿ→Mⁿ⁺¹(c) has constant scalar curvature. The immersionxis stable if

d²J(0) dt² (f) = 2

Z

M

[L1(f)−(S1S2−3S3)f−f(n−1)cS1]f dM ≤0, (3.16)

(7)

for all volume-presering variations ofx. IfMⁿ is noncompact,xis stable if for every conpact submanifoldsM⁰⊂Mⁿ with boundary, the restrictionx|M⁰ 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) 2f = 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

h²_klf.

(3.20)

For the second term in (3.19), we have hAek,∇e_kVi=X

j

hkjhej,∇e_kVi=λkhek,∇e_kVi=λkψ, (3.21)

where in the last equality we use the fact thatV is conformal vector having conformal factorψ, and we have

(8)

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,∇e_k∇e_kVi=−hek,∇e_k∇NVi.

(3.23)

On the other hand, noting that

[N, e_k](p) =−∇_Ne_k(p)− ∇_e_kN(p) =−λ_ke_k(p), so we have

hR(N, e_k)V, e_ki = h∇_N∇_e_kV − ∇_e_k∇_NV − ∇_[N,e_k_]V, e_ki_p

= Nh∇ekV, eki+hN,∇ek∇ekVi − h∇λkekV, eki

= hN,∇e_k∇e_kVi+N(ψ)−λkψ.

For the third term in (3.19), we have

hN,∇e_k∇e_kVi=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)

Substituting (3.20), (3.21) and (3.25) into (3.19) we can get

(9)

fkk = ekek(f) =−X

l

αlek(hkl) +X

l

h²_klf−2λkψ+ X

l

(αlek(hkl)−αlel(hkk)) +fR(N, ek, N, ek)−N(ψ) +λkψ

= (X

l

h²_kl+ R(N, ek, N, ek))f− hV^>,∇hkki −N(ψ)−λkψ.

(3.26) So we have

4f = (kAk²+ Ric(N, N))f− hV^>,∇S1i −nN(ψ)−S1ψ, (3.27)

and X

k

λkfkk = −2X

k

λ²_kψ−X

k,l

λkek(hkl) +X

k

λkh²_klf+X

k

λkαl(ek(hkl)

−el(hkk)) +X

k

f λkR(N, ek, N, ek)−X

k

λkN(ψ) +X

k

λ²_kψ

= −X

k

λ²_kψ+X

k

λ³_kf −X

k,l

αlλkel(λk) +f cS1−S1N(ψ).

(3.28)

Note that (2.9) and (2.10) X

k

λ²_k =S₁²−2S2, X

k

λ³_k =S³₁−3S1S2+ 3S3, substituting (3.27) and (3.28) into (2.13), so we can get

L1(f) = X

i,j

(S1δij−hij)fij=X

i

S1fii−X

i

hiifii

= (S1S2−3S3)f+f(n−1)cS1−(n−1)S1N(ψ)−2S2ψ− hV^>,∇S2i.

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 :Mⁿ⁺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=φ⁰. Now we have the follow corollary

(10)

Corollary 4.1.IfMⁿis a closed spacelike hypersurface having constant normalized 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_I)_∂t^∂ andf =hV, Ni, then

(4.1) L1(f) = (S1S2−3S3)f+f(n−1)cS1+ (n−1)S1φ⁰⁰hN, ∂

∂ti −2S2ψ.

Proof. Since we have

(4.2) ∇φ⁰=−h∇φ⁰, ∂

∂ti∂

∂t =−φ⁰⁰∂

∂t, then

(4.3) N(φ⁰) =hN,∇φ⁰i=−φ⁰⁰h∂

∂t, Ni.

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 Hφ⁰⁰ ≥ max{(R−c)φ⁰,0}, and Mⁿ is stable, then

(I)R=c onM, or

(II)M is spacelike sliceMt0=t0×Fⁿ, for some t0∈I, satisfying Hφ⁰⁰= (R−c)φ⁰.

Proof.Using Proposition 3.5 and Corollary 4.1, we can get J⁰⁰(0)(f) = 2

Z

M

[(n−1)S1φ⁰⁰hN, ∂

∂ti −2S2φ⁰]f dM

= 2 Z

M

[(n−1)S1φ⁰⁰hN, ∂

∂ti −n(n−1)(c−R)φ⁰]φhN, ∂

∂tidM.

Let hN,_∂t^∂i = −coshθ, where θ denotes the hyperbolic angle between the timelike verctor fieldsN and _∂t^∂. SinceMⁿ is stable, so

0 ≥2 Z

M

[(n−1)S1φ⁰⁰hN, ∂

∂ti −n(n−1)(c−R)φ⁰]φhN, ∂

∂tidM

≥2 Z

M

(n−1)S1φφ⁰⁰coshθ(coshθ−1)dM ≥0, and hence

Hφ⁰⁰(coshθ−1) = 0 and Hφ⁰⁰= (R−c)φ⁰ (4.4)

holds onMⁿ. IfR6=candφ⁰6= 0 thenHφ⁰⁰6= 0, and coshθ= 1, soM is an umbilical

leaf satisfyingHφ⁰⁰= (R−c)φ⁰. 2

Acknowledgements. The first author is supported by RFDP((20050141011), NCET(06-0276) and MXDUT073011.

(11)

References

[1] H. Alencar, M. do Carmo and A. G. Colares, Stable hyersurface with constant scalar curvature, Math. Z. 213(1993), 117-131.

[2] J. L. Barbosa and M. do Carmo, Stability of hyersurface with constant mean curvature, Math. Z. 185(1984), 339-353.

[3] J. L. Barbosa, M. do Carmo and J. Eeschenburg, Stability of hyersurface with constant mean curvature in Riemannian manifolds, Math. Z. 197(1988), 123-138.

[4] J. L. Barbosa and A. G. Colares, Stability of hyersurfaces with constant r-mean curvature, Ann. Global Anal. Geom., 15(1997), 277-297.

[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, Aca- demic Press, New York, 1983.

[11] R. C. Reilly,Variational properties of functions of constant mean curvature hy- ersurface in space forms, J. Diff. Geom. 8(1973), 465-477.

[12] H. Rosenberg, Hpyersurfaces of constant curvature in space forms, Bull. Sc.

Math., 2ê 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]