In this paper we investigate spacelike hypersurfaces with constantr-th mean curvature and two distinct principal curvatures in anti-de Sitter spaceHn+11 (c)

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Vol. LXXIX, 2(2010), pp. 165–174

SPACELIKE HYPERSURFACES WITH CONSTANT

r

^{-TH MEAN}

CURVATURE IN ANTI-DE SITTER SPACES

BIAO-GUI YANG and XI-MIN LIU

Abstract. In this paper we investigate spacelike hypersurfaces with constantr-th mean curvature and two distinct principal curvatures in anti-de Sitter spaceHⁿ⁺¹1 (c).

We give some characterizations of the hyperbolic cylinders inHⁿ⁺¹₁ (c).

1. Introduction

LetMⁿ⁺¹₁ (c) be an (n+ 1)-dimensional Lorentzian space form with constant sectional curvaturec and index 1. When c > 0, Mⁿ⁺¹₁ (c) = Sⁿ⁺¹1 (c) is called an (n+ 1)-dimensional de Sitter space; when c = 0, Mⁿ⁺¹₁ (c) = Lⁿ⁺¹ is called an (n+ 1)-dimensional Minkowski space; whenc <0, Mⁿ⁺¹₁ (c) =Hⁿ⁺¹1 (c) is called an (n+ 1)-dimensional anti-de Sitter space. A hypersurface Mⁿ is said to be spacelike if the induced metric onMⁿfrom that of the ambient space is Riemann- ian metric. Spacelike hypersurfaces with constant mean curvature in Lorentzian space forms are very interesting geometrical objects which have been investigated by many geometers.

S. Montiel [4, 5] gave a characterization of hyperbolic cylinders or totally umbil- ical hypersurfaces in the de Sitter space. T. Ishihara [3] studied ann-dimensional (n≥ 2) complete maximal spacelike hypersurface M in the anti-de Sitter space Hⁿ⁺¹1 (−1) and proved the norm square of the second fundamental form ofM satisfies S ≤ n. Moreover, S =n if and only if Mⁿ =H^m(−_mⁿ)×H^n−m(−_n−mⁿ ), (1≤m≤n−1).

Cao and Wei [1] studied maximal spacelike hypersurfaces with two distinct principal curvatures in the anti-de Sitter spaceHⁿ⁺¹1 (−1) and gave a characterization of the hyperbolic cylinders in the anti-de Sitter space:

Theorem 1.1. LetMⁿ be ann-dimensional (n≥3) complete maximal spacelike hypersurface with two distinct principal curvatures λ and µ in an anti-de Sitter spaceHⁿ⁺¹1 (−1). Ifinf(λ−µ)²>0, then Mⁿ=H^m(−_mⁿ)×H^n−m(−_n−mⁿ ), (1≤m≤n−1).

Received January 2, 2009; revised March 23, 2010.

2000Mathematics Subject Classification. Primary 53B30, 53C50.

Key words and phrases. spacelike hypersurface; r-th mean curvature; principal curvature;

isoparametric hypersurface.

This work is supported by NSFC (10771023 and 10931005).

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In [8], we extended the above result to complete spacelike hypersurfaces with constant mean curvature and two distinct principal curvatures in an anti-de Sitter space. In fact, we proved the following result.

Theorem 1.2. Let Mⁿ be ann-dimensional (n≥3) complete spacelike hypersurface with constant mean curvature immersed in an anti-de Sitter spaceHⁿ⁺¹1 (c).

Suppose in addition that Mⁿ has two distinct principal curvaturesλ and µ with the multiplicities(n−1) and1, respectively. Satisfying inf(λ−µ)²>0, thenMⁿ is a hyperbolic cylinderHⁿ⁻¹(c1)×H¹(c2)

In this paper we will investigate spacelike hypersurfaces with constantr-th mean curvature and with two distinct principal curvatures in the anti-de Sitter spaces and obtain the following result

Theorem 1.3. Let Mⁿ be a complete spacelike hypersurface of Hⁿ⁺¹1 (c) for n≥3. Assume thatMⁿ has constantr-th mean curvature and two distinct principal curvatures such that for one of them, the associated space of principal curvature vectors has dimension1. Then:

(i) Hr= 0, and therefore,Mⁿ is an(r−1)-maximal hypersurface, or

(ii) Mⁿ is the locus of a family of moving (n−1)-dimensional submanifolds M₁ⁿ⁻¹(s). The principal curvatureλof multiplicityn−1 is constant along each of the submanifolds M₁ⁿ⁻¹(s). The manifolds M₁ⁿ⁻¹(s)have constant curvature((log|λ^r−H_r|^1/n)⁰)²+c−λ², which does not change sign. Here the parameter s is the arc length of an orthogonal trajectory of the family M₁ⁿ⁻¹(s), and λ=λ(s)satisfies the ordinary second order differential equation

w⁰⁰−w{Hr(Hr+w⁻ⁿ)^2−r^r −n−r

r (Hr+w⁻ⁿ)^2−r^r w⁻ⁿ−c}= 0, (1.1)

or

w⁰⁰−w{H_r(H_r−w⁻ⁿ)^2−r^r +n−r

r (H_r−w⁻ⁿ)^2−r^r w⁻ⁿ−c}= 0, (1.2)

wherew=|λ^r−Hr|⁻¹ⁿ.

In particular, we also study spacelike hypersurfaces with vanishingr-th mean curvature and with two distinct principal curvatures in an anti-de Sitter space and give some characterizations of hyperbolic cylinders.

2. Preliminaries

LetMⁿ be a complete spacelike hypersurface in an anti-de Sitter spaceHⁿ⁺¹1 (c) of constant sectional curvature c < 0. For any p ∈ M, we can choose a local orthonormal frame field e1, . . . , en, en+1 in a neighborhood U of Mⁿ such that e1, . . . , en are tangential toMⁿ and en+1 is normal to Mⁿ. We use the following convention for the indices

1≤A, B, C, D≤n+ 1, 1≤i, j, k, l≤n.

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SPACELIKE HYPERSURFACES IN ANTI-DE SITTER SPACES

Denote{ωA} the corresponding dual coframe and{ωAB} the connection forms of Hⁿ⁺¹1 (c) so that the semi-Riemannian metric and structure equation of Hⁿ⁺¹1 (c) are given, respectively, by

ds²=X

i

ω_i²−ω_n+1² , (2.1)

dω_A=X

i

ω_Ai∧ω_i−ω_An+1∧ω_n+1, ω_AB+ω_BA= 0, (2.2)

dω_AB=X

C

ε_Cω_AC∧ω_CB+ Ω_AB,Ω_AB=−1 2

X

C,D

ε_Cε_DK_ABCDω_C∧ω_D, (2.3)

KABCD=cεAεB(δACδBD−δADδBC).

(2.4)

A well-known argument shows that the formsωin+1may be expressed as

(2.5) ωin+1=X

j

hijωj, hij =hji. The second fundamental form is given by A =P

jh_ijω_i⊗ω_j. Furthermore, the mean curvature is given byH =_n¹P

ihii. The structure equations ofMⁿ are given by

dωi=X

i

ωij∧ωj, ωij+ωji= 0, (2.6)

dωij =X

k

ωik∧ωkj+ Ωij, Ωij =−1 2

X

k,l

Rijklωk∧ωl. (2.7)

Using the structure equation, we can obtain the Gauss equation (2.8) Rijkl =c(δikδjl−δilδjk)−(hikhjl−hilhjk).

The Ricci curvature and the normalized scalar curvature ofM are given, respectively, by

Rij=c(n−1)δij−nHhij+X

k

hikhkj, (2.9)

R= 1 n(n−1)

X

i

R_ii. (2.10)

From (2.9) and (2.10) we obtain

(2.11) n(n−1)(R−c) =−n²H²+S, whereS=|A|²=P

i,jh²_ij is the square of the second fundamental form A.

The Codazzi equation is

(2.12) h_ijk =h_ikj,

where the covariant derivative ofhij is defined by

(2.13) X

k

h_ijkω_k =dh_ij+X

k

h_kjω_ki+X

k

h_ikω_kj.

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Associated to the second fundamental formA ofMⁿ one hasninvariantsSr, 1≤r≤n, given by the equality

det(tI−A) =

n

X

k=0

(−1)^kS_kt^n−k.

Ifp∈M andek is the basis ofTpM formed by eigenvectors of the shape operator Ap, with corresponding eigenvaluesλk, one immediately sees that

Sr=σr(λ1, . . . , λn),

where σ_r ∈ R[x₁, . . . , x_n] is the r-th elementary symmetric polynomial on the indeterminatesx₁, . . . , x_n. Ther-th mean curvature ofM is given by

Hr= 1

n r

Sr. In particular, whenr= 1

H1= 1 n

X

i

λi = 1

nS1=H is nothing but the mean curvature ofM.

A spacelike hypersurface Mⁿ in Lorentzian space forms Mⁿ⁺¹₁ (c) is called r-maximal ifH_r+1≡0.

3. Spacelike hypersurfaces of constantr-th curvature with two distinct principal curvatures

In this section, we will study hypersurfaces with constant r-th mean curvature and with two distinct principal curvatures in the anti-de Sitter space Hⁿ⁺¹1 (c).

Suppose that the multiplicities of the principal curvatures λ and µ are m and n−m, respectively. In the following if there is no special statement, we further adopt the notational convention that indicesa, b, crange from 1 tomand indices α, β, γ from m+ 1 to n. We may choose {eA}1≤A≤n+1 such that hab = λδab, h_αβ=µδ_αβ,h_aα= 0.

Firstly, we state a theorem that can be proved using the method of Otsuki [7].

Theorem 3.1. Let Mⁿ be a spacelike hypersurface with two distinct principal curvatures in Hⁿ⁺¹1 (c) such that the multiplicities of principal curvatures are all constant. Then the distribution of the space of principal vectors corresponding to each principal curvature is completely integrable. In particular, if the multiplicity of a principal curvature is greater than1, then this principal curvature is constant on each integral submanifold of the corresponding distribution of the space of principal vectors.

Proof. We suppose thatMⁿ is a spacelike hypersurface with two distinct principal curvatures inHⁿ⁺¹1 (c) such that the multiplicities of the principal curvatures λandµarem andn−m, respectively. We may choose local orthonormal frame fielde1, . . . , en, en+1 in a neighborhoodU ofMⁿ such that

ωin+1=λiωi, (3.1)

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whereλi are principal curvatures satisfyingλa=λ,λα=µ. Hence we have dωin+1=dλi∧ωi+λidωi =dλi∧ωi+λi

X

j

ωij∧ωj. (3.2)

On the other hand, by means of (2.3), (2.4) and (3.1), we get dω_in+1=X

j

λ_jω_ij∧ω_j onMⁿ. (3.3)

Hence, we get

dλi∧ωi+X

j

(λi−λj)ωij∧ωj= 0.

(3.4)

In particular,

dλ∧ωa+ (λ−µ)X

α

ωaα∧ωα= 0.

(3.5)

Puttingdλ =P

iλ,iωi =P

aλ,aωa+P

αλ,αωα, where λ,i =ei(λ), (3.5) can be written as

X

b

λ_,bω_b∧ω_a+X

α

λ_,αω_α∧ω_a+ (λ−µ)X

α

ω_aα∧ω_α= 0.

(3.6)

It implies thatλ,b= 0 for b6=aand X

α

ωaα∧ωα= 1 λ−µ

X

α

λ,αωa∧ωα. (3.7)

Therefore,

dω_a=X

b

ω_ab∧ω_b+X

α

ω_aα∧ω_α

=X

b

ωab∧ωb+ 1 λ−µ

X

α

λ,αωa∧ωα. (3.8)

This means that

dωa ≡0 mod (ω1, . . . , ωm), (3.9)

for any 1≤a≤m. Therefore, the system of Pfaff equationsωa= 0 (1≤a≤m) is completely integrable. In particular, ifm >1, thenλ,a= 0 for any 1≤a≤m.

Hence, along the integral submanifold corresponding distribution of the space of principal vectors Span{ea}_1≤a≤m,λis a constant.

In the following we separate our discussion into two cases.

Case 1: 2≤m≤n−2. Using Theorem 3.1, we can obtain the following result.

Theorem 3.2. Let Mⁿ (n > 3) be a spacelike hypersurface in Hⁿ⁺¹1 (c) with constant r-th mean curvature and with two nonzero distinct principal curvatures.

If the multiplicitiesm andn−m of the principal curvaturesλand µare greater than1, then we have

(i) Both λ and µ are constants and they satisfy λµ =c. In addition, Mⁿ is locally the hyperbolic cylinder H^m(c1)×H^n−m(c2);

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(ii) IfMⁿis assumed to be complete inHⁿ⁺¹1 (c), thenMⁿ=H^m(c1)×H^n−m(c2), wherec1,c2<0are constants.

Proof. Let us denote the integral submanifold through x∈Mⁿ, corresponding toλandµ, byM₁^m(x) andM₂^n−m(x), respectively. We write

dλ=X

i

λ_,iω_i, dµ=X

i

µ_,iω_i. (3.10)

Then Theorem 3.1 implies

λ,a= 0, µ,α = 0.

(3.11) Since

S_r= n

r

H_r= X

0≤s≤r

m s

n−m r−s

λ^sµ^r−s, (3.12)

from (3.11) and (3.12), we have X

1≤s≤r

s m

s

n−m r−s

λ^s−1µ^r−sλ,α = 0.

(3.13)

For some α, if p∈ M satisfying λ_,α(p)6= 0 exists, then the open set U ={p∈ M |λ_,α(p)6= 0} is nonempty. From (3.13), it implies

(3.14) X

1≤s≤r

s m

s

n−m r−s

λ^s−1µ^r−s= 0 onU. Similarly, from (3.14) it implies inductivelyr! ^m_r

= 0 onU form≥rwhich is a contradiction, or

(3.15) m!

n−m r−m

µ^r−m= 0 onU m < r.

Thusµ≡0 onU by (3.15) which is a contradiction with λµ6= 0. Therefore, for anyα, we haveλ,α≡0. This meansλis a nonzero constant. It can force thatµ is a constant from (3.12). Hence,Mⁿis an isoparametric hypersurface ofHⁿ⁺¹(1) with two distinct principal curvatures. Therefore, we can obtain our result from

[3, Theorem 1].

Case 2. m =n−1, m ≥2. If Hr = 0, thenMⁿ is a (r−1)-maximal. In the following we assume H_r 6= 0 from Theorem 3.1 and S_r = ⁿ_r

H_r = ⁿ⁻¹_r λ^r+

n−1 r−1

λ^r−1µ6= 0 is a nonzero constant, we haveλ6= 0 and λ,a= 0, µ,a= 0,

(3.16)

µ=nH_r−(n−r)λ^r

rλ^r−1 , λ−µ= n(λ^r−H_r) rλ^r−1 6= 0.

(3.17)

From (2.13) and (3.16), we can obtain X

k

habkωk=dhab+X

k

hkbωka+X

k

hakωkb

=dhab=δabdλ=δabλ,nωn,

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and

X

k

hnnkωk= dhnn=dµ=µ,nωn. It follows that

h_abc= 0, h_abn=δ_abλ_,n; h_nna= 0, h_nnn=µ_,n. (3.18)

Therefore, combining (2.12), (2.13) and (3.18), we have ω_an= λ,n

λ−µω_a. (3.19)

Thus, using structure equation (2.6), we havedωn =P

aωna∧ωa = 0.For this reason, we may putωn=ds, wheresis the arc length of an orthogonal trajectory of the family of the integral submanifoldsM₁ⁿ⁻¹(s) corresponding toλ. Then for λ=λ(s), we haveλ,n=λ⁰(s).

Combining (3.17) and (3.19), we get ω_an= λ_,n

λ−µω_a = rλ^r−1λ_,n

n(λ^r−Hr)ω_a= (log|λ^r−H_r|ⁿ¹)⁰ω_a=−w⁰ wω_a, (3.20)

wherew=|λ^r−H_r|⁻ⁿ¹ >0.

Using this formula, on the one hand, from (2.5), (2.6) and (2.7), we have dωa = X

b

ωab∧ωb+ωan∧ωn =X

b

ωab∧ωb−w⁰

wωa∧ωn, and

dω_an=X

b

ω_ab∧ω_bn−R_ananω_a∧ω_n

=−w⁰ w

X

b

ωab∧ωb+ (λµ−c)ωa∧ωn. (3.21)

On the other hand,

dω_an=d(−w⁰ wω_a)

=w⁰⁰w−w⁰²

w² ds∧ωa−w⁰ wdωa

=w⁰⁰

w ω_a∧ω_n−w⁰ w

X

b

ω_ab∧ω_b. (3.22)

By comparing (3.21) and (3.22), we obtain w⁰⁰−w(λµ−c) = 0.

(3.23)

Integrating (3.23), we have

w⁰²=w²[(Hr±w⁻ⁿ)²^r −c] +C, (3.24)

whereC is an integration constant.

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Similarly, we have dωab−X

c

ωac∧ωcb=ωan∧ωnb−Rababωa∧ωb

=− (

w⁰ w

2

−(λ²−c) )

ωa∧ωb. (3.25)

Therefore we see thatM₁ⁿ⁻¹(s) is of constant sectional curvature

w⁰ w

2

+c−λ² which has the same sign for alls.

We can choose the orthonormal frame{x;e1, . . . , en, en+1, en+2} in Rⁿ⁺²2 with e_n+2=xand put

N =e1∧. . .∧en−1∧Z, Z=−w⁰

wen−λen+1+en+2. (3.26)

Then, using structure equations, by straightforward calculation we can show that hZ, Zi=

w⁰ w

2

+c−λ², dZ =−w⁰

wZω_n, dN =−w⁰ wN ω_n. (3.27)

From (3.27) it shows that the n-vector N in Rⁿ⁺²2 is constant along M₁ⁿ⁻¹(s).

Hence, there exists ann-dimensional linear spaceEⁿ(s) inRⁿ⁺²2 containingM₁ⁿ⁻¹(s).

Moreover,n-vectorN depends only onsand integration gives N = w(s0)

w N(s₀).

(3.28)

Thus, we can see thatEⁿ(s) is parallel to Eⁿ(s0).

Hence, we have the following result.

Theorem 3.3. Let Mⁿ a complete spacelike hypersurface of Hⁿ⁺¹1 (c) for n≥3. Assume thatMⁿ has constantr-th mean curvature and two distinct principal curvatures such that for one of them, the associated space of principal curvature vectors has dimension1. Then:

(i) Hr= 0, and therefore,Mⁿ is an(r−1)-maximal hypersurface, or

(ii) Mⁿ is the locus of a family of moving (n−1)-dimensional submanifolds M₁ⁿ⁻¹(s). The principal curvatureλof multiplicityn−1 is constant along each of the submanifolds M₁ⁿ⁻¹(s). The manifolds M₁ⁿ⁻¹(s)have constant curvature((log|λ^r−H_r|^1/n)⁰)²+c−λ², which does not change sign. Here the parameter s is the arc length of an orthogonal trajectory of the family M₁ⁿ⁻¹(s), and λ=λ(s)satisfies the ordinary second order differential equation (1.1)or (1.2)wherew=w=|λ^r−Hr|⁻ⁿ¹.

Corollary 3.4. In Hⁿ⁺1 (c), there exist infinitely many spacelike hypersurfaces with constantr-th mean curvature that are not congruent to each other.

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4. (r−1)-maximal spacelike hypersurfaces in the anti-de Sitter spaces

In this section, we will investigate spacelike hypersurfaces with vanishing r-th mean curvature in the anti-de Sitter spaces which is called (r−1)-Maximal spacelike hypersurfaces. Let Mⁿ be an (r−1)-maximal spacelike hypersurface with two distinct principal curvatures in the anti-de Sitter spaceHⁿ⁺¹1 (c). In addition suppose the multiplicities of the principal curvaturesλ andµ arem and n−m, respectively.

Firstly, using Theorem 3.2, it is easy to prove the following result.

Proposition 4.1. LetMⁿ (n >3)be an(r−1)-maximal spacelike hypersurface inHⁿ⁺¹1 (c)with two distinct principal curvatures. Suppose the multiplicitiesmand n−mof the principal curvaturesλand µare greater than1, then we have:

(i) If λµ= 0, thenR≤c;

(ii) If λµ6= 0, thenMⁿ is locally the hyperbolic cylinderH^m(c1)×H^n−m(c2).

Proof. (i) Since λµ= 0, thenλ= 0 or µ= 0. Without loss of generality, we supposeµ= 0, soλ6= 0. From (2.11), we have

n(n−1)(R−c) =−n²H²+S= (m−m²)λ²≤0.

It can implyR≤c. Moreover, in view of S_r=

n r

H_r= X

0≤s≤r

m s

n−m r−s

λ^sµ^r−s= 0, thenm < r.

(ii) This is a direct result from Theorem 3.2.

Corollary 4.2. Let Mⁿ (n >3) be an(r−1)-maximal spacelike hypersurface inHⁿ⁺¹1 (c)with two distinct principal curvatures. If the multiplicitiesmandn−m of the principal curvaturesλand µare greater than r−1 (4 ≤2r≤n), then we haveλµ6= 0 andMⁿ is locally the hyperbolic cylinder H^m(c1)×H^n−m(c2).

Secondly, we proved the following result in [9].

Proposition 4.3 ([9]). Let M be an n-dimensional (n≥3) (r−1)-maximal spacelike hypersurface immersed in an anti-de Sitter space Hⁿ⁺¹1 (c). Suppose in addition thatM has two distinct principal curvaturesλandµwith the multiplici- tiesn−1 and1, respectively. Then we have:

(i) If λ≡0, thenR=cand therefore, Mⁿ is1-maximal, or (ii) If inf(λ−µ)²>0, then

S ≥n(r²−2r+n) r(n−r) , (4.1)

and S = ^n(r_r(n−r)²^−2r+n) if and only if M is a hyperbolic cylinder Hⁿ⁻¹(c1)× H¹(c2).

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Biao-Gui Yang, School of Mathematics and Computer Sciences, Fujian Normal University, Fuzhou 350108, People’s Republic China,

e-mail:[email protected]

Xi-Min Liu, Department of Mathematics, South China University of Technology, Guangzhou, 510641, People’s Republic China,

e-mail:[email protected]