On the complete linear Weingarten spacelike hypersurfaces with two distinct principal curvatures in Lorentzian space forms

J. Math. Anal. Appl. 418 (2014) 248–263

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On the complete linear Weingarten spacelike hypersurfaces with two distinct principal curvatures in Lorentzian space forms José N. Gomes a , Henrique F. de Lima b,∗ , Fábio R. dos Santos b , Marco Antonio L. Velásquez b a Departamento de Matemática, Universidade Federal do Amazonas, 69.077-070 Manaus, Amazonas, Brazil b Departamento de Matemática, Universidade Federal de Campina Grande, 58.429-970 Campina Grande, Paraíba, Brazil

a r t i c l e

i n f o

Article history: Received 27 December 2013 Available online 3 April 2014 Submitted by H.R. Parks Keywords: Lorentzian space forms Mean curvature Normalized scalar curvature Complete spacelike hypersurfaces Isoparametric hypersurfaces

a b s t r a c t We deal with complete linear Weingarten spacelike hypersurfaces immersed in a Lorentzian space form, having two distinct principal curvatures. In this setting, we show that such a spacelike hypersurface must be isometric to a certain isoparametric hypersurface of the ambient space, under suitable restrictions on the values of the mean curvature and of the norm of the traceless part of its second fundamental form. Our approach is based on the use of a Simons type formula related to an appropriated Cheng–Yau modified operator jointly with some generalized maximum principles. © 2014 Elsevier Inc. All rights reserved.

1. Introduction Let Ln+1 be an (n + 1)-dimensional Lorentz space, that is, a semi-Riemannian manifold of index 1. When 1 the Lorentz space Ln+1 is simply connected and has constant sectional curvature c, it is called a Lorentz 1 space form and we will denote it by Ln+1 (c). The Lorentz–Minkowski space Ln+1 , the de Sitter space Sn+1 1 1 n+1 and the anti-de Sitter space H1 are the standard Lorentz space forms of constant sectional curvature 0, 1 and −1, respectively. We also recall that a hypersurface M n immersed in a Lorentz space Ln+1 is said to 1 be spacelike if the metric on M n induced from that of the ambient space Ln+1 is positive definite. 1 The last few decades have seen a steadily growing interest in the study of spacelike hypersurfaces of Lorentz manifolds. Apart from physical motivations, from the mathematical point of view this is mostly due to the fact that such hypersurfaces exhibit nice Bernstein-type properties, and one can truly say that the first remarkable results in this direction were the rigidity theorems of Calabi in [4] and Cheng and Yau * Corresponding author. E-mail addresses: [email protected] (J.N. Gomes), [email protected] (H.F. de Lima), [email protected] (F.R. dos Santos), [email protected] (M.A.L. Velásquez). http://dx.doi.org/10.1016/j.jmaa.2014.03.090 0022-247X/© 2014 Elsevier Inc. All rights reserved.

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in [12], who showed (the former for n  4, and the latter for general n) that the only maximal (that is, with zero mean curvature) complete, noncompact, spacelike hypersurfaces of the Minkowski space Ln+1 are the spacelike hyperplanes. As for the case of the de Sitter space, Goddard [15] conjectured that every complete spacelike hypersurface should be totally umbilical. Although the conjecture turned out to with constant mean curvature H in Sn+1 1 be false in its original statement, it motivated a great deal of work of several authors trying to find a positive answer to the conjecture under appropriate additional hypotheses. For instance, in [2] Akutagawa showed that Goddard’s conjecture is true when H 2  1 in the case n = 2, and when H 2 < 4(n − 1)/n2 in the case n  3. Afterwards, Montiel [20] solved Goddard’s problem in the compact case proving that the only closed with constant mean curvature are the totally umbilical hypersurfaces. spacelike hypersurfaces in Sn+1 1 Furthermore, Montiel also exhibited in [20] examples of complete spacelike hypersurfaces in Sn+1 with 1 4(n−1) 2 constant H satisfying H  n2 and being non-totally umbilical, the so-called hyperbolic cylinders, which are isometric to the Riemannian product Sn−1 (c1 ) × H1 (c2 ), where c1 > 0, c2 < 0 and c11 + c12 = 1. Later, Montiel [21] characterized such hyperbolic cylinders as the only complete noncompact spacelike √ 2 n−1 n+1 with constant mean curvature H = and having at least two ends. In [3], hypersurfaces in S1 n Brasil, Colares and Palmas obtained extensions of Montiel’s result, showing that the hyperbolic cylinders with constant mean curvature, nonnegative Ricci are the only complete spacelike hypersurfaces in Sn+1 1 curvature and having at least two ends. They also characterized all complete spacelike hypersurfaces of constant mean curvature with two distinct principal curvatures as rotation hypersurfaces or generalized hyperbolic cylinders Sn−p (c1 ) × Hp (c2 ). Another Goddard-like problem is to study complete spacelike hypersurfaces immersed in a Lorentz space with with constant scalar curvature. In [26], Zheng proved that a compact spacelike hypersurface in Sn+1 1 constant normalized scalar curvature R, R < 1 and nonnegative sectional curvatures is totally umbilical. Later, Cheng and Ishikawa [11] showed that Zheng’s result in [26] is also true without additional assumptions on the sectional curvatures of the hypersurface. When M n is a complete spacelike hypersurface immersed , n  3, with bounded mean curvature H and constant normalized scalar curvature R, as an in Sn+1 1 application of the so-called generalized maximum principle of Omori [22], Camargo, Chaves and Sousa [5] n extended a technique due to Cheng and Yau [13] to show that if n−2 n  R  1, then M is totally umbilical. Moreover, they also proved that if R  1 and the square of the length of the second fundamental form B √ of M n satisfies |B|2 < 2 n − 1, then M n is totally umbilical. Proceeding with this branch, an interesting question is to describe the spacelike hypersurfaces immersed in a Lorentzian space with constant scalar curvature and with two distinct principal curvatures. For inwith stance, Hu, Scherfner and Zhai in [17], and Shu in [24] proved that the spacelike hypersurface in Sn+1 1 constant scalar curvature and with two distinct principal curvatures whose multiplicities are greater than 1 is isometric to a generalized hyperbolic cylinder. Also in [17], Hu, Scherfner and Zhai investigated the similar problem for the case of the multiplicity of one of the principal curvatures is n − 1. Furthermore, in [19] Liu and Gao treated this problem in the context of the higher order mean curvatures. When the ambient space is the anti-de Sitter space, Cao and Wei [7] showed that, if n  3, then every n-dimensional complete with exactly two principal curvatures everywhere is isometric to maximal spacelike hypersurface in Hn+1 1 some hyperbolic cylinder under an additional condition on these curvatures. Later, Perdomo [23] studied the 2-dimensional case and constructed new examples of complete maximal surfaces in H31 . More recently, with either conChaves, Sousa and Valério [10] studied complete maximal spacelike hypersurfaces in Hn+1 1 stant scalar curvature or constant non-zero Gauss–Kronecker curvature. In this setting, they characterized the hyperbolic cylinders as the only such hypersurfaces with (n − 1) principal curvatures with the same sign everywhere. Meanwhile, Hou and Yang [16] have extended the ideas of Li, Suh and Wei [18] in order to get charac, that is, spacelike hypersurfaces terization results for linear Weingarten spacelike hypersurfaces into Sn+1 1 n+1 of S1 whose mean and scalar curvatures are linearly related. In [14], the second and fourth authors applied

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a suitable extension of a maximum principle at the infinity of Yau [25] due to Caminha in [6], in order to establish another characterizations results concerning complete noncompact linear Weingarten spacelike hypersurfaces in Sn+1 . Furthermore, Chao [9] used the generalized maximum principle of Omori to obtain a 1 rigidity theorem also related to complete linear Weingarten spacelike hypersurfaces in Sn+1 , which extends 1 the main result of [5]. Here, motivated by the works above described, our aim is to study complete linear Weingarten spacelike (c), n  2, having two distinct principal curvatures. hypersurfaces immersed in a Lorentzian space form Ln+1 1 For this, in Section 2, we develop a suitable Simons type formula. Afterwards, in Section 3, we establish some key lemmas which constitute our analytical and algebraic machineries. In particular, we use our Simons type formula to get an appropriated lower estimate related to a Cheng–Yau modified operator acting on the mean curvature of a linear Weingarten spacelike hypersurface (cf. Lemma 4). Finally, Section 4 is devoted to present our characterization results concerning such spacelike hypersurfaces (cf. Theorems 1, 2, 3 and 4, when the ambient space is the de Sitter space Sn+1 , and Theorem 5, when the ambient space is either the 1 n+1 Lorentz–Minkowski space L or the anti-de Sitter space Hn+1 ). 1 2. Preliminaries In this section we will introduce some basic facts and notations that will appear along the paper. In what follows, we will suppose that all considered spacelike hypersurfaces are connected. Let M n be an n-dimensional spacelike hypersurface immersed in a Lorentz space form Ln+1 (c), n  2, 1 with constant sectional curvature c ∈ {−1, 0, 1}. We choose a local field of semi-Riemannian orthonormal frame {e1 , . . . , en+1 } in Ln+1 (c), with dual coframe {ω1 , . . . , ωn+1 }, such that, at each point of M n , e1 , . . . , en 1 n are tangent to M and en+1 is normal to M n . We will use the following convention for the indices: 1  A, B, C, . . .  n + 1,

1  i, j, k, . . .  n.

In this setting, the Lorentz metric of Ln+1 (c) is given by 1 ds2 =



2 A ω A =



2 ωi2 − ωn+1 ,

i

A

where i = 1 and n+1 = −1. Denoting by {ωAB } the connection forms of Ln+1 (c), we have that the 1 (c) are given by: structure equations of Ln+1 1 dωA =



B ωAB ∧ ωB ,

ωAB + ωBA = 0,

(2.1)

C ωAC ∧ ωCB −

1 KABCD ωC ∧ ωD , 2

(2.2)

B

dωAB =

 C

C,D

where KABCD = A B c(δAC δBD − δAD δBC ). Next, we restrict all the tensors to M n . First of all, ωn+1 = 0 on M n ; so by Cartan’s Lemma [8] we can write ωn+1i =

 j

hij ωj ,

hij = hji .

 i

ωn+1i ∧ ωi = dωn+1 = 0 and

(2.3)

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 This gives the second fundamental form of M n , B = ij hij ωi ωj en+1 . Furthermore, the mean curvature H  of M n is defined by H = n1 i hii . The structure equations of M n are given by dωi =



ωij ∧ ωj ,

ωij + ωji = 0,

(2.4)

ωik ∧ ωkj −

1 Rijkl ωk ∧ ωl , 2

(2.5)

j

dωij =

 k

k,l

where Rijkl are the components of the curvature tensor of M n . Moreover, using the previous structure equations, we obtain the Gauss equation Rijkl = c(δik δjl − δil δjk ) − (hik hjl − hil hjk ).

(2.6)

The Ricci curvature and the normalized scalar curvature of M n are given, respectively, by Rij = c(n − 1)δij − nHhij +



hik hkj

(2.7)

k

and R=

 1 Rii . n(n − 1) i

(2.8)

From (2.7) and (2.8) we obtain |B|2 = n2 H 2 + n(n − 1)(R − c),

(2.9)

 where |B|2 = i,j h2ij is the square of the length of the second fundamental form B of M n . The components hijk of the covariant derivative ∇B satisfy 

hijk ωk = dhij +



k



hik ωkj +

k

hjk ωki .

(2.10)

k

The Codazzi equation and the Ricci identity are, respectively, given by hijk = hikj

(2.11)

and 

hijkl − hijlk =

hmj Rmikl +

m



him Rmjkl ,

(2.12)

m

where hijk and hijkl denote the first and the second covariant derivatives of hij .  The Laplacian Δhij of hij is defined by Δhij = k hijkk . From Eqs. (2.11) and (2.12), we obtain that Δhij =



hkkij +



k

hkl Rlijk +

k,l



hli Rlkjk .

k,l

Since 2



Δ|B| = 2

i,j

hij Δhij +

 i,j,k

 h2ijk

,

(2.13)

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from (2.13) we get    1 Δ|B|2 = |∇B|2 + hij hkkij + hij hlk Rlijk + hij hil Rlkjk . 2 i,i,k

i,j,k,l

(2.14)

i,j,k,l

Consequently, taking a (local) orthonormal frame {e1 , . . . , en } on M n such that hij = λi δij , from Eq. (2.14) we obtain the following Simons type formula  1 1 Δ|B|2 = |∇B|2 + λi (nH),ii + Rijij (λi − λj )2 . 2 2 i i,j

(2.15)

3. Key lemmas In order to establish our next results, it will be necessary to use some key lemmas. The first one is an extension of Lemma 2.2 of [16] (see also Lemma 2.1 of [9] or Lemma 3.1 of [14]). Lemma 1. Let M n be a linear Weingarten spacelike hypersurface immersed in a Lorentzian space form (c), such that R = aH + b for some a, b ∈ R. Suppose that b = c and Ln+1 1 (n − 1)a2 + 4n(c − b)  0.

(3.1)

|∇B|2  n2 |∇H|2 .

(3.2)

Then,

Moreover, if the inequality (3.1) is strict and the equality holds in (3.2) on M n , then H is constant on M n . Proof. Since we are supposing that R = aH + b, from Eq. (2.9) we get 2



  hij hijk = 2n2 H + n(n − 1)a H,k .

i,j

Thus, 4

 k

2 hij hijk

 2 = 2n2 H + n(n − 1)a |∇H|2 .

i,j

Consequently, using Cauchy–Schwartz inequality, we obtain that 4|B| |∇B| = 4 2

2



h2ij

 

i,j

4

 k

 h2ijk

i,j,k

2

hij hijk

i,j

 2 = 2n2 H + n(n − 1)a |∇H|2 .

(3.3)

On the other hand, since R = aH + b, from Eq. (2.9) we easily see that 

2n2 H + n(n − 1)a

2

  = n2 (n − 1) (n − 1)a2 + 4n(c − b) + 4n2 |B|2 .

(3.4)

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Thus, from (3.3) we have   4|B|2 |∇B|2  n2 (n − 1) (n − 1)a2 + 4n(c − b) |∇H|2 + 4n2 |B|2 |∇H|2 .

(3.5)

Therefore, since (n − 1)a2 + 4n(c − b)  0, we obtain |B|2 |∇B|2  |B|2 n2 |∇H|2 . On the other hand, if there exists p ∈ M n such that |B|(p) = 0, then H(p) = 0. But, taking into account (3.4), such a situation cannot occur when b = c. Thus, |∇B|2  n2 |∇H|2 . Moreover, if (n − 1)a2 + 4n(c − b) > 0 and if assume in addition that the equality holds in (3.2) on M n , then from (3.5) we conclude that H is constant. 2 Now, let Ψ =

n i,j=1

ψij ωi ⊗ ωj be the symmetric tensor on M n defined by ψij = nHδij − hij .

In [13], Cheng and Yau have introduced an operator  associated to Ψ acting on any smooth function f , which is defined by f =

n 

ψij fij =

i,j=1

n 

(nHδij − hij )fij ,

(3.6)

i,j=1

where fij stands for a component of the Hessian of f . Here, in order to study linear Weingarten hypersurfaces, we will consider, for each a ∈ R, an appropriated Cheng–Yau’s modified operator, given by L=+

n−1 aΔ. 2

(3.7)

We can reason as in Lemma 2.1 of [16] to get the following sufficient criteria of ellipticity for the operator L. Lemma 2. Let M n be a linear Weingarten spacelike hypersurface immersed in a Lorentzian space form (c), such that R = aH + b with b < c. Then, H does not vanish on M n and L is elliptic. Ln+1 1 Proof. From Eq. (3.4), since b < c, we easily see that H cannot vanish on M n and, by choosing the appropriate Gauss mapping, we may assume that H > 0 on M n . Thus, from Eq. (2.9) we get that a=

 2  1 |B| − n2 H 2 + n(n − 1)(c − b) . n(n − 1)H

Consequently, for every i ∈ {1, . . . , n}, with a straightforward algebraic computation we verify that nH − λi +

 n−1 1  2 a = nH − λi + |B| − n2 H 2 + n(n − 1)(c − b) 2 2nH    1 2 2 = (nH − λi ) + λj + n(n − 1)(c − b) . 2nH j=i

Therefore, since b < c, we conclude that L is elliptic.

2

The next lemma, which guarantees the existence of an Omori-type sequence related to the operator L, is an extension of Proposition 2.3 of [9].

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Lemma 3. Let M n be a complete linear Weingarten spacelike hypersurface in a Lorentzian space form Ln+1 (c), such that R = aH + b for some a, b ∈ R, with a  0. Suppose that (3.1) is valid. If H is bounded 1 on M n , then there is a sequence of points {qk }k∈N ⊂ M n such that lim nH(qk ) = sup nH,

k→+∞

M

  lim ∇nH(qk ) = 0

k→+∞

and   lim sup L nH(qk )  0. k→∞

Proof. Let us choose a local orthonormal frame {e1 , . . . , en } on M n such that hij = λi δij . From (3.7) we have that   n−1 nH + a − λi (nH)ii . L(nH) = (3.8) 2 i On the other hand, we observe that, if H vanishes identically on M n , the lemma is obvious. So, let us suppose that H is not identically zero. By changing the orientation of M n if necessary, we may assume sup H > 0. Thus, for all i = 1, . . . , n, from (2.9) and with a straightforward computation we get (λi )2  |B|2 = n2 H 2 + n(n − 1)(aH + b − c) 2   n−1 n − 1 a − (n − 1)a2 + 4n(c − b) = nH + 2 4  2 n−1  nH + a , 2 where we have used our assumption that (n−1)a2 +4n(c−b)  0 to obtain the last inequality. Consequently, for all i = 1, . . . , n, we have    n − 1   (3.9) a. |λi |  nH + 2 Thus, from (2.6) and (3.9) we obtain, for i = j, 2 n−1 a . = (c − λi λj )  c − nH + 2 

Rijij

(3.10)

Hence, since we are supposing that H is bounded on M n , it follows from (3.10) that the sectional curvatures of M n are bounded from below. Therefore, we may apply the well known generalized maximum principle of Omori [22] to the function nH, obtaining a sequence of points {qk }k1 in M n such that lim nH(qk ) = n sup H,

k→∞

  lim ∇nH(qk ) = 0,

k→∞

lim sup k→∞



(nH)ii (qk )  0.

(3.11)

i

Since supM H > 0, taking subsequences if necessary, we can arrive to a sequence {qk }k1 in M n which satisfies (3.11) and such that H(qk )  0. Moreover, H is bounded, hence we infer from (3.9) that nH(qk ) + n−1 n 2 a − λi (qk ) is nonnegative and bounded on M . Therefore, from (3.8) and the last inequality in (3.11), we obtain that  

   n−1 a − λi (qk )(nH)ii (qk )  0. lim sup L(nH)(qk )  lim sup nH + 2 2 k→∞ k→∞ i

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We set Φij = hij − Hδij and consider the following symmetric tensor n 

Φ=

Φij ωi ⊗ ωj .

i,j=1

n Let |Φ|2 = i,j=1 Φ2ij be the square of the length of Φ. It is easy to check that Φ is traceless and, from (2.9), we get the following relation |Φ|2 = |B|2 − nH 2 = n(n − 1)H 2 + n(n − 1)(R − c).

(3.12)

Proceeding, we establish a lower estimate for the operator L applied on the mean curvature of a linear Weingarten spacelike hypersurface. Such an estimate will be essential in the proofs of our results in the next section. Lemma 4. Let M n be a linear Weingarten spacelike hypersurface in a Lorentzian space form Ln+1 (c), such 1 that R = aH + b for some a, b ∈ R, having two distinct principal curvatures with multiplicity p and n − p, where 1  p  n2 . If (3.1) is valid and b = c, then   L(nH)  |Φ|2 PH,p,c |Φ| ,

(3.13)

where   n(n − 2p) PH,p,c (x) = x2 −  |H|x − n H 2 − c . np(n − p) Proof. Let us choose a (local) orthonormal frame {ei }1in on M n such that hij = λi δij and Φij = μi δij . Since M n has two distinct principal curvatures with multiplicity p and n − p, then there are μ and ν such that ⎧ μ1 = · · · = μp = μ, ⎪ ⎪ ⎪ ⎨μ = · · · = μ = ν, p+1

n

λ1 = · · · = λp = μ + H, ⎪ ⎪ ⎪ ⎩ λp+1 = · · · = λn = ν + H. Then, with a straightforward computation we obtain 0=

n 

μi = pμ + (n − p)ν,

(3.14)

i=1

|Φ| = 2

n 

μ2i = pμ2 + (n − p)ν 2

(3.15)

i=1

and n    tr Φ3 = μ3i = pμ3 + (n − p)ν 3 .

(3.16)

i=1

From (3.14) and (3.15), we also get n−p ν, μ=− p

 ν=±

p |Φ|. n(n − p)

(3.17)

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Thus, from (3.16) and (3.17) we have that n    tr Φ3 = μ3i =

 (n − p) −

i=1

= n(n − p)(2p − n)

 (n − p)3 3 ν p2

ν3 (2p − n) = ± |Φ|3 . 2 p pn(n − p)

(3.18)

Now, setting f = nH in (3.6) and taking a local frame field {e1 , . . . , en } on M n such that hij = λi δij , from Eq. (2.9) we obtain the following: (nH) = nHΔ(nH) −

n 

λi (nH),ii

i=1

=

n  1 Δ(nH)2 − n2 |∇H|2 − λi (nH),ii 2 i=1

=

n  1 n(n − 1) Δ|B|2 − n2 |∇H|2 − ΔR − λi (nH),ii . 2 2 i=1

Consequently, taking into account Eq. (2.15), we get (nH) = |∇B|2 − n2 |∇H|2 −

n 1  n(n − 1) ΔR + Rijij (λi − λj )2 . 2 2 i,j=1

(3.19)

Hence, from (3.19), (3.7) and R = aH + b we have that L(nH) = |∇B|2 − n2 |∇H|2 +

n 1  Rijij (λi − λj )2 . 2 i,j=1

(3.20)

Since (3.1) is valid and b = c, we can use Lemma 1 and from (3.20) we deduce that L(nH) 

n 1  Rijij (λi − λj )2 . 2 i,j=1

(3.21)

Thus, replacing the Gauss equation Rijij = (c − λi λj )(1 − δij ) in (3.21) we get    L(nH)  nc |B|2 − nH 2 + |B|4 − nH λ3i 

 nc|Φ|2 + |Φ|2 + nH

 2 2

i

− nH



λ3i .

i

On the other hand, since Φij = μi δij = λi δij − Hδij , we have that 

μ3i =

i



(λi − H)3 =

i

=

 i



λ3i − 3H|B|2 + 3nH 3 − nH 3

i

λ3i

− 3H|Φ| − nH 3 . 2

(3.22)

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That is, 

λ3i =

i



μ3i + 3H|Φ|2 + nH 3 .

(3.23)

i

Consequently, from (3.22) and (3.23) we get L(nH)  −nH

n 

  μ3i + |Φ|2 |Φ|2 − nH 2 + nc

i=1

  n       3 μi  + |Φ|2 |Φ|2 − nH 2 + nc .  −nH  

(3.24)

i=1

Therefore, from (3.18) and (3.24), we obtain   n − 2p L(nH)  −n|H|  |Φ|3 + |Φ|2 |Φ|2 − nH 2 + nc np(n − p)     n(n − 2p) = |Φ|2 |Φ|2 −  |H||Φ| − n H 2 − c . np(n − p)

2

We will also use the following result obtained by Caminha (Proposition 2.1 of [6]), which extends a result of Yau [25] on a version of Stokes theorem for an n-dimensional, complete noncompact Riemannian manifold. In what follows, let L1 (M ) denote the space of Lebesgue integrable functions on M n . Lemma 5. Let X be a smooth vector field on the n-dimensional complete noncompact oriented Riemannian manifold M n , such that divM X does not change sign on M n . If |X| ∈ L1 (M ), then divM X = 0. 4. Characterizing linear Weingarten hypersurfaces in Ln+1 (c) 1 In this section, we will use the analytical and algebraic machineries of Section 3 to characterize complete linear Weingarten spacelike hypersurfaces immersed in a Lorentzian space form. Initially, we will consider . the case when the ambient space is the de Sitter space Sn+1 1 Theorem 1. Let M n be a complete linear Weingarten spacelike hypersurface in Sn+1 , n  3, such that 1 R = aH + b, and having two distinct principal curvatures with multiplicity p and n − p, where 1  p < n2 . Suppose that H 2  4p(n−p) . If one of the following conditions is satisfied: n2 (i) b < 1 and H attains a maximum on M n ; (ii) (n − 1)a2 + 4n(1 − b) > 0, b = 1 and |∇H| ∈ L1 (M ), then M n is isometric to Sn−p (c1 ) × Hp (c2 ), with c1 =

n−2p n−p

and c2 = − n−2p p .

Proof. From Lemma 4, we get   L(nH)  |Φ|2 PH,p,1 |Φ| ,

(4.1)

  n(n − 2p) PH,p,1 (x) = x2 −  |H|x − n H 2 − 1 . np(n − p)

(4.2)

where

Moreover, from our restriction on H, we have that PH,p,1 (|Φ|)  0.

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Assume, firstly, that b < 1 and H attains a maximum on M n . From Lemma 2 we conclude that H is constant. Thus, from (3.20) and (4.1) we get 0 = L(nH) = |∇B|2 − n2 |∇H|2 +

n 1  Rijij (λi − λj )2 2 i,j=1

n   1   Rijij (λi − λj )2  |Φ|2 PH,p,1 |Φ|  0. 2 i,j=1

Hence, we have |∇B|2 = n2 |∇H|2 = 0, that is, M n is an isoparametric hypersurface of Sn1 . Suppose that (n − 1)a2 + 4n(1 − b) > 0 and |∇H| ∈ L1 (M ). From (3.6) and (3.7), it is not difficult to verify that   L(nH) = divM P (∇H) ,

(4.3)

where P = (n2 H + n(n−1) a)I − nB and I denotes the identity in the algebra of smooth vector fields on M n . 2 Moreover, since R = aH + b and H is bounded on M n , from Eq. (2.9) we have that B is bounded on M n . Consequently, the operator P is bounded and, since we are also assuming that |∇H| ∈ L1 (M ), we obtain that   P (∇H) ∈ L1 (M ).

(4.4)

So, taking into account once more (4.1), from Lemma 5 we obtain that L(nH) = 0 on M n . Thus, from (3.20) we obtain |∇B|2 = n2 |∇H|2 . Hence, in this case, from Lemma 1 we guarantee that H is constant. Consequently, we also conclude that M n must be an isoparametric hypersurface of Sn1 . Now, from (4.1) we deduce that PH,p,1 (|Φ|) = 0. Hence, with a direct computation we verify that (up to a choice of orientation)  2 p(n − p) H= n

and |Φ| =

n − 2p √ . n

On the other hand, from a classical congruence theorem due to Abe, Koike and Yamaguchi (cf. [1, Theorem 5.1]) we conclude that M n must be isometric to Sn−k (c1 ) × Hk (c2 ), where k ∈ {p, n − p}, c1 > 0, c2 < 0 and c11 + c12 = 1. Moreover, it is not difficult to see that, for a suitable choice of the normal vector field, we have that the principal curvatures of Sn−k (c1 ) × Hk (c2 ) → Sn+1 are given by 1 λ1 = · · · = λn−k =

√

1 − c1

and λn−k+1 = · · · = λn =

√

1 − c2 .

Consequently, √ √ nH = (n − k) 1 − c1 + k 1 − c2

(4.5)

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and |B|2 = (n − k)(1 − c1 ) + k(1 − c2 ).

(4.6)

Thus, inserting (4.5) and (4.6) into (3.12), with a straightforward computation we get |Φ|2 =

  2 n (n − 2k)H ± n2 H 2 − 4k(n − k) . 4k(n − k)

Therefore, taking into account once more that H = n−2p with c1 = n−2p n−p and c2 = − p . 2

 2 p(n−p) , n

(4.7)

M n must be isometric to Sn−p (c1 ) × Hp (c2 ),

Theorem 2. There are not exist complete linear Weingarten spacelike hypersurfaces in S2m+1 , satisfying 1 R = aH + b with (2m − 1)a2 + 8m(1 − b)  0, a  0 and b = 1, having two distinct principal curvatures with the same multiplicity and such that H 2  1. Proof. Suppose by contradiction that there exists such a complete linear Weingarten spacelike hypersurface M 2m . From Lemma 3 applied to the function H, we obtain a sequence of points {qk }k∈N ⊂ M 2m such that lim H(qk ) = sup H,

k→+∞

and

M

lim sup L(H)(qk )  0.

(4.8)

k→+∞

Thus, from (4.1) and (4.8) we have   0  lim sup L(2mH)(qk )  sup |Φ|2 Psup H,m,1 sup |Φ| . M

k→∞

M

Hence, since M 2m is supposed to have two distinct principal curvatures, we conclude that   Psup H,m,1 sup |Φ|  0. M

Consequently, taking into account our restriction on H, from (4.2) we get   0  sup |Φ|2  2m sup H 2 − 1  0. M

M

Therefore, we must have |Φ| = 0 on M 2m and, hence, we reach at a contradiction. 2 From Theorem 2 and taking into account the classification of the totally umbilical spacelike hypersurfaces of Sn+1 due to Montiel in [20], we also get the following: 1 Corollary 1. Let M 2 be a complete linear Weingarten spacelike surface in S31 , such that R = aH + b with a2 + 8(1 − b)  0, a  0 and b = 1. If supM H 2  1, then M 2 is isometric either to R2 or S2 , up to scaling. With a suitable boundedness for the norm of the traceless part of the second fundamental form, we obtain another characterization theorem in the de Sitter space.

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Theorem 3. Let M n be a complete linear Weingarten spacelike hypersurface in Sn+1 , n  3, such that 1 R = aH + b, having two distinct principal curvatures with multiplicity p and n − p, where 1  p < n2 . Suppose that 4p(n−p)  H 2 < 1 and n2 √

   n |Φ|   (n − 2p)|H| − n2 H 2 − 4p(n − p) . 2 p(n − p)

(4.9)

If one of the following conditions is satisfied: (i) b < 1 and H attains a maximum on M n ; (ii) (n − 1)a2 + 4n(1 − b) > 0, b = 1 and |∇H| ∈ L1 (M ), then M n is isometric to Sn−p (c1 ) × Hp (c2 ), with 0 < c1 <

n(n−2p) (n−p)2 ,

− n(n−2p) < c2 < 0 and p2

1 c1

+

1 c2

= 1.

Proof. From our restriction √  on H and on |Φ|, it is not difficult to verify that PH,p,1 (|Φ|)  0√nfor |Φ|   n ((n − 2p)|H| − n2 H 2 − 4p(n − p) ), with PH,p,1 (|Φ|) = 0 if, and only if, |Φ| = 2p(n−p) ((n − 2 p(n−p)  2p)|H| − n2 H 2 − 4p(n − p) ). Therefore, in a similar way of the proof of Theorem 1, we conclude that M n must be an isoparametric and, hence, we can apply Theorem 5 of [1] to get that M n is isometric to Sn−k (c1 ) × hypersurface of Sn+1 1 k H (c2 ), where k ∈ {p, n − p}, c1 > 0, c2 < 0 and c11 + c12 = 1. Furthermore, taking into account once more our assumption on H jointly with hypothesis (4.9), from Eq. (4.5) we obtain that k = p, 0 < c1 <

n(n−2p) (n−p)2

and − n(n−2p) < c2 < 0. 2 p2 We can reason as in the proof of Theorem 3 to also get the following: Theorem 4. Let M n be a complete linear Weingarten spacelike hypersurface in Sn+1 , n  3, such that 1 R = aH + b, having two distinct principal curvatures with multiplicity p and n − p. Suppose that either , or 1  p < n2 and H 2  4p(n−p) , and that 1  p  n2 and H 2 > 4p(n−p) n2 n2 √    n |Φ|   (n − 2p)|H| + n2 H 2 − 4p(n − p) . 2 p(n − p) If one of the following conditions is satisfied: (i) (n − 1)a2 + 4n(1 − b)  0, a  0, b = 1 and H is bounded on M n ; (ii) b < 1 and H attains a maximum on M n ; (iii) (n − 1)a2 + 4n(1 − b) > 0, b = 1, H is bounded and |∇H| ∈ L1 (M ), then M n is isometric to Sn−p (c1 ) × Hp (c2 ), with c1 > 0, c2 < 0 and

1 c1

+

1 c2

= 1.

Proof. Let us suppose that occurs the situation of item (i). In this case, we can apply Lemma 3 to the function H, obtaining a sequence of points {qk }k∈N ⊂ M n such that lim H(qk ) = sup H,

k→+∞

and

M

lim sup L(H)(qk )  0.

(4.10)

k→+∞

Thus, from (4.1) and (4.10) we have   0  lim sup L(nH)(qk )  sup |Φ|2 Psup H,p,1 sup |Φ| . k→∞

M

M

(4.11)

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Hence, since M n is supposed to have two distinct principal curvatures, from (4.11) we conclude that   Psup H,p,1 sup |Φ|  0. (4.12) M

On the other hand, from ours restrictions on H and on |Φ|, it is not difficult to verify that PH,p,1 (|Φ|)  0, with PH,p,1 (|Φ|) = 0 if, and only if, √    n |Φ| =  (n − 2p)|H| + n2 H 2 − 4p(n − p) . 2 p(n − p) Consequently, from (4.12) we get that √    n sup |Φ| =  (n − 2p)|H| + n2 H 2 − 4p(n − p) 2 p(n − p) M and, taking into account once more our restriction on |Φ|, we have that |Φ| is constant on M n . Thus, since M n is a linear Weingarten hypersurface, from (3.12) we have that H is also on M n . At this point, the proof proceed as in that one of Theorem 1. Finally, if occurs either item (ii) or (iii), we can also reason in an analogous way of the proof of Theorem 1. 2 Finally, let us consider the case when the ambient space Ln+1 is either the Lorentz–Minkowski space or . the anti-de Sitter space Hn+1 1 Theorem 5. Let M n be a complete linear Weingarten spacelike hypersurface immersed in a Lorentzian space (c), where c ∈ {0, −1}, such that R = aH + b, having two distinct principal curvatures with form Ln+1 1 multiplicity p and n − p, where 1  p  n2 . Suppose that √

   n |Φ|   (n − 2p)|H| + n2 H 2 − 4p(n − p)c . 2 p(n − p)

(4.13)

If one of the following conditions is satisfied: (i) (n − 1)a2 + 4n(c − b)  0, a  0, b = c and H is bounded on M n ; (ii) b < c and H attains a maximum on M n ; (iii) (n − 1)a2 + 4n(c − b) > 0, b = c, H is bounded and |∇H| ∈ L1 (M ), then M n is isometric to: (a) Rn−p × Hp (c2 ), where c2 < 0, when c = 0; (b) Hn−p (c1 ) × Hp (c2 ), where c1 < 0, c2 < 0 and

1 c1

+

1 c2

= −1, when c = −1.

Proof. We can proceed in an analogous way of the proof of Theorem 3 in order to assure that such a complete (c). linear Weingarten spacelike hypersurface M n must be an isoparametric spacelike hypersurface of Ln+1 1 n Hence, since M is supposed to have two distinct principal curvatures with multiplicity p and n − p, where 1  p  n2 , we use once more Theorem 5.1 of [1] to conclude that M n is isometric to: (a) Rn−k × Hk (c2 ), where c2 < 0, when c = 0; (b) Hn−k (c1 ) × Hk (c2 ), where c1 < 0, c2 < 0 and where k ∈ {p, n − p}.

1 c1

+

1 c2

= −1, when c = −1,

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Furthermore, when c = 0, for a suitable choice of the normal vector field, we have that the principal curvatures of Rn−k × Hk (c2 ) → Ln+1 are given by λ1 = · · · = λn−k = 0

and λn−k+1 = · · · = λn =

√

−c2 .

Consequently, √ nH = k −c2

(4.14)

|B|2 = −kc2 .

(4.15)

and

Consequently, inserting (4.14) and (4.15) into (3.12), with a straightforward computation we get |Φ|2 =

n(n − k) 2 H . k

(4.16)

Hence, taking into account our hypothesis (4.13), from (4.16) we have that M n must be isometric to Rn−p × Hp (c2 ), with c2 < 0. Finally, when c = −1, for another suitable choice of the normal vector field, we have that the principal are given by curvatures of Hn−k (c1 ) × Hk (c2 ) → Hn+1 1 λ1 = · · · = λn−k =

√

−1 − c1

√ and λn−k+1 = · · · = λn = − −1 − c2 .

Consequently, √ √ nH = (n − k) −1 − c1 − k −1 − c2

(4.17)

|B|2 = (n − k)(−1 − c1 ) + k(−1 − c2 ).

(4.18)

and

Consequently, inserting (4.17) and (4.18) into (3.12), with a straightforward computation we get |Φ|2 =

  2 n (n − 2k)H + n2 H 2 + 4k(n − k) . 4k(n − k)

(4.19)

Hence, taking into account once more our hypothesis (4.13), from (4.19) we have that M n must be isometric to Hn−p (c1 ) × Hp (c2 ), with c1 < 0, c2 < 0 and c11 + c12 = −1. 2 Acknowledgments The first author is partially supported by PRONEX/CNPq/FAPEAM, Brazil, grant 716.UNI52.1769. 03062009. The second author is partially supported by CNPq, Brazil, grant 300769/2012-1. The third author is partially supported by CAPES, Brazil. The second and fourth authors are partially supported by CAPES/CNPq, Brazil, grant Casadinho/Procad 552.464/2011-2. The authors would like to thank the referee for giving some valuable comments and suggestions which improved the paper.

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