Height estimates and half-space theorems for spacelike hypersurfaces in generalized Robertson–Walker spacetimes

Differential Geometry and its Applications 32 (2014) 46–67

Contents lists available at ScienceDirect

Differential Geometry and its Applications www.elsevier.com/locate/difgeo

Height estimates and half-space theorems for spacelike hypersurfaces in generalized Robertson–Walker spacetimes Sandra C. García-Martínez a,1 , Debora Impera b,∗ a Instituto de Matemática e Estatística, Universidade de São Paulo, Rua do Matão, 1010, Cidade Universitária, São Paulo, Brazil b Dipartimento di Matematica e Applicazioni, Università degli studi di Milano-Bicocca, via Cozzi 53, I-20125 Milano, Italy

a r t i c l e

i n f o

Article history: Received 5 July 2013 Received in revised form 23 October 2013 Available online 20 November 2013 Communicated by P.B. Gilkey MSC: primary 53C42 secondary 53B30, 53C50

a b s t r a c t In this paper, we obtain height estimates for spacelike hypersurfaces Σ n of constant k-mean curvature, 1  k  n, in a generalized Robertson–Walker spacetime −I×ρ Pn and with boundary contained in a slice {s} × Pn . As an application, we obtain some information on the topology at infinity of complete spacelike hypersurfaces of constant k-mean curvature properly immersed in a spatially closed generalized Robertson–Walker spacetime. Finally, using a version of the Omori–Yau maximum principle for the Laplacian and for more general elliptic trace-type differential operators, some non-existence results are also obtained. © 2013 Elsevier B.V. All rights reserved.

1. Introduction In the seventies, with the works of Calabi [13], Cheng and Yau [14], Brill and Flaherty [12], ChoquetBruhat [16,17], began the mathematical interest for the study of spacelike constant mean curvature hypersurfaces in Lorentzian manifolds. This interest is also motivated by their relevance from a physical point of view (see for instance [25] for more details). Recently the study of a priori estimates for the height of constant mean curvature compact spacelike graphs or, more generally, compact spacelike hypersurfaces with boundary, has become the subject of a rapidly increasing research. This is motivated by the fact that these estimates turn out to be a very useful tool in order to investigate existence and uniqueness results for complete spacelike hypersurfaces with constant mean curvature, as well as to obtain information on the topology at infinity of such hypersurfaces. * Corresponding author. E-mail addresses: [email protected] (S.C. García-Martínez), [email protected] (D. Impera). The first author was supported by FAPESP (Fundação de Amparo á Pesquisa do Estado de São Paulo, Brazil) Process 2012/22490-7, MINECO (Ministerio de Economía y Competitividad) and FEDER (Fondo Europeo de Desarrollo Regional) project MTM2012-34037 and Fundación Séneca project 04540/GERM/06, Spain. This research is a result of the activity developed within the framework of the Program in Support of Excellence Groups of the Region de Murcia, Spain, by Fundamental Séneca, Regional Agency for Science and Technology (Regional Plan for Science and Technology 2007–2010). 1

0926-2245/$ – see front matter © 2013 Elsevier B.V. All rights reserved. http://dx.doi.org/10.1016/j.difgeo.2013.10.017

S.C. García-Martínez, D. Impera / Differential Geometry and its Applications 32 (2014) 46–67

47

A priori estimates for the height of constant mean curvature compact spacelike hypersurfaces in the Lorentz–Minkowski spacetime Ln+1 and with boundary on a spacelike hyperplane, were first obtained by López in [24] in case n = 2. Later on, in [27], Montiel obtained height estimates for compact spacelike graphs in the steady state spacetime, which is an open region of the de Sitter spacetime (see Remark 5), and he applied them to prove some existence and uniqueness theorems for complete spacelike hypersurfaces in the de Sitter spacetime with constant mean curvature H > 1 and prescribed asymptotic future boundary. In [19], Lima extended the height estimates proved by López to any n and he also obtained sharp height estimates for compact spacelike hypersurfaces with some positive constant higher order mean curvature in the Lorentz–Minkowski spacetime and with boundary contained in a hyperplane. Later on, Colares and Lima, [18], following the techniques used in the Riemannian setting by Cheng and Rosenberg, [15], were able to generalize these estimates to the case of compact spacelike hypersurfaces of positive constant k-mean curvature in Lorentzian product spaces −R × Pn satisfying some energy condition and with boundary contained in a slice. These estimates have the important feature that they only depend on the k-mean curvature of the hypersurface and on a bound on the hyperbolic angle between the future-pointing unit normal vector field and the coordinate vector field induced by the universal time on −R × Pn . Due to this feature, Colares and Lima were able to apply them to the study of topological properties of complete spacelike hypersurfaces of positive constant mean curvature. In this paper we aim at completing the picture described above by considering compact spacelike hypersurfaces with boundary immersed in a wider family of spacetimes, that is, the generalized Robertson–Walker (GRW) spacetimes. Roughly speaking (see Section 2 for a detailed definition), a GRW spacetime is a warped product of a definite negative one-dimensional base, and having a Riemannian manifold as a fiber. Notice that the family of GRW spacetimes includes, for instance, the Lorentzian spaceforms, the steady state spacetime mentioned above and the Lorentzian products. In any GRW spacetime there is a distinguished family of spacelike hypersurfaces, the so-called slices, which are defined as level hypersurfaces of the time coordinate of the spacetime. Moreover, any slice is totally umbilical and has constant k-mean curvature for any 1  k  n. Controlling the mean curvature of the hypersurface in terms of the mean curvature of the slices and imposing suitable conditions on the geometry of the ambient spacetime (that is, on the warping function and on the curvature of the fiber), we are able to extend the results in [18] to compact spacelike hypersurfaces with constant k-mean curvature (1  k  n) in a GRW spacetime and with boundary contained in a slice (see Theorems 8, 9 in Section 3 and Theorem 16 in Section 4). In the same spirit of [18], we also aim at applying these results to deduce topological properties of spacelike hypersurfaces properly immersed in spatially closed GRW spacetimes. However, an important difference between this more general setting and the product case, is that the height estimates, in general, do not involve only the k-mean curvature of the hypersurface and a bound on the angle function, but, in some cases, they strongly depend on the values of the height function. This reflects on the character of the applications to the study of complete spacelike hypersurfaces in spatially closed GRW spacetimes. When the estimates do not depend on the values of the height function, we can perform the same reasoning as in the product case and deduce topological properties of complete spacelike hypersurfaces with constant mean curvature (see Theorem 22 in Section 5). On the other hand, when the estimates depend also on the values of the height function, the applications take the form of non-existence results (see Theorems 21 and 26 in Section 5). In a final section, we also prove further non-existence results, in the form of half-space theorems, in the more general setting of non-spatially closed GRW spacetimes (see Theorems 32 and 35 in Section 6). Note that, since the height estimates do not apply in this more general case, we adopt a different technique based on a generalized version of the Omori–Yau maximum principle. Finally, since the steady state spacetimes belong to the class of GRW spacetimes, we also deduce some topological and non-existence results for complete spacelike hypersurfaces in this spacetime.

48

S.C. García-Martínez, D. Impera / Differential Geometry and its Applications 32 (2014) 46–67

2. Preliminaries In what follows we consider an n-dimensional Riemannian manifold Pn , an open interval I of the real line and we let ρ : I → (0, ∞) be a positive smooth function. Let −I ×ρ Pn be the warped product endowed with the Lorentzian metric     , = −πI∗ dt2 + ρ2 (πI )πP∗ ,P , where πI and πP are the corresponding projections on I and P respectively. Following the terminology used in [7] we will refer to −I ×ρ Pn as a GRW spacetime. ∂ It is well known that the vector field ρ(t) ∂t is a closed conformal timelike vector field on −I ×ρ Pn which determines a foliation t → Pt := {t} × P of −I ×ρ Pn by totally umbilical spacelike hypersurfaces with ∂ constant mean curvature H(t) = (log ρ) (t) with respect to T := ∂t . n n Consider now a spacelike hypersurface f : Σ → −I ×ρ P . In this case, since T is a unitary timelike vector field globally defined on −I ×ρ Pn , there exists a unique unitary timelike normal field N globally defined on Σ. Throughout the paper we will always assume that the orientation is chosen so that N is future-directed, that is − cosh θ := N, T   −1 < 0. Furthermore, we will refer to the function Θ : Σ → (−∞, −1], Θ := − cosh θ, as the angle function. Let A : T Σ → T Σ be the second fundamental form of the immersion f . Its eigenvalues k1 , . . . , kn are the principal curvatures of the hypersurface Σ. Their elementary symmetric functions S0 = 1,

Sk =



ki1 · · · kik ,

k = 1, . . . , n,

i1 <···
define the k-mean curvatures of the immersion via the formula   n Hk = (−1)k Sk . k Thus H1 = −1/n Tr(A) is the mean curvature and n(n − 1)H2 = S − S + 2Ric(N, N ), where S is the scalar curvature of Σ, while S and Ric are, respectively, the scalar curvature and the Ricci tensor of the GRW spacetime. Even more, when k is even, it follows from the Gauss equation that Hk is a geometric quantity which is related to the intrinsic curvature of Σ n . We introduce the Newton transformations Pk : T Σ → T Σ which are inductively defined by P0 = I,

  n Pk = Hk I + APk−1 , k

k = 1, . . . , n.

It is not difficult to see that the Newton transformations satisfy the following properties: (a) Tr(Pk ) = ck Hk , (b) Tr(A ◦ Pk ) = −ck Hk+1 ,  n  (nH1 Hk+1 − (n − k − 1)Hk+2 ), (c) Tr(A2 ◦ Pk ) = k+1    n  where ck = (n − k) nk = (k + 1) k+1 . We refer the reader to [3] for more details. Let ∇ be the Levi-Civita connection of Σ and let g ∈ C ∞ (Σ). We define the second order linear differential operator Lk : C ∞ (Σ) → C ∞ (Σ) associated to Pk by Lk g = Tr(Pk ◦ hess g),

(1)

S.C. García-Martínez, D. Impera / Differential Geometry and its Applications 32 (2014) 46–67

49

where hess g is the linear operator metrically equivalent to 

 hess g(X), Y = ∇X ∇g, Y .

It follows by Eq. (1) that the operator Lk is elliptic if and only if Pk is positive definite. Let us state two useful lemmas in which geometric conditions are given in order to guarantee the ellipticity of Lk when k  1 (recall that L0 = Δ is always elliptic). Lemma 1. Let Σ be a spacelike hypersurface in a GRW spacetime. If H2 > 0 on Σ, then L1 is an elliptic operator (for an appropriate choice of the Gauss map N ). For a proof of Lemma 1 see Lemma 3.2 in [4], where Alías and Colares proved it as a consequence of Lemma 3.10 in [20]. The next lemma concerning the ellipticity of the operators Lk in case 2  k  n, is a consequence of Proposition 3.2 in [8] (see also Lemma 3.3 in [4]). Lemma 2. Let Σ be a spacelike hypersurface in a GRW spacetime. If there exists an elliptic point of Σ, with respect to an appropriate choice of the Gauss map N , and Hk > 0 on Σ, 3  k  n, then the operator Lj is elliptic for any 1  j  k − 1. Recall that by an elliptic point in a spacelike hypersurface we mean a point of Σ where all the principal curvatures are negative, with respect to a suitable choice of the Gauss map N . Note that the existence of an elliptic point implies that Hk is positive at that point, and, if it does not change sign, applying Garding ˙ inequalities, [21], we have 1/2

H 1  H2

1/(k−1)

 · · ·  Hk−1

1/k

 Hk

> 0,

(2)

with equality at any stage only for an umbilical point. Finally, we conclude this section with the following technical proposition that will be essential for the proofs of our main results (for more details and a complete proof see Sections 4 and 8 in [4]). Proposition 3. Let f : Σ n → −I ×ρ Pn be a spacelike hypersurface with angle function Θ and height function h = πI ◦ f . Then the following formulas hold: (a) Let σ(t) be a primitive of ρ(t). Then   Lk−1 h = −(log ρ) (h) ck−1 Hk−1 + Pk−1 ∇h, ∇h − Θck−1 Hk ,   Lk−1 σ(h) = −ck−1 ρ (h)Hk−1 + Θρ(h)Hk .

(3) (4)

= ρ(h)Θ. Then (b) Let Θ = Lk−1 Θ

  n ρ(h)∇h, ∇Hk  + ρ (h)ck−1 Hk k     n nH1 Hk − (n − k)Hk+1 +Θ k n 

2  

Θ μk−1,i KP N ∗ ∧ Ei∗ N ∗ ∧ Ei∗

2 ρ (h) i=1   2 − Θ(log ρ) (h) ∇h ck−1 Hk−1 − Pk−1 ∇h, ∇h ,

+

(5)

50

S.C. García-Martínez, D. Impera / Differential Geometry and its Applications 32 (2014) 46–67

where ∇h = −T − ΘN , {Ei }ni=1 is an orthonormal frame on Σ and, for any vector field X in −I ×ρ Pn we set X ∗ = πP∗ X. 3. Height estimates for constant mean curvature hypersurfaces We begin this section with the following proposition stating that any compact spacelike hypersurface in a GRW spacetime with non-empty boundary contained in a slice must lie entirely on one of the two regions of the spacetime bounded by the slice. We point out that, to prove this result we are not assuming that the mean curvature of the spacelike hypersurface Σ is constant. Proposition 4. Let Σ n be a compact spacelike hypersurface in a GRW spacetime −I ×ρ Pn . Assume that ∂Σ ⊂ {s} × P for some s ∈ I. (i) If H1  min{0, inf I (log ρ) }, then h  s; (ii) If H1  max{0, supI (log ρ) }, then h  s. Proof. Let us prove part (i) first. Note that, in this case, H1 is a non-positive function and H1  inf I (log ρ)  (log ρ) (h). Using Eq. (4) with k = 0 we then obtain   Δσ(h) = −nρ(h) (log ρ) (h) + ΘH1  −nρ(h)H1 (1 + Θ)  0. It follows then by the classical maximum principle that σ(h) must attain its minimum on ∂Σ, that is, σ(h)  σ(s). Since σ is an increasing function, this implies that h  s. For what concern part (ii), observe that, in this case, H1 is a non-negative function and H1  supI (log ρ)  (log ρ) (h). Hence   Δσ(h) = −nρ(h) (log ρ) (h) + ΘH1  −nρ(h)H1 (1 + Θ)  0 and we conclude again by the classical maximum principle that h must attain its maximum on ∂Σ, that is h  s. 2 Remark 5. Let Rn+2 be the (n + 2)-dimensional Lorentz–Minkowski space and consider the hyperquadric 1  Sn+1 = p ∈ Rn+2 : p, p = 1 . 1 1 This hyperquadric is known as the de Sitter space and it is a model of spacetime of sectional curvature 1. let us consider on Sn+1 the closed conformal vector field For any a ∈ Rn+1 1 1 Ta (p) = a − a, pp,

p ∈ Sn+1 . 1

It is not difficult to see that Ta is a timelike vector field if we restrict on certain subsets of Sn+1 and it generates different foliations depending on the causal character of a (see [26] for more details). In particular, if a is a null vector, the (closed conformal) vector field Ta is timelike on the open set : p, a = 0 . p ∈ Sn+1 1



Let Hn+1 be the connected component of this set characterized by p, a > 0, which is called the (n + 1)-dimensional steady state spacetime. In this case the corresponding foliation has as leaves the spacelike hypersurfaces

S.C. García-Martínez, D. Impera / Differential Geometry and its Applications 32 (2014) 46–67

51

 Hτ = p ∈ Hn+1 : p, a = τ . It is well known that the Hτ ’s are totally geodesic spacelike hypersurfaces of constant mean curvature 1 with respect to the future-pointing Gauss map Nτ (p) = p −

1 a τ

which are isometric to the Euclidean space Rn . Furthermore, as well-explained in [2], the steady state spacetime is isometric to the GRW spacetime −R ×et Rn . Namely, if b ∈ Rn+2 is a null vector satisfying 1 a, b = 1, it can be easily checked that the map Φ : Hn+1 → −R ×et Rn defined by  Φ(p) =

  p − p, ab − p, ba log p, a , p, a



is an isometry which preserves the time orientation. Finally, note that, for a spacelike hypersurface ψ : Σ n → −R ×et Rn the height function h can be written as    h(p) = log u Φ−1 ψ(p) ,

p ∈ Σ,

where u(p) = Φ−1 (ψ(p)), a. Moreover, the umbilical hypersurfaces Hτ correspond to the slices {log τ }×Rn . With this preparation the following corollary is straightforward. Corollary 6. Let f : Σ n → Hn+1 be a compact spacelike hypersurface in the steady state space Hn+1 . Assume that ∂Σ ⊂ Hτ for some 0 < τ ∈ R. (i) If H1  0 then u  τ ; (ii) If H1  1 then u  τ , where u(p) = f (p), a. This last result is comparable with Proposition 3 proved by Montiel in [27], although we do not need the hypothesis of H1 being constant. In order to avoid confusion note that, in [27, Proposition 3], Montiel used the fact there is an (orientation reversing) isometry between the steady state spacetime and the upper half space endowed with the Lorentzian metric g given by    1 2 2 Rn+1 , g = dx − dx . (x,xn+1 ) n+1 + x2n+1 Recall that a spacetime obeys the null convergence condition (NCC ) if its Ricci tensor is non-negative on null (or lightlike) directions (see [22] for more details on this physical conditions). That is, Ric(V, V )  0 for all V = 0 satisfying V, V  = 0. A direct computation using the general relationship between the curvature tensor of a warped product and the curvature tensor of its base and its fiber, as well as the derivatives of the warping function (see for instance [28, Proposition 42]) implies that  2   R(U, V )W = RP (U ∗ , V ∗ )W ∗ + (log ρ) (πR ) U, W V − V, W U   + (log ρ) (πR )W, T  V, T U − U, T V   − (log ρ) (πR ) U, W V, T  − V, W U, T  T

(6)

S.C. García-Martínez, D. Impera / Differential Geometry and its Applications 32 (2014) 46–67

52

for every U, V, W on −I ×ρ Pn , where T = ∂∂t and we are using the notation U ∗ to denote πP ∗ U for an arbitrary U in −I ×ρ Pn . In particular, the Ricci tensor of the warped product has the expression     2  Ric(U, V ) = RicP U ∗ , V ∗ + n (log ρ) + (log ρ) U, V  − (n − 1)(log ρ) U, T V, T .

(7)

Here RicP denotes the Ricci tensor of P. Using Eq. (7) it is easy to see that a GRW spacetime obeys the NCC if and only if  RicP  (n − 1) sup ρ2 (log ρ) .

(8)

I

Keeping this in mind it is straightforward to prove the following Proposition 7. Let −I ×ρ Pn be a GRW spacetime obeying the NCC. Assume that Σ n is a spacelike hyper is superharmonic. surface of constant mean curvature in −I ×ρ Pn . Then the function φ = H1 σ(h) + Θ Proof. Using Eqs. (4), (5) and the fact that H1 is constant, we get     A 2 − nH 2 + RicP N ∗ , N ∗ − (n − 1)(log ρ) (h) ∇h 2 . Δφ = Θ 1 2

As a consequence of the Cauchy–Schwartz inequality, one has A  nH12 . Moreover, using the decomposition N = N ∗ − ΘT, one easily gets N ∗ , N ∗ P = ∇h /ρ2 (h) and hence the conclusion follows from the NCC. 2 2

Propositions 4 and 7, combined with the classical maximum principle, allow us to obtain one of the main theorems of this section. Theorem 8. Let −I ×ρ Pn be a GRW spacetime obeying the NCC and let Σ n be a compact spacelike hypersurface in −I ×ρ Pn with non-vanishing constant mean curvature satisfying H1  max{0, supI (log ρ) }. Assume that ∂Σ ⊂ {s} × Pn for some s ∈ I and that ρ is a monotone function on I. (i) If ρ is non-decreasing, then Σ n ⊂ [s − α1 , s] × Pn , where α1 =

ρ(s) ρ(minΣ h)

max∂Σ (−Θ) − 1 H1

(ii) If ρ is non-increasing, then Σ n ⊂ [s − α2 , s] × Pn , where α2 =

max∂Σ (−Θ) − 1  0. H1

 0.

S.C. García-Martínez, D. Impera / Differential Geometry and its Applications 32 (2014) 46–67

53

Proof. Since H1  max{0, supI (log ρ) }, it follows by Proposition 4 that h  s. is subharmonic, it must attain its Moreover, since, by Proposition 7, the function −φ = −H1 σ(h) − Θ maximum on ∂Σ. Then  −H1 σ(s) + ρ(s) max(−Θ). −H1 σ(h) + ρ(h)  −H1 σ(h) − Θ ∂Σ

(9)

Note that, for any t  s it holds t σ(s) − σ(t) =



ρ(u)du  inf

u∈(t,s)

 ρ(u) (s − t).

s

Hence, for any x ∈ Σ, 





σ h(x) − σ(s) 

ρ(h(x))(h(x) − s)

if ρ  0,

ρ(s)(h(x) − s)

if ρ  0.

In case ρ  0, we then obtain   H1 ρ(h)(h − s)  H1 σ(h) − σ(s)  ρ(h) − ρ(s) max(−Θ). ∂Σ

In particular, H1 (h − s)  1 −

ρ(s) ρ(s) max(−Θ)  1 − max(−Θ), ρ(h) ∂Σ ρ(minΣ h) ∂Σ

(10)

proving part (i) of the theorem. As for part (ii), note that, in case ρ  0,     H1 ρ(s)(h − s)  H1 σ(h) − σ(s)  ρ(h) − ρ(s) max(−Θ)  ρ(s) 1 − max(−Θ) ∂Σ

∂Σ

and the conclusion follows. 2 Reasoning in an analogous way we can also prove the next Theorem 9. Let −I ×ρ Pn be a GRW spacetime obeying the NCC and let Σ n be a compact spacelike hypersurface of non-vanishing constant mean curvature satisfying H1  min{0, inf I (log ρ) }. Assume that ∂Σ ⊂ {s} × Pn for some s ∈ I and that ρ is a monotone function on I. (i) If ρ is non-decreasing, then Σ n ⊂ [s, s + β1 ] × Pn , where β1 =

1 − max∂Σ (−Θ)  0. H1

S.C. García-Martínez, D. Impera / Differential Geometry and its Applications 32 (2014) 46–67

54

(ii) If ρ is non-increasing, then Σ n ⊂ [s, s + β2 ] × Pn , where β2 =

1−

ρ(s) ρ(maxΣ h)

max∂Σ (−Θ)

H1

 0.

Remark 10. We point out that when the GRW spacetime is static, that is, ρ(t) ≡ 1 (i.e. −I × Pn is a Lorentzian product), the NCC is equivalent to the assumption that the fiber Pn has non-negative Ricci curvature. If we consider a compact spacelike hypersurface Σ in −I × Pn with positive constant mean curvature and ∂Σ ⊂ {s} × Pn , as an application of Theorem 8 we obtain α1 = α2 =

max∂Σ (−Θ) − 1 max∂Σ |Θ| − 1 = := α H1 H1

and we recover the height estimate s − α  h  s proved by Lima and Colares in [18, Corollary 3.6] for the case k = 1, but without requiring that the sectional curvature of the fiber P is constant. Moreover, note that in the Lorentz–Minkowski spacetime Ln+1 = −R × Rn , we have that max |Θ|  cosh r ∂Σ

where r > 0 is the radius of a geodesic ball of Hn which contains the image of the Gauss map of Σ. Hence with Theorem 8 we recover the estimate obtained by Lima in [19, Corollary 4.1], who also proved that this estimate is sharp. As a nice consequence of Theorems 8 and 9, keeping in mind Remark 5, we also have the validity of the following Corollary 11. Let f : Σ n → Hn+1 be a compact spacelike hypersurface of constant mean curvature in the steady state spacetime Hn+1 . Assume that ∂Σ ⊂ Hτ for some 0 < τ ∈ R. (i) If H1 < 0, then τ  u  τ eβ , where u(p) = f (p), a and β=

1 − max∂Σ (−Θ)  0. H1

(ii) If H1  1, then τ e−α  u  τ, where u(p) = f (p), a and α=

τ max∂Σ (−Θ) − minΣ u  0. H1 minΣ u

S.C. García-Martínez, D. Impera / Differential Geometry and its Applications 32 (2014) 46–67

55

Remark 12. As seen in Remark 5, the steady state spacetime Hn+1 admits a representation as warped product space of the form −R×et Pn . Motivated by this fact, one can consider the wider family of spacetimes of the form −R×et Pn . Following the terminology in [2], we will refer to them as steady state-type spacetimes. For instance, a significant example of a spacetime belonging to this family is the so-called de Sitter cusp space. This was defined by Hawking and Ellis in [22] as −R ×et Tn , where Tn is the n-dimensional flat torus, and it is obtained as a quotient of a region in de Sitter space. As a consequence of Theorem 8, for spacelike hypersurfaces of constant mean curvature in a steady state-type spacetime one can prove the following Corollary 13. Let Σ n be a compact spacelike hypersurface of constant mean curvature H1  1 in a steady state-type spacetime −R ×et Pn satisfying RicP  0. Assume that ∂Σ ⊂ {τ } × Pn for some τ ∈ R. Then −α  h − τ  0, where α=

eτ −minΣ h max∂Σ (−Θ) − 1  0. H1

4. Height estimates for hypersurfaces of constant k-mean curvature The estimates of the previous section can be generalized to spacelike hypersurfaces of constant k-mean curvature, 2  k  n. We begin with the following version of Proposition 4 in the higher order mean curvature case. Proposition 14. Let Σ n be a compact spacelike hypersurface in a GRW spacetime −I ×ρ Pn with positive k-mean curvature for some 2  k  n. Assume that ∂Σ ⊂ {s} × P for some s ∈ I and that, if k  3, there exists an elliptic point p ∈ Σ. If either (i) ρ (h)  0 or 1/k (ii) ρ (h)  0 and Hk  supI (log ρ) , then h  s. Proof. Assume first that ρ (h)  0. Since Hk > 0 and, if k  3, there exists an elliptic point, it follows that (k−1)/k

Hk−1  Hk

>0

and we conclude by Lemmas 1 and 2 that Lk−1 is an elliptic operator. Now, using Proposition 3 we obtain   Lk−1 σ(h) = −ck−1 ρ (h)Hk−1 + Θρ(h)Hk  0 and the conclusion follows immediately by the classical maximum principle for elliptic operators.

S.C. García-Martínez, D. Impera / Differential Geometry and its Applications 32 (2014) 46–67

56

1/k

Finally, in case (ii), Hk

 (log ρ) (h)  0 and we introduce the differential operator L defined as

k−1    ck−1  (log ρ) (h) k−1−i − L = Tr Pi ◦ hess ci Θ i=0 =

k−1  i=0

 k−1−i (log ρ) (h) ck−1 − Li . ci Θ

(11)

Note that, since (log ρ)  0 and the operators Li , 0  i  k − 1, are elliptic, L is an elliptic operator. Moreover, it is not difficult to prove by induction on k (see [5] for more details) that Lσ(h) =

 k  ck−1 ρ(h) − (log ρ) (h) + (−Θ)k Hk . k−1 (−Θ)

(12)

  ck−1 ρ(h)Hk (−Θ)k − 1  0 k−1 (−Θ)

(13)

Hence Lσ(h) 

and the conclusion follows again by the maximum principle. 2 We are now ready to present a version of Theorem 8 for constant higher order mean curvature spacelike hypersurfaces. However, in order to do that we need to replace the null convergence condition on the spacetime with a stronger curvature assumption, that is  KP  sup ρ2 (log ρ) ,

(14)

I

where KP denotes the sectional curvature of the fiber P. Following the terminology in [4] we will refer to (14) as the strong null convergence condition (strong NCC ). Note that important examples of spacetimes obeying the strong NCC are, for instance, the Lorentzian spaceforms, the Robertson–Walker spacetimes (i.e. GRW spacetimes with Riemannian factor of constant sectional curvature) obeying the NCC, the steady state spacetime, the de Sitter cusp space described in Remark 12 and, more generally, any steady state-type spacetime of non-negative sectional curvature. Proposition 15. Let −I ×ρ Pn be a GRW spacetime obeying the strong NCC and let Σ n be a spacelike hypersurface in −I ×ρ Pn with positive constant k-mean curvature Hk for some 2  k  n. Moreover, if 1/k If ρ (h)  0, then k  3, assume that there exists an elliptic point on Σ. Let φ = Hk σ(h) + Θ. Lk−1 φ  0 on Σ. We refer to [4] or [5] for a proof of the previous proposition. Combining Propositions 14 and 15 we can then prove the following Theorem 16. Let −I ×ρ Pn be a GRW spacetime obeying the strong NCC and with monotone warping function ρ. Let Σ n be a spacelike hypersurface in −I ×ρ Pn with positive constant k-mean curvature satisfying 1/k Hk  supI (log ρ) , for some 2  k  n. Suppose that ∂Σ ⊂ {s} × Pn for some s ∈ I and, if k  3, that there exists an elliptic point on Σ. Then Σ n ⊂ [s − α1 , s] × Pn ,

S.C. García-Martínez, D. Impera / Differential Geometry and its Applications 32 (2014) 46–67

57

where α1 =

ρ(s) ρ(minΣ h)

max∂Σ (−Θ) − 1 1/k

Hk

 0.

Proof. Note first that, as a consequence of Proposition 14 we have h  s. Moreover, since the function φ defined as in Proposition 15 satisfies Lk−1 (−φ)  0 and Σ is compact, we can apply the classical maximum principle for the elliptic operator Lk−1 to conclude that 1/k  max(−φ). −Hk σ(h) − Θ ∂Σ

Since 1/k

max(−φ) = −Hk σ(s) + ρ(s) max(−Θ), ∂Σ

∂Σ

we then obtain 1/k 

Hk

 σ(h) − σ(s)  ρ(h) − ρ(s) max(−Θ), ∂Σ

and we conclude reasoning as in Theorem 8. 2 Remark 17. Note that, if the GRW spacetime is static we recover again the estimates proved by Colares and Lima in [18, Corollary 3.6]. In the particular situation of spacelike hypersurfaces of constant k-mean curvature in the steady state space, Theorem 16 reads as follows. Corollary 18. Let f : Σ n → Hn+1 be a compact spacelike hypersurface of constant k-mean curvature for some 2  k  n, in the steady state spacetime Hn+1 . If k  3 suppose that there exists an elliptic point on Σ. Furthermore, assume that ∂Σ ⊂ Hτ for some 0 < τ ∈ R. If Hk  1 then τ e−α  u  τ, where u(p) = f (p), a and α=

τ max∂Σ (−Θ) − minΣ u 1/k

minΣ u Hk

 0.

More generally, for spacelike hypersurfaces in a steady state-type spacetime, the next corollary holds true. Corollary 19. Let Σ be a compact spacelike hypersurface in a steady state-type spacetime −R×et Pn satisfying 1/k KP  0. Suppose that Σ has constant k-mean curvature Hk , 2  k  n, satisfying Hk  1 and, if k  3, suppose that there exists an elliptic point on Σ. Assume that ∂Σ ⊂ {τ } × Pn for some τ ∈ R. Then τ − α  h  τ,

S.C. García-Martínez, D. Impera / Differential Geometry and its Applications 32 (2014) 46–67

58

where α=

eτ −minΣ h max∂Σ (−Θ) − 1 1/k

Hk

 0.

5. Half-space theorems and topology at infinity of spacelike hypersurfaces in spatially closed GRW spacetimes In the following we will consider complete spacelike hypersurfaces in spatially closed GRW spacetimes −R ×ρ Pn , that is GRW spacetimes whose Riemannian fiber Pn is compact. We begin introducing the next Definition 20. Let Σ be a spacelike hypersurface in a GRW spacetime −R ×ρ Pn . We say that Σ lies in an upper or lower half-space if it is respectively contained in a region of −R ×ρ Pn of the form

τ, +∞) × Pn

 or (−∞, τ × Pn ,

for some real number τ . Applying the height estimates obtained in the previous section, we can prove the following Theorem 21. Let Σ n be a complete spacelike hypersurface properly immersed in a spatially closed GRW spacetime −R ×ρ Pn satisfying the NCC and with monotone warping function. Suppose that either (i) ρ is non-decreasing and Σ has non-vanishing mean curvature satisfying H1  supR (log ρ) or (ii) ρ is non-increasing and Σ has non-vanishing mean curvature satisfying H1  inf R (log ρ) . Then (a) If (i) holds, Σ cannot lie in a lower half-space. In particular, Σ must have at least one top end; (b) If (ii) holds, Σ cannot lie in an upper half-space. In particular, Σ must have at least one bottom end. Proof. Suppose by contradiction that Σ lies in a half-space of the form (−∞, τ ] × Pn , τ ∈ R and that assumptions in (i) hold. For any s ∈ R, s < τ , denote by Σs+ the hypersurface   Σs+ = (t, x) ∈ Σ  t  s . Note that, since P is compact and the immersion is proper, Σs+ is a compact spacelike hypersurface with boundary contained in Ps . Furthermore, it satisfies the assumptions of Proposition 4 and we conclude that the height function of Σs+ satisfies h  s, leading to a contradiction since s is arbitrary. Finally, assume by contradiction that Σ lies in a half-space of the form [τ, +∞) × Pn , τ ∈ R, and that the assumptions in (ii) hold. For any s ∈ R, s > τ , the hypersurface   Σs− = (t, x) ∈ Σ  t  s is a compact spacelike hypersurface with boundary contained in Ps . Furthermore, it satisfies the assumptions of Proposition 4 and we reach again a contradiction since, in this case, h  s. 2 Theorem 22. Let Σ n be a complete spacelike hypersurface properly immersed in a spatially closed GRW spacetime −R ×ρ Pn satisfying the NCC and with monotone warping function. Suppose that either

S.C. García-Martínez, D. Impera / Differential Geometry and its Applications 32 (2014) 46–67

59

(i) ρ is non-increasing and Σ has positive constant mean curvature or (ii) ρ is non-decreasing and Σ has negative constant mean curvature. If |Θ| is bounded, then Σ cannot lie in a half-space. In particular, Σ must have at least one top and one bottom end. Proof. Let us prove part (i) first. Suppose by contradiction that Σ lies in a half-space of the form [τ, +∞) × Pn , τ ∈ R. For any s ∈ R, s > τ , consider the hypersurface Σs− defined in the previous theorem. Again, Σs− is a compact spacelike hypersurface contained in a slab of width s − τ and with boundary contained in Ps . Furthermore, it satisfies the assumptions of Theorem 8 and we conclude that s−

supΣ |Θ| − 1  s − α2  h  s. H1

Hence Σs− is contained in a slab of width (supΣ |Θ| − 1)/H1  0. Choosing s sufficiently big we may violate this estimate, reaching a contradiction. On the other hand, if Σ is contained in a half-space of the form (−∞, τ ] × Pn , for any s ∈ R, s < τ , the spacelike hypersurface Σs+ defined as in Theorem 21 is compact, with boundary contained in Ps and it satisfies the assumptions of Theorem 8. We obtain then a contradiction since Theorem 8 implies that the height function on Σs+ satisfies h  s and s is arbitrary. Finally, as for part (ii), the proof proceeds in the same way using Theorem 9 instead of Theorem 8. 2 Remark 23. Note that the assumption on the boundedness of the angle function seems quite natural since, for instance, when the GRW spacetime is the Lorentz–Minkowski spacetime, it is automatically satisfied by the hyperbolic caps, that realize the height estimates. In the more general case, this assumption has a physical meaning as it can be interpreted, following Latorre and Romero, [23], in terms of maximum speeds for comoving observers. To be more precise, recall that, given a GRW spacetime M , a comoving observer is nothing but an integral curve of the timelike vector field T . Moreover, if p ∈ M , we say that T (p) is an instantaneous comoving observer. If p is a point of a spacelike-hypersurface Σ in M , timelike vectors T (p) and N (p) appear naturally among all the instantaneous observers at p. In this setting, the quantity θ(p) = cosh Θ(p) corresponds to the energy that T (p) measures for the normal observer N (p) and the speed |v(p)| of the Newtonian velocity v(p) that T (p) measures for N (p) satisfies |v(p)|2 = tanh2 (|θ(p)|). Hence, from a physical point of view, the boundedness of the angle function means that the speed of the Newtonian velocity that the instantaneous observer measure for the normal observer do not approach the speed of light 1. For spacelike hypersurfaces in spatially closed steady state-type spacetimes we deduce the following Corollary 24. Let Σ n be a complete spacelike hypersurface properly immersed in a spatially closed steady state-type spacetime −R ×et Pn satisfying RicP  0. (i) If H1  1, then Σ cannot lie in a lower half-space. In particular, Σ must have at least one top end. (ii) If 0 > H1 ≡ const. and |Θ| is bounded, then Σ cannot lie in a half-space. In particular, Σ must have at least one top and one bottom end. Note the previous theorems remain true if we replace the assumption of P being compact with that of Σ being cylindrically bounded. Recall that Σ is cylindrically bounded if Σ ⊂ R × K,

60

S.C. García-Martínez, D. Impera / Differential Geometry and its Applications 32 (2014) 46–67

where K is a compact set. Under this assumption, proceeding as in the proof of Theorems 21 and 26 the next corollaries are straightforward. Recall that a spacelike hypersurface in a steady state spacetime Hn+1 is said to be bounded away from the future infinity if it lies in a lower half-space. Analogously, Σ is said to be bounded away from the past infinity if it lies in an upper half-space. Corollary 25. Let Σ n be a complete spacelike hypersurface properly immersed in Hn+1 and cylindrically bounded. (i) If H1  1, then Σ cannot be bounded away from the future infinity. In particular, Σ must have at least one top end. (ii) If 0 > H1 ≡ const. and |Θ| is bounded, then Σ cannot be bounded away neither from the past nor from the future infinity. In particular, Σ must have at least one top and one bottom end. As an application of Proposition 14, reasoning as in Theorem 21, we can also prove the following result for hypersurfaces of positive constant k-mean curvature. Theorem 26. Let Σ n be a complete spacelike hypersurface properly immersed in a spatially closed GRW spacetime −R ×ρ Pn satisfying the strong NCC and with non-decreasing warping function. Suppose that Σ n 1/k has positive k-mean curvature satisfying Hk  supR (log ρ) for some 2  k  n and that, if k  3, there exists an elliptic point on Σ. Then Σ cannot lie in a lower half-space. In particular, Σ must have at least one top end. Finally, as an application of Theorem 26, we obtain the following corollaries concerning spacelike hypersurfaces in a spatially closed steady state-type spacetime and cylindrically bounded spacelike hypersurfaces in a steady state spacetime respectively. Corollary 27. Let Σ n be a complete spacelike hypersurface properly immersed in a spatially closed steady 1/k state-type spacetime −R×et Pn satisfying KP  0. Suppose that Σ n has k-mean curvature satisfying Hk  1 for some 2  k  n and that, if k  3, there exists an elliptic point on Σ. Then Σ cannot lie in a lower half-space. In particular, Σ must have at least one top end. Corollary 28. Let Σ n be a complete spacelike hypersurface properly immersed in Hn+1 and cylindrically bounded. Suppose that Σ n has k-mean curvature satisfying Hk  1 for some 2  k  n and, if k  3, that there exists an elliptic point on Σ. Then Σ cannot be bounded away from the future infinity. In particular, Σ must have at least one top end. 6. Further half-space theorems for spacelike hypersurfaces in GRW spacetimes In this section we will see how, under suitable conditions on the spacetime, half-space theorems in the form of non-existence results can still be proved, even dropping the assumption of Σ being cylindrically bounded. The main tool to obtain them is a version of the Omori–Yau maximum principle for the Laplacian and for more general trace-type differential operators that we are going to present. Let Σ be a Riemannian manifold and let L = Tr(P ◦ hess) be a semi-elliptic operator, where P : T Σ → T Σ is a positive semi-definite symmetric tensor. Following the terminology introduced in [6], we say that the Omori–Yau maximum principle holds on Σ for the operator L if, for any function u ∈ C 2 (Σ) with u∗ = supΣ u < +∞, there exists a sequence {pj }j∈N ⊂ Σ with the properties (i)

1 u(pj ) > u∗ − , j

(ii)



∇u(pj ) < 1 , j

(iii)

Lu(pj ) <

1 j

(15)

S.C. García-Martínez, D. Impera / Differential Geometry and its Applications 32 (2014) 46–67

61

for every j ∈ N. Equivalently, for any function u ∈ C 2 (Σ) with u∗ = inf Σ u > −∞, there exists a sequence {pj }j∈N ⊂ Σ with the properties (i)

1 u(pj ) < u∗ + , j

(ii)



∇u(pj ) < 1 , j

(iii)

Lu(pj ) > −

1 j

(16)

for every j ∈ N. Clearly the Laplacian belongs to this class of operators. In this case, Pigola, Rigoli and Setti showed in [29] that a condition of the form Ric(∇r, ∇r)  −C 2 G(r),

(17)

is sufficient to guarantee the validity of the Omori–Yau maximum principle for the Laplacian. Here r is the distance function on Σ from a fixed reference point and G : [0, +∞) → R is a smooth function satisfying (i) (iii)

G(0) > 0,

(ii)

/ L1 (+∞), G(t)− 2 ∈ 1

(iv)

G (t)  0

on [0, +∞), √ tG( t) < +∞. lim sup G(t) t→∞

(18)

Analogously, in [5], Alías, Rigoli and the second author showed that the condition K(∇r ∧ X)  −G(r),

(19)

where X is any vector field tangent to Σ and G satisfies (18), is sufficient to guarantee the validity of the Omori–Yau maximum principle on Σ for semi-elliptic trace-type operators L as above satisfying supΣ Tr P < +∞. Remark 29. Note that in [6, Theorem 1], it was proved the validity of the Omori–Yau maximum principle for the operator L as above on every Riemannian manifold Σ admitting a non-negative C 2 function γ satisfying the following requirements: (1) γ(x) → +∞ as x → ∞; √ (2) ∃A > 0 such that |∇γ|  A γ outside a compact set;  √ (3) ∃B > 0 such that Lγ  B γG( γ) outside a compact set, where G : [0, +∞) → R is a smooth function satisfying (18). It is worth to point out that it was proved by Borbély, [11] and [10], that actually condition (iv) in (18) is not needed (see also [9] and [1] for an adaptation of the same result for diffusion operators and more general semi-elliptic operators). Moreover, if L = Tr(P ◦ hess) and Tr P > 0, condition supΣ Tr P < +∞ can be dropped by substituting L with the = Tr(P ◦ hess), where P = 1 P . operator L Tr P For spacelike hypersurfaces in a GRW spacetime, the curvature conditions (17) and (19) can be replaced by curvature conditions on the fiber P and on the warping function ρ, as shown in the next Proposition 30. Let −I ×ρ Pn be a GRW spacetime satisfying the strong NCC. Let f : Σ n → −I ×ρ Pn be a  2 (h) > −∞. Then the sectional curvature complete spacelike hypersurface with supΣ A < +∞ and inf Σ ρρ(h) of Σ is bounded from below and the Omori–Yau maximum principle holds on Σ for every semi-elliptic operator L = Tr(P ◦ hess) with supΣ Tr P < +∞.

62

S.C. García-Martínez, D. Impera / Differential Geometry and its Applications 32 (2014) 46–67

Proof. Recall the Gauss equation  T R(X, Y )Z = R(X, Y )Z − AX, ZAY + AY, ZAX, for all vector fields X, Y, Z tangent to Σ, where R and R are the curvature tensors of Σ and −I ×ρ Pn , respectively. Then, if {X, Y } is an orthonormal basis for an arbitrary 2-plane tangent to Σ, we have KΣ (X ∧ Y ) = K(X ∧ Y ) − AX, XAY, Y  + AX, Y 2  K(X ∧ Y ) − AX

AY

 K(X ∧ Y ) − A 2 ,

(20)

where the last inequality follows from the fact that   2 2

AX  Tr A2 X = A 2 for every unit vector X tangent to Σ. Using Eq. (6), we find that for the orthonormal basis {X, Y } it holds K(X ∧ Y ) =

1 ρ2 (h)

2   

2 KP X ∗ ∧ Y ∗ X ∗ ∧ Y ∗ + (log ρ) (h)

  − (log ρ) (h) X, ∇h2 + Y, ∇h2 .

(21)

On the other hand,

∗



 

X ∧ Y ∗ 2 = X ∗ 2 Y ∗ 2 − X ∗ , Y ∗ 2 = 1 + X, ∇h2 + Y, ∇h2 .

(22)

Therefore, if −I ×ρ Pn satisfies the strong NCC, we deduce  2 ρ (h) , K(X ∧ Y )  (log ρ) (h) + (log ρ) (h) = ρ(h) and the conclusion follows immediately. 2 Remark 31. From the equality 2

A = n2 H12 − n(n − 1)H2 it follows that if we assume that either H2 = const. > 0 or Hk = const, 3  k  n and there exists an 2 elliptic point on Σ, the condition supΣ A < +∞ is equivalent to supΣ |H1 | < +∞. Moreover, using Gauss equation it is not difficult to prove that Ric(∇r, ∇r) 

n 

K(∇r ∧ Ei ) −

i=1

n2 2 H . 4 1

Hence, the validity of the Omori–Yau maximum principle for the Laplacian is guaranteed if, in the previous 2 proposition, the assumption supΣ A < +∞ is replaced by supΣ |H1 | < +∞. With this preparation we are ready to prove the next

S.C. García-Martínez, D. Impera / Differential Geometry and its Applications 32 (2014) 46–67

63

Theorem 32. Let Σ n be a complete spacelike hypersurface of constant mean curvature in a GRW spacetime  (h) −R ×ρ Pn which satisfies the strong NCC. Suppose that inf Σ ρρ(h) > −∞. (i) If H1 > supR (log ρ) , then Σ cannot lie in a lower half-space. (ii) If H1 < inf R (log ρ) , then Σ cannot lie in an upper half-space. Proof. Let us prove part (i) first. Assume that H1 > supR (log ρ) and, by contradiction, that Σ lies in a lower half-space, that is, supΣ h := h∗ < +∞. By Proposition 30 the Omori–Yau maximum principle holds on Σ for the Laplacian and there exists a sequence {pj }j∈N satisfying conditions (15), that is lim h(pj ) = h∗ ,

j→∞

Δh(pj ) <



∇h(pj ) 2 = Θ2 (pj ) − 1 < 1 j2

and

1 . j

By Eq. (3) we have

2   1 > Δh(pj ) = −nH1 Θ(pj ) − (log ρ) (h)(pj ) n + ∇h(pj ) . j Letting j → ∞ and using the fact that limj→∞ Θ(pj ) = −1 we obtain    0  H1 − (log ρ) h∗ . Thus, part (i) is proved since H1  (log ρ) (h∗ )  supR (log ρ) , contradicting the initial assumption on H1 . As for part (ii), we assume H1 < inf R (log ρ) and suppose by contradiction that Σ lies in an upper half-space, that is, inf Σ h := h∗ > −∞. Again, we can find a sequence {pj }j∈N satisfying conditions (16), that is lim h(pj ) = h∗ ,

j→∞



∇h(pj ) 2 = Θ2 (pj ) − 1 < 1 j2

and

1 Δh(pj ) > − . j Hence −



2   1 < Δh(pj ) = −nH1 Θ(pj ) − (log ρ) (h)(pj ) n + ∇h(pj ) , j

and we reach again a contradiction letting j → +∞.

2

For spacelike hypersurfaces in the steady state spacetime the following corollary follows immediately. Corollary 33. Let Σ n be a complete spacelike hypersurface of constant mean curvature in Hn+1 . (i) If H1 > 1, then Σ cannot be bounded away from the future infinity; (ii) If H1 < 1, then Σ cannot be bounded away from the past infinity. Note that the previous corollary could also be deduced from Theorem 3 in [2]. More generally, Corollary 34. Let Σ n be a complete spacelike hypersurface of constant mean curvature in a steady state-type spacetime −R ×et Pn satisfying KP  0.

S.C. García-Martínez, D. Impera / Differential Geometry and its Applications 32 (2014) 46–67

64

(i) If H1 > 1, then Σ cannot lie in a lower half-space; (ii) If H1 < 1, then Σ cannot lie in an upper half-space. We conclude this section with some non-existence results for complete spacelike hypersurfaces of constant k-mean curvature in case 2  k  n. Theorem 35. Let Σ n be a complete spacelike hypersurface of constant k-mean curvature for some 2  k  n, in a GRW spacetime −R×ρ Pn which satisfies the strong NCC and whose warping function is non-decreasing.  (h) > −∞. Moreover, if k  3 suppose that there exists an elliptic Suppose that supΣ |H1 | < +∞ and inf Σ ρρ(h) point on Σ. (i) If Hk > supR (log ρ) , then Σ cannot lie in a lower half-space. 1/k (ii) If 0 < Hk < inf R (log ρ) , then Σ cannot lie in an upper half-space. 1/k

Proof. Assume first that Hk > supR (log ρ) and, by contradiction, that Σ lies in a lower half-space, i.e. supΣ h := h∗ < +∞. Consider the operator L defined as in (11), i.e., 1/k

L = Tr(P ◦ hess) =

 k−1−i (log ρ) (h) ck−1 − Li . ci Θ

k−1  i=0

Note that, since the Li ’s are elliptic, 0  i  k − 1, and −(log ρ) (h)/Θ  0, the differential operator L is semi-elliptic. Moreover sup Tr P = ck−1 sup Σ

Σ

k−1  i=0

(log ρ) (h) − Θ

k−1−i Hi < +∞,

where the last inequality follows from (log ρ) (h)  supR (log ρ) < Hk

1/k

and from the fact that, by Eq. (2),

sup Hi  sup H1i < +∞. Σ

Σ

Moreover, the assumptions in Proposition 30 are met and hence the Omori–Yau maximum principle holds on Σ for the operator L. Hence, since σ(h)  σ(h∗ ) < +∞, there exists a sequence {pj }j∈N such that   lim σ(h)(pj ) = σ h∗ ,

j→∞

Lσ(h)(pj ) <





 

∇σ(h)(pj ) = ρ2 h(pj ) ∇h(pj ) < 1 j

and

1 . j

Using Eq. (12) we then have   k  k   ck−1 1 > Lσ(h)(pj ) = ρ h(pj ) − (log ρ) h(pj ) + −Θ(pj ) Hk , j (−Θ(pj ))k−1 and, letting j → ∞, we obtain    k  0  ck−1 ρ h∗ − (log ρ) h∗ + Hk . 1/k

Thus, Hk

 (log ρ) (h∗ )  supR (log ρ) , giving a contradiction.

S.C. García-Martínez, D. Impera / Differential Geometry and its Applications 32 (2014) 46–67

65

Finally, assume that Hk < inf R (log ρ) and suppose by contradiction that Σ lies in an upper half-space, that is, inf Σ h := h∗ > −∞. Note that, since Hk > 0, it follows that (log ρ) (h)  inf I (log ρ) > 0 and 1/k

Tr P = ck−1

k−1 

−

i=0

(log ρ) (h) Θ

k−1−i Hi > 0.

Hence the differential operator ◦ hess) = Tr L = Tr(P



 1 P ◦ hess Tr P

= 1 < +∞ and, since σ(h)  σ(h∗ ) > −∞, we can find a is an elliptic trace-type operator with supΣ Tr P sequence {pj }j∈N such that lim σ(h)(pj ) = σ(h∗ ),

j→∞





 

∇σ(h)(pj ) = ρ2 h(pj ) ∇h(pj ) < 1 j

and

1 Lσ(h)(p j) > − . j Hence −

  k  k  1 ck−1 ρ(h(pj )) − (log ρ) h(pj ) + −Θ(pj ) Hk < Lσ(h)(p j) = k−1 j Tr P(pj )(−Θ(pj )) <

  k  k  ρ(h(pj )) − (log ρ) h(pj ) + −Θ(pj ) Hk . k−1 i i=0 (−Θ(pj ))

(k−1)/k Hk

In the last inequality we used the fact that Tr P = ck−1

k−1 

−

i=0

 ck−1

k−1 

(log ρ) (h) Θ

k−1 

Hi

 k−1−i i/k (−Θ)−k+i+1 inf (log ρ) Hk R

i=0

> ck−1

k−1−i

k−1−i k

(−Θ)−k+i+1 Hk

i

Hkk

i=0 k−1

= ck−1 Hk k

k−1 

(−Θ)−k+i+1 .

i=0

Letting j → ∞ we then obtain 0

ρ(h∗ ) (k−1)/k kHk

 k  − (log ρ) (h∗ ) + Hk ,

leading again to a contradiction. 2 Finally, for hypersurfaces of constant higher mean curvature in the steady state spacetime and, more generally, in a steady state-type spacetime, the following corollaries are straightforward.

66

S.C. García-Martínez, D. Impera / Differential Geometry and its Applications 32 (2014) 46–67

Corollary 36. Let Σ n be a complete spacelike hypersurface in Hn+1 . Suppose that Σ n has constant k-mean curvature 2  k  n, that supΣ |H1 | < +∞ and, if k  3, that there exists an elliptic point on Σ. (i) If Hk > 1, then Σ cannot be bounded away from the future infinity; (ii) If 0 < Hk < 1, then Σ cannot be bounded away from the past infinity. Corollary 37. Let Σ n be a complete spacelike hypersurface in a steady state-type spacetime −R ×et Pn satisfying KP  0. Suppose that Σ n has constant k-mean curvature 2  k  n, that supΣ |H1 | < +∞ and, if k  3, that there exists an elliptic point on Σ. (i) If Hk > 1, then Σ cannot lie in a lower half-space; (ii) If 0 < Hk < 1, then Σ cannot lie in an upper half-space. References [1] G. Albanese, L.J. Alías, M. Rigoli, A general form of the weak maximum principle and some applications, Rev. Mat. Iberoam. 29 (4) (2013) 1437–1476. [2] A.L. Albujer, L.J. Alías, Spacelike hypersurfaces with constant mean curvature in the steady state space, Proc. Am. Math. Soc. 137 (2) (2009) 711–721. [3] L.J. Alías, J.A. Brasil, A.G. Colares, Integral formulae for spacelike hypersurfaces in conformally stationary spacetimes and applications, Proc. Edinb. Math. Soc. (2) 46 (2) (2003) 465–488. [4] L.J. Alías, A.G. Colares, Uniqueness of spacelike hypersurfaces with constant higher order mean curvature in generalized Robertson–Walker spacetimes, Math. Proc. Camb. Philos. Soc. 143 (3) (2007) 703–729. [5] L.J. Alías, D. Impera, M. Rigoli, Spacelike hypersurfaces of constant higher order mean curvature in generalized Robertson– Walker spacetimes, Math. Proc. Camb. Philos. Soc. 152 (2) (2012) 365–383. [6] L.J. Alías, D. Impera, M. Rigoli, Hypersurfaces of constant higher order mean curvature in warped products, Trans. Am. Math. Soc. 365 (2) (2013) 591–621. [7] L.J. Alías, A. Romero, M. Sánchez, Uniqueness of complete spacelike hypersurfaces of constant mean curvature in generalized Robertson–Walker spacetimes, Gen. Relativ. Gravit. 27 (1) (1995) 71–84. [8] J.L. Barbosa, A.G. Colares, Stability of hypersurfaces with constant r-mean curvature, Ann. Glob. Anal. Geom. 15 (3) (1997) 277–297. [9] G.P. Bessa, S. Pigola, A.G. Setti, Spectral and stochastic properties of the f -Laplacian, solutions of PDEs at infinity and geometric applications, Rev. Mat. Iberoam. 29 (2) (2013) 579–610. [10] A. Borbély, Immersion of manifolds with unbounded image and a modified maximum principle of Yau, Bull. Aust. Math. Soc. 78 (2) (2008) 285–291. [11] A. Borbély, A remark on the Omori–Yau maximum principle, Kuwait J. Sci. Eng. 39 (2A) (2012) 45–56. [12] D. Brill, F. Flaherty, Isolated maximal surfaces in spacetime, Commun. Math. Phys. 50 (2) (1976) 157–165. [13] E. Calabi, Examples of Bernstein problems for some nonlinear equations, in: Global Analysis, Proc. Sympos. Pure Math., vol. XV, Berkeley, Calif., 1968, 1970, pp. 223–230. [14] S.Y. Cheng, S.T. Yau, Maximal space-like hypersurfaces in the Lorentz–Minkowski spaces, Ann. Math. (2) 104 (3) (1976) 407–419. [15] X. Cheng, H. Rosenberg, Embedded positive constant r-mean curvature hypersurfaces in M m × R, An. Acad. Bras. Ciênc. 77 (2) (2005) 183–199. [16] Y. Choquet-Bruhat, Quelques propriétés des sousvariétés maximales d’une variété lorentzienne, C. R. Acad. Sci. Paris Sér. A-B 281 (14) (1975) A577–A580, Aii. [17] Y. Choquet-Bruhat, Maximal submanifolds and submanifolds with constant mean extrinsic curvature of a Lorentzian manifold, Ann. Sc. Norm. Super. Pisa Cl. Sci. (4) 3 (3) (1976) 361–376. [18] A.G. Colares, H.F. de Lima, Space-like hypersurfaces with positive constant r-mean curvature in Lorentzian product spaces, Gen. Relativ. Gravit. 40 (10) (2008) 2131–2147. [19] H.F. de Lima, A sharp height estimate for compact spacelike hypersurfaces with constant r-mean curvature in the Lorentz– Minkowski space and application, Differ. Geom. Appl. 26 (4) (2008) 445–455. [20] M.F. Elbert, Constant positive 2-mean curvature hypersurfaces, Ill. J. Math. 46 (1) (2002) 247–267. [21] L. Garding, ˙ An inequality for hyperbolic polynomials, J. Math. Mech. 8 (1959) 957–965. [22] S.W. Hawking, G.F.R. Ellis, The Large Scale Structure of Space–Time, Cambridge Monographs on Mathematical Physics, vol. 1, Cambridge University Press, London, 1973. [23] J.M. Latorre, A. Romero, Uniqueness of noncompact spacelike hypersurfaces of constant mean curvature in generalized Robertson–Walker spacetimes, Geom. Dedic. 93 (2002) 1–10. [24] R. López, Area monotonicity for spacelike surfaces with constant mean curvature, J. Geom. Phys. 52 (3) (2004) 353–363. [25] J.E. Marsden, F.J. Tipler, Maximal hypersurfaces and foliations of constant mean curvature in general relativity, Phys. Rep. 66 (3) (1980) 109–139.

S.C. García-Martínez, D. Impera / Differential Geometry and its Applications 32 (2014) 46–67

67

[26] S. Montiel, Uniqueness of spacelike hypersurfaces of constant mean curvature in foliated spacetimes, Math. Ann. 314 (3) (1999) 529–553. [27] S. Montiel, Complete non-compact spacelike hypersurfaces of constant mean curvature in de Sitter spaces, J. Math. Soc. Jpn. 55 (4) (2003) 915–938. [28] B. O’Neill, Semi-Riemannian Geometry. With Applications to Relativity, Pure and Applied Mathematics, vol. 103, Academic Press Inc., Harcourt Brace Jovanovich Publishers, New York, 1983. [29] S. Pigola, M. Rigoli, A.G. Setti, Maximum principles on Riemannian manifolds and applications, Mem. Am. Math. Soc. 174 (822) (2005), x+99.