Weak maximum principles and geometric estimates for spacelike hypersurfaces in generalized Robertson–Walker spacetimes

Linear Algebra Applications Nonlinear Analysis and 129 its (2015) 119–142 466 (2015) 102–116

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Linear Algebra and its Applications Nonlinear Analysis www.elsevier.com/locate/laa www.elsevier.com/locate/na

Inverse eigenvalue problem of Jacobi matrix Weak maximum principles and geometric estimates for spacelike mixed data hypersurfaces with in generalized Robertson–Walker spacetimes✩ Ying Rigoli Wei 1 b , Simona Scoleri b Luis J. Al´ıas a,∗ , Marco a b

Department of Mathematics, Nanjing University of Aeronautics and Astronautics, Departamento de Matem´ aticas, Universidad de Murcia, E-30100 Espinardo, Murcia, Spain Nanjing 210016, PR China Dipartimento di Matematica, Universit` a degli Studi di Milano, Via Saldini 50, I-20133, Milano, Italy

article

a r t i c l e info

i n f o abstract

a b s t r a c t

Article history: In this paper, the inverse eigenvalue problem of reconstructing Article history: this paper we deal with complete spacelike hypersurfaces in a generalized Received 16 January In 2014 a Jacobi matrix from its eigenvalues, its leading principal Received 25 March 2015Accepted 20 September Robertson–Walker spacetime. Using aspart mainof analytical tool a new local form of 2014 submatrix and the eigenvalues of its submatrix Accepted 26 August 2015 Available online 22 October 2014 the weak maximum principle for a class of operators including the Lorentzian mean is considered. The necessary and sufficient conditions for Communicated by EnzoSubmitted Mitidieri by Y. Wei curvature operator, we obtain some mean curvature estimates and height estimates MSC: 53C40 53C42

MSC: 15A18 15A57

Keywords: Keywords: Generalized Robertson–Walker Jacobi matrix spacetimes Eigenvalue Weak maximum principle Inverse problem Spacelike graphs Submatrix Lorentzian mean curvature operator

the existence and uniqueness of the solution are derived. for spacelike graphs with nice Bernstein type consequences. Weand also some give height estiFurthermore, a numerical algorithm numerical mates for spacelike hypersurfaces with constant higher order mean curvature, which examples are given. are obtained via the local form of the weak maximum givenInc. in © 2014 principle Publishedrecently by Elsevier Al´ıas et al. (in press) for a class of operators including those constructed from the Newton tensors of a hypersurface. © 2015 Elsevier Ltd. All rights reserved.

1. Introduction The study of constant mean curvature spacelike hypersurfaces in Lorentzian manifolds is motivated by physical and mathematical interests; indeed, Lorentzian geometry is the geometric language of general relativity. In this setting spacelike hypersurfaces play a central role being involved, for example, in the initial value formulation of the field equations. This latter consists, roughly speaking, in prescribing quantities determining the spacetime structure. To be more precise, one considers as initial data a triple (Σ , g, K), where Σ is a smooth manifold, g a Riemannian metric and K a symmetric tensor field on Σ , and one looks for a spacetime (M, ⟨,E-mail ⟩) satisfying Einstein’s equation and possessing a spacelike hypersurface isometric address: [email protected]. 1 Tel.: +86 13914485239. to (Σ , g) and with extrinsic curvature (in physical language; that is, second fundamental form) K. This http://dx.doi.org/10.1016/j.laa.2014.09.031 0024-3795/© 2014 Published by Elsevier Inc. This work was partially supported by MINECO (Ministerio de Econom´ıa y Competitividad) and FEDER (Fondo Europeo de Desarrollo Regional) project MTM2012-34037, Spain. M. Rigoli was partially supported by Fundaci´ on S´ eneca Grant 18883/IV/13, Programa Jim´ enez de la Espada, Regi´ on de Murcia, Spain. ∗ Corresponding author. ✩

E-mail addresses: [email protected] (L.J. Al´ıas), [email protected] (M. Rigoli), [email protected] (S. Scoleri).

http://dx.doi.org/10.1016/j.na.2015.08.018 0362-546X/© 2015 Elsevier Ltd. All rights reserved.

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spacetime is foliated by spacelike hypersurfaces Σt that represent, in some sense, the time evolution of Σ0 := Σ . See [21,14] for details and physical interpretations. In 1968 Calabi showed that spacelike hypersurfaces in the Lorentz–Minkowski space Rn+1 have a nice 1 Bernstein-type property. More precisely in [9] he proved that the only complete maximal spacelike hypersurfaces in Rn+1 , with n ≤ 4, are spacelike hyperplanes, later on Cheng and Yau [12] extended the theorem to 1 any dimension. In another direction, this result can be generalized to prove that spacelike hyperplanes are with image of the Gauss map contained the only complete constant mean curvature hypersurfaces in Rn+1 1 in a geodesic ball of the hyperbolic space (see [15,1,22]). In this paper we deal with complete spacelike hypersurfaces in a generalized Robertson–Walker spacetime, that is, following the terminology introduced in [6], in a Lorentzian warped product manifold M n+1 = −I ×ϱ Pn . In the special case when the Riemannian factor Pn has constant sectional curvature we obtain the classical Robertson–Walker spacetimes. The warped manifold M n+1 is foliated by the slices Mt := {t} × Pn , that constitute a family of totally umbilical leaves with constant k-mean curvature, for each k = 1, . . . , n. Thus, the Calabi–Bernstein problem in this general setting becomes natural: it amounts to investigate under which circumstances a complete constant rth-mean curvature spacelike hypersurface is a spacelike slice Mt . In [6,3,13] the authors consider the compact case, that is the case of compact spacelike hypersurfaces in spatially closed general Robertson–Walker spacetimes. They show that, with some additional hypothesis on the ambient space, namely the null convergence condition, immersed compact spacelike hypersurfaces with constant k-mean curvature must be slices, unless other very special cases occur. Recall that a spacetime obeys the null convergence condition if its Ricci curvature is non-negative on lightlike directions. In [4] the authors extend this result to the complete case. To achieve their goal, they assume the sectional curvature of the Riemannian factor to be bounded from below and the hypersurface to be contained in a slab [a, b] × Pn ⊆ M n+1 , a condition automatically satisfied in the compact case. An interesting approach to these uniqueness questions is via “a priori” height estimates for constant k-mean curvature spacelike hypersurfaces. Roughly speaking, estimates of this type give a quantitative measure of the deviation of the hypersurface from being a slice and they can also be used to get informations on the topology of the immersion at infinity. Consider, now, a spacelike hypersurface Σ → −R ×ϱ Pn ; call ∂t the coordinate vector field along R and N the unique timelike normal globally defined on Σ with the same orientation of ∂t . Define Θ := ⟨N, ∂t ⟩. The assumption on Σ to be spacelike implies the bound Θ ≤ −1, that enables us to introduce the hyperbolic angle θ > 0 defined by Θ = − cosh θ. In some of our results we assume that Θ is bounded. Although this hypothesis seems only to have a technical role in our arguments, we show, by way of a simple counterexample, that it is indeed necessary. Furthermore, this assumption has a nice physical interpretation as follows. In a spacetime (M, ⟨, ⟩) the exponential map provides an efficient tool to formalize the concept of “observer”. An observer on M is forced to move into the future, and thus it can be modeled by a future pointing timelike curve. Often it suffices to consider an instantaneous observer, that is, a couple (p, X), where p is a point of M and X ∈ Tp M is a future pointing timelike direction. Along Σ we can consider two relevant observers: (p, Np ) and (p, ∂t| p ). Consider the orthogonal decomposition N = −Θ∂t + w, where w ∈ T Pn . By definition v := |w|/|Θ| = tanh θ is the Newtonian speed of N observed by ∂t . So, the assumption of the boundedness of θ assures that v does not approach the speed of light in vacuum. See [19, pp. 41–45] for further details. The paper is organized as follows. In Section 2 we present our main analytical tool to study the problem, that is, a local form of the weak maximum principle in a Riemannian manifold (M, ⟨, ⟩) for a class of operators including the Lorentzian mean curvature operator. This version of the maximum principle, called the open weak maximum principle, has been introduced in [5] for operators of the form Lϕ u = div(|∇u|−1 ϕ(|∇u|)∇u),

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where ϕ ∈ C 0 ([0, ∞)) ∩ C 1 ((0, ∞)) satisfies some structural conditions and u has a suitable regularity; for example u can be supposed of class C 1 and correspondingly the divergence has to be understood in the weak sense. Sufficient conditions for its validity can be found in [2] and in [17]; we only recall here that when Lϕ = ∆, the Laplace–Beltrami operator, its validity is equivalent to that of stochastic completeness of the manifold, as proven in [16]. However, this class of operators does not include the Lorentzian mean curvature operator   Du , Lu := div  1 − |Du|2  defined on A1 (P) = {v ∈ Liploc (P) : |Dv| < 1 and 1/ 1 − |Dv|2 ∈ L1loc (P)}, that obviously plays a central role in the study of spacelike graphs. Thus, in Proposition 2.3 we present a volume growth condition ensuring the validity of the weak maximum principle for a large class of operators containing L. Furthermore, in Theorem 2.2 we show that the open form is equivalent to the classical version of the principle. In Section 3 we study spacelike hypersurfaces Σ n in generalized Robertson–Walker spacetimes −R ×ϱ Pn . Assuming the hyperbolic angle of Σ n to be bounded, we estimate the mean curvature of the hypersurface by means of the Cheeger constant of the fiber Pn . This result implies, in particular, that a CMC spacelike hypersurface in a Lorentzian product −R × Pn is maximal provided Pn has zero Cheeger constant, a result first due to Salavessa [20]. Then, we assume on Pn the validity of the weak maximum principle for the Lorentzian mean curvature operator and we use its local form to prove that the maximum and the minimum of the function u : Pn → R defining an entire maximal graph different from a slice are related to the sign of ϱ′ . This fact has a nice Bernstein type consequence. Specifically, if the graph Σ (u) ⊂ −R ×ϱ Pn is confined in a slab (a, b) × Pn and the warping function ϱ has exactly a maximum at t0 ∈ (a, b), then Σ (u) is the slice determined by u(x) ≡ t0 (see Corollary 3.8 for the precise statement). We conclude Section 3 with another geometric application of the open weak maximum principle for the Lorentzian mean curvature operator. In particular, we give a height estimate for a spacelike, bounded above graph whose mean curvature is not everywhere negative. Finally, in Section 4 we consider constant k-mean curvature spacelike hypersurfaces in a generalized Robertson–Walker spacetime. The estimates we obtain are the Lorentzian analogue of height estimates presented in [5] in the Riemannian case and they are deduced via the open weak maximum principle for the Laplacian and for some more general differential operators constructed from the Newton tensors of the immersion. To illustrate these latter results we report the following, easy to read, consequence of Theorem 4.10. Theorem 1.1. Let F : Σ n → −R × Pn be a complete spacelike hypersurface with constant (future) mean curvature H > 0. Assume that the height function h = πR ◦ F : Σ → R satisfies lim h(x) = −∞.

x→∞

Suppose that RicP ≥ 0.

(1)

Let Ω ⊂ Σ be a relatively compact open set with ∂Ω ̸= ∅ such that F (∂Ω ) ⊂ {0} × Pn , and let β = sup |Θ|.

(2)

Ω

Then 

 (1 − β) , 0 × Pn . F (Ω ) ⊂ H

(3)

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2. The open weak maximum principle for the Lorentzian mean curvature operator In [5] the authors gave an equivalent form of the weak maximum principle valid for a very general class of operators on a Riemannian manifold (M, ⟨, ⟩), including in particular, those of the following form:   Lϕ u = div |∇u|−1 ϕ(|∇u|)∇u (4) + for u ∈ C 1 (M ), and to be understood in the weak sense. Here, setting R+ 0 = [0, +∞) and R = (0, +∞), the + function ϕ ∈ C 0 (R0 ) ∩ C 1 (R+ ) satisfies the structural conditions   (i) ϕ(0) = 0; (5) (ii) ϕ(t) > 0 on R+ ;  (iii) ϕ(t) ≤ Atδ on R+

for some constants A, δ > 0. Specifically, the following is a particular case of Theorem 2.5 in [5] that shall be used in Section 4. Theorem 2.1. The WMP holds on M for the operator Lϕ if and only if for each f ∈ C 0 (R), for each open set Ω ⊂ M with ∂Ω ̸= ∅, and for each v ∈ C 0 (Ω ) ∩ C 1 (Ω ) satisfying  (i) Lϕ v ≥ f (v) on Ω ; (6) (ii) sup v < +∞, Ω

we have that either sup v = sup v Ω

(7)

∂Ω

or f (sup v) ≤ 0.

(8)

Ω

In this paper, motivated by the differential equation arising from the mean curvature of a spacelike graph in Lorentzian geometry (see Section 3), we are interested in the family of operators that we now describe. Let φ ∈ C 0 ([0, ω)) ∩ C 1 ((0, ω)) for some 0 < ω < +∞, and suppose   (i) φ(0) = 0; (9) (ii) φ(t) > 0 on (0, ω);  (iii) φ(t) ≤ Atδ on (0, ω) for some constants A, δ > 0. For u ∈ Aω (M ) = {u ∈ Liploc (M ) : |∇u| < ω and |∇u|−1 φ(|∇u|) ∈ L1loc (M )}, define   Lφ u = div |∇u|−1 φ(|∇u|)∇u

(10)

in the weak sense. Following [5], for q(x) ∈ C 0 (M ), q(x) > 0, we say that the q-weak maximum principle (shortly q-WMP) holds on M for the operator Lφ if for each u ∈ Aω (M ) bounded from above and for each γ < u∗ = supM u we have inf {q(x)Lφ u} ≤ 0 Ωγ

(11)

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in the weak sense, where Ωγ = {x ∈ M : u(x) > γ}. In case q(x) is a positive constant we will simply say that Lφ satisfies the WMP (the weak maximum principle). Recall that (11) in the weak sense means that for each ε > 0 there exists ψ ∈ Cc∞ (Ωγ ), ψ ≥ 0, ψ ̸≡ 0, such that   ε −1 ψ. − |∇u| φ(|∇u|)⟨∇u, ∇ψ⟩ ≤ q(x) Ωγ Ωγ On the other hand, again following [5], we say that the open q-WMP holds on M for the operator Lφ if for each f ∈ C 0 (R), for each open set Ω ⊂ M with ∂Ω ̸= ∅ and for each v ∈ C 0 (Ω ) ∩ Aω (Ω ) satisfying  (i) q(x)Lφ v ≥ f (v) on Ω ; (12) (ii) sup v < +∞, Ω

we have that either sup v = sup v Ω

(13)

∂Ω

or f (sup v) ≤ 0.

(14)

Ω

Note that (i) in (12) has to be understood in the weak sense, that is, for each ψ ∈ Cc∞ (Ω ), ψ ≥ 0,   f (v) −1 ψ. − |∇u| φ(|∇u|)⟨∇u, ∇ψ⟩ ≥ Ωγ Ωγ q(x) In this paper we will need the following version of the equivalence of the two forms of the maximum principle for this family of operators. Theorem 2.2. In the above assumptions, the validity of the q-WMP for the operator Lφ is equivalent to that of the open q-WMP. Proof. Assume that the q-WMP holds for the operator Lφ on M and let Ω , f , v be as above, with v satisfying (12). We suppose sup∂Ω v < supΩ v and we claim f (v ∗ ) ≤ 0. Fix sup∂Ω v < γ < v ∗ and define Ωγ := {x ∈ Ω : v(x) > γ}. In our setting Ωγ ⊂ Ω . Consider the function  v on Ωγ u := γ on M \ Ωγ and observe that u ∈ Aω (M ) and that u∗ = supM u = v ∗ = supΩ v. Choose γ < σ < u∗ = v ∗ . Since we are supposing the validity of the q-WMP, then for each ε > 0 there exists ψ ∈ Cc∞ (Ωσ ), ψ ≥ 0 and ψ ̸≡ 0, such that   ε −1 − |∇u| φ(|∇u|)⟨∇u, ∇ψ⟩ ≤ ψ. (15) q(x) Ωσ Ωσ On the other hand, since supp(ψ) ⊂ Ωσ ⊂ Ω , and since we are assuming q(x)Lφ v ≥ f (v) on Ω in the weak sense, we also have that   f (v) ψ≤− |∇v|−1 φ(|∇v|)⟨∇v, ∇ψ⟩. (16) q(x) Ωσ Ωσ Note that u = v on Ωσ and therefore from (15) and (16) we deduce   f (v) ε ψ≤ ψ. Ωσ q(x) Ωσ q(x)

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Now, fix ε > 0 and, recalling that f in continuous, consider σ sufficiently close to v ∗ so that f (v) > f (v ∗ )−ε on Ωσ . Hence from the above we deduce   ψ ψ ∗ ≤ε , (f (v ) − ε) q(x) q(x) Ωσ Ωσ  ψ > 0, we deduce f (v ∗ ) < 2ε. But the choice of ε is where ψ depends on the choice of ε and σ. Since Ωσ q(x) ∗ arbitrary, so that f (v ) ≤ 0. Assume, now, the validity of the open q-WMP and consider u ∈ Aω (M ) bounded above. Fix γ < u∗ ; we claim that inf Ωγ {q(x)Lφ u} ≤ 0 in the weak sense. By contradiction, suppose that there exists ε > 0 such that for each ψ ∈ Cc∞ (Ωγ ), ψ ≥ 0   ε −1 − |∇u| φ(|∇u|)⟨∇u, ∇ψ⟩ > ψ. q(x) Ωγ Ωγ This means q(x)Lφ u ≥ ε weakly on Ωγ . Note that γ = sup u < sup u = u∗ < ∞.

(17)

Ωγ

∂Ωγ

Applying the open q-WMP with Ω = Ωγ , v = u|Ωγ ∈ Aω and f = ε, inequality (17) yields the desired contradiction.  Since it turns out that the q-WMP for Lφ is a very useful tool in the study of spacelike hypersurfaces in Lorentzian manifolds we give a geometric condition on the volume-growth of geodesic balls that guarantees its validity. Proposition 2.3. Let (M, ⟨, ⟩) be a complete Riemannian manifold. Assume that lim inf r→∞

log Vol Br = d0 < +∞. r1+δ

(18)

Let u ∈ Aω (M ) such that u∗ := supM u < ∞. Then for all γ < u∗ we have inf Lφ u ≤ 0

(19)

Ωγ

in the weak sense, where Ωγ = {x ∈ M : u(x) > γ}. In other words, under assumption (18) the WMP holds on M for the operator Lφ . Proof. We follow the proof of Theorem 1.1 in [18] (see also the proof of Theorem 4.2 in [17]). If ν ∈ R and we set uν := u + ν, we have Lφ (uν ) = Lφ (u) in the weak sense, so we can replace u with uν , where ν > 0 is such that u∗ν > 0. With abuse of notation we are going to omit the subscript ν. Fix γ < u∗ and note that Ωt ⊂ Ωs if t > s, so that we may suppose without lost of generality that γ ≥ 0. Next we let K := inf Lφ u Ωγ

= sup {a ∈ R : Lφ u ≥ a}  = sup a ∈ R : ∀ψ ∈

Cc∞ (Ωγ ),

 ψ ≥ 0, −

−1

|∇u| Ωγ



 φ(|∇u|)⟨∇u, ∇ψ⟩ ≥ a

ψ

,

(20)

Ωγ

and assume by contradiction that K > 0. Observe that, in this case, u is nonconstant on any connected component of Ωγ .

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We fix θ ∈ (1/2, 1) and choose R0 > 0 such that BR0 ∩ Ωγ ̸= ∅. Given R > R0 , let ψ ∈ C ∞ (M ) be a cut-off function such that (i) 0 ≤ ψ ≤ 1; (ii) ψ ≡ 1 on BθR ; (iii) ψ ≡ 0 on M \ BR ; 2 (iv) |∇ψ| ≤ . R(1 − θ) Let also ξ ∈ C ∞ (R) be such that 0 ≤ ξ ≤ 1, supp(ξ) = (γ, +∞) and ξ ′ ≥ 0. Consider the test function f = ψ 1+δ ξ(u) exp(g) ≥ 0 where g := −q(α − u)r1+δ for some q > 0 and α > u∗ to be chosen later. Denoting with Ω := Ωγ ∩ BR , observe that f ∈ Liploc (Ω ) and supp(f ) ⊂ Ω , and therefore it is an admissible test function for (20). Then we have   − |∇u|−1 φ(|∇u|)⟨∇u, ∇f ⟩ = − |∇u|−1 φ(|∇u|)⟨∇u, ∇f ⟩ Ω

Ωγ

 ≥K

 f =K

Ωγ

f.

(21)

Ω

Computing ∇f = (1 + δ)ψ δ ξ(u) exp(g)∇ψ + ψ 1+δ ξ ′ (u) exp(g)∇u − (1 + δ)q(1 + r)δ ψ 1+δ ξ(u) exp(g)(α − u)∇r + qψ 1+δ ξ(u) exp(g)(1 + r)1+δ ∇u and substituting it into (21), using ξ ′ ≥ 0 and the Cauchy–Schwarz inequality, we obtain the estimate    −φ(|∇u|)(1 + δ)ψ δ ξ(u)eg |∇ψ| + ψ 1+δ ξ(u)eg q(1 + r)1+δ B(|∇u|, r) ≤ 0 (22) Ω

where B(|∇u|, r) =

K + |∇u|φ(|∇u|) − (1 + δ)(α − u)(1 + r)−1 φ(|∇u|) q(1 + r)1+δ

≥

K 1 + 1/δ φ(|∇u|)1+1/δ − (1 + δ)α(1 + r)−1 φ(|∇u|) q(1 + r)1+δ A

(23)

on Ω , since γ ≥ 0. Observe that in the last inequality we have used that tφ(t) ≥ A−1/δ φ(t)1+1/δ , which follows from the structural condition φ(t) ≤ Atδ . At this time we need to estimate the right hand side of (23) so as to have B(|∇u|, r) ≥ Λφ(|∇u|)1+1/δ

(24)

for some positive constant Λ independent of |∇u| and r. Towards this end, we apply Lemma 4.2 in [17] with the choices (following the notation of Lemma 4.2): ω = 1/A1/δ , ρ = (K/q)(1 + r)−(1+δ) , and β = α(1 + δ)(1 + r)−1 . Applying Lemma 4.2 with s = φ(|∇u|) and r ≥ 0 fixed, it is easy to verify the validity of (24) provided 1 δα1+1/δ q 1/δ − . (25) 1/δ A K 1/δ Since the right hand side of the above inequality is independent of r, for every such Λ (24) holds. In particular, if τ ∈ (0, 1) and we choose Λ≤

q=

τ δK Aδ δ α1+δ

(26)

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then Λ=

1−τ >0 A1/δ

and it satisfies (25). We insert now (24) into (22) to obtain   qΛ 1+δ g 1+δ 1+1/δ ψ ξ(u)e (1 + r) φ(|∇u|) ≤ ψ δ ξ(u)eg |∇ψ|φ(|∇u|). 1+δ Ω Ω Applying H¨ older’s inequality with conjugate exponents 1 + δ and 1 + 1/δ to the integral on the right hand side and simplifying, we obtain 1+δ    qΛ ψ 1+δ ξ(u)eg (1 + r)1+δ φ(|∇u|)1+1/δ ≤ ξ(u)eg (1 + r)−δ(1+δ) |∇ψ|1+δ . (27) 1+δ Ω Ω Recall that Ω = Ωγ ∩ BR . By the volume growth assumption (18), for every d > d0 there exists a diverging sequence Rk ↑ +∞ with R1 > 2R0 such that log Vol BRk ≤ dRk1+δ . Noting that θRk > Rk /2 > R0 , we apply (27) with R = Rk , and use the bound for |∇ψ| and the fact that ξ ≤ 1 to get  1+δ  qΛ E = ξ(u)eg φ(|∇u|)1+1/δ 1+δ Ωγ ∩BR0  1+δ  qΛ ≤ ψ 1+δ ξ(u)eg (1 + r)δ(1+δ) φ(|∇u|)1+1/δ 1+δ Ωγ ∩BR0  ≤ ξ(u)eg (1 + r)−δ(1+δ) |∇ψ|1+δ Ωγ ∩BRk

 ≤

Ωγ ∩(BRk \BθRk )

ξ(u)eg (1 + r)−δ(1+δ) |∇ψ|1+δ

21+δ ≤ (1 + θRk )δ(1+δ) (1 − θ)1+δ Rk1+δ

 Ωγ ∩(BRk \BθRk )

eg .

It follows from here that −(1+δ)2

E ≤ CRk

 Ωγ ∩(BRk \BθRk )

eg ,

(28)

where C > 0 is a constant independent of k. Now we observe that since |∇u| ̸≡ 0 on Ωγ ∩ BR0 , then E > 0. On the other hand, taking into account that ∗

eg ≤ e−q(α−u

)(1+θRk )1+δ

on Ωγ ∩ (BRk \ BθRk ), inserting this into (28) we obtain the inequality −(1+δ)2

0 < E ≤ CRk

exp(dRk1+δ − q(α − u∗ )(1 + θRk )1+δ ).

In order for this inequality to hold for every k, we must have d ≥ q(α − u∗ )θ1+δ , and letting θ → 1, d ≥ q(α − u∗ ).

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Set α = tu∗ , t > 1, and insert the definition of q (26) in the above inequality, solve with respect to K and let τ → 1 to obtain K ≤ Ad(u∗ )δ δ δ

t1+δ . t−1

Taking into account that mint>1

t1+δ (1 + δ)1+δ = t−1 δδ

and letting d → d0 we obtain K ≤ Ad0 (u∗ )δ (1 + δ)1+δ . Now fix ε > 0. As we observe at the beginning of the proof K does not depend on adding a constant to u, and therefore we can suppose that u∗ = ε. Since ε is arbitrary, this yields K ≤ 0, contradiction.  3. Spacelike graphs in generalized Robertson–Walker spacetimes Let (Pn , ⟨, ⟩P ) be a complete n-dimensional Riemannian manifold and ϱ : I → R+ a smooth function defined on the open interval I ⊆ R. We denote by M n+1 := −I ×ϱ Pn the Lorentzian warped product endowed with the Lorentzian metric ⟨, ⟩ = −πI∗ (dt2 ) + ϱ2 (πI )(πP∗ (⟨, ⟩P )), where πI and πP are the projections, respectively, on the I and P factors of the product and t is the global coordinate on I. Following [6], we will refer to M n+1 as a generalized Robertson–Walker (GRW) spacetime. Consider a spacelike hypersurface F : Σ n → M n+1 and let N denote the unique timelike normal globally defined on Σ with the same time orientation of ∂t . We set Θ := ⟨N, ∂t ⟩ = − cosh θ ≤ −1,

θ ≥ 0,

and from now on we refer to θ as to the hyperbolic angle between Σ and ∂t . We also refer to N as the future-pointing Gauss map of the hypersurface. Let A : T Σ → T Σ be the second fundamental tensor of the immersion in the direction of N . Its eigenvalues κ1 , . . . , κn are called the principal curvatures of the hypersurface, and the mean curvature of the immersion with respect to N is defined by n

1 H=− κi . n i=1 We will refer to H as the future mean curvature function of Σ . Given u ∈ C ∞ (Pn ), u : Pn → I, we denote by Σ (u) the graph of u in the GRW spacetime, that is, Σ (u) := {(u(x), x) ∈ I × Pn : x ∈ Pn } ⊂ −I ×ϱ Pn . The metric induced on P from the Lorentzian metric of the ambient space via Σ (u) is given by ⟨, ⟩ = −du2 + ϱ(u)2 ⟨, ⟩P . Therefore, Σ (u) is spacelike if and only if |Du|2 < ϱ(u)2 everywhere on P, where Du denotes the gradient of u in P and |Du| denotes its norm, both with respect to the original metric on P. If Σ (u) is spacelike, then it is not difficult to see that the vector field 1  N= (ϱ(u)2 ∂t + Du) 2 ϱ(u) ϱ (u) − |Du|2

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defines the future-pointing Gauss map of Σ (u), for which we have ϱ(u) . cosh θ = −Θ =  2 ϱ (u) − |Du|2 In particular, |Du| sinh θ =  . 2 ϱ (u) − |Du|2 The corresponding mean curvature H(u), with respect to our choice of N , is given by the following differential equation     ϱ′ (u) |Du|2 Du  = nH(u), (29) divP + n+ 2 ϱ (u) ϱ(u) ϱ2 (u) − |Du|2 ϱ2 (u) − |Du|2 where divP denotes the divergence operator on the original (Pn , ⟨, ⟩P ). Observe that the above equation can also be written in the following equivalent form   nϱ′ (u) 1 Du divP  + = nH(u). (30) ϱ(u) ϱ2 (u) − |Du|2 ϱ2 (u) − |Du|2 Recall that for a complete and non-compact manifold as Pn , the Cheeger constant is defined by Area(∂Ω ) , Vol(Ω )

hPn := inf Ω

where Ω ranges over all open relatively compact subsets of Pn with smooth boundary. Note that in the compact case the Cheeger constant is defined in a slightly different way. Cheeger proved in [10] that λ1 (P) ≥

1 2 h , 4 P

where λ1 (P) is the spectral radius of Pn . On the other hand, P. Buser in [8] found an analogous upper bound for λ1 (P). Namely he proved that if RicP ≥ −(n − 1)δ 2 , δ ≥ 0, then λ1 (P) ≤ 2δ(n − 1)hP + 10h2P . Thus λ1 and hPn can be considered equivalent in some sense; for example, under the above bound, λ1 = 0 if and only if hPn = 0. One may ask under which geometric conditions λ1 (Pn ) = 0. It turns that if Pn is complete and non-compact, and it has subexponential volume growth then λ1 (Pn ) = 0 (see [11,7]). Extending a simple but clever idea of Salavessa [20], we obtain the following. Theorem 3.1. Let (Pn , ⟨, ⟩P ) be an n-dimensional complete, non-compact Riemannian manifold, and let u ∈ C ∞ (Pn ), with u : P → I = (a, b), be such that Σ (u) is a spacelike graph in −I ×ϱ Pn with bounded hyperbolic angle. If ϱ′ ≤ 0 then inf H ≤ P

sinh θ∗ hP , nϱ(u∗ )

where θ∗ = supP θ < +∞ and u∗ = supP u ≤ b. Analogously if ϱ′ ≥ 0 then sup H ≥ − P

sinh θ∗ hP . nϱ(u∗ )

Note that it could be either ϱ(u∗ ) = 0 or ϱ(u∗ ) = 0; in these cases the corresponding inequalities are trivial.

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129

Proof. Consider the case ϱ′ ≤ 0 and let Ω ⊂⊂ Pn be a relatively compact domain with smooth boundary. Integrating (29) over Ω and using the divergence theorem we obtain     ϱ′ (u) |Du|2 ⟨Du, ν⟩   . + n+ 2 n inf H Vol(Ω ) ≤ 2 2 Ω ϱ (u) ϱ2 (u) − |Du|2 Ω ∂Ω ϱ(u) ϱ (u) − |Du| Since ϱ′ ≤ 0, ϱ(u) ≥ ϱ(u∗ ) and  n inf H Vol(Ω ) ≤ Ω

∂Ω

|Du|  = ϱ(u) ϱ2 (u) − |Du|2

 ∂Ω

sinh θ sinh θ∗ ≤ Area(∂Ω ). ϱ(u) ϱ(u∗ )

Therefore, for any such Ω ⊂ P we have inf H ≤ P

sinh θ∗ Area(∂Ω ) nϱ(u∗ ) Vol(Ω )

and hence inf H ≤ P

sinh θ∗ hP . nϱ(u∗ )

′

The case where ϱ ≥ 0 follows from a similar argument.



From Theorem 3.1 we immediately obtain: Corollary 3.2. Let (Pn , ⟨, ⟩P ) be an n-dimensional complete, non-compact Riemannian manifold with vanishing Cheeger constant, and let u ∈ C ∞ (Pn ), with u : P → I = (a, b), be such that Σ (u) is a spacelike graph in −I ×ϱ Pn with bounded hyperbolic angle. If ϱ′ ≤ 0 and u∗ < b then inf P H ≤ 0. Analogously if ϱ′ ≥ 0 and u∗ > a then supP H ≥ 0. In the Lorentzian product case ϱ ≡ 1 and Theorem 3.1 becomes Corollary 3.3. Let (Pn , ⟨, ⟩P ) be an n-dimensional Riemannian manifold. Let u ∈ C ∞ (Pn ) such that Σ (u) is a spacelike graph of the Lorentzian product −R × Pn for which the hyperbolic angle θ is bounded. Then inf H ≤ P

sinh θ∗ hP n

and

sup H ≥ − P

sinh θ∗ hP n

∗

where θ = supP θ < +∞ and hP is the Cheeger constant of P. In the special case of a constant mean curvature spacelike graph in a Lorentzian product, we recover a result of Salavessa [20]. Corollary 3.4. Let (Pn , ⟨, ⟩P ) be an n-dimensional Riemannian manifold with vanishing Cheeger constant. Let u ∈ C ∞ (Pn ) such that Σ (u) is a spacelike constant mean curvature hypersurface of the Lorentzian product −R × Pn for which the hyperbolic angle θ is bounded. Then Σ (u) is maximal. Note that the hypothesis on the hyperbolic angle to be bounded is necessary. Indeed, consider the function  u(x, y) = 1 + x2 + y 2 defined on R2 . Its graph is one sheet of the hyperboloid z 2 − x2 − y 2 = 1 immersed in the Minkowski space R31 . Its hyperbolic angle is not bounded since   x y Du =  , 1 + x2 + y 2 1 + x2 + y 2 |Du|2 =

x2 + y 2 →1 1 + x2 + y 2

if |(x, y)| → +∞.

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Computing the mean curvature of the graph of u we obtain H = 2 ̸= 0. Since hR2 = 0 we have an example of spacelike non-maximal constant mean curvature graph in a product −R × R2 where the fiber has zero Cheeger constant. Theorem 3.1 can be generalized to the case of spacelike hypersurfaces as follows. Theorem 3.5. Let F : Σ n → −I ×ϱ Pn be a complete, non-compact spacelike hypersurface for which the hyperbolic angle is bounded. If ϱ′ ≤ 0 then inf H ≤ Σ

sinh θ∗ hΣ . n

′

Analogously, if ϱ ≥ 0 then sup H ≥ − Σ

sinh θ∗ hΣ , n

where θ∗ = supΣ θ < +∞ and hΣ is the Cheeger constant of Σ . Proof. Consider the height function h = πI ◦ F : Σ → I ⊂ R, whose gradient is given by ∇h = −∂t − ΘN = −∂t + cosh θ N. In particular, |∇h|2 = Θ 2 − 1 = sinh2 θ. Observe that ∇X ∇h = −

ϱ′ (h) (⟨X, ∇h⟩∇h + X) + ΘAX ϱ(h)

for every X ∈ T Σ . Therefore, ∆h = −

 ϱ′ (h)  |∇h|2 + n − nΘH. ϱ(h)

(31)

Consider the case where ϱ′ ≤ 0. Then from the above we obtain ∆h ≥ −nΘH = n cosh θ H. If inf Σ H < 0 there is nothing to prove. So, without loss of generality we can assume that inf Σ H ≥ 0. Fix Ω ⊂⊂ Σ a relatively compact domain with smooth boundary. Integrating over Ω and using divergence theorem we obtain    n inf H Vol(Ω ) ≤ nH cosh θ ≤ ∆h ≤ |∇h| ≤ sinh θ∗ Area(∂Ω ). Ω

Ω

Ω

∂Ω

Therefore, inf H ≤ Σ

The case where ϱ′ ≥ 0 is obtained in a similar way.

sinh θ∗ hΣ . n 

As a consequence Corollary 3.6. Let F : Σ n → −R × Pn be a complete, non-compact, constant mean curvature spacelike hypersurface for which the hyperbolic angle is bounded. If hΣ = 0 then Σ is maximal.

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131

In the statement of the next results we will assume the validity on the Riemannian manifold (Pn , ⟨, ⟩P ) of the weak maximum principle for the Lorentzian mean curvature operator   Dv , Lv := divP  1 − |Dv|2 with v in the class A1 (P) = {v ∈ Liploc (P) : |Dv| < 1 and 1/

 1 − |Dv|2 ∈ L1loc (P)}.

According to Proposition 2.3, this is guaranteed by the volume growth condition (18) with δ = 1, that is, log Vol Br < +∞. r2 In terms of curvature, note that condition (32) is implied by

(32)

lim inf r→∞

RicP ≥ −c(1 + r2 )⟨, ⟩P for some constant c > 0. Theorem 3.7. Consider a generalized Robertson–Walker spacetime −I ×ϱ Pn , and assume on Pn the validity of the weak maximum principle for the Lorentzian mean curvature operator. Let Σ (u) be an entire spacelike maximal graph in −I ×ϱ Pn , with I = (a, b), −∞ ≤ a < b ≤ +∞, which is not a slice. Then either u∗ = b or

u∗ ≤ inf{λ ∈ I : ϱ′ (t) < 0 on [λ, b)}.

(33)

either u∗ = a

u∗ ≥ sup{µ ∈ I : ϱ′ (t) > 0 on (a, µ]}.

(34)

Similarly, or

Proof. Let us prove (33). The proof of (34) is analogous. Suppose that u∗ < b and, by contradiction, assume that u∗ > inf{λ ∈ I : ϱ′ (t) < 0 on [λ, b)}. Choose λ < u∗ such that ϱ′ (t) < 0 on [λ, b) and sufficiently near to u∗ so that if Λλ = {x ∈ Σ : u(x) > λ}, then ∂Λλ ̸= ∅. We fix an origin o ∈ Pn and for u0 := u(o) we consider the function  s dt 1 ψ(s) := , for which ψ ′ = > 0. ϱ(t) ϱ u0 Setting v(x) = ψ(u(x)), a calculation using (30) with H = 0 shows that   Dv ϱ′ (ψ −1 (v)) divP  = −n  . 1 − |Dv|2 1 − |Dv|2 Let γ = ψ(λ) and observe that Ωγ = {x ∈ Σ : v(x) > γ} = {x ∈ Σ : u(x) > λ} = Λλ . Since ϱ′ (ψ −1 (v)) < 0 on Ωγ , we have  divP

Dv

 1 − |Dv|2

 ≥ −nϱ′ (ψ −1 (v))

Now observe that sup v = ψ(u∗ ) > γ = sup v. Ωγ

∂Ωγ

on Ωγ .

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Therefore, by the open form of the weak maximum principle we conclude that −nϱ′ (u∗ ) ≤ 0, 

yielding the desired contradiction.

As a nice application of Theorem 3.7 we have the following. Corollary 3.8. Consider a generalized Robertson–Walker spacetime −I ×ϱ Pn , and assume on Pn the validity of the weak maximum principle for the Lorentzian mean curvature operator. For a, b ∈ I, a < b, let (a, b) × P = {(t, x) : a < t < b, x ∈ P} be an open slab in −I ×ϱ Pn , and assume that there exists t0 ∈ (a, b) with the property that ϱ′ (t) > 0 on [a, t0 ) and ϱ′ (t) < 0 on (t0 , b]. Then the only entire maximal graph contained in (a, b) × P is the slice u ≡ t0 . Proof. Choose ε > 0 sufficiently small such that ϱ′ (t) > 0 on (a − ε, t0 ) and ϱ′ (t) < 0 on (t0 , b + ε). Set α = a − ε and β = b + ε, and let J = (α, β). By contradiction, suppose that Σ (u) is not a slice. By applying Theorem 3.7 to the generalized Robertson–Walker spacetime −J ×ϱ Pn and taking into account that u∗ < β and u∗ > α we conclude that u∗ ≤ t0 ≤ u∗ . That is u ≡ t0 , contradiction. Therefore Σ (u) must be a maximal slice in the slab (a, b) × P, and the only maximal slice contained in that slab is u ≡ t0 .  We give now another height estimate for spacelike graphs in a generalized Robertson–Walker spacetime. Theorem 3.9. Let Pn be an n-dimensional Riemannian manifold and assume the validity of the WMP for the Lorentzian mean curvature operator on it. Consider a function u ∈ C ∞ (Pn ) bounded above on Pn . Assume the graph Σ (u) is a spacelike hypersurface of −I ×ϱ Pn such that H∗ := inf Σ (u) H ≤ 0. Then either Σ (u) is a slice Pu0 with H(u0 ) = H∗ ≡ H or H(u∗ ) ≥ H∗ , with u∗ := supP u. Proof. If Σ (u) is a slice Pu0 then, since Du ≡ 0, from (29) it follows directly H(u0 ) = H∗ ≡ H. So, assume u is non-constant. We reason by contradiction and we suppose H(u∗ ) < H∗ . Define Ωγ = {x ∈ Pn : u(x) > γ} having chosen γ < u∗ such that ∂Ωγ ̸= ∅ and H(u) < H∗ on Ωγ . Note that this is always possible since H is continuous. Reasoning as in Theorem 3.7, we define  u(x) ds v(x) := ψ(u(x)) = . ϱ(s) u0 Then from (30) v satisfies  divP

Dv 

1 − |Dv|2



 = nϱ(u) H(u) − 

Since H∗ ≤ 0 we have H(u) ≥ H∗ ≥ 

H∗ 1 − |Dv|2

,

H(u) 1 − |Dv|2

 .

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133

so that  divP





Dv  1 − |Dv|2

H(u)



≥ nϱ(u) H∗ −  1 − |Dv|2   ≥ nϱ(u) (H∗ − H(u)) ≥ C H∗ − H(ψ −1 (v))

on Ωγ , where C := nmin[γ,u∗ ] ϱ and we have used the fact that H(u) < H∗ on Ωγ . Since ψ is strictly increasing, sup v = v ∗ = ψ(u∗ ) > ψ(γ) = sup v. Ωγ

Ωγ

Then, because of the open form of the WMP we have H∗ − H(ψ −1 (v ∗ )) = H∗ − H(u∗ ) ≤ 0, contradicting the fact that H∗ − H(u∗ ) > 0.



Remark 3.10. We note that in Theorem 3.9 conclusion H(u∗ ) ≥ H∗ gives interesting informations only if H(u∗ ) < 0, that is ϱ′ (u∗ ) < 0. 4. Height estimates for spacelike hypersurfaces In this section we consider spacelike hypersurfaces in generalized Robertson–Walker spacetimes. Let F : Σ → −I ×ϱ Pn be a spacelike hypersurface and let A be the second fundamental form of the immersion with respect to the future-pointing Gauss map N . As we already observed in the previous section, the eigenvalues of A are the principal curvatures of the hypersurface, κ1 , . . . , κn . Their normalized elementary symmetric functions (−1)k Hk := n k



κi1 . . . κik

1≤i1 <···
define the future k-mean curvatures of the immersion. The Newton tensors Pk : T Σ → T Σ , k = 0, . . . , n, associated to the immersion F are inductively defined by  P0 := I;   n Pk := Hk I + A ◦ Pk−1 . k    n  Note that Tr(Pk ) = ck Hk , where ck := (n − k) nk = (k + 1) k+1 . Via the Newton tensors we introduce the second order linear differential operators Lk := C ∞ (Σ ) → C ∞ (Σ ), associated to each Pk , by Lk u = Tr(Pk ◦ hess(u)). Note that Lk is elliptic if and only if Pk is positive definite. Recall that for spacelike hypersurfaces the existence of an elliptic point jointly with the positivity of Hk+1 , for some 1 ≤ k ≤ n − 1, guarantees the ellipticity of Lj for all 1 ≤ j ≤ k. See [3, Section 3].

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We recall the following results that will be used in our computations (see [3,4]). Proposition 4.1. Let F : Σ n → −I ×ϱ Pn be a spacelike hypersurface and let σ be defined on I by σ(t) = t ϱ(s)ds, for some fixed t0 ∈ I. Then t0  Lk h = −H(h)(ck Hk + ⟨Pk ∇h, ∇h⟩) − Θck Hk+1 Lk σ(h) = −ck (ϱ′ (h)Hk + Θϱ(h)Hk+1 ), where Θ = ⟨N, ∂t ⟩. By an inductive procedure, from Proposition 4.1 we obtain Lemma 4.2. Define, for 2 ≤ k ≤ n, the operator Lk−1 =

k−1  i=0

ck−1 H(h)k−1−i (−Θ)i Li = Tr(Pk−1 ◦ hess), ci

where Pk−1 =

k−1  i=0

ck−1 H(h)k−1−i (−Θ)i Pi . ci

We then have Lk−1 σ(h) = −ck−1 ϱ(h)(H(h)k − (−Θ)k Hk ). Proof. Observe that, when k = 2, L1 = (n − 1)H(h)∆ − ΘL1 . Therefore, Proposition 4.1 implies directly L1 σ(h) = −c1 ϱ(h)(H(h)2 − (−Θ)2 H2 ). For the general case, proceeding by induction we get k−2  ck−2 ck−1 ′ H (h) H′ (h)k−2−i (−Θ)i Li σ(h) + (−Θ)k−1 Lk−1 σ(h) ck−2 c i i=0 ck−1 ′ H (h)Lk−2 σ(h) + (−Θ)k−1 Lk−1 σ(h) = ck−2

Lk−1 σ(h) =

= −ck−1 ρ(h)H′ (h)k + ck−1 ρ(h)H′ (h)(−Θ)k−1 Hk−1   − ck−1 ρ(h) H′ (h)(−Θ)k−1 Hk−1 − (−Θ)k Hk   = −ck−1 ρ(h) H′ (h)k − (−Θ)k Hk .  We start with an observation on the sign of H analogous to Corollary 3.2. Proposition 4.3. Consider F : Σ n → −I ×ϱ Pn a spacelike hypersurface, where I = (a, b) with −∞ ≤ a < b ≤ +∞, and suppose the validity of the WMP on Σ for the Laplacian. If ϱ′ ≤ 0 and h∗ < b then H∗ ≤ 0; similarly, if ϱ′ ≥ 0 and h∗ > a then H ∗ ≥ 0. Proof. We focus our attention on the first case. If h is constant then there is nothing to prove because, in this case, Σ is a slice {h∗ } × P with constant mean curvature H = H∗ = H(h∗ ) ≤ 0. If h is non-constant we reason by contradiction and assume that H∗ > 0. Let γ < h∗ such that ∂Ωγ ̸= ∅, where Ωγ = {x ∈ Σ : h(x) > γ}.

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135

Now, recall that h satisfies the equation ∆h = −

ϱ′ (h) (n + |∇h|2 ) − nΘH, ϱ(h)

so that, since ϱ′ ≤ 0, we get ∆h ≥ −nΘH ≥ nH ≥ nH∗ > 0. Hence applying Theorem 2.1 on Ωγ , with f ≡ nH∗ a positive constant, we get a contradiction, since h∗ = supΩγ h > sup∂Ωγ h = γ. The case where ϱ′ ≥ 0 follows in a similar way.



We now give a height estimate for spacelike hypersurfaces with constant k-mean curvature Hk in a generalized Robertson–Walker spacetime, under the assumption H′ = (log ϱ)′′ ≤ 0. Observe that this condition is closely related to the timelike convergence condition (TCC) and the null convergence condition (NCC). Recall that a spacetime obeys TCC if its Ricci curvature is nonnegative on timelike directions. It is not difficult to see that a generalized Robertson–Walker spacetime −I ×ϱ Pn obeys TCC if and only if RicP ≥ (n − 1) sup((log ϱ)′′ ϱ2 )⟨, ⟩P ,

(35)

ϱ′′ ≤ 0,

(36)

I

and

where RicP and ⟨, ⟩P are respectively the Ricci and metric tensors of the Riemannian manifold P. On the other hand, NCC is nothing but (35). In our next result we work under the more general condition H′ = (log ϱ)′′ ≤ 0. Theorem 4.4. Let F : Σ n → −I ×ϱ Pn , I = (a, b) with −∞ ≤ a < b ≤ +∞, be a spacelike hypersurface with non-zero, constant k-mean curvature for some k ≥ 2, and h∗ < b. Assume the existence of an elliptic point with respect to the future-pointing Gauss map, and the validity on Σ of the WMP for the operator Lk−1 . If H > 0 and H′ ≤ 0 on I = (a, b), then H(h∗ )k ≥ Hk .

(37)

The proof of Theorem 4.4 (and also that of Theorem 4.11) is based on the following particular case of Theorem 2.5 in [5] for trace operators. Here by a trace operator we mean an operator of the form LT (u) = trace(T ◦ hess(u)) = div(T (∇u)) − ⟨divT, ∇u⟩, where T is a positive definite, symmetric endomorphism on T M . Theorem 4.5. The WMP holds on M for the operator LT if and only if for each f ∈ C 0 (R), for each open set Ω ⊂ M with ∂Ω ̸= ∅, and for each v ∈ C 0 (Ω ) ∩ C 1 (Ω ) satisfying  (i) LT v ≥ f (v) on Ω ; (38) (ii) sup v < +∞, Ω

we have that either sup v = sup v Ω

(39)

∂Ω

or f (sup v) ≤ 0. Ω

(40)

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Remark 4.6. Observe that, under our assumptions, the operator Pk−1 is positive definite; equivalently, the operator Lk−1 is elliptic. In fact, the existence of an elliptic point on Σ and the fact that Hk is a non-zero constant imply that Hk > 0. Then, by the discussion before Proposition 4.1, we have that Pj is positive definite for all 1 ≤ j ≤ k − 1. Since H(h) > 0 and Θ ≤ −1 < 0, this implies that Pk−1 itself is positive definite. Proof of Theorem 4.4. If h is constant then there is nothing to prove because, in this case, Σ is a slice {h∗ } × P with constant k-mean curvature Hk = H(h∗ )k . If h is non-constant we reason by contradiction and assume that H(h∗ )k < Hk . Let γ < h∗ be such that ∂Ωγ ̸= ∅ and H(γ)k < Hk , where Ωγ = {x ∈ Σ : h(x) > γ}. Define v := σ(h), where σ(t) :=

t t0

ϱ(s)ds. Note that, since σ is an increasing function, σ(h)∗ = σ(h∗ ) < +∞.

Recalling that H is non-increasing and Θ ≤ −1 we have H(h)k − (−Θ)k Hk ≤ H(γ)k − Hk

on Ωγ .

Therefore, since ϱ is non-increasing and (H(γ)k − Hk ) < 0, we get Lk−1 v = −ck−1 ϱ(h)(H(h)k − (−Θ)k Hk ) ≥ −ck−1 ϱ(h)(H(γ)k − Hk ) ≥ −ck−1 ϱ(γ)(H(γ)k − Hk ) > 0 on Ωγ , with supΩγ v < +∞. Observe that σ(h∗ ) = sup v > sup v = σ(γ). Ωγ

∂Ωγ

Then, applying Theorem 4.5 on Ωγ to the elliptic trace operator Lk−1 , with f ≡ −ck−1 ϱ(γ)(H(γ)k − Hk ) a positive constant, we get −ck−1 ϱ(γ)(H(γ)k − Hk ) ≤ 0 which is a contradiction.



4.1. Height estimates for spacelike hypersurfaces in Lorentzian products Consider now the case of hypersurfaces in Lorentzian products −R × Pn . Theorem 4.7. Let F : Σ n → −R × Pn be a stochastically complete spacelike hypersurface with constant mean curvature H > 0. Suppose that for some α > 0 RicP ≥ −nα.

(41)

Let Ω ⊂ Σ be an open set with ∂Ω ̸= ∅ for which F (Ω ) is contained in a slab and F (∂Ω ) ⊂ {0} × Pn . Assume β 2 = sup Θ 2 < Ω

α + H2 . α

(42)

Then 

 (1 − β)H F (Ω ) ⊂ , 0 × Pn . H 2 − α(β 2 − 1)

(43)

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Proof. If β = 1 then there is nothing to prove because, in this case, Θ ≡ −1 is constant on Ω , or equivalently h is constant on Ω . Thus, F (Ω ) is contained in the slice {0} × P. Let β > 1. From (42), we can choose δ > 0 sufficiently small such that (α + δ)(β 2 − 1) < H 2 . We consider the function α+δ 2 H 2 − (α + δ)(β 2 − 1) (β − 1)h = Θ + h, H H where φ := Θ + Hh. From Proposition 4.1 (with k = 0) we know that ψδ := φ −

∆h = −nΘH, and by Eq. (8.10) in [3] we also have that ∆Θ = Θ|A|2 + ΘRicP (N ∗ , N ∗ ), where N ∗ denotes the projection of N onto the fiber Pn . Therefore, using |A|2 = n2 H 2 − n(n − 1)H2 we obtain ∆ψδ = Θ{n(n − 1)(H 2 − H2 ) + RicP (N ∗ , N ∗ ) + n(α + δ)(β 2 − 1)}.

(44)

From (41), RicP (N ∗ , N ∗ ) ≥ −nα|N ∗ |2 = −nα(Θ 2 − 1) ≥ −nα(β 2 − 1)

on Ω .

Thus using the basic inequality H 2 ≥ H2 , (44) implies ∆ψδ ≤ nΘδ(β 2 − 1) ≤ −nδ(β 2 − 1) < 0

on Ω ,

where the last inequality is due to β > 1. We define w = ψδ |Ω . Since F (Ω ) is contained in a slab we have  ∆w ≤ −nδ(β 2 − 1) on Ω ; inf w > −∞. Ω

Stochastic completeness of Σ and Theorem 2.1 give inf w = inf w. Ω

∂Ω

n

By assumption F (Ω ) ⊂ {0} × P and thus h ≡ 0 on ∂Ω , so that w = ψδ = Θ ≥ −β on ∂Ω . We then have H 2 − (α + δ)(β 2 − 1) H 2 − (α + δ)(β 2 − 1) h ≤ −1 + h. H H That is, dividing by the positive quantity H 2 − (α + δ)(β 2 − 1), −β ≤ Θ +

h≥

H2

(1 − β)H . − (α + δ)(β 2 − 1)

Taking the limit as δ ↓ 0 we deduce h≥

(1 − β)H . H 2 − α(β 2 − 1)

On the other hand 

∆h = −nHΘ ≥ nH > 0; sup h < +∞. Ω

Using again Theorem 2.1 we deduce supΩ h = sup∂Ω h = 0, that is h ≤ 0 on Ω . This completes the proof of the theorem. 

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Remark 4.8. The statement of Theorem 4.7 corresponds to the case of positive constant mean curvature H. We recall that, by definition, H is the future mean curvature of the hypersurface. The corresponding statement for the case of negative constant mean curvature is the same, up to replacing (43) with   (1 − β)H × Pn . (45) F (Ω ) ⊂ 0, 2 H − α(β 2 − 1) The proof is analogous; it is enough to substitute the function ψδ with −ψδ . We observe that also the limit case α = 0, in other words, RicP ≥ 0, can be easily treated. Indeed, fix α ˆ > 0 sufficiently small such that (42) holds. Then (41) is obviously true with α ˆ instead of α. Applying Theorem 4.7 and letting α ˆ ↓ 0 we obtain the following corollary. Corollary 4.9. Let F : Σ n → −R × Pn be a stochastically complete spacelike hypersurface with constant mean curvature H > 0. Suppose that RicP ≥ 0.

(46)

Let Ω ⊂ Σ be an open set with ∂Ω ̸= ∅ for which F (Ω ) is contained in a slab and F (∂Ω ) ⊂ {0} × Pn , and assume β = sup |Θ| < +∞.

(47)

Ω

Then  1−β , 0 × Pn . F (Ω ) ⊂ H 

(48)

As observed in Remark 4.8, the corresponding statement of Corollary 4.9 for the case of negative constant mean curvature is the same, up to replace (48) with   1−β F (Ω ) ⊂ 0, × Pn . H

(49)

Here is a geometric way to guarantee stochastic completeness. Theorem 4.10. Let F : Σ n → −R × Pn be a complete spacelike hypersurface with constant mean curvature H > 0. Assume that the height function h = πR ◦ F : Σ → R satisfies lim h(x) = −∞.

x→∞

Suppose that for some α > 0 RicP ≥ −nα.

(50)

Let Ω ⊂ Σ be a relatively compact open set with ∂Ω ̸= ∅ such that F (∂Ω ) ⊂ {0} × Pn . Assume β 2 = sup Θ 2 < Ω

α + H2 . α

(51)

Then  F (Ω ) ⊂

 (1 − β)H , 0 × Pn . H 2 − α(β 2 − 1)

(52)

L.J. Al´ıas et al. / Nonlinear Analysis 129 (2015) 119–142

139

Proof. The result follows from Theorem 4.7 once we show the validity of the WMP on Σ for the Laplacian. Towards this end we let γ = −h, so that it satisfies ∆γ = nΘH ≤ −nH < 0 and γ(x) → +∞ as x → ∞. We then apply Theorem A of [2] to get the desired conclusion.



We now generalize Theorem 4.7 to the case of higher order mean curvatures. Theorem 4.11. Let F : Σ → −R×Pn be an immersed hypersurface with constant, non-zero k-mean curvature Hk , for some k = 2, . . . , n and with an elliptic point with respect to the future-pointing Gauss map. Suppose that the sectional curvature of Pn satisfies KPn > −α, for some α > 0 and assume the validity of the WMP for the operator Lk−1 on Σ . Let Ω ⊂ Σ be an open set with ∂Ω ̸= ∅ for which F (Ω ) is contained in a slab and F (∂Ω ) ⊂ {0} × Pn . Assume β 2 = sup Θ 2 < Ω

(k+1)/k

∗ αHk−1 + Hk ∗ αHk−1

,

∗ where Hk−1 := supΩ Hk−1 . Then

 F (Ω ) ⊆ 

(1 − β)Hk k+1 k

Hk

−

α(β 2

−

∗ 1)Hk−1

 , 0 .

∗ Remark 4.12. Observe that, under our assumptions, Hk−1 := supΩ Hk−1 > 0. In fact, the existence of an elliptic point on Σ and the fact that Hk is a non-zero constant imply that Hk > 0. Then, by Lemma 3.3 in [3], Pk−1 is positive definite and, in particular, Hk−1 (x) > 0 for every x ∈ Σ .

Proof. As in the proof of Theorem 4.7 we may assume that β > 1. Otherwise, F (Ω ) is contained in the slab {0} × Pn and there is nothing to prove. 1

Let us consider the function φ := Hkk h + Θ. We know that Lk−1 h = −ck−1 ΘHk . On the other hand Hk is constant, so by Corollary 8.5 in [3] we have   n  n Lk−1 Θ = (nH1 Hk − (n − k)Hk+1 )Θ + Θ µk−1,i KP (N ∗ ∧ Ei∗ )|N ∗ ∧ Ei∗ |2 , k i=1

(53)

(54)

where {Ei }n1 is a local orthonormal frame that diagonalizes Pk−1 , the µk−1,i ’s are the eigenvalues of this latter and star denotes projection onto Pn . We then get    n  k+1 n Lk−1 φ = nH1 Hk − (n − k)Hk+1 − kHk k Θ +Θ µk−1,i KP (N ∗ ∧ Ei∗ )|N ∗ ∧ Ei∗ |2 . (55) k i=1 Using G˚ arding inequalities we obtain k+1

H1 Hk ≥ Hk k ,

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and therefore   k+1 k+1 nH1 Hk − (n − k)Hk+1 − kHk k ≥ (n − k) Hk k − Hk+1 ≥ 0.

(56)

From the decompositions N = N ∗ − ⟨N, ∂t ⟩∂t ,

Ei = Ei∗ − ⟨Ei , ∂t ⟩∂t ,

and ∂t = −∇h − ⟨N, ∂t ⟩N,

it follows easily that |Ei∗ ∧ N ∗ |2 = |∇h|2 − ⟨Ei , ∇h⟩2 . In particular, |Ei∗ ∧ N ∗ |2 ≤ |∇h|2 = Θ 2 − 1. Now, recall that the existence of an elliptic point on Σ and the assumption that Hk is a non-zero constant implies that Hk > 0 and Pk−1 is positive definite. So, using this latter fact and the assumption on KP , we have µk−1,i KP (N ∗ ∧ Ei∗ )|N ∗ ∧ Ei∗ |2 ≥ −αµk−1,i (Θ 2 − 1).

(57)

Inserting (56) and (57) into (55), we estimate Lk−1 φ ≤ −Θα(Θ 2 − 1) Tr(Pk−1 ) = −Θα(Θ 2 − 1)ck−1 Hk−1 .

(58)

In particular Lk−1 φ ≤ −Θα(β 2 − 1)ck−1 Hk−1

on Ω .

Now, choose δ > 0 satisfying k+1

∗ (αHk−1 + δ)(β 2 − 1) < Hk k

and define ψδ = φ −

∗ αHk−1 +δ 2 (β − 1)h Hk k+1

∗ + δ)(β 2 − 1) H k − (αHk−1 h. =Θ+ k Hk

We let w = ψδ |Ω . Using (53) and (59) we obtain ∗ Lk−1 w ≤ ck−1 (β 2 − 1)Θ{α(Hk−1 − Hk−1 ) + δ}

≤ ck−1 (β 2 − 1)Θδ ≤ −ck−1 (β 2 − 1)δ < 0 on Ω , where the last inequality is due to β > 1. Since F (Ω ) is contained in a slab we also have inf w > −∞. Ω

Using Theorem 4.5 for the elliptic trace operator Lk−1 , we deduce inf w = inf w. Ω

∂Ω

Therefore, k+1

k+1

∗ ∗ + δ)(β 2 − 1) H k − (αHk−1 + δ)(β 2 − 1) H k − (αHk−1 −1 + k h≥Θ+ k h Hk Hk ≥ −β,

(59)

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and letting δ → 0 we finally obtain h≥

(1 − β)Hk k+1 k

Hk

on Ω ,

∗ − α(β 2 − 1)Hk−1

(60)

k+1

∗ since Hk k − α(β 2 − 1)Hk−1 > 0. On the other hand, by (53) we also have  Lk−1 h = −ck−1 ΘHk ≥ ck−1 hk > 0 on Ω sup h < +∞. Ω

Reasoning as above we deduce sup h = sup h, Ω

∂Ω

that implies h ≤ 0 and, combining this inequality with (60), we get the desired conclusion.



Similarly to Theorem 4.10, we finish with a geometric way to guarantee the validity of the WMP for the operator Lk−1 on Σ . Theorem 4.13. Let F : Σ → −R×Pn be an immersed hypersurface with constant, non-zero k-mean curvature Hk , for some k = 2, . . . , n and with an elliptic point with respect to the future-pointing Gauss map. Assume that the height function h = πR ◦ F : Σ → R satisfies lim h(x) = −∞.

x→∞

Suppose that the sectional curvature of Pn satisfies KPn > −α, for some α > 0. Let Ω ⊂ Σ be a relatively compact open set with ∂Ω ̸= ∅ such that F (∂Ω ) ⊂ {0} × Pn . Assume β 2 = sup Θ 2 < Ω

(k+1)/k

∗ αHk−1 + Hk ∗ αHk−1

,

∗ := supΩ Hk−1 . Then where Hk−1

 F (Ω ) ⊆ 



(1 − β)Hk k+1 k

Hk

∗ − α(β 2 − 1)Hk−1

, 0 .

Proof. The result follows from Theorem 4.11 once we show the validity of the WMP on Σ for the operator Lk−1 . Towards this end we let γ = −h, so that it satisfies Lk−1 γ = ck−1 ΘHk ≤ −ck−1 Hk < 0 and γ(x) → +∞ as x → ∞. We then apply Theorem A of [2] to get the desired conclusion.



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