Stable maximal hypersurfaces in Lorentzian spacetimes
aa r X i v : . [ m a t h . DG ] M a r STABLE MAXIMAL HYPERSURFACES IN LORENTZIAN SPACETIMES
GIULIO COLOMBO, JOS´E A. S. PELEGR´IN, AND MARCO RIGOLI
Abstract.
We study the geometry of stable maximal hypersurfaces in a variety of spacetimessatisfying various physically relevant curvature assumptions, for instance the Timelike Conver-gence Condition (TCC). We characterize stability when the target space has constant sectionalcurvature as well as give sufficient conditions on the geometry of the ambient spacetime (e.g.,the validity of TCC) to ensure stability. Some rigidity results and height estimates are alsoproven in GRW spacetimes. In the last part of the paper we consider k -stability of spacelikehypersurfaces, a concept related to mean curvatures of higher orders. MSC 2010
Primary: 53C24, 53C42, 35J20; Secondary: 35P15, 53C50, 53C80, 83C99
Keywords
Stable maximal hypersurface · k -stability · Spacetime of constant sectional curva-ture · Generalized Robertson-Walker spacetime1.
Introduction
In the last decades, maximal hypersurfaces in spacetimes have attracted a great deal of mathe-matical and physical interest. The importance of this family of spacelike hypersurfaces in GeneralRelativity is well-known and a summary of several reasons justifying this opinion can be found,for instance, in [30]. Among them, we emphasize the key role they play in the study of theCauchy problem [19, 28] as well as their importance in the proof of the positivity of the grav-itational mass [41]. Furthermore, maximal hypersurfaces describe, in some relevant cases, thetransition between the expanding and contracting phases of a relativistic universe. Finally, theexistence of constant mean curvature (and in particular maximal) hypersurfaces is useful in thestudy of the structure of singularities in the space of solutions of the Einstein equations [8]. Atlast, we should also mention their use in numerical relativity for integrating forward in time [27].From a mathematical point of view, maximal hypersurfaces in a spacetime M are (locally)critical points for a natural variational problem, namely, that of the area functional (see, forinstance, [12]) and their study is helpful for understanding the structure of M [9]. In particu-lar, for some asymptotically flat spacetimes, maximal hypersurfaces produce a foliation of thespacetime, defining a time function [14]. Classical papers dealing with uniqueness of maximalhypersurfaces are, for instance, [14, 18], although a previous relevant result in this directionwas the proof given by Cheng and Yau [17] of the Bernstein-Calabi conjecture [15]: spacelikeaffine hyperplanes are the only complete maximal hypersurfaces in the the ( m + 1)-dimensionalLorentz-Minkowski spacetime. Nishikawa [32] extended their result by proving that any completemaximal hypersurface immersed in a spacetime M is totally geodesic when M belongs to a familyof locally symmetric Lorentzian manifolds that includes spacetimes of nonnegative constant cur-vature. Ishihara [26] showed that this property is not shared by spacetimes of negative constantcurvature by exhibiting an example of a complete maximal hypersurface with constant nonzeronorm of the shape operator in the ( m + 1)-dimensional anti-de Sitter spacetime of curvature − G. COLOMBO, J. A. S. PELEGR´IN, AND M. RIGOLI bound on the norm of the shape operator first obtained by Ishihara in the case of ambient space-times of constant curvature. More recently, new uniqueness results for maximal hypersurfaceshave been found in a large variety of spacetimes by means of different techniques [6, 35, 37].In this paper we will focus on a particular family of maximal hypersurfaces, namely, stablemaximal hypersurfaces, that is, critical points of the volume functional for compactly supportednormal variations with non-positive second variation. A mild condition on the curvature of theambient spacetime is enough to ensure stability of maximal hypersurfaces.
Theorem A.
Let M be a spacetime with nonnegative Ricci curvature on timelike vectors. If ψ : M −→ M is a (not necessarily complete) oriented maximal hypersurface, then ψ is stable. If M is also compact, then ψ is totally geodesic. Note that in an oriented spacetime M , the time orientation of M ensures that every spacelikehypersurface is oriented. In General Relativity, a spacetime with nonnegative Ricci curvature ontimelike vectors is said to obey the Timelike Convergent Condition (TCC). It is usually arguedthat the TCC is the mathematical way to express that gravity, on average, attracts (see [33]).Theorem A generalizes Corollary 5.6 of [6] and Theorem 1 of [35], where the authors show thatcompact maximal hypersurfaces in a spacetime M obeying the TCC are totally geodesic by alsoassuming the existence of certain infinitesimal symmetries in M . As a corollary, we also have analternative proof of Theorem 4.1 of [14], a uniqueness result for vacuum spacetimes. Corollary B.
Let M be a compact maximal hypersurface in a spacetime that obeys the Einsteinvacuum equations without cosmological constant. Then, M is totally geodesic. If a maximal hypersurface ψ : M → M is unstable, then there exist spacelike hypersurfaces oflarger volume in M nearby ψ . This happens, for instance, for the equator of de Sitter spacetime,which is a saddle point for the volume functional. In fact, we have the following Theorem C.
Let M be an ( m + 1) -dimensional spacetime of constant curvature κ and let ψ : M −→ M be a complete oriented maximal hypersurface.i) If κ > then M is compact and the immersion ψ is totally geodesic and unstable.ii) If κ = 0 then ψ is totally geodesic and stable.iii) If κ < then ψ is stable and the shape operator A and the scalar curvature S of M satisfy trace( A ) ≤ − mκ, S ≤ ( m − mκ. If M is also compact, then ψ is totally geodesic. In case the ambient manifold M is a 3-dimensional spacetime of constant curvature, we are ableto provide more information on the topology of orientable complete maximal surfaces. Theorem D.
Let M be a -dimensional spacetime of constant curvature κ , M a completemaximal oriented surface in M .i) If κ > then M is a totally geodesic, unstable round sphere of constant curvature κ .ii) If κ = 0 then M is totally geodesic, stable and it is either a Euclidean plane, or a flatcylinder or a flat torus.iii) If κ < then M is stable and has non-positive Gaussian curvature. If M is compactthen it is totally geodesic and its Euler characteristic satisfies χ ( M ) = κ π Vol( M ) . If M is non-compact but its total curvature and its Euler characteristic are finite, then χ ( M ) ≥ κ π Vol( M ) . TABLE MAXIMAL HYPERSURFACES IN LORENTZIAN SPACETIMES 3
Stable maximal hypersurfaces have been previously studied in [14] and [20], where the au-thors introduced the relative variational formulas and some characterizations in certain ambientspacetimes. More recently, sufficient conditions to ensure stability, in some physically relevantspacetimes, have been given in [21]. In fact, a maximal hypersurface ψ : M −→ M , withunit normal vector N and shape operator A , is stable if and only if the differential operator L = ∆ − Ric(
N, N ) − trace( A ) has non-negative first eigenvalue on M . More generally, a max-imal hypersurface is said to have finite index if the stability operator L has finite index. In a3-dimensional spacetime with nonnegative Ricci curvature on spacelike directions, we prove thatany complete maximal surface with finite index has either finite or positive infinite total curva-ture, provided it is well defined. We remark that when M is an oriented surface with Gaussiancurvature K, its total curvature is defined as Z M K = Z M K + − Z M K − , where K + and K − are the positive and negative parts of K. Hence, when M is noncompact, thetotal curvature is well defined only if at least one of the integrals on the right side is finite. Theorem E.
Let M be a -dimensional spacetime with nonnegative Ricci curvature on spacelikevectors, ψ : M −→ M a complete maximal oriented surface immersed in M . If M has finiteindex and its total curvature is well defined, then Z M K > −∞ . A physically relevant family of Lorentzian manifolds is that of Generalized Robertson-Walker(GRW) spacetimes. They can be defined as product manifolds M = I × F , where I ⊆ R is an openinterval with the standard negative definite metric − dt and ( F, g F ) is a Riemannian manifold.On M we put a Lorentzian warped product metric of the form g = − dt + ρ ( t ) g F , with ρ asmooth positive function on I . In these ambient manifolds we give the following generalization ofthe first part of Theorem D, suggested by the work of Albujer and Al´ıas, [1], where the authorsconsider maximal surfaces in Lorentzian products, that is, the case ρ ≡ Theorem F.
Let M be a -dimensional GRW spacetime M = I × ρ F with nonnegative sectionalcurvatures on spacelike -planes and let ψ : M −→ M be a complete maximal surface. Then ψ is totally geodesic and one of the following cases occurs:i) ψ ( M ) is a spacelike slice { t } × F for some t ∈ I such that ρ ′ ( t ) = 0 ,ii) F is a Riemann surface with a complete, flat metric g F and M is the product manifold R × F with the flat metric − dt + g F ,iii) F is a compact Riemann surface with a metric g F of constant positive Gaussian curva-ture, M is a round sphere and the spacetime M has constant positive curvature in thesmallest slab I ′ × F ⊆ M , I ′ ⊆ I , such that ψ ( M ) ⊆ I ′ × F . In a GRW spacetime with sectional curvatures bounded below, the image of a complete maximalhypersurface whose projection on the I factor is relatively compact in I must always intersect atleast one totally geodesic spacelike slice of the ambient spacetime. More precisely, we have thefollowing Theorem G.
Let M = I × ρ F be a GRW spacetime whose sectional curvatures on spacelike -planes are bounded below. Let ψ : M −→ M be a complete maximal hypersurface, I ′ ⊆ I thesmallest interval closed in I such that ψ ( M ) ⊆ I ′ × F . If I ′ = [ t ∗ , t ∗ ] for some t ∗ , t ∗ ∈ I , then ρ ′ ( t ∗ ) ≥ and ρ ′ ( t ∗ ) ≤ . In particular, if ψ ( M ) is contained in a slab [ a, b ] × F , a, b ∈ I , thenthere exists t ∈ [ a, b ] such that ρ ′ ( t ) = 0 and G. COLOMBO, J. A. S. PELEGR´IN, AND M. RIGOLI i) if ρ ′ < on [ a, t ) and ρ ′ > on ( t , b ] , then ψ ( M ) must intersect the spacelike slice { t } × F ,ii) if ρ ′ > on [ a, t ) and ρ ′ < on ( t , b ] , then ψ ( M ) = { t } × F . As an application of Theorems G and C, we give a simple proof of the following Frankel typeresult.
Corollary H.
Let M = S m +11 ( κ ) be the ( m + 1) -dimensional de Sitter spacetime of constantcurvature κ > and let ψ : M → M , ψ : M → M be two complete maximal hypersurfaces.Then ψ ( M ) ∩ ψ ( M ) = ∅ . If ρ satisfies ρ ′′ ≤ I , then every maximal hypersurface in M = I × ρ F is stable, as observedin Theorem 9 of [21]. On the other hand, if a compact maximal hypersurface M is stable in M and ρ ′′ ≥ I , then ρ ′′ ≡ M . More precisely, we have the following Theorem I.
Let M be a complete oriented stable maximal hypersurface in a GRW spacetime M = I × ρ F and let I ′ ⊆ I be the smallest interval such that ψ ( M ) ⊆ I ′ × F .i) If M is compact then either ρ ′′ ≡ or ρ ′′ attains both positive and negative values on I ′ .ii) If M is non-compact and, for some o ∈ M , the normal vector field N of M satisfies lim r → + ∞ Z ra Z ∂ Br g ( T, N ) ! − = + ∞ , for some (hence any) a > , where B r is the geodesic ball of ( M, g ) with radius r centeredat o and T = ρ ( t ) ∂ t , then either ρ ′′ ≡ on I ′ or there exists t ∈ I ′ such that ρ ′′ ( t ) < . In this work we also obtain new results for higher order mean curvatures. In particular, westudy the k -stability of spacelike hypersurfaces with zero ( k + 1)-th mean curvature in spacetimesof constant curvature. The notion of k -stability has been previously studied in the Lorentziansetting in [13] as well as in [16]. The following two results are somehow companions of TheoremsA, E and G in this context. Note that the requirements on the sign of the k -th mean curvaturefunction H k and on the rank of the shape operator A are minimal to guarantee the ellipticity ofthe k -stability operator of ψ , as defined in Section 2.2. Theorem J.
Let ψ : M −→ M be a complete oriented spacelike hypersurface with zero ( k + 1) -thmean curvature, for some ≤ k ≤ m − , in a spacetime M of constant curvature κ . Supposethat the k -th mean curvature function H k is positive and that the shape operator A has rank > k on M .i) If κ ≤ , then M is non-compact and ψ is k -stable.ii) If κ > and M is compact, then ψ is not k -stable.iii) If κ > and we assume that M is non-compact and that, for some o ∈ M and for some(hence any) a ∈ R , lim r → + ∞ Z ra (cid:18)Z ∂B r H k (cid:19) − = + ∞ , lim r −→ + ∞ Z B r (cid:0) κ trace( P k ) − trace( A P k ) (cid:1) = + ∞ , where B r is the geodesic ball of ( M, g ) with radius r centered at o and P k is the k -thNewton operator associated to ψ (as defined in Section 2.2), then for every compactsubset K ⊆ M the hypersurface ψ | M \ K : M \ K −→ M is not k -stable. TABLE MAXIMAL HYPERSURFACES IN LORENTZIAN SPACETIMES 5
We remark that the quantity κ trace( P k ) − trace( A P k ) appearing in the statement of TheoremJ can be expressed in terms of the higher order mean curvature functions H , . . . , H m of ψ as κ trace( P k ) − trace( A P k ) = (cid:18) mk (cid:19) ( m − k ) κH k + (cid:18) mk + 1 (cid:19) (( m − k − H k +2 − mH H k +1 ) . (1) Theorem K.
Let M = I × ρ F be a Robertson-Walker spacetime of constant sectional curvatureand let ψ : M −→ M be a complete oriented spacelike hypersurface with zero ( k + 1) -th meancurvature. Suppose that there exists C > such that C − < H k < C , that rank( A ) > k on M and that for some o ∈ M and for some (hence any) a ∈ R , one of the following conditions issatisfied: ( i ) lim r → + ∞ Z ra dt Vol( ∂B r ) = + ∞ and Vol( M ) = + ∞ , or ( ii ) lim r → + ∞ Z ra dt Vol( ∂B r ) < + ∞ and lim r → + ∞ log(Vol( ∂B r )) r = 0 , where B r is the geodesic ball of ( M, g ) with radius r centered at o . If ψ ( M ) is contained in a slab [ a, b ] × F , then there exists t ∈ [ a, b ] such that ρ ′ ( t ) = 0 and ψ ( M ) must intersect the spacelikeslice { t } × F . Our paper is organized as follows. In Section 2 we recall the notion of stability and k -stabilityfor spacelike hypersurfaces with vanishing mean curvature functions together with some generalproperties of Schr¨odinger differential operators that we will need in the subsequent sections.In Section 3 we mainly deal with maximal hypersurfaces in locally symmetric spacetimes andwe prove Theorems A, C, D and E above (see Theorems 9, 10, 11 and 12, respectively). InSection 4 we study maximal hypersurfaces in GRW spacetimes and we prove Theorems F, G, Iand Corollary H (see Theorems 22, 24 and Corollaries 20, 21), also giving a characterization ofGRW spacetimes with spacelike sectional curvatures bounded below (Lemma 16). In section 5we consider hypersurfaces with zero ( k + 1)-th order mean curvature in spacetimes of constantcurvature and we prove Theorems J and K (see Theorems 27 and 28).2. Preliminaries
We devote this section to introduce the basic concepts concerning the stability of maximalhypersurfaces in general ambient spacetimes as well as their natural generalization to the caseof k -stable spacelike hypersurfaces with zero ( k + 1)-th mean curvature.2.1. Stability of maximal hypersurfaces.
Let ψ : M −→ M be a spacelike hypersurfaceimmersed in a spacetime ( M , g ) and let g = ψ ∗ g be the Riemannian metric induced on M . Wewill denote by N a chosen unit normal vector field to M and by A the shape operator in thedirection of N , determined by the validity of the Weingarten formula AX = −∇ X N for any X ∈ T M , with ∇ the Levi-Civita connection of ( M , g ). The linear operator A is self-adjointat each tangent space and its eigenvalues κ , . . . , κ m are, by definition, the principal curvaturesof the hypersurface. The mean curvature function H of ψ in the direction of N is given by thenormalized trace of − A , that is,(2) H = − m trace( A ) = − κ + · · · + κ m m . A spacelike hypersurface is called a maximal hypersurface if it is a critical point of the vol-ume functional for compactly supported normal variations of the immersion. This condition is
G. COLOMBO, J. A. S. PELEGR´IN, AND M. RIGOLI equivalent to the hypersurface having zero mean curvature. If ψ is a maximal hypersurface andwe take a normal variation driven by a variational vector field φN , with φ a smooth functionsupported in a relatively compact domain Ω ⊆ M , then the second variation of the volume of Ωis (see Theorem 2.1 of [14] and Theorem 1 of [20])(3) Z Ω (cid:0) ∆ φ − (cid:0) Ric(
N, N ) + trace( A ) (cid:1) φ (cid:1) φ dV, where ∇ is the Levi-Civita connection of ( M, g ), ∆ is the corresponding Laplacian and Ric isthe Ricci tensor of (
M , g ). In view of the identity(4) S = S + 2Ric(
N, N ) + trace( A )satisfied by the scalar curvatures S, S of M and M (see formula (28) in Section 2.4), we have(5) Ric( N, N ) + trace( A ) = S − (cid:0) S + Ric(
N, N ) (cid:1) = m X i =1 (cid:0) Ric( E i , E i ) − Ric( E i , E i ) (cid:1) for any local orthonormal frame { E , . . . , E m } on T M , where Ric is the Ricci tensor of (
M, g ).The immersion ψ is said to be stable if the second variation of the volume of M is non-positivefor every compactly supported normal variation. Stability is detected by the sign of the firsteigenvalue λ L ( M ) of the stability operator L defined by(6) Lu = ∆ u − (cid:0) Ric(
N, N ) + trace( A ) (cid:1) u for every u ∈ C ( M ) , analogously to what happens in the Riemannian case for minimal hypersurfaces (see for instance[23, 42]). In fact, since λ L ( M ) is variationally characterized by(7) λ L ( M ) = inf φ ∈ C ∞ c ( M ) φ =0 R M |∇ φ | + (cid:0) Ric(
N, N ) + trace( A ) (cid:1) φ R M φ , by applying the divergence theorem to (3) we see that ψ is stable if and only if λ L ( M ) ≥
0. Moregenerally, M is said to have finite index if the operator L has finite index. When M is compact,this is always the case, while for a complete, non-compact hypersurface this happens if and onlyif there exists a relatively compact open set Ω ⊆ M such that the second variation of the volumeof M is non-positive for every normal variation compactly supported in M \ Ω. In this case, wealso say that M is stable at infinity.2.2. k -stability of spacelike hypersurfaces with zero ( k + 1) -th mean curvature. Wecan generalize the concepts above to study the k -stability of spacelike hypersurfaces with zero( k +1)-th mean curvature in spacetimes of constant curvature. In order to do so, let ψ : M −→ M be a spacelike hypersurface in a spacetime M with constant curvature κ and let A be the shapeoperator of the immersion with respect to a unit normal timelike vector N . We associate to A the algebraic invariants S , . . . , S m and the mean curvature functions H , . . . , H m of orders1 , . . . , m in the direction of N by setting(8) S k = X ≤ i < ···
TABLE MAXIMAL HYPERSURFACES IN LORENTZIAN SPACETIMES 7
Lemma 1.
For ≤ k ≤ m − , let c k = ( m − k ) (cid:0) mk (cid:1) . Then, for every X ∈ X ( M )trace( P k ) = c k H k , trace( P k ◦ A ) = − c k H k +1 , trace( P k ◦ ∇ X A ) = − (cid:18) mk + 1 (cid:19) ∇ X H k +1 . (10)Note that the identity (1) stated in the Introduction follows by applying relations (9) and (10).For 0 ≤ k ≤ m , let L k be the second order linear differential operator given by(11) L k u = trace ( P k ◦ Hess( u )) ≡ div( P k ( ∇ u )) for every u ∈ C ( M ) , where the second equality holds because P k is divergence-free as long as M has constant cur-vature. Note that H = H and that L = ∆. The operator L k is elliptic if and only if P k ispositive definite. For 1 ≤ k ≤ m −
2, if H k +1 = 0 at some point then P k is positive definite thereif and only if H k > A ) > k , see Corollary 2.3 of [25].Now, for 0 ≤ k ≤ m − k -volume functional for a relatively compact domainΩ ⊂ M by setting Vol k (Ω) = Z Ω F k ( S , S , . . . , S k ) dV, where S , . . . , S m are the invariants defined above and the functions F , . . . , F m − are recursivelydefined by F = 1 , F = − S , F k = ( − k S k − κ ( m − k + 1) k − F k − for 2 ≤ k ≤ m − . According to [16], if M has zero ( k + 1)-th mean curvature for some 1 ≤ k ≤ m − M given by φN , with φ supported in a relatively compact domain Ω ⊆ M ,then the second variation of the k -volume functional of M is(12) ( k + 1) Z Ω (cid:0) L k φ + κ trace( P k ) φ − trace( A P k ) φ (cid:1) φ dV. As a generalization of (5), for every local orthonormal frame { E , . . . , E m } on T M we have − κ trace( P k ) + trace( A P k ) = m X i =1 (cid:0) Ric( P k E i , E i ) − Ric( P k E i , E i ) (cid:1) , as a consequence of formula (29) of Section 2.4 and of Lemma 1. The immersion ψ is said tobe k -stable if the second variation of the k -volume of M is non-positive for every compactlysupported normal variation. Analogously to the maximal case, k -stability is detected by the signof the first eigenvalue λ e L k ( M ) of the k -stability operator e L k defined by(13) e L k u = L k u − (cid:0) trace( A P k ) − κ trace( P k ) (cid:1) u for every u ∈ C ( M ) . Since λ e L k ( M ) is variationally characterized by(14) λ e L k ( M ) = inf φ ∈ C ∞ c ( M ) φ =0 R M g ( P k ( ∇ φ ) , ∇ φ ) + (cid:0) trace( A P k ) − κ trace( P k ) (cid:1) φ R M φ , we see that a spacelike hypersurface M with zero ( k + 1)-th mean curvature is k -stable if andonly if λ e L k ( M ) ≥
0. Similarly to what happens with the usual stability operator, a complete,non-compact, spacelike hypersurface with zero ( k + 1)-th mean curvature is said to be k -stableat infinity if the k -stability operator has finite index. G. COLOMBO, J. A. S. PELEGR´IN, AND M. RIGOLI
General facts on Schr¨odinger operators.
Consider a Riemannian manifold (
M, g ), afunction F ( x ) ∈ L ∞ loc ( M ) and let P : T M −→ T M be a positive definite, self-adjoint, endomor-phism of class C . Define the second order linear elliptic operator L by setting(15) Lu = div( P ( ∇ u )) − F ( x ) u for every u ∈ C ( M ) . For every open set Ω ⊆ M , let λ L (Ω) be the first eigenvalue of L on Ω, given by(16) λ L (Ω) = inf φ ∈ C ∞ c (Ω) φ =0 R Ω g ( P ( ∇ φ ) , ∇ φ ) + F ( x ) φ R Ω φ = inf φ ∈ C ∞ c (Ω) φ =0 R Ω − φLφ R Ω φ If Ω is a relatively compact domain with sufficiently regular boundary, say of class C , then theinfimum in the RHS of (16) is achieved by the non-zero solutions of the Dirichlet problem Lu + λ L (Ω) u = 0 on Ω ,u = 0 on ∂ Ω , which belong to C ,α (Ω) ∩ H (Ω) ∩ C (Ω) for some 0 < α < Proposition 2.
Let ( M, g ) be a Riemannian manifold and for F ( x ) ∈ L ∞ loc ( M ) , P : T M −→ T M let L be the operator defined in (15). Let Ω , Ω be two relatively compact domains in M suchthat Ω ⊆ Ω . Then (17) λ L (Ω ) ≥ λ L (Ω ) . If Ω and Ω have C boundaries and the interior of Ω \ Ω is nonempty, then (17) holds withstrict inequality sign.Proof . Observe that (17) is a trivial consequence of the definition (16). To prove the last state-ment we will proceed as follows. Consider an open subset Ω ⊆ M and two functions u, v ∈ C (Ω),with v = 0 on Ω: since P is positive definite and self-adjoint, a direct computation yields thefollowing extension of the classic Picone’s identity(18) 0 ≤ g (cid:16) P (cid:16) ∇ u − uv ∇ v (cid:17) , ∇ u − uv ∇ v (cid:17) = g ( P ( ∇ u ) , ∇ u ) − g (cid:18) ∇ (cid:18) u v (cid:19) , P ( ∇ v ) (cid:19) . In particular, g (cid:16) P (cid:16) ∇ u − uv ∇ v (cid:17) , ∇ u − uv ∇ v (cid:17) ≡ u = Cv for some constant C ∈ R . Now, let us suppose that Ω and Ω have C boundaries and let u and v be non-zero solutions of the Dirichlet problems Lu + λ L (Ω ) u = 0 in Ω , Lv + λ L (Ω ) v = 0 in Ω ,u = 0 on ∂ Ω , v = 0 on ∂ Ω . Note that we can suppose v > . Taking (18) into account, since u = 0 on ∂ Ω we get0 ≤ Z Ω g ( P ( ∇ u ) , ∇ u ) − g (cid:18) ∇ (cid:18) u v (cid:19) , P ( ∇ v ) (cid:19) = (cid:0) λ L (Ω ) − λ L (Ω ) (cid:1) Z Ω u . We now reason by contradiction assuming that ◦ Ω \ Ω = ∅ and λ L (Ω ) = λ L (Ω ). From theabove inequalities it follows that on the connected components of Ω , u = Cv for some C ∈ R .Choose one of the component, say e Ω , with ∂ e Ω ∩ Ω = ∅ (this is possible since Ω \ Ω has non-empty interior). Since u ≡ ∂ Ω , we have v ≡ ∂ e Ω ∩ Ω = ∅ , reaching a contradiction. (cid:3) TABLE MAXIMAL HYPERSURFACES IN LORENTZIAN SPACETIMES 9
The following generalization of Barta’s theorem also holds.
Proposition 3.
Let ( M, g ) be a Riemannian manifold and for F ( x ) ∈ L ∞ loc ( M ) , P : T M −→ T M let L be the operator defined in (15). If u ∈ C ( M ) is a positive function, we have (19) λ L ( M ) ≥ inf M (cid:18) − Luu (cid:19) . Proof . Let φ ∈ C ∞ c ( M ) and consider the vector field Z = − φ u P ( ∇ u ) . Taking the divergence and integrating we get(20) 2 Z M φv g ( P ( ∇ u ) , ∇ φ ) = − Z M φ div (cid:18) P ( ∇ u ) u (cid:19) . Since P is positive definite and self-adjoint, by Cauchy-Schwarz and Young’s inequalities2 φu g ( P ( ∇ u ) , ∇ φ ) ≤ | φ | u p g ( P ( ∇ u ) , ∇ u ) p g ( P ( ∇ φ ) , ∇ φ ) ≤ φ u g ( P ( ∇ u ) , ∇ u ) + g ( P ( ∇ φ ) , ∇ φ ) . (21)Using (21), (20) and the divergence theorem again we obtain Z M − φLφ = Z M g ( P ( ∇ φ ) , ∇ φ ) + Z M F ( x ) φ ≥ − Z M φ u g ( P ( ∇ u ) , ∇ u ) + 2 Z M φv g ( P ( ∇ u ) , ∇ φ ) + Z M F ( x ) φ = − Z M (cid:18) g (cid:18) ∇ uu , P ( ∇ u ) (cid:19) + div (cid:18) P ( ∇ u ) u (cid:19) − F ( x ) (cid:19) φ = − Z M (cid:18) div( P ( ∇ u )) u − F ( x ) (cid:19) φ = Z M − Lvv φ ≥ inf M (cid:18) − Lvv (cid:19) Z M ϕ (22)and by definition (16) of λ L ( M ) we obtain inequality (19). (cid:3) We conclude this paragraph with the following characterization of non-negativity of λ L (Ω) foran open subset Ω ⊆ M . For P = I : T M −→ T M and F ( x ) ∈ C ∞ ( M ), it is given as Theorem1 in [23]. For P = I and F ( x ) ∈ L ∞ loc ( M ) it is proved as Lemma 3.10 of [39]. The proof for ageneral self-adjoint, positive definite C endomorphism P is a straightforward extension of theproof given in [39] for P = I . Lemma 4.
Let ( M, g ) be a Riemannian manifold, Ω ⊆ M an open set with possibly non-compactclosure. For F ( x ) ∈ L ∞ loc ( M ) , P : T M −→ T M as above, let λ L (Ω) be the first eigenvalue on Ω of the operator L defined in (15). Then, the following conditions are equivalent:(1) λ L (Ω) ≥ .(2) There exists u ∈ C (Ω) , u > , weak solution of Lu = 0 on Ω .(3) There exists u ∈ H (Ω) , u > , weak solution of Lu ≤ on Ω . We remark that when P and F ( x ) are smooth, which will always be the case in the following,standard elliptic regularity ensures that solutions of Lu = 0 are also smooth. Gauss equations.
Our main reference is O’Neill’s book [33]. However, we remark thatwe adopt the convention of defining the Riemann curvature operator R of a semi-Riemannianmanifold (
M, g ) by setting R ( X, Y ) Z = ∇ X ( ∇ Y Z ) − ∇ Y ( ∇ X Z ) − ∇ [ X,Y ] Z for every X, Y, Z ∈ X ( M ) , so we have R ( X, Y ) Z = − R XY Z = R Y X Z for every X, Y, Z ∈ X ( M ), where R ( · , · ) ( · ) is thenotation used in [33]. The (0 , W, Z, X, Y ) = g ( R ( X, Y ) Z, W ) for every
X, Y, Z, W ∈ X ( M ) , the sectional curvature of any non-degenerate 2-plane X ∧ Y ⊆ T M spanned by a couple ofvectors
X, Y ∈ T M is(23) Sect( X ∧ Y ) = Riem( X, Y, X, Y ) g ( X, X ) g ( Y, Y ) − g ( X, Y ) . For every p ∈ M and for every choice of a g -orthonormal basis { E i , . . . , E dim M } of T p M , thevalues of the Ricci tensor Ric and of the scalar curvature S are given byRic( X, Y ) = dim M X i =1 g ( E i , E i )Riem( X, E i , Y, E i ) for every X, Y ∈ T p M, (24) S( p ) = dim M X i =1 g ( E i , E i )Ric( E i , E i ) . (25)Let ψ : M −→ M be a spacelike hypersurface immersed in a spacetime M , with unit timelikevector field N and shape operator A in the direction of N . For every X, Y, Z, W ∈ X ( M ) wehave the validity of Gauss equations (see Theorem 4.5 and Lemma 4.19 of [33])(26) Riem( X, Z, Y, W ) = Riem(
X, Z, Y, W ) − g ( AX, Y ) g ( AZ, W ) + g ( AX, W ) g ( AZ, Y ) , for any X, Y, Z, W ∈ T M , where Riem and Riem are the Riemann curvature tensors of (
M, g ) =(
M, ψ ∗ g ) and ( M , g ), respectively. For every p ∈ M , X, Z ∈ T p M and for any orthonormal basis { E , . . . , E m } of T p M we have, by (24) and (26),Ric( X, Y ) = m X i =1 Riem(
X, E i , Y, E i ) − m X i =1 g ( AX, Y ) g ( AE i , E i )+ m X i =1 g ( AX, E i ) g ( AY, E i )= Ric( X, Y ) + Riem(
X, N, Y, N ) + mHg ( AX, Y ) + g ( AX, AY ) , (27)recalling that mH = − trace( A ). Since { E , . . . , E m , N } is a g -orthonormal basis of T ψ ( p ) M , by(25), (24) and (27) we getS = m X i =1 (cid:0) Ric( E i , E i ) + Riem( N, E i , N, E i ) + mHg ( AE i , E i ) + g ( A E i , E i ) (cid:1) = m X i =1 Ric( E i , E i ) + Ric( N, N ) − m H + trace( A )= S + 2Ric( N, N ) − m H + trace( A ) . (28) TABLE MAXIMAL HYPERSURFACES IN LORENTZIAN SPACETIMES 11 If M has constant sectional curvature κ , then, for every X, Y, Z, W ∈ X ( M ),Riem( X, Z, Y, W ) = κ ( g ( X, Y ) g ( Z, W ) − g ( X, W ) g ( Y, Z )) , Ric(
X, Y ) = mκ g ( X, Y ) , S = m ( m + 1) κ and for every self-adjoint endomorphism P : T M −→ T M it follows that m X i =1 Ric(
P E i , E i ) = m X i =1 Ric(
P E i , E i ) + m X i =1 κ g ( N, N ) g ( P E i , E i )+ m X i =1 mHg ( AP E i , E i ) + m X i =1 g ( AP E i , AE i )= m X i =1 Ric(
P E i , E i ) − κ trace( P ) + mH trace( AP ) + trace( A P ) . (29) 3. Maximal hypersurfaces in locally symmetric spacetimes
In this section we first prove Theorems A and C of the Introduction, then we focus ourattention to the case of maximal surfaces and we give proofs of Theorems D and E. We startwith a slight generalization of results obtained by Nishikawa (Theorem B of [32]) and Ishihara(Theorems 1.1 and 1.2 of [26]), whose proof relies on Theorem 6 below, a consequence of a moregeneral result which is proved as Theorem 3.6 in [4].
Definition 5 (Definitions 2.1 and 2.3 of [4]) . Let M be a Riemannian manifold. The Omori-Yaumaximum principle for the Laplacian is said to hold on M if, for any function u ∈ C ( M ) with u ∗ = sup M u < + ∞ , there exists a sequence of points { x k } k ∈ N ⊆ M satisfying(i) u ( x k ) > u ∗ − k , (ii) |∇ u ( x k ) | < k and (iii) ∆ u ( x k ) < k ∀ k ∈ N . The weak maximum principle for the Laplacian is said to hold on M if, for any function u asabove, there exists a sequence of points { x k } k ∈ N ⊆ M such that (i) and (iii) hold. Theorem 6.
Let ( M, g ) be a Riemannian manifold on which the Omori-Yau maximum principlefor the Laplacian holds, a ∈ R and F a positive continuous function on [ a, + ∞ ) satisfying Z + ∞ a + ε (cid:18)Z ta F ( s ) ds (cid:19) − / dt < + ∞ and lim sup t → + ∞ tF ( t ) Z ta F ( s ) ds < + ∞ for some (hence, any) ε > . If u ∈ C ( M ) , f ∈ C ( R ) are such that ∆ u ≥ f ( u ) on M and lim inf t → + ∞ f ( t ) F ( t ) > , then u ∗ := sup M u is finite and f ( u ∗ ) ≤ . When M is a complete Riemannian manifold, a sufficient condition for the validity of the Omori-Yau maximum principle for the Laplacian on M is the existence of a constant C ∈ R such thatRic( X, X ) ≥ C | X | for every X ∈ T M , see the book [4]. We are now ready to state and provethe following result. Recall that a semi-Riemannian manifold is said to be locally symmetric ifits Riemannian curvature tensor is parallel. Semi-Riemannian manifolds of constant curvatureprovide the simplest examples of such manifolds.
Theorem 7.
Let M be a locally symmetric spacetime of dimension m + 1 whose Ricci andsectional curvatures satisfy Ric(
Z, Z ) ≥ c for all unit timelike vectors Z ∈ T M (30) Sect(Π) ≥ c for all non-degenerate tangent -planes Π ⊆ T M (31) for some constants c , c ∈ R and let ψ : M −→ M be a complete maximal hypersurface. Thenthe shape operator A satisfies (32) trace( A ) ≤ max { , − ( c + 2 mc ) } on M. In particular, if c ≥ − mc then ψ is totally geodesic.Proof . Let q ∈ M be a given point and let { E , . . . , E m } be an orthonormal basis of T q M given bythe principal directions of curvature, that is, eigenvectors of A ( q ) corresponding to the principalcurvatures κ , . . . , κ m . Since M is locally symmetric and ψ is maximal, following Nishikawa [32]we have Simons’ formula12 ∆trace( A ) = |∇ A | + Ric( N, N )trace( A )+ 2 X ≤ i N, N )trace( A ) ≥ c trace( A ) . So, the function u = trace( A ) satisfies ∆ u ≥ c + 2 mc + u ) u . For any given unit vector X ∈ T q M , by choosing an orthonormal basis { e , . . . , e m − } of X ⊥ ⊆ T q M we deduce fromGauss equations (see formulas (26) and (27) in Section 2.4) thatRic( X, X ) = m − X i =1 Sect( X ∧ e i ) + | AX | ≥ ( m − c . By bilinearity, it follows that Ric( X, X ) ≥ ( m − c | X | for every X ∈ T M . Since M is complete,the Omori-Yau maximum principle for the Laplacian on M . Applying Theorem 6 with the choice F ( t ) = t , we deduce that u is bounded above and that u ∗ = sup M u satisfies( c + 2 mc + u ∗ ) u ∗ ≤ , that is, u ∗ ≤ max { , − ( c + 2 mc ) } . (cid:3) Remark 8. We remark that in Theorem B of [32] the general estimate (32) is not stated and itis only proved that ψ is totally geodesic when c + 2 mc ≥ Theorem 9. Let M be a spacetime such that Ric( Z, Z ) ≥ for every timelike vector Z ∈ T M .If ψ : M −→ M is a maximal hypersurface, then ψ is stable. If M is also compact, then ψ istotally geodesic.Proof . The unit normal vector N on M is timelike so Ric( N, N ) ≥ Z M |∇ φ | + (cid:0) Ric( N, N ) + trace( A ) (cid:1) φ ≥ φ ∈ C ∞ c ( M ) . TABLE MAXIMAL HYPERSURFACES IN LORENTZIAN SPACETIMES 13 Therefore ψ is stable by definition and from Lemma 4 we deduce the existence of a positivefunction u satisfying ∆ u − (cid:0) Ric( N, N ) + trace( A ) (cid:1) u = 0. If M is compact we have Z M (cid:0) Ric( N, N ) + trace( A ) (cid:1) u = Z M ∆ u = 0by the divergence theorem. But Ric( N, N ) ≥ 0, trace( A ) ≥ u > 0, so trace( A ) ≡ 0, thatis, ψ is totally geodesic. (cid:3) From Theorems 7 and 9 we easily deduce the next Theorem 10. Let M be a spacetime of dimension m + 1 and constant curvature κ and let ψ : M −→ M be a complete maximal hypersurface.i) If κ > then M is compact and the immersion ψ is totally geodesic and unstable.ii) If κ = 0 then ψ is totally geodesic and stable.iii) If κ < then ψ is stable and the shape operator A and the scalar curvature S of M satisfy (33) trace( A ) ≤ − mκ, S ≤ ( m − mκ. If M is also compact, then ψ is totally geodesic.Proof . We have Ric( Z, Z ) = − mκ for all unit timelike vectors Z ∈ T M , so conditions (30) and(31) are satisfied with c = − mκ , c = κ . Hence, c + 2 mc = mκ and by Theorem 7 we havetrace( A ) ≤ max { , − mκ } . If κ ≥ ψ is totally geodesic, while for κ < A ) ≤ − mκ , that by (4) is equivalent to saying that the scalar curvature S of M satisfies S = ( m − mκ + trace( A ) ≤ ( m − mκ. If κ > M has constant positive sectional curvature κ and therefore it must be compactby the Bonnet-Myers theorem. Since ψ is totally geodesic, Ric( N, N ) + trace( A ) = − mκ < M and the constant, compactly supported function φ ≡ R M |∇ φ | + (cid:0) Ric( N, N ) + trace( A ) (cid:1) φ R M φ = − mκ < , implying λ L ( M ) < κ ≤ Z, Z ) = − mκ ≥ Z ∈ T M , so the other statements are direct consequences of Theorem 9. (cid:3) The following Theorem 11 is a refinement of Theorem 10 for maximal surfaces in 3-dimensionalspacetimes of constant sectional curvature. Let us recall from (4) that the Gaussian curvatureK of such a surface satisfies(34) 2K = S = S + 2 Ric( N, N ) + trace( A ) , where S denotes the scalar curvature of the ambient spacetime M . Theorem 11. Let ψ : M → M be a complete maximal oriented surface in a -dimensionalspacetime M of constant sectional curvature κ .i) If κ > then M is a totally geodesic, unstable round sphere of constant curvature κ .ii) If κ = 0 then M is totally geodesic, stable and it is either a Euclidean plane, a flatcylinder, or a flat torus.iii) If κ < then M is stable and has non-positive Gaussian curvature. If M is compactthen it is totally geodesic and its Euler characteristic satisfies (35) χ ( M ) = κ π Vol( M ) . If M is non-compact but its total curvature and its Euler characteristic are finite, then (36) χ ( M ) ≥ κ π Vol( M ) . Proof . By formula (34), the Gaussian curvature K of M always satisfiesK = 12 S + Ric( N, N ) + 12 trace( A ) = κ + 12 trace( A ) ≥ κ. If κ > M is a topological sphere. Since M is a compact surface of constant Gaussian curvature K = κ , by the Gauss-Bonnet theoremthe Euler characteristic χ ( M ) satisfies χ ( M ) = 12 π Z M K = κ π Vol( M ) > . Since χ ( M ) = 2 − g with g the topological genus of M , we conclude that g = 0 and M is atopological sphere.If κ = 0 then ψ is totally geodesic by Theorem 10 and M is a flat surface. Note that all of thethree cases described in point ii ) of the statement of the theorem can occur, for example when M is a spacelike slice of a Lorentzian product M = R × F with metric g = − dt + g F and ( F, g F )is a flat surface of one of the three above types.If κ < M is compact, we obtain (35) by applying again the Gauss-Bonnet theorem. If M is non-compact but has finite total curvature and finite Euler characteristic, inequality (36)follows by using the Cohn-Vossen’s inequality (see page 86 in [34]) χ ( M ) ≥ π Z M K , which is valid under our assumptions. (cid:3) We conclude this section by restating and proving Theorem E from the Introduction underslightly more general hypotheses, see Remark 14 below. Theorem 12. Let M be a -dimensional spacetime satisfying S + Ric( Z, Z ) ≥ for every unittimelike vector Z ∈ T M and let ψ : M −→ M be a complete stable oriented maximal surface.If ψ is stable at infinity and K + ∈ L ( M ) , then also K − ∈ L ( M ) , where K + and K − are thepositive and negative parts of the Gaussian curvature K of M . To prove Theorem 12 we need the following result due to Fischer-Colbrie, see Theorem 1 of [22]. Lemma 13. Let ( M, g ) be a complete Riemann surface with Gaussian curvature K . If u ∈ C ∞ ( M ) is a positive function such that ∆ u ≤ K u on M \ Ω for some relatively compact open set Ω ⊆ M , then ( M, u g ) is complete.Proof of Theorem 12. If M is compact, then it has finite total curvature and we are done. Hence,suppose that M is complete and non-compact. By (34) and since S+Ric( N, N ) ≥ 0, the Gaussiancurvature of M satisfies 2K ≥ Ric( N, N ) + trace( A ). Let Ω ⊆ M be a relatively compact openset such that the stability operator L = ∆ − (cid:0) Ric( N, N ) + trace( A ) (cid:1) satisfies λ L ( M \ Ω ) ≥ φ ∈ C ∞ ( M \ Ω ) we have Z M \ Ω |∇ φ | + K φ ≥ Z M \ Ω |∇ φ | + 2K φ ≥ Z M \ Ω |∇ φ | + (cid:0) Ric( N, N ) + trace( A ) (cid:1) φ ≥ − K satisfies λ ∆ − K1 ( M \ Ω ) ≥ u of ∆ u = K u on M \ Ω . By standard elliptic regularity results, u is smooth. Let TABLE MAXIMAL HYPERSURFACES IN LORENTZIAN SPACETIMES 15 Ω ⊆ M be a relatively compact open set such that Ω ⊆ Ω and let u ∈ C ∞ ( M ) be a positivefunction such that u = u on M \ Ω, so that(37) ∆ u = K u on M \ Ω . By Lemma 13, M is complete in the conformally deformed metric ˜ g = u g and the Gaussiancurvature ˜K of ( M, ˜ g ) is nonnegative on M \ Ω since(38) ˜K = 1 u (K − ∆ log u ) = K u − ∆ uu + |∇ u | u . Let o ∈ M be a given point and let B r denote the geodesic ball of ( M, ˜ g ) centered at o with radius r . Completeness of ( M, ˜ g ) by the Hopf-Rinow theorem enables us to choose R > ⊆ B R . As ˜ K ≥ M \ B R , by the volume comparison theoremthere exists C > ∂B r ) ≤ Cr for a.e. r ≥ R , with Vol( ∂B r ) the 1-dimensionalHausdorff measure of ∂B r induced by ˜ g . Note that this is well defined for a.e. r > 0. Thus(39) 1Vol( ∂B r ) / ∈ L (+ ∞ ) . We let w ( r ) = Vol( ∂B r ) and we set(40) A ( r ) = 1 w ( r ) Z ∂B r − K u d Vol ˜ g . We then consider the Cauchy problem(41) ( ( w ( r ) z ′ ) ′ + A ( r ) w ( r ) z = 0 on R + z (0 + ) = 1 , ( wz ′ )(0 + ) = 0 . Applying Proposition 4.2 of [11] we deduce the existence of a weak solution z of (41). We nowreason by contradiction and we suppose that K + ∈ L ( M, d Vol g ) while K − / ∈ L ( M, d Vol g ).Then(42) lim r → + ∞ Z B r K d Vol g = −∞ . By the coarea formula,(43) Z r A ( s ) w ( s ) ds = Z B r − K u d Vol ˜ g = Z B r − K d Vol g and therefore(44) lim r → + ∞ Z r A ( s ) w ( s ) ds = + ∞ . This, together with (39) implies, by Corollary 2.9 of [29], that z is oscillatory. Let R ≤ R < R be two consecutive zeros of z such that z > R , R ). Define a function ϕ ∈ Lip ( B R \ B R )by setting ϕ ( x ) := z ( r ( x )) for each x ∈ B R \ B R , with r ( x ) the distance from x to o in themetric ˜ g . By the coarea formula and (40) Z B R \ B R | ˜ ∇ ϕ | g d Vol ˜ g + Z B R \ B R K u ϕ d Vol ˜ g == Z R R w ( s ) z ′ ( s ) ds − Z R R A ( s ) w ( s ) z ( s ) ds. (45)Since z is a weak solution of (41), using z χ [ R ,R ] ∈ Lip ([ R , R ]) as a test function we get(46) Z R R w ( s ) z ′ ( s ) ds = Z R R A ( s ) w ( s ) z ( s ) ds. Collecting (45) and (46), by the monotonicity property of eigenvalues we obtain λ ˜ L ( M \ Ω) < λ ˜ L ( B R \ B R ) ≤ R B R \ B R | ˜ ∇ ϕ | g d Vol ˜ g + R B R \ B R K u ϕ d Vol ˜ g R B R \ B R ϕ = 0 , where ˜ L is the operator defined by ˜ L = ˜∆ − K u , with ˜∆ the Laplace-Beltrami operator of ( M, ˜ g ). Since u ˜∆ f = ∆ f on M for every f ∈ C ( M ),we see from (37) that u is a positive solution of ˜ Lu = 0 on M \ Ω, so λ ˜ L ( M \ Ω) ≥ (cid:3) Remark 14. If { E , E , Z } is a local Lorentz orthonormal frame on T M and Z is timelike thenS + Ric( Z, Z ) = Ric( E , E ) + Ric( E , E ), so Theorem E is indeed a consequence of Theorem12. 4. Maximal hypersurfaces in GRW spacetimes Let ( F, g F ) be an m -dimensional (connected) Riemannian manifold, m ≥ I an open intervalin R endowed with the metric − dt and ρ a positive smooth function defined on I . The Gener-alized Robertson-Walker (GRW) spacetime M = I × ρ F , with fiber ( F, g F ), base ( I, − dt ) andwarping function ρ , is the product manifold M = I × F endowed with the Lorentzian metric(47) g = − π ∗ I ( dt ) + ρ ( π I ) π ∗ F ( g F ) , where, respectively, π I and π F denote the projections from M onto I and F . If the fiber hasconstant sectional curvature, M is simply called a Robertson-Walker spacetime.In any GRW spacetime M = I × ρ F , the coordinate vector field ∂ t := ∂/∂t is a unit timelikevector field and hence M is time-orientable. With a slight abuse of notation, we write ρ ( t ), ρ ′ ( t ), ρ ′′ ( t ) to denote ρ ◦ π I , ρ ′ ◦ π I , ρ ′′ ◦ π I . If we consider the timelike vector field T := ρ ( t ) ∂ t , from the relation between the Levi-Civita connection of M and those of the base and the fiber(see Corollary 7.35 of [33]) it follows that(48) ∇ X T = ρ ′ ( t ) X, for any X ∈ X ( M ), where ∇ is the Levi-Civita connection of the Lorentzian metric (47). Thus, T is conformal and its metrically equivalent 1-form is closed, that is, T is a closed conformalvector field. The curvature tensors of M are given by the following formulas. Lemma 15. The GRW spacetime M = I × ρ F has Riemann and Ricci curvature tensors givenby Riem = ρ ( t ) π F ∗ Riem F + ρ ( t ) (cid:0) ρ ′ ( t ) − ρ ( t ) ρ ′′ ( t ) (cid:1) π F ∗ ( g F (cid:13)∧ g F ) + ρ ′′ ( t ) ρ ( t ) ( g (cid:13)∧ g ) , (49) Ric = π F ∗ Ric F − ( m − (cid:0) ρ ( t ) ρ ′′ ( t ) − ρ ′ ( t ) (cid:1) π F ∗ g F + m ρ ′′ ( t ) ρ ( t ) g, (50) where Riem F and Ric F are the Riemann and Ricci tensors of ( F, g F ) and (cid:13)∧ denotes Kulkarni-Nomizu product. TABLE MAXIMAL HYPERSURFACES IN LORENTZIAN SPACETIMES 17 Proof. The gradient and the Hessian of the warping function ρ in the base ( I, − dt ) are given by ∇ I ρ = − ρ ′ ∂ t and Hess( ρ ) = ρ ′′ dt ⊗ dt . Since ∇ ∂ t ∂ t = 0 on M , the lift − ρ ′ ( t ) ∂ t ∈ X ( M ) of ∇ I ρ satisfies ∇ ∂ t ( − ρ ′ ( t ) ∂ t ) = − ρ ′′ ( t ) ∂ t on M . Let p = ( t, x ) ∈ M be given and let X, Y, U, V, W ∈ T p M be such that π F ∗ ( X ) = π F ∗ ( Y ) = 0 ∈ T x F, π I ∗ ( U ) = π I ∗ ( V ) = π I ∗ ( W ) = 0 ∈ T t I. From formulas (2)-(5) of Proposition 7.42 of [33] we getRiem( V, W, U, · ) = ρ ( t ) π ∗ F Riem F ( V, W, U, · )+ ρ ′ ( t ) ρ ( t ) ( g ( V, U ) g ( W, · ) − g ( W, U ) g ( V, · )) , Riem( V, X, W, · ) = ρ ′′ ( t ) ρ ( t ) g ( V, W ) g ( X, · ) , Riem( V, X, Y, · ) = − ρ ′′ ( t ) ρ ( t ) g ( X, Y ) g ( V, · )and from formulas (1)-(3) of Corollary 7.43 of [33] we also haveRic( V, · ) = π ∗ F Ric F ( V, · ) + (cid:18) ρ ′′ ( t ) ρ ( t ) + ( m − ρ ′ ( t ) ρ ( t ) (cid:19) g ( V, · ) , Ric( X, · ) = m ρ ′′ ( t ) ρ ( t ) g ( X, · )A direct computation shows that the RHS’s of (49) and (50) also satisfy the identities above. Bythe symmetry properties of Riem, these identities uniquely determine its action on T p M . (cid:3) As we see from (49), M has constant curvature κ if and only if the fiber F has constantcurvature κ F and the warping function ρ satisfies(51) κ = ρ ′′ ρ and κ F = ρρ ′′ − ( ρ ′ ) on I. These equations are not independent. In fact, if there exists C ∈ R such that ρρ ′′ − ( ρ ′ ) = C on an interval I ⊆ I , then (cid:18) ρ ′′ ρ (cid:19) ′ = Cρ + (cid:18) ρ ′ ρ (cid:19) ! ′ = − Cρ ′ ρ + 2 ρ ′ ρ (cid:18) ρ ′ ρ (cid:19) ′ = − Cρ ′ ρ + 2 ρ ′ ρ ρρ ′′ − ( ρ ′ ) ρ = 0 , that is, ρ ′′ /ρ is constant on I . We also characterize GRW spacetimes with spacelike sectionalcurvatures bounded below. Lemma 16. Let M = I × ρ F be a GRW spacetime. For every C ∈ R , the following areequivalent:i) Sect(Π) ≥ C for every spacelike -plane Π ⊆ T M ,ii) there exists C ∈ R such that Sect F (Π ) ≥ C for every -plane Π ⊆ T FC − ρρ ′′ + ( ρ ′ ) ρ ≥ max (cid:26) C − ρ ′′ ρ , (cid:27) on I. Proof . For every t ∈ I , define the (0 , V t : T F ⊗ T F ⊗ T F ⊗ T F → C ∞ ( F ) by setting V t ( X, Y, Z, W ) = ρ ( t ) Riem F ( X, Y, Z, W )+ ρ ( t ) (cid:0) ρ ′ ( t ) − ρ ( t ) ρ ′′ ( t ) (cid:1) ( g F (cid:13)∧ g F )( X, Y, Z, W )(52) for every X, Y, Z, W ∈ T x F , x ∈ F .Assume that i) holds. Let ( t, x ) ∈ M , α ∈ R , X, Y be given, with(53) X, Y ∈ T x F such that g F ( X, X ) = g F ( Y, Y ) = 1 , g F ( X, Y ) = 0 . The vectors(54) E = 1 ρ ( t ) X, E = cosh αρ ( t ) Y + sinh α ∂ t belong to T ( t,x ) M and satisfy g ( E , E ) = g ( E , E ) = 1, g ( E , E ) = 0, so, by (49), (52) and(54),(55) C ≤ Sect( E ∧ E ) = Riem( E , E , E , E ) = cosh αρ ( t ) V t ( X, Y, X, Y ) + ρ ′′ ( t ) ρ ( t ) , that is,(56) Sect F ( X ∧ Y ) − ρ ( t ) ρ ′′ ( t ) + ρ ′ ( t ) ρ ( t ) = V t ( X, Y, X, Y ) ρ ( t ) ≥ α (cid:18) C − ρ ′′ ( t ) ρ ( t ) (cid:19) . For α = 0 and α → + ∞ we respectively getSect F ( X ∧ Y ) − ρ ( t ) ρ ′′ ( t ) + ρ ′ ( t ) ρ ( t ) ≥ C − ρ ′′ ( t ) ρ ( t ) , Sect F ( X ∧ Y ) − ρ ( t ) ρ ′′ ( t ) + ρ ′ ( t ) ρ ( t ) ≥ . (57)For any fixed t , these inequalities must hold for every x ∈ F and for every X, Y as in (53), so C = inf { Sect F (Π ) : Π ⊆ T F is a 2-plane } is finite and ii) follows with C = C .Vice versa, assume that ii) holds. Let ( t, x ) ∈ M and Π ⊆ T ( t,x ) M a spacelike 2-plane begiven. We can find a g -orthonormal basis { E , E } for Π of the form (54), with α ∈ R and X, Y as in (53). Since inequalities (57) hold by assumption, we have (56) and therefore (55), that is,Sect(Π) ≥ C . As Π ⊆ T M is arbitrarily given, we obtain i). (cid:3) Let ψ : M −→ M be a spacelike hypersurface immersed in the GRW spacetime M = I × ρ F .Consider the unit timelike vector N normal to M with the same time orientation as ∂ t and let A and H be the shape operator and the mean curvature of ψ in the direction of N as describedin Section 2. The height function τ of the immersion ψ onto the factor I and the amplitude θ ofthe hyperbolic angle between N and ∂ t are given by(58) τ = π I ◦ ψ, cosh θ = − g ( N, ∂ t ) . Note that θ is well defined (up to a sign) by the wrong-way Cauchy Schwarz inequality, since N and ∂ t are unit timelike vectors with the same time-orientation. As above, we write ρ ( τ ), ρ ′ ( τ ), ρ ′′ ( τ ) to denote ρ ◦ τ , ρ ′ ◦ τ , ρ ′′ ◦ τ . For a fixed t ∈ I , we set(59) η ( t ) = Z tt ρ ( s ) ds ∀ t ∈ I, η = η ◦ τ. Since ρ > η is strictly increasing on I . We also consider the positive function(60) v = − g ( T, N ) = ρ ( τ ) cosh θ. In the sequel we will make extensive use of these auxiliary functions. Denoting by ∇ and ∆the Levi-Civita connection and the Laplace-Beltrami operator of ( M, g ), we have the followingcomputational result. TABLE MAXIMAL HYPERSURFACES IN LORENTZIAN SPACETIMES 19 Lemma 17. Let ψ : M −→ M be a spacelike hypersurface immersed in a GRW spacetime M = I × ρ F and let T , N , A , H , θ , τ , η and v be as above. Then |∇ τ | = sinh θ, (61) |∇ η | = ρ ( τ ) sinh θ = v − ρ ( τ ) , (62) |∇ v | ≤ trace( A ) |∇ η | = trace( A )( v − ρ ( τ ) ) , (63) ∆ η = − mρ ′ ( τ ) + mHv, (64) ∆ v = (cid:18) Ric( N, N ) + trace( A ) + m ρ ′′ ( τ ) ρ ( τ ) (cid:19) v − mg ( T ⊤ , ∇ H ) − mHρ ′ ( τ ) , (65) where T ⊤ is the tangential part of T along ψ . If ψ is maximal, then |∇ v | ≤ m − m trace( A )( v − ρ ( τ ) ) , (66) ∆ η = − mρ ′ ( τ ) , (67) ∆ v = (cid:18) Ric( N, N ) + trace( A ) + m ρ ′′ ( τ ) ρ ( τ ) (cid:19) v. (68) Proof . As dt = g ( · , ∂ t ) in M , we have that ∇ τ = − ∂ ⊤ t on M , where ∂ ⊤ t is the tangential partof ∂ t along ψ . From (59) it follows that(69) ∇ η = ρ ( τ ) ∇ τ = − T ⊤ . Using the orthogonal decomposition ∂ t = ∂ ⊤ t + cosh θN and g ( ∂ t , ∂ t ) = g ( N, N ) = − |∇ τ | = g ( ∂ ⊤ t , ∂ ⊤ t ) = g ( ∂ t , ∂ t ) − cosh θ g ( N, N ) = sinh θ, |∇ η | = ρ ( τ ) sinh θ, proving (61) and (62) in view of (60). Since the tangential component of T along ψ is T ⊤ = T + g ( T , N ) N = T − vN , a direct computation using (48) gives(70) ∇ v = AT ⊤ = − A ∇ η. Denoting by λ , . . . , λ m the eigenvalues of A : T p M → T p M at a given point p ∈ M , we have(71) | AX | = g ( A X, X ) ≤ | X | max ≤ i ≤ m λ i ≤ trace( A ) | X | for each vector X ∈ T p M . So, (63) follows from (70) and (62). If ψ is maximal, thentrace( A ) ≡ ≤ i ≤ m we have λ i = − P ≤ j ≤ m,j = i λ j and using Cauchy inequality we get λ i + 1 m − λ i = λ i + 1 m − X ≤ j ≤ mj = i λ j ≤ λ i + m − m − X ≤ j ≤ mj = i λ j = trace( A ) . Hence, λ i ≤ m − m trace( A ) for each 1 ≤ i ≤ m , thus | AX | ≤ m − m trace( A ) | X | for every X ∈ T p M , proving the refined version (66).In order to prove (64) and (65), we recall that Gauss and Weingarten formulas for the immer-sion ψ are respectively given by(72) ∇ X Y = ∇ X Y − g ( AX, Y ) N and AX = −∇ X N for any X, Y ∈ X ( M ) and that the covariant derivative ∇ A of A , defined by ( ∇ X A ) Y = ∇ X ( AY ) − A ( ∇ X Y ) for every X, Y ∈ X ( M ), satisfies Codazzi equation(73) g (( ∇ X A ) Y, Z ) = g (( ∇ Y A ) X, Z ) − g (R( X, Y ) N, Z ) for every X, Y, Z ∈ X ( M ). Taking the tangential component in (48) and using (72) and (73)together with the definition of ∇ A , we get ∇ X T ⊤ = − ρ ( τ ) g ( N, ∂ t ) AX + ρ ′ ( τ ) X = vAX + ρ ′ ( τ ) X, (74) ∇ X ( AT ⊤ ) = ( ∇ X A ) T ⊤ + A ( ∇ X T ⊤ )(75) = ( ∇ T ⊤ A ) X − (R( X, T ⊤ ) N ) ⊤ + vA X + ρ ′ ( τ ) AX (76)for any X ∈ X ( M ). By definition, for every function u ∈ C ∞ ( M ),∆ u = trace( ∇ ( ∇ u )) = m X i =1 g ( ∇ E i ∇ u, E i )for any choice of a local orthonormal frame { E , . . . , E m } on T M . By (69), (70), (74), (75) andsince trace( A ) = − mH and trace( ∇ T ⊤ A ) = ∇ T ⊤ (trace( A )) = − m ∇ T ⊤ H , we obtain∆ η = − mρ ′ ( τ ) − mHv, ∆ v = − Ric( T ⊤ , N ) + trace( A ) v − m ∇ T ⊤ H − mHρ ′ ( τ ) , and (64) is proved. Writing T ⊤ = T − vN , by (50) we have(77) − Ric( T ⊤ , N ) = − Ric( T, N ) + Ric( N, N ) v = (cid:18) m ρ ′′ ( τ ) ρ ( τ ) + Ric( N, N ) (cid:19) v, as π F ∗ Ric F ( ∂ t , · ) = 0 and ( dt ⊗ dt )( T, N ) = − g ( T, N ) = ρ ( τ ) cosh θ = v . This concludes theproof of (65). If ψ is maximal, then H ≡ (cid:3) Remark 18. The spacelike slices { t } × F , t ∈ I of M are totally umbilical hypersurfaces, inother words they satisfy trace( A ) = mH , and they have mean curvature H = ρ ′ ( t ) /ρ ( t ) in thedirection of the future-pointing normal. This is a consequence of (64) and (65), as the image ψ ( M ) of an immersed hypersurface ψ : M −→ M is contained in a spacelike slice if and only if τ is constant on M , in which case N = ∂ t and v ≡ ρ ( t ).As a first application of equation (67), we prove Theorem G of the Introduction as a corollaryof the following result, which generalizes Theorem 3.7 of [7]. Theorem 19. Let ψ : M −→ M be a maximal hypersurface in a GRW spacetime M = I × ρ F .If the weak maximum principle for the Laplacian holds on M and ψ ( M ) is contained in a slab [ a, b ] × F ⊆ I × F , then ρ ′ ( τ ∗ ) ≥ and ρ ′ ( τ ∗ ) ≤ , where τ ∗ = sup M τ , τ ∗ = inf M τ .Proof . Let η be defined as in (59). Since τ ( M ) ⊆ [ a, b ], we have(78) − ∞ < Z at ρ ( s ) ds = η ( a ) ≤ η ≤ η ( b ) = Z bt ρ ( s ) ds < + ∞ on M. By the weak maximum principle applied to η and − η , see Definition 5, we can find two sequencesof points { x k } k ∈ N , { y k } k ∈ N ⊆ M such thatlim k → + ∞ η ( x k ) = sup M η, lim sup k → + ∞ ∆ η ( x k ) ≤ , lim k → + ∞ η ( y k ) = inf M η, lim inf k → + ∞ ∆ η ( y k ) ≥ . We recall that η is a strictly monotonic function of τ and that ∆ η = − mρ ′ ( τ ) on M . Hence, − mρ ′ ( τ ∗ ) = lim k → + ∞ − mρ ′ ( τ ( x k ))) = lim k → + ∞ ∆ η ( x k ) ≤ − mρ ′ ( τ ∗ ) ≥ (cid:3) TABLE MAXIMAL HYPERSURFACES IN LORENTZIAN SPACETIMES 21 Corollary 20. Let M = I × ρ F be a GRW spacetime whose sectional curvatures on spacelike -planes are bounded below and let ψ : M −→ M be a complete maximal hypersurface. If ψ ( M ) is contained in a slab [ a, b ] × F , then ρ ′ ( τ ∗ ) ≥ , ρ ′ ( τ ∗ ) ≤ , where τ ∗ = sup M τ , τ ∗ = inf M τ . Inparticular, if there exist t , t ∈ [ a, b ] such that ρ ′ = 0 on [ t , t ] , then the following implicationshold: i) if ρ ′ < on [ a, t ) and ρ ′ > on ( t , b ] , then ψ ( M ) must intersect every spacelike slice { t } × F with t ≤ t ≤ t ;ii) if ρ ′ > on [ a, t ) and ρ ′ < on ( t , b ] , then ψ ( M ) ⊆ [ t , t ] × F .Proof . As in the proof of Theorem 7, since M is maximal and there exists C ∈ R such thatSect(Π) ≥ C for every spacelike 2-plane Π ⊆ T M , the Ricci curvature of M is bounded below by(dim M − C . Since ( M, g ) is complete, the weak maximum principle for the Laplacian holdson M and the main statement is a direct consequence of Theorem 19. If t ∈ [ a, b ] is such that ρ ′ < ρ ′ > 0) on [ a, t ), then τ ∗ ≥ t (resp., τ ∗ ≥ t ). Similarly, if t ∈ [ a, b ] issuch that ρ ′ > ρ ′ < 0) on ( t , b ], then τ ∗ ≤ t (resp., τ ∗ ≤ t ). This concludes the proof. (cid:3) The following consequence is a Frankel type result. Corollary 21. Let S m +11 ( κ ) be the ( m + 1) -dimensional de Sitter spacetime of constant cur-vature κ > and let ψ : M −→ S m +11 ( κ ) , ψ : M −→ S m +11 ( κ ) be two complete maximalhypersurfaces. Then ψ ( M ) ∩ ψ ( M ) = ∅ .Proof . Let F = S m ( κ ) be the standard m -sphere of constant curvature κ > 0, set ρ ( t ) =cosh( √ κt ) for every t ∈ R and let M = R × ρ F . The GRW spacetime M is isometric to S m +11 ( κ ) (see, for instance, page 339 of [44]). By Theorem 10, M is compact and ψ is totallygeodesic, so there exists an isometry Ψ : S m +11 ( κ ) −→ M such that Ψ ◦ ψ : M → M sends M into the totally geodesic spacelike slice { } × F . Since M is compact and F is connected,(Ψ ◦ ψ )( M ) = { } × F . By Theorem 10 again, M is compact. Let ψ := Ψ ◦ ψ : M → M .The projection of ψ ( M ) on the R -factor of M is compact. Moreover, ρ ′ < −∞ , 0) and ρ ′ > , + ∞ ). So, we apply point i) of Corollary 20 to obtain that ( { } × F ) ∩ ψ ( M ) = ∅ .Since ψ ( M ) = Ψ − ( { } × F ), ψ ( M ) = Ψ − ( ψ ( M )), we obtain ψ ( M ) ∩ ψ ( M ) = ∅ . (cid:3) We are now ready to prove Theorem F of the Introduction. Theorem 22. Let ψ : M −→ M be a complete maximal surface in a -dimensional GRWspacetime M = I × ρ F . Suppose that M has nonnegative sectional curvatures on spaceliketangent -planes. Then ψ is totally geodesic and one of the following cases occurs:i) ψ ( M ) is a spacelike slice { t } × F for some t ∈ I such that ρ ′ ( t ) = 0 ,ii) ( F, g F ) is a flat, complete Riemann surface and M is the product manifold R × F endowedwith the flat metric − dt + g F ,iii) ( F, g F ) is a compact Riemann surface of constant positive Gaussian curvature, M is around sphere and the spacetime M has constant positive sectional curvature in the slab τ ( M ) × F .Proof . As already remarked, the assumption that M has nonnegative sectional curvatures onspacelike 2-planes implies that Ric ≥ M in the sense of quadratic forms, that is, theGaussian curvature K of M is nonnegative. Moreover, denoting by K F the Gaussian curvatureof F , by Lemma 16 there exists C ∈ R such that, for each ( t, x ) ∈ M ,K F ( x ) ≥ C , C − ρ ( t ) ρ ′′ ( t ) + ρ ′ ( t ) ρ ( t ) ≥ ,C − ρ ( t ) ρ ′′ ( t ) + ρ ′ ( t ) ρ ( t ) ≥ − ρ ′′ ( t ) ρ ( t ) . (79) Let v = − g ( T, N ) on M . By (68), for any α ∈ R the function v − α satisfies∆ v − α = − αv − α − ∆ v + α ( α + 1) v − α − |∇ v | = − αv − α (cid:18) Ric( N, N ) + trace( A ) + 2 ρ ′′ ( τ ) ρ ( τ ) − ( α + 1) |∇ v | v (cid:19) (80)on M . Since π ∗ F g F ( N, N ) = ρ ( t ) − ( g + dt ⊗ dt )( N, N ) = ρ ( t ) − ( − θ ) = ρ ( t ) − sinh θ ,from (50) and the first two inequalities in (79) we have(81) Ric( N, N ) + 2 ρ ′′ ( t ) ρ ( t ) = sinh θ (cid:18) K F ( x ) ρ ( t ) − ρ ′′ ( t ) ρ ( t ) + ρ ′ ( t ) ρ ( t ) (cid:19) ≥ α ∈ [ − , A ) − ( α + 1) |∇ v | v ≥ − α A ) ≥ . Inserting these inequalities into (80) we find that for every α ∈ [0 , 1] the positive function v − α issuperharmonic on M . In particular, 1 /v is a positive superharmonic function on M . If ( M, g ) iscomplete, then it is parabolic because of its nonnegative Gaussian curvature, so 1 /v is constanton M . Therefore v is also constant and from (68) we obtain(83) Ric( N, N ) + 2 ρ ′′ ( τ ) ρ ( τ ) ≡ , trace( A ) ≡ M in view of (81). Hence, ψ is totally geodesic.Suppose that ψ ( M ) is not a spacelike slice: then τ is not constant on M and by (61) thehyperbolic angle θ is not identically null, so M := { q ∈ M : sinh θ ( q ) = 0 } is a nonemptyopen subset of M . Since η is a strictly increasing function on I , the function η defined in (59) isnonconstant. Moreover, equation (67) reads(84) ∆ η = − ρ ′ ◦ η − )( η )and the function − ρ ′ ◦ η − ) is of class C on its domain; hence the unique continuation propertyholds for equation (84), that is, η is constant on some nonempty open subset of M if and onlyif it is constant on M (see Theorem A.5 of [39]). Therefore, M is a dense open subset of M .Finally, v = ρ ( τ ) cosh θ ≥ ρ ( τ ) implies that ρ is bounded on τ ( M ) ⊆ I , as v is constant. We set(85) I = { t ∈ I : ρ ( t ) ρ ′′ ( t ) − ρ ′ ( t ) = C } , F = { x ∈ F : K F ( x ) = C } . Let q ∈ M be given and set ( t, x ) = ψ ( q ) ∈ M . From (83) it follows that (81) holds withthe equality sign. Since sinh θ ( q ) = 0, the same is true for the first two inequalities in (79).Therefore, τ ( M ) ⊆ I and ( π F ◦ ψ )( M ) ⊆ F . Note that I and F are closed in I and F ,respectively, and that τ is constant on each connected component of M \ M . Hence, τ ( M ) = τ ( M ) ⊆ I , ( π F ◦ ψ )( M ) = ( π F ◦ ψ )( M ) ⊆ F . As observed at the beginning of this section, ρ ′′ /ρ is constant on every interval contained in I . Since M is connected, τ ( M ) is an intervaland there exists κ ∈ R such that(86) ρ ′′ ( t ) ρ ( t ) = κ for each t ∈ τ ( M ) . By the third inequality in (79), κ ≥ 0. So far, we have proved that M has constant curvature κ ≥ τ ( M ) × ( π F ◦ ψ )( M ) ⊆ I × F ⊆ I × F. We conclude the proof by showing that ( π F ◦ ψ )( M ) = F and that in case where κ = 0 it mustbe τ ( M ) = R and ρ ′ ≡ R . TABLE MAXIMAL HYPERSURFACES IN LORENTZIAN SPACETIMES 23 First suppose that κ = 0. Then, ρ ′ is constant on τ ( M ). If ρ ′ ≡ C on τ ( M ) for a positiveconstant C > 0, then η is a nonconstant superharmonic function on M because it satisfiesequation (84), so it cannot be bounded above on the parabolic surface M . This implies that τ ( M ) is not bounded above, otherwise we would havesup M η ≤ Z sup τ ( M ) t ρ ( s ) ds = (sup τ ( M ) − t ) ρ ( t ) + C τ ( M ) − t ) < + ∞ . Since ρ ′ ≡ C > τ ( M ) and τ ( M ) is not bounded above, we obtain that ρ is not bounded on τ ( M ) and we reach a contradiction. Similarly, we conclude that ρ ′ cannot be identically equalto a negative constant on τ ( M ), so we are left with the case where ρ ′ ≡ τ ( M ). In thiscase, by (84) we have that η is a nonconstant harmonic function on M and therefore it is notbounded above nor below. Arguing as above we can show that τ ( M ) is not bounded above norbelow, so R = τ ( M ) ⊆ I ⊆ R and we conclude that I = R . By (85) and (87), since ρ ′ ≡ τ ( M ) we deduce C = 0 and K F = 0 on F . If we set F ′ = ( π F ◦ ψ )( M ) and we endow thesurface M with the metric σ := ( π F ◦ ψ ) ∗ g F , then ( M, σ ) is complete because σ = g + dτ ≥ g and ( π F ◦ ψ ) : M → F ′ is a local Riemannian isometry, so ( F ′ , ( g F ) | F ′ ) is also complete and weconclude that F = F ′ ⊆ F ⊆ F . In particular, F = F and K F ≡ F .Now, suppose that (86) holds with κ > ψ ( M ) is contained in the cylinder Σ and ψ istotally geodesic, so M has constant positive Gaussian curvature κ and therefore it is compact byBonnet theorem. In particular, from Theorem 11 it follows that ( M, g ) is a round sphere. Themap π F ◦ ψ is continuous and open (being a local diffeomorphism), so ( π F ◦ ψ )( M ) is compactand open in F . Since F is connected, we conclude ( π F ◦ ψ )( M ) = F . Moreover, since the secondof (79) holds with the equality sign for every t ∈ τ ( M ), we have (log ρ ) ′′ = C /ρ on τ ( M ).Suppose, by contradiction, that C ≤ 0. Then log ρ is concave and the sign of ρ ′ is nonincreasingon τ ( M ). As τ ( M ) is compact, we can apply the last statement of Corollary 20 to deduce that ρ ′ ( τ ) ≡ M and by (84) we get that η is a nonconstant harmonic function on the compactsurface M , contradiction. Therefore, C > F = F has constantpositive Gaussian curvature. (cid:3) We conclude this section with the following two results. Theorem 23. Let M = I × ρ F be a GRW spacetime with ρ ′′ ≤ . Let ψ : M −→ M be acomplete, non-compact, maximal hypersurface in M such that, for some o ∈ M and for some(hence, any) a > , (88) lim r → + ∞ Z ra dt Vol( ∂B t ) = + ∞ , where B t is the geodesic ball of ( M, g ) centered at o with radius t . Then (89) lim sup r → + ∞ Z B r (cid:0) Ric( N, N ) + trace( A ) (cid:1) ≥ . Proof . From (68) we know that the positive function v = − g ( T , N ) satisfies(90) Lv = ∆ v − (cid:0) Ric( N, N ) + trace( A ) (cid:1) v = m ρ ′′ ( τ ) ρ ( τ ) v ≤ . Therefore, by Lemma 4, the operator L satisfies λ L ( M ) ≥ 0. Set(91) w ( r ) = Vol( ∂B r ) , A ( r ) = − w ( r ) Z ∂B r (cid:0) Ric( N, N ) + trace( A ) (cid:1) and consider the weak solution z ∈ Lip loc ( R +0 ) of the Cauchy problem(92) ( ( w ( r ) z ′ ) ′ + A ( r ) w ( r ) z = 0 on R + z (0 + ) = 1 , ( wz ′ )(0 + ) = 0 . We have z > R +0 . If not, let R be the first positive zero of z and set ϕ ( x ) = z ( r ( x ))for x ∈ B R , with r ( x ) the distance from x to o in ( M, g ). Since z solves problem (92), byProposition 2 and using the coarea formula as in the proof of Theorem 12 we get0 ≤ λ L ( M ) < λ L ( B R ) ≤ R R w ( r ) z ′ ( r ) dr + R R A ( r ) w ( r ) z ( r ) dr R R w ( r ) z ( r ) = 0 , contradiction. So, problem (92) has a positive solution and (88) holds. By Theorem 2.8 of [29],(93) 0 ≥ lim inf R −→ + ∞ Z R A ( s ) w ( s ) ds = − lim sup R −→ + ∞ Z B R (cid:0) Ric( N, N ) + trace( A ) (cid:1) . (cid:3) Theorem 24. Let ψ : M −→ M be a complete stable maximal hypersurface in a GRW spacetime M = I × ρ F and let τ , v be as above.i) If M is compact, then either ρ ′′ ( τ ) ≡ on M or ρ ′′ ( τ ) attains both positive and negativevalues on M .ii) If M is non-compact and for some o ∈ M the function v satisfies (94) lim r → + ∞ Z ra (cid:18)Z ∂B s v (cid:19) − ds = + ∞ for some (hence any) a > , where B s is the geodesic ball of ( M, g ) centered at o withradius s , then either ρ ′′ ( τ ) ≡ on M or ρ ′′ ( τ ) attains negative values at some points of M .Proof . Since M is stable, from Lemma 4 there exists a positive function u satisfying ∆ u = (cid:0) Ric( N, N ) + trace( A ) (cid:1) u on M . Set ϕ = u/v . By (68), a direct computation shows that(95) div( v ∇ ϕ ) v = ∆ ϕ + 2 g (cid:18) ∇ vv , ∇ ϕ (cid:19) = ∆ uv − u ∆ vv = − m ρ ′′ ( τ ) ρ ( τ ) ϕ on M. i) Suppose that M is compact. We apply the divergence theorem to obtain0 = Z M div( v ∇ ϕ ) = − m Z M ρ ′′ ( τ ) ρ ( τ ) v ϕ. Since v ϕ/ρ ( τ ) > M , if ρ ′′ ( τ ) ≥ ρ ′′ ( τ ) ≤ M then it must be ρ ′′ ( τ ) ≡ M is non-compact. (94) ensures that any positive function w ∈ C ( M ) suchthat div( v ∇ w ) ≤ M must be constant, see Theorem 4.14 of [4]. Since ϕ > 0, if ρ ′′ ( τ ) ≥ M then from (95) we deduce that ϕ is constant and therefore ρ ′′ ( τ ) ≡ (cid:3) Remark 25. Let ψ : M −→ M = I × ρ F be a maximal hypersurface such that τ ( M ) ⊆ T = { t ∈ I : ρ ′′ ( t ) ≤ } . Since v > M , by condition 3 of Lemma 4 and (68) we immediatelydeduce that M is stable. Similarly, if τ ( M \ P ) ⊆ T for some relatively compact open set P ,then ψ is stable at infinity. For T = I , stability of ψ is observed in Theorem 7 of [21]. TABLE MAXIMAL HYPERSURFACES IN LORENTZIAN SPACETIMES 25 Higher order mean curvatures in Robertson-Walker spacetimes In this section we will consider spacelike hypersurfaces in spacetimes of constant sectionalcurvature. Let ψ : M −→ M be such a hypersurface and suppose that the ( k + 1)-th meancurvature function H k +1 vanishes on M , for some 1 ≤ k ≤ m − 2. Then, the k -th Newton tensor P k corresponding to the shape operator A of ψ is positive definite if and only if H k > A ) > k on M , see Proposition 6.27 of [11]. Hence, the differential operator L k defined in(11) is elliptic if and only if these conditions are satisfied. Furthermore, since M has constantcurvature, L k can be written in divergence form, that is,(96) L k u = div( P k ( ∇ u )) for every u ∈ C ( M ) . If the ambient spacetime has a Robertson-Walker spacetime structure M = I × ρ F , then we canconsider the functions η and v on M as in the previous section. The action of L k on η and v isgiven by identities (97) and (98) below, also proved in Lemma 4.1 of [3] and Lemma 3.1 of [16]. Lemma 26. Let ψ : M −→ M be a spacelike hypersurface in a Robertson-Walker spacetime M = I × ρ F of constant curvature and let η and v be defined in (59) and (60). For ≤ k ≤ m − , L k η = − c k ρ ′ ( τ ) H k + c k H k +1 v, (97) L k v = trace( A P k ) v − (cid:18) mk + 1 (cid:19) g ( T ⊤ , ∇ H k +1 ) − c k H k +1 ρ ′ ( τ ) . (98) Proof . In the proof of Lemma 17 we have already calculated the second covariant derivatives of η and v . More precisely, from (74) we can write ∇ X ∇ η = vAX + ρ ′ ( τ ) X, (99) ∇ X ∇ v = ( ∇ T ⊤ A ) X − (cid:0) R (cid:0) X, T ⊤ (cid:1) N (cid:1) ⊤ + vA X + ρ ′ ( τ ) AX (100)for every X ∈ X ( M ). Recalling the definition (11) of L k , formulas (97) and (98) follow fromLemma 1 and from the fact that g (R ( X, Y ) N, Z ) = Riem( Z, N, X, Y ) = 0 for every X, Y, Z ∈ X ( M ) since M has constant curvature. (cid:3) The next theorem collects some observations about the k -stability of hypersurfaces with zero( k + 1)-th mean curvature and positive definite k -th Newton tensor. Theorem 27. Let ψ : M −→ M be a complete spacelike hypersurface with zero ( k + 1) -th meancurvature, for some ≤ k ≤ m − , in a spacetime M of dimension m + 1 and constant curvature κ . Suppose that H k > on M and that rank( A ) > k on M .i) If κ ≤ , then M is non-compact and ψ is k -stable.ii) If κ > and M is compact and simply connected, then ψ is not k -stable.iii) If κ > and we assume that M is non-compact and that, for some o ∈ M and for some(hence any) a > , lim r → + ∞ Z ra (cid:18)Z ∂B r H k (cid:19) − = + ∞ , lim r −→ + ∞ Z B r (cid:0) κ trace( P k ) − trace( A P k ) (cid:1) = + ∞ , (101) where B r is the geodesic ball of ( M, g ) with radius r centered at o , then e L k has infiniteindex.Proof . Firstly, note that H k +1 ≡ A ) > k on M guarantee that the self-adjointoperator P k is positive definite (see Proposition 6.27 of [11]). Since A and P k are simultaneously diagonalizable, A P k is also self-adjoint and positive definite and therefore(102) trace( P k ) > , trace( A P k ) > . Moreover, the operator L k is elliptic and it can be put in divergence form due to the fact that M has constant sectional curvature κ , that is, we have(103) L k u = div( P k ( ∇ u )) for every u ∈ C ( M ) . i) Suppose that κ ≤ 0. By (102) it follows that (cid:0) κ trace( P k ) − trace( A P k ) (cid:1) ≤ 0, so Z M − φ e L k φ = Z M g ( P k ( ∇ φ ) , ∇ φ ) − (cid:0) κ trace( P k ) − trace( A P k ) (cid:1) φ ≥ φ ∈ C ∞ c ( M ), and ψ is k -stable by definition. By Lemma 4, there exists a positivefunction u ∈ C ∞ ( M ) such that L k u = (trace( A P k ) − κ trace( P k )) u on M . Suppose, bycontradiction, that M is compact. By (103) and the divergence theorem,0 = Z M L k u = Z M (trace( A P k ) − κ trace( P k )) u. From (102) and κ ≤ u ≡ M , contradiction.ii) Suppose, by contradiction, that κ > M is compact and simply connected and ψ is k -stable. Let π : f M −→ M be the Lorentzian universal covering of M . f M is isometric to de Sitterspacetime of dimension m + 1 and curvature κ , which in turn is isometric to the GRW spacetime R × cosh( √ κt ) S m ( κ ). M is simply connected, so for every p ∈ M , ˜ p ∈ π − ( p ) there exists a uniqueimmersion e ψ : M −→ f M such that e ψ ( p ) = ˜ p and π ◦ e ψ = ψ . The shape operator induced by e ψ isequal to A , up to a change of sign, because π is a local isometry. Then, by Lemma 26, M supportsa positive smooth functions v satisfying L k v = trace( A P k ) v . Since ψ is k -stable, by Lemma 4, M also supports a positive smooth function u satisfying L k u = (trace( A P k ) − κ trace( P k )) u . Adirect computation shows that the positive function ϕ = u/v satisfiesdiv( v P k ( ∇ ϕ )) v = L k ϕ + 2 g (cid:18) ∇ vv , P k ( ∇ ϕ ) (cid:19) = − κ trace( P k ) ϕ. Since M is compact, we have0 = Z M div( v P k ( ∇ ϕ )) = − Z M κ trace( P k ) v ϕ. From (102) and κ > v ϕ ≡ M , contradiction.iii) Suppose, by contradiction, that κ > 0, that condition (101) holds for some o ∈ M andthat e L k has finite index. Then there exists a relatively compact open set Ω ⊆ M such that(104) λ e L k ( M \ Ω) ≥ . Since ( M, g ) is complete, there exists R > ⊆ B R . Next we define(105) v k ( r ) = Z ∂B r c k H k , A ( r ) = 1 v k ( r ) Z ∂B r (cid:0) κ trace( P k ) − trace( A P k ) (cid:1) . We consider the Cauchy problem(106) ( ( v k ( r ) z ′ ) ′ + A ( r ) v k ( r ) z = 0 on R + z (0 + ) = Z > , w (0 + ) z ′ (0 + ) = 0 ∈ R . TABLE MAXIMAL HYPERSURFACES IN LORENTZIAN SPACETIMES 27 Since v k ∈ L ∞ loc ( R +0 ), there exists a solution z of (106) with z ∈ Lip loc ( R +0 ) due to Proposition4.2 of [11]. Moreover, from the coarea formula and (101) we obtain(107) lim inf r −→ + ∞ Z r A ( s ) v k ( s ) ds = lim inf r −→ + ∞ Z B r (cid:0) κ trace( P k ) − trace( A P k ) (cid:1) = + ∞ . This condition and the fact that v − k ∈ L ∞ loc ( R + ) and v − k / ∈ L (+ ∞ ) enable us to use Corollary2.9 of [29] to obtain that any solution z of (106) is oscillatory. Taking now R ≤ R < R twoconsecutive zeros of z such that z > R , R ) we define the function ϕ ( x ) := z ( r ( x )), where r ( x ) is the distance from x to o in ( M, g ), and compute Z B R \ B R − ϕ e L k ϕ = Z B R \ B R g ( P k ( ∇ ϕ ) , ∇ ϕ ) − (cid:0) κ trace( P k ) − trace( A P k ) (cid:1) ϕ ≤ Z B R \ B R trace( P k ) |∇ ϕ | − (cid:0) κ trace( P k ) − trace( A P k ) (cid:1) ϕ . (108)With the aid of the coarea formula, integrating by parts and using (106) we obtain(109) Z B R \ B R − ϕ e L k ϕ ≤ − Z R R [( v k z ′ ) ′ ( s ) + A ( s ) v k ( s ) z ( s )] z ( s ) ds = 0 . Therefore, from (109) and Proposition 2 we deduce(110) λ e L k ( M \ Ω) < λ e L k ( B R \ B R ) ≤ R B R \ B R − ϕ e L k ϕ R B R \ B R ϕ ≤ , contradicting (104). (cid:3) Theorem 28. Let M = I × ρ F be a Robertson-Walker spacetime of constant sectional curvatureand let ψ : M −→ M be a complete non-compact spacelike hypersurface with zero ( k + 1) -th meancurvature. Suppose that H k > , sup M H k < + ∞ , rank( A ) > k on M and that, for some o ∈ M and for some (hence any) a ∈ R , one of the following conditions is satisfied: ( i ) lim r → + ∞ Z ra (cid:18)Z ∂B r H k (cid:19) − dt = + ∞ and lim r → + ∞ Z B r H k = + ∞ , or ( ii ) lim r → + ∞ Z ra (cid:18)Z ∂B r H k (cid:19) − dt < + ∞ and lim r → + ∞ r log (cid:18)Z ∂B r H k (cid:19) = 0 , where B r is the geodesic ball of ( M, g ) with radius r centered at o . If ψ ( M ) is contained in aslab [ a, b ] × F , then inf M | H k ρ ′ ( τ ) | = 0 . In particular, if inf M H k > , then inf M | ρ ′ ( τ ) | = 0 andtherefore there exists t in the closure of τ ( M ) ⊆ [ a, b ] such that ρ ′ ( t ) = 0 .Proof . First observe that H k +1 ≡ H k > A ) > k guarantee that P k is positivedefinite and therefore L k is elliptic (see Proposition 6.27 of [11]). Set w ( r ) = R ∂B r trace( P k ) forevery r > M H k .Let R > r → + ∞ Z ra dtw ( t ) = lim r → + ∞ Z ra w ( t ) dt = + ∞ , since trace( P k ) = c k H k . Let ε > z ∈ Lip loc ([ R, + ∞ )) of the Cauchy problem(111) ( ( w ( r ) z ′ ) ′ + εw ( r ) z = 0 on [ R, + ∞ ) z ( R ) = 1 , ( wz ′ )( R + ) = 0 is oscillatory. Let R < R < R be two consecutive zeros of z such that z > R , R ) andlet ϕ ( x ) = z ( r ( x )), where r ( x ) is the distance from x to o in ( M, g ). By the coarea formula, theinequality Vol( ∂B r ) ≥ w ( r ) / Λ and the fact that z solves (111), we have λ L k ( B R \ B R ) ≤ R B R \ B R g ( P k ( ∇ ϕ ) , ∇ ϕ ) R B R \ B R ϕ = R R R w ( r ) z ′ ( r ) R R R Vol( ∂B r ) z ( r ) ≤ R R R εw ( r ) z ( r ) R R R w ( r )Λ z ( r ) = ε Λ , and by Proposition 2 we get(112) 0 ≤ λ L k ( M \ B R ) < ε Λ . Suppose now that (ii) is satisfied. Then lim r → + ∞ R ra dtw ( t ) = 0 and for every ε > C ε > w ( r ) ≤ C ε e εr for every r > 0. Let 0 < ε < r > ε > ε (cid:18) lim s → + ∞ (cid:18) − s − r ) log Z + ∞ s dtC ε e εt (cid:19)(cid:19) ≥ (cid:18) inf s>r (cid:18) − s − r ) log Z + ∞ s dtC ε e εt (cid:19)(cid:19) and by Proposition 6.9 of [11] this condition is sufficient to deduce that for every R > z of the Cauchy problem (111) is oscillatory, so inequality (112) follows again.Letting ε → + and R → + , we deduce that λ L k ( M ) = 0. We have ψ ( M ) ⊆ [ a, b ] × F , with a, b ∈ I . Choose t = a and let η , η be as in (59). Then 0 = η ( a ) ≤ η ≤ η ( b ) < + ∞ on M . Fix ε > 0. By Proposition 3 applied to the positive functions η + ε , η ( b ) + ε − η and by (97) we have0 ≥ inf M (cid:18) − L k ηη + ε (cid:19) = inf M c k H k ρ ′ ( τ ) η + ε , ≥ inf M (cid:18) L k ηη ( b ) + ε − η (cid:19) = inf M − c k H k ρ ′ ( τ ) η ( b ) + ε − η . Since η + ε > ε , η ( b ) + ε − η > ε and c k > 0, we deduce inf M | H k ρ ′ ( τ ) | = 0. (cid:3) Remark 29. Under the hypothesis C − < H k < C we have c k C − Vol( ∂B r ) ≤ w ( r ) ≤ c k C Vol( ∂B r ) for every r > Acknowledgements The second author is supported by Spanish MINECO and ERDF project MTM2016-78807-C2-1-P. References [1] A.L. Albujer, L.J. Al´ıas, Calabi-Bernstein results for maximal surfaces in Lorentzian product spaces, J.Geom. Phys. , (2009), 620–631.[2] L.J. Al´ıas, A. Brasil and A.G. Colares, Integral formulae for spacelike hypersurfaces in conformally stationaryspacetimes and applications, Proc. Edinb. Math. Soc. (2) , (2003), 465–488.[3] L.J. Al´ıas and A.G. Colares, Uniqueness of spacelike hypersurfaces with constant higher order mean curvaturein generalized Robertson-Walker spacetimes, Math. Proc. Cambridge , (2007), 703–729.[4] L.J. Al´ıas, P. Mastrolia and M. Rigoli, Maximum principles and geometric applications , Springer (2016).[5] L.J. Al´ıas, A. Romero and M. S´anchez, Spacelike hypersurfaces of constant mean curvature and Calabi-Bernstein type problems, Tˆohoku Math. J. , (1997), 337–345.[6] L.J. Al´ıas, A. Romero and M. S´anchez, Uniqueness of complete spacelike hypersurfaces of constant meancurvature in Generalized Robertson-Walker spacetimes, Gen. Relativ. Gravit. , (1995), 71–84.[7] L.J. Al´ıas, M. Rigoli and S. Scoleri, Weak maximum principles and geometric estimates for spacelike hyper-surfaces in generalized Robertson-Walker spacetimes, Nonlinear Anal. (2015), 119–142. TABLE MAXIMAL HYPERSURFACES IN LORENTZIAN SPACETIMES 29 [8] J.M. Arms, J.E. Marsden and V. Moncrief, The structure of the space of solutions of Einstein’s equations.II. Several Killing fields and the Einstein-Yang-Mills equations, Ann. Phys. , (1982), 81–106.[9] R. Bartnik, Existence of maximal surfaces in asymptotically flat spacetimes, Commun. Math. Phys. , (1984), 155–175.[10] B. Bianchini, L. Mari and M. Rigoli, Spectral radius, index estimates for Schr¨odinger operators and geometricapplications, J. Funct. Anal. , (2009), 1769–1820.[11] B. Bianchini, L. Mari and M. Rigoli, On some aspects of oscillation theory and geometry, Mem. Am. Math.Soc. , (2013), vi+195.[12] A. Brasil and A.G. Colares, On constant mean curvature spacelike hypersurfaces in Lorentz manifolds, Mat.Contemp. , (1999) 99–136.[13] A. Brasil and A.G. Colares, Stability of spacelike hypersurfaces with constant r -mean curvature in de Sitterspace, Proceedings of the XII Fall Workshop on Geometry and Physics , Publ. R. Soc. Mat. Esp. , (2004),139–145.[14] D. Brill and F. Flaherty, Isolated maximal surfaces in spacetime, Commun. Math. Phys. (1984), 157–165.[15] E. Calabi, Examples of Bernstein problems for some nonlinear equations, P. Symp. Pure Math. , (1970),223–230.[16] F. Camargo, A. Caminha, M. da Silva and H. de Lima, On the r-stability of spacelike hypersurfaces, J.Geom. Phys. , (2010), 1402-1410.[17] S.Y. Cheng and S.T. Yau, Maximal spacelike hypersurfaces in the Lorentz-Minkowski spaces, Ann. of Math. , (1976), 407–419.[18] Y. Choquet-Bruhat, Quelques propri´et´es des sousvari´et´es maximales d’une vari´et´e lorentzienne, Cr. Acad.Sci. A Math. (Paris) Serie A , (1975), 577–580.[19] Y. Choquet-Bruhat and R. Geroch, Global Aspects of the Cauchy Problem in General Relativity, Commun.Math. Phys. , (1969), 329–335.[20] T. Frankel, Applications of Duschek’s formula to cosmology and minimal surfaces, B. Am. Math. Soc. , (1975), 579–583.[21] D. de la Fuente, R.M. Rubio and J.J. Salamanca, Stability of maximal hypersurfaces in spacetimes: newgeneral conditions and applications to relevant spacetimes, Gen. Relativ. Gravit. , (2017), 129–143.[22] D. Fischer-Colbrie, On complete minimal surfaces with finite Morse index in three-manifolds, Invent. Math. , (1985), 121–132.[23] D. Fischer-Colbrie and R. Schoen, The structure of complete stable minimal surfaces in 3-manifolds of non-negative scalar curvature, Commun. Pur. Appl. Math. , (1980), 199–211.[24] D. Gilbarg, N. Trudinger, Elliptic partial differential equations of second order , Springer (2001).[25] J. Hounie, M.L. Leite, The maximum principle for hypersurfaces with vanishing curvature functions, J.Differential Geom. , (1995), 247–258.[26] T. Ishihara, Maximal spacelike submanifolds of a pseudoriemannian space of constant curvature, MichiganMath. J. , (1988), 345–352.[27] J.L. Jaramillo, J.A.V. Kroon and E. Gourgoulhon, From geometry to numerics: interdisciplinary aspects inmathematical and numerical relativity, Classical Quant. Grav. , (2008), 093001.[28] A. Lichnerowicz, L’integration des ´equations de la gravitation relativiste et le probl`eme des n corps, J. Math.Pure Appl. , (1944), 37–63.[29] L. Mari, P. Mastrolia and M. Rigoli, A note on Killing vector fields and CMC hypersurfaces, J. Math. Anal.Appl. , (2015), 919–934.[30] J.E. Marsden and F.J. Tipler, Maximal hypersurfaces and foliations of constant mean curvature in GeneralRelativity, Phys. Rep. , (1980), 109–139.[31] P. Mastrolia, M. Rigoli and A.G. Setti, Yamabe-type equations on complete, noncompact manifolds , Springer,(2012).[32] S. Nishikawa, On maximal spacelike hypersurfaces in a Lorentzian manifold, Nagoya Math. J. , (1984),117–124.[33] B. O’Neill, Semi-Riemannian Geometry with applications to Relativity , Academic Press, (1983).[34] R. Osserman, A survey of minimal surfaces , Dover Publications, (1986).[35] J.A.S. Pelegr´ın, A. Romero and R.M. Rubio, On maximal hypersurfaces in Lorentz manifolds admitting aparallel lightlike vector field, Classical Quant. Grav. , (2016), 055003(1–8).[36] J.A.S. Pelegr´ın, A. Romero and R.M. Rubio, On uniqueness of the foliation by comoving observers restspacesof a Generalized Robertson-Walker spacetime, Gen. Relativ. Gravit. , (2017), Art. 16, 14pp.[37] J.A.S. Pelegr´ın, A. Romero and R.M. Rubio, Uniqueness of complete maximal hypersurfaces in spatially open( n + 1)-dimensional Robertson-Walker spacetimes with flat fiber, Gen. Relativ. Gravit. , (2016), 1–14.[38] S. Pigola, M. Rigoli and A.G. Setti, Vanishing theorems on Riemannian manifolds, and geometric applica-tions, J. Funct. Anal. , (2005), 424–461. [39] S. Pigola, M. Rigoli and A.G. Setti, Vanishing and finiteness results in geometric analysis , Birkh¨auser (2008).[40] M. Rigoli and A.G. Setti, Liouville type theorems for φ -subharmonic functions, Rev. Mat. Iberoam , (2001),471–520.[41] R. Schoen and S.T. Yau, On the proof of the positive mass conjecture in General Relativity, Comm. Math.Phys. (1979), 45–76.[42] J. Simons, Minimal varieties in riemannian manifolds, Ann. Math. , (1968), 62–105.[43] P. Tolksdorf, Regularity of a more general class of quasilinear elliptic equations, J. Differential Equations , (1984), 126–150.[44] J.A. Wolf, Spaces of constant curvature , McGraw-Hill (1967). Dipartimento di Matematica, Universit`a degli Studi di Milano, 20133 Milano, Italy E-mail address : [email protected] Departamento de Geometr´ıa y Topolog´ıa, Universidad de Granada, 18071 Granada, Spain Current address : Departamento de Matem´atica Aplicada y Estad´ıstica, Universidad CEU San Pablo, 28003Madrid, Spain E-mail address : [email protected], [email protected] Dipartimento di Matematica, Universit`a degli Studi di Milano, 20133 Milano, Italy E-mail address ::