Electronic Journal of Differential Equations, Vol. 2020 (2020), No. 83, pp. 1–19. ISSN: 1072-6691. URL: http://ejde.math.txstate.edu or http://ejde.math.unt.edu SOLUTIONS TO MEAN CURVATURE EQUATIONS IN WEIGHTED STANDARD STATIC SPACETIMES HENRIQUE F. DE LIMA, ANDRÉ F. A. RAMALHO, MARCO ANTONIO L. VELÁSQUEZ Communicated by Giovanni Molica Bisci Abstract. In this article, we study the solutions for the mean curvature equa- tion in a weighted standard static spacetime, Pnf ×ρR1, having a warping func- tion ρ whose weight function f does not depend on the parameter t ∈ R. We establish a f -parabolicity criterion to study the rigidity of spacelike hypersur- faces immersed in Pnf ×ρ R1 and, in particular, of entire Killing graphs con- structed over the Riemannian base Pn. Also we give applications to weighted standard static spacetimes of the type Gn ×ρ R1, where Gn is the Gaussian space. 1. Introduction Standard static spacetimes are part of the so called stationary spacetimes. Let us recall that a stationary spacetime is a time-orientable Lorentzian manifold (M n+1 , g) where there exists an infinitesimal symmetry given by a timelike Killing vector field Y (see [27]). The existence of Y enables us to define around each point a coordinate system (t, x1, . . . , xn) such that Y coincides with the coordinate vector field ∂/∂t on its domain of definition and such that the components of the metric tensor in these coordinates are independent of t. When we normalize Y we obtain an observers vector field Z = Y/ √ −g(Y, Y ). These observers measure a metric tensor that does not change with time. Furthermore, if this timelike Killing vector field is also irro- tational (that is, the distribution Y ⊥ of all smooth vector fields on M n+1 that are orthogonal to Y is involutive), then a local warped product structure appears and the spacetime is called static (for more details see, for instance, [1]). In fact, when this structure is global this spacetime is known as a standard static spacetime. More precisely, a standard static spacetime (M n+1 , g) endowed with a globally defined timelike Killing vector field Y is isometric to the warped product (Pn ×ρ R1 , π ∗ Pn(g̃) + (ρ ◦ πPn)2π∗R(−dt2) ) where πPn and πR denote the canonical projections from Pn ×R1 onto each factor, g̃ is the Riemannian metric on the base Pn, R1 is the manifold R endowed with the metric −dt2 and ρ = √ −g(Y, Y ) is the warping function. In this context, it 2010 Mathematics Subject Classification. 53C42, 53B30, 53C50. Key words and phrases. Standard static spacetimes with density; Gaussian space; f -parabolic spacelike hypersurface; entire Killing graphs; mean curvature equation. c©2020 Texas State University. Submitted June 15, 2020. Published July 30, 2020. 1 2 H. F. DE LIMA, A. F. A. RAMALHO, M. A. L. VELÁSQUEZ EJDE-2020/83 is known that any static spacetime is locally isometric to a standard static one (see [23, Proposition 12.38]). Conversely, Sánchez [29] and more recently Aledo, Romero and Rubio [1] obtained some sufficient conditions for a static spacetime to be standard. Other properties on the geometry of standard static spacetimes were studied by Sánchez [28, 29, 30]. The importance of standard static spacetimes also comes from the fact that they include some classical spacetimes, such as the (n+1)-dimensional Lorentz-Minkowski space Ln+1, Einstein static universe as well as models that describe an universe where there is only a spherically symmetric non-rotating mass, as a star or a black hole, like exterior Schwarzschild spacetime and some regions of Reissner-Nordström spacetime (see, for example, [5, 17]). On the other hand, the study of spacelike hypersurfaces immersed with constant mean curvature in a spacetime has attracted the interest of a considerable group of geometers as evidenced by the amount of works that it has generated in the last decades. This is due not only to its mathematical interest, but also to its relevance in General Relativity. For example, constant mean curvature spacelike hypersurfaces are particularly suitable for studying the propagation of gravitational radiation. See, for instance, [21, 31] for a summary of several reasons justifying this interest. From the mathematical point of view, the study of the geometry of constant mean curvature spacelike hypersurfaces is mostly due to the fact that they exhibit nice Calabi-Bernstein type properties. More precisely, this study had its beginnings when Bernstein [6] proved that the only entire minimal graphs in the 3-dimensional Euclidean space R3 are planes. In the Lorentzian setting, there is an analogue result to Bernstein’s theorem, which states that the only entire maximal graphs in the 3-dimensional Lorentz-Minkowski space L3 are the spacelike planes. This result was firstly proved by Calabi [7], and extended to the general n-dimensional case by Cheng and Yau [9]. A natural extension to the Calabi-Bernstein problem is to determine a reason- able set of sufficient conditions which guarantee the uniqueness (or nonexistence) of complete spacelike hypersurfaces immersed into a certain ambient spacetime. When such a spacetime is a standard static spacetime Pn×ρ R1, there is a remark- able family of spacelike hypersurfaces, namely, the spacelike slices Pn × {t0}, with t0 ∈ R, which are totally geodesics constituting a foliation for the ambient space- time. Therefore, it is natural to approach Calabi-Bernstein problems in a standard static spacetime. In this branch, the first author together with Lima Jr, de Lima and Medeiros [12] extended a technique due to Romero et al. [25] to establish suffi- cient conditions to guarantee the parabolicity of complete spacelike hypersurfaces in Pn ×ρ R1 whose Riemannian base Pn has parabolic universal Riemannian covering and, as applications, they obtain uniqueness results concerning these hypersurfaces. Afterwards, Pelegŕın, Romero and Rubio [24] also studied complete spacelike hy- persurfaces in spatially parabolic standard static spacetimes. In this context, they used a similar parabolicity criterion to give new uniqueness and nonexistence results for these spacelike hypersurfaces and to solve new Calabi-Bernstein-type problems. At this point, we recall that a weighted manifold Mn+1 f is a semi-Riemannian manifold (Mn+1, g) endowed with a weighted volume form dµ = e−fdM, where the weight function f is a real-valued smooth function on Mn+1 and dM is the volume element induced by the metric g (for details see, for instance, [4, 22]). Concerning the weighted product space Gn × R1, where Gn stands for the so-called Gaussian space which is nothing but that the Euclidian space Rn endowed with the Gaussian EJDE-2020/83 SOLUTIONS TO MEAN CURVATURE EQUATIONS 3 probability density e−f(x) = (2π)−(n+1)/2e−|x| 2/2, x ∈ Rn, An et al. [2] extended the classical Bernstein’s theorem showing that the only weighted maximal graphs Σ(z) of smooth functions z(x) = t over Gn, with supΣ(z) |Dz|G < 1, are the affine hyperplanes t = constant. Motivated by the works described above, our purpose in this paper is to obtain uniqueness results related to the mean curvature equation for entire Killing graphs constructed over the Riemannian base Pn of a weighted standard static spacetime Pnf ×ρ R1 having warping function ρ and whose weight function f does not depend on the parameter t ∈ R (see Section 5). For this, in Section 2 we recall some basic facts about spacelike hypersurfaces immersed in a weighted standard static spacetime. Afterwards, in Section 3 we establish a suitable f -parabolicity criterion and, under appropriate constraints on the Bakry-Émery Ricci tensor and on the f -mean curvature, in Section 4 we study the rigidity of spacelike hypersurfaces immersed in Pnf ×ρ R1. Finally, we point out that, in Section 5, applications of our main results to weighted standard static spacetimes of the type Gn ×ρ R1 are also given. 2. Weighted standard static spacetimes Along this paper, we will consider an (n+1)-dimensional Lorentz manifold M n+1 with Lorentzian metric g = g(·, ·) and endowed with a Killing timelike vector field Y . Here timelike referred to a vector field means that Yp ∈ TpM is a timelike (and so nonzero) vector for each p ∈ Mn+1 . On the other hand, Killing mean that the LY g = 0, where LY stands for the Lie derivative of g in the direction of Y . We observe that the distribution D of all smooth vector fields of M n+1 that are orthogonal to Y , defined at each point by M n+1 3 p 7→ D(p) = {v ∈ TpM : g(v, Yp) = 0}, is of constant rank and integrable. Given a Riemannian integral leaf Pn of that distribution D, let Ψ : I × Pn → M n+1 be the flow generated by Y with initial values in Pn, where I is a maximal interval of definition. Without loss of generality, in what follows we will consider I = R. In this setting, our space M n+1 can be regarded as the standard static spacetime Pn ×ρ R1 (cf. [23, Proposition 12.38]), that is, the Lorentzian product manifold Pn×R1 endowed with the warping metric 〈·, ·〉 = π∗Pn(〈·, ·〉Pn) + (ρ ◦ πPn)2π∗R(−dt2), (2.1) where πPn and πR denote the canonical projections from Pn ×R1 onto each factor, 〈·, ·〉Pn is the induced Riemannian metric on the base Pn, R1 is the manifold R endowed with the metric −dt2 and ρ = |Y | = √ −〈Y, Y 〉 > 0 is the warping function. We mean by C∞(Pn ×ρ R1) the ring of real functions of class C∞ on Pn ×ρ R1 and by X(Pn ×ρ R1) the C∞(Pn ×ρ R1)-module of vector fields of class C∞ on Pn ×ρ R1. The Levi-Civita connections of Pn ×ρ R1 and Pn will be denoted by ∇ and ∇̃, respectively. Now, in the configuration described above, let (Pn×ρR1)f be a weighted standard static spacetime, namely, a standard static spacetime Pn ×ρ R1 endowed with a weighted volume form dσ = e−fdv, where f ∈ C∞(Pn ×ρ R1) is a real-valued function, called weight function (or density function), and dv is the volume element 4 H. F. DE LIMA, A. F. A. RAMALHO, M. A. L. VELÁSQUEZ EJDE-2020/83 induced by the warping metric 〈·, ·〉 defined in (2.1). For ( Pn ×ρ R1 ) f , we recall that the Bakry-Émery-Ricci tensor Ricf is defined by Ricf = Ric + Hessf, (2.2) where Ric and Hess stand for the Ricci tensor and the Hessian operator in Pn×ρR1, respectively. Throughout this work, we will deal with complete spacelike hypersurfaces ψ : Σn ↪→ (Pn ×ρ R1)f , namely, isometric immersions from a (connected) n-dimensional Riemannian mani- fold Σn into weighted static spacetime (Pn×ρR1)f . In this setting, the Levi-Civita connection of Σn will be denoted by ∇. As (Pn ×ρ R1)f is time-oriented by the timelike vector field Y and x : Σn ↪→ (Pn ×ρ R1)f is a spacelike hypersurface, then Σn is orientable (cf. [23, Proposition 5.26]) and one can choose a globally defined unit normal vector field N on Σn having the same time-orientation of (Pn ×ρ R1)f (cf. [23, Proposition 5.29]), that is, 〈Y,N〉 < 0. (2.3) Such N is said the future-pointing Gauss map of Σn. Let A denote the shape operator of Σn with respect to N . So that at each p ∈ Σn A restricts to a self- adjoint linear map Ap : TpΣ→ TpΣ given by Apv = −∇vN. According to Gromov [16], the weighted mean curvature (or simply the f -mean curvature) Hf of Σn is given by nHf = nH − 〈∇f,N〉, (2.4) where H = − 1 n tr(A) denotes the standard mean curvature of Σn with respect to its orientation N . Moreover, we say that ψ : Σn ↪→ (Pn ×ρ R)f is f -maximal when its f -mean curvature vanishes identically. The f -divergence on Σn is divf : C∞(Σn)→ C∞(Σn), defined by divf (X) = divX − 〈∇f,X〉, where div(·) denotes the standard divergence on Σn. We define the f -Laplacian (also called the drift Laplacian) of Σn by ∆f : C∞(Σn)→ C∞(Σn), as ∆f (u) = divf (∇u) = ∆u− 〈∇f,∇u〉 (2.5) where ∆ is the standard Laplacian on Σn. In what follows, associated with a spacelike hypersurface ψ : Σn ↪→ (Pn ×ρ R1)f , we will consider two particular smooth functions, namely, the (vertical) height function h = (πR) ∣∣ Σn : Σn → R (2.6) and the angle function Θ : Σn → R defined as Θ(p) = 〈N(p), Y (p)〉, (2.7) where N is the future-pointing Gauss map of Σn and Y is the Killing vector field on (Pn ×ρ R1)f . From (2.3), we note that Θ will be always a negative function on Σn. EJDE-2020/83 SOLUTIONS TO MEAN CURVATURE EQUATIONS 5 We have that ∇h = − 1 ρ2 Y >, (2.8) where (·)> denote the projections of a smooth vector field in X(Pn×ρR1) on X(Σn). Moreover, N∗ = N + 1 ρ2 ΘY, (2.9) where (·)∗ denote the projections of a smooth vector field in X(Pn×ρR1) on X(Pn). Hence, from (2.8) and (2.9) it is not difficult to verify that |∇h|2 = 1 ρ2 |N∗|2Pn . (2.10) 3. An f-parabolicity criterion for spacelike hypersurfaces in (Pn ×ρ R1)f Romero, Rubio and Salamanca [26] investigated the parabolicity of complete spacelike hypersurfaces in GRW spacetimes whose Riemannian fiber has a para- bolic universal Riemannian covering. In this setting, they were able to guarantee the parabolicity of complete spacelike hypersurfaces, under suitable boundedness assumptions on the warping function and on the hyperbolic angle function of these hypersurfaces. Our aim in this section is just, following the ideas of [11], to ob- tain an extension of this parabolicity criterion to the context of standard static spacetimes. A smooth function u on a weighted manifold Σnf is said to be f -superharmonic if ∆fu ≤ 0. Taking this into account, the weighted manifold Σnf is called f -parabolic if there is no nonconstant, nonnegative, f -superharmonic function on Σn. On the other hand, given a weighted manifold Σnf we define, for any compact subset K ⊂ Σn, the f -capacity of K as capf (K) = inf {∫ Σ |∇u|2e−fdΣ : u ∈ Lip0(Σ)andu|K ≡ 1 } , where Lip0(Σ) is the set of all compactly supported Lipschitz functions on Σn. The following statement relates the notion of f -capacity to the concept of f -parabolicity (see [15, Proposition 2.1]). Lemma 3.1. The weighted manifold Σnf is f -parabolic if and only if capf (K) = 0 for any compact set K ⊂ Σn. Let us recall that given two Riemannian manifolds (Σ′, g′) and (Σ, g), a diffeo- morphism ϕ from Σ′ onto Σ is called a quasi-isometry if there exists a constant c ≥ 1 such that c−1|v|g′ ≤ |dϕ(v)|g ≤ c|v|g′ for all v ∈ TpΣ ′, p ∈ Σ′ (see [19] for more details). In this case, given a smooth function f : Σ → R, we can reason as in [14, Section 5] to verify that the (f ◦ ϕ)- capacity of the compact subsets in Σ′ changes under a quasi-isometry at most by a constant factor of the f -capacity of the compact subsets in Σ. The following statement corresponds to [11, Lemma 1]. Lemma 3.2. Keeping the same notation above, we have: (a) Given a quasi-isometry ϕ : Σ′ → Σ, Σ is f -parabolic if and only if Σ′ is (f ◦ ϕ)-parabolic; 6 H. F. DE LIMA, A. F. A. RAMALHO, M. A. L. VELÁSQUEZ EJDE-2020/83 (b) Let Σ̃ be the universal Riemannian covering of Σ with canonical projection πΣ : Σ̃→ Σ. If Σ̃ is (f ◦ πΣ)-parabolic, then Σ is f -parabolic. Recall that every connected manifold Σ has an universal covering, that is, there exist a simply connected manifold Σ̃ (called a universal covering of Σ) and a smooth map π̃ : Σ̃→ Σ (called a covering map) such that each point p ∈ Σ has a connected neighborhood U that is evenly covered by π̃, that is, π̃ maps each component of π̃−1(U) diffeomorphically onto U (for more details, see [23, Appendix A]). Moreover, if Σ is a Riemannian manifold, then it is possible to give Σ̃ a Riemannian structure such that the covering map π̃ : Σ̃→ Σ is a local isometry. In this case, Σ̃ is said a universal Riemannian covering of Σ (see [13, pg. 152]). From now on, we will denote by P̃ the universal Riemannian covering of base Pn with projection π̃ : P̃ → Pn and f̃ will denote the composition f ◦ π̃. In this setting, a standard static spacetime (Pn ×ρ R1)f will be said spattialy f -parabolic if the universal Riemannian covering P̃ of its base Pn is f̃ -parabolic. Proposition 3.3. Let (Pn ×ρ R1)f be a weighted standard static spacetimes which is spatially f̃ -parabolic. If ψ : Σn ↪→ Pn+1 is a spacelike hypersurface such that the function η := Θ ρ is bounded on it, then Σn is f -parabolic. Proof. From Lemma 3.2 we have that (i) f -parabolicity is invariant under a quasi-isometry; (ii) if the universal Riemannian covering Σ̃ of Σn is (f ◦πΣ)-parabolic, then Σn is also f -parabolic. Denoting π = πP ◦ ψ : Σn → Pn, for any tangent vector v ∈ TΣ we have 〈v, v〉 = 〈π∗v, π∗v〉P − ρ2〈h∗v, h∗v〉R ≤ c〈π∗v, π∗v〉P, where c = supΣ η 2 ≥ 1. In particular, by previous inequality we see that π∗,p : TpΣ → Tπ(p)M is a isomorphism for every p ∈ Σn. Then, from inverse function theorem we obtain that π is a local diffeomorphism and applying [13, Lemma 7.3.3] (see also [20, Lemma 8.8.1]) we can conclude that π is a covering map and that Pn is complete. On the other hand, using the Cauchy-Schwartz inequality we see that 〈∇h, v〉2 ≤ 〈∇h,∇h〉〈v, v〉 and, consequently, since h∗v = dh(v) = 〈∇h, v〉, we have 〈v, v〉 = 〈π∗v, π∗v〉P − ρ2〈h∗v, h∗v〉R = 〈π∗v, π∗v〉P − ρ2〈∇h, v〉2 ≥ 〈π∗v, π∗v〉P − ρ2|∇h|2〈v, v〉; that is, 〈v, v〉(1 + ρ2|∇h|2) ≥ 〈π∗v, π∗v〉P. By definition of the function η and from (2.10) we obtain 〈v, v〉 ≥ 1 η2 〈π∗v, π∗v〉P From our hypothesis we conclude that c−1〈π∗v, π∗v〉P ≤ 〈v, v〉 ≤ c〈π∗v, π∗v〉P. (3.1) EJDE-2020/83 SOLUTIONS TO MEAN CURVATURE EQUATIONS 7 So, let Σ̃ be the universal Riemannian covering of Σn with projection πΣ : Σ̃→ Σn. Then, the map π0 = π ◦ πΣ : Σ̃ → Pn is a covering map. Now, if M̃ is the universal Riemannian covering of Pn with projection π̃ : M̃ → Pn, then there exists a diffeomorphism ϕ : Σ̃→ M̃ such that π̃◦ϕ = π0. Moreover, ϕ is a quasi-isometry. Indeed, if v ∈ T Σ̃, we have from (3.1) that 〈ϕ∗v, ϕ∗v〉M̃ = 〈π̃∗(ϕ∗v), π̃∗(ϕ∗v)〉M = 〈(π0)∗v, (π0)∗v〉M = 〈π∗((πΣ)∗v), π∗((πΣ)∗v)〉M ≤ c〈(πΣ)∗v, (πΣ)∗v〉Σ = c〈v, v〉Σ̃. Analogously, we obtain 〈ϕ∗v, ϕ∗v〉M̃ ≥ c−1〈v, v〉Σ̃. Therefore, since the universal Riemannian covering of Pn is parabolic, it follows that the universal Riemannian covering of Σn is parabolic and, hence, Σn must be also parabolic. � 4. Rigidity results for spacelike hypersurfaces in Pnf ×ρ R1 It follows from [8] that in a weighted timelike geodesically complete spacetime M n+1 f that contains a timelike line, with Ricf (X,X) ≥ 0 for all timelike vector field X and whose weight function f is bounded, the weight function f must be constant along timelike line ofM n+1 f . Consequently, in any weighted standard static spacetime (Pn ×ρ R1)f having nonnegative Bakry-Émery-Ricci tensor for timelike vector fields and with bounded weight function f , we have that f does not depend on the parameter of the flow associated with the Killing vector field ∂ ∂t ≡ Y . Motivated by this fact, we will consider standard static spacetimes Pn ×ρ R1 endowed with a weight function f not depending on the parameter t ∈ R, that is, 〈∇f, Y 〉 = 0. For sake of simplicity, we will denote such an ambient space by Pnf ×ρ R1. In this section, we will apply the Proposition 3.3 in order to obtain rigidity results for spacelike hypersurfaces in Pn ×ρ R1. For this, we will need of the following key proposition, which provides an explicit formula for the drift Laplacian of the angle function Θ defined in (2.7). Proposition 4.1. Let ψ : Σn ↪→ Pnf ×ρ R1 be an immersed spacelike hypersurface and let Θ ∈ C∞(Σn) be the angle function defined in (2.7). Then ∆fΘ = nY >(Hf ) + ( R̃icf (N∗, N∗)− 1 ρ H̃essρ(N∗, N∗) + Θ2 ∆̃f (ρ) ρ3 + |A|2 ) Θ. Proof. Firstly, since Y is a Killing vector field, for any X ∈ X(Σn), we have 〈∇Θ, X〉 = X(Θ) = X(〈N,Y 〉) = 〈∇XN,Y 〉+ 〈N,∇XY 〉 = 〈−A(Y >)−∇NY,X〉, which assures that ∇Θ = −A(Y >)− (∇NY )>. (4.1) On the other hand, from (2.4) we note that nY >(H) = Y >(nHf + 〈∇f,N〉) = nY >(Hf ) + Y >(〈∇f,N〉) = nY >(Hf ) + 〈Y,Hessf(N)〉+ ΘHessf(N,N)− 〈A(Y >),∇f〉, (4.2) 8 H. F. DE LIMA, A. F. A. RAMALHO, M. A. L. VELÁSQUEZ EJDE-2020/83 where we used the decomposition Y = Y > −ΘN . Moreover, since f is supposed to be invariant along the flow determinate by Y , from (4.1) we obtain that 〈∇Θ,∇f〉 = −〈A(Y >) + (∇NY )>,∇f〉 = −〈A(Y >),∇f〉 − 〈∇NY,∇f〉 = −〈A(Y >),∇f〉+ 〈Y,∇N∇f〉 = −〈A(Y >),∇f〉+ 〈Y,Hessf(N)〉. (4.3) Substituting (4.3) in (4.2) we obtain nY >(H) = nY >(Hf ) + ΘHessf(N,N) + 〈∇Θ,∇f〉. (4.4) From [3, Proposition 2.12] we have ∆Θ = nY >(H) + Θ(Ric(N,N) + |A|2), (4.5) Thus, from (2.2), (2.5), (4.5) and (4.4) we obtain ∆fΘ = nY >(Hf ) + (Ricf (N,N) + |A|2)Θ. (4.6) Now, if we consider the decomposition N = N∗ + N⊥ of N , where (·)⊥ denote the projection of a vector field in X(Pn ×ρ R1) on X(R1), we have Hessf(N,N) = 〈∇N∇f,N〉 = 〈∇N ∇̃f,N∗ +N⊥〉 = H̃essf(N∗, N∗) + 1 ρ 〈∇̃f, ∇̃ρ〉|N⊥|2 = H̃essf(N∗, N∗)− 1 ρ3 〈∇̃f, ∇̃ρ〉Θ2. (4.7) From [23, Corollary 7.43] we obtain Ric(N,N) = R̃ic(N∗, N∗)− 1 ρ H̃essρ(N∗, N∗) + Θ2 ∆̃(ρ) ρ3 . (4.8) Hence, from (2.2), (4.7) and (4.8), we have Ricf (N,N) = R̃icf (N∗, N∗)− 1 ρ H̃essρ(N∗, N∗) + Θ2 ∆̃f (ρ) ρ3 (4.9) Therefore, from (4.9) and (4.6) we obtain the desired result. � Now, we are in position to present our first rigidity theorem. Theorem 4.2. Let Pnf ×ρ R1 be a weighted standard static spacetimes which is spatially f̃ -parabolic. Suppose that R̃icf ≥ 0, the warping function ρ is convex and 〈∇̃f, ∇̃ρ〉 ≤ 0. Let ψ : Σn ↪→ Pn+1 be an immersed spacelike hypersurface with constant f -mean curvature Hf such that its angle function Θ is bounded and infΣ ρ > 0. Then, Σn is totally geodesic and ρ is a positive constant. In addition, if R̃icf is positive at some point p0 ∈ Σn, then Σn is contained in a slice Pn×{t0}, for some t0 ∈ R. EJDE-2020/83 SOLUTIONS TO MEAN CURVATURE EQUATIONS 9 Proof. Since Hf is constant, from Proposition 4.1, we have the formula ∆fΘ = ( R̃icf (N∗, N∗)− 1 ρ H̃essρ(N∗, N∗) + Θ2 ∆̃f (ρ) ρ3 + |A|2 ) Θ. (4.10) Let us observe that at points where N∗ is different from zero we have 1 ρ H̃essρ(N∗, N∗) = |N∗|2 ρ H̃essρ( N∗ |N∗| , N∗ |N∗| ) = Θ2 − ρ2 ρ3 H̃essρ( N∗ |N∗| , N∗ |N∗| ) . Taking a local orthonormal frame {E1 = N∗ |N∗| , E2, . . . , En} tangent to Pn, we also have Θ2 ρ3 ∆̃(ρ) = Θ2 ρ3 H̃essρ( N∗ |N∗| , N∗ |N∗| ) + Θ2 ρ3 n∑ i=2 H̃essρ(Ei, Ei). Then −1 ρ H̃essρ(N∗, N∗) + Θ2 ρ3 ∆̃(ρ) = 1 ρ H̃essρ( N∗ |N∗| , N∗ |N∗| ) + Θ2 ρ3 n∑ i=2 H̃essρ(Ei, Ei) and, from (2.5), we obtain − 1 ρ H̃essρ(N∗, N∗) + Θ2 ρ3 ∆̃f (ρ) = 1 ρ H̃essρ( N∗ |N∗| , N∗ |N∗| ) + Θ2 ρ3 n∑ i=2 H̃essρ(Ei, Ei)− Θ2 ρ3 〈∇̃f, ∇̃ρ〉 ≥ 0, (4.11) where in the last step we use the convexity of ρ and the hypothesis 〈∇̃f, ∇̃ρ〉 ≤ 0. Using our constraint on R̃icf and equation (4.11), it follows that Θ is a bounded f -superharmonic function on Σn. From Proposition 3.3, Σn is f -parabolic and, thus, Θ is constant on it. So, returning to (4.10), we obtain |A|2 = 0, that is, Σn is totally geodesic. Now we claim that ρ is a positive constant. Indeed, for any X ∈ TΣ, we can write X = X∗ − 〈X,Y 〉 ρ2 Y, where X∗ denotes the orthogonal projection of X onto TP. Since Σn is totally geodesic, from [23, Proposition 7.35], we have X(Θ) = 〈N,∇XY 〉 = 〈N,∇X∗Y 〉 − 〈X,Y 〉 ρ2 〈N,∇Y Y 〉 = 1 ρ 〈X,∇ρ〉〈N,Y 〉 − 1 ρ 〈X,Y 〉〈N,∇ρ〉. Thus, from the above equation, we conclude that ∇Θ = 1 ρ (Θ∇ρ− 〈N,∇ρ〉Y ). Since Θ is constant, taking into account that ∇ρ and Y are linearly independent, it follows that ρ is a positive constant. 10 H. F. DE LIMA, A. F. A. RAMALHO, M. A. L. VELÁSQUEZ EJDE-2020/83 Furthermore, we have, again from (4.10), that R̃icf (N∗, N∗)(p0) = 0. So, if R̃icf is positive at some point p0 ∈ Σn, then N∗(p0) = 0. Consequently, using (2.10) it is not difficult to see that |∇h|2 = 1 ρ2 |N∗|2P = 1 ρ2 (Θ2 ρ2 − 1 ) = 0, which means that Σn is contained in a slice Pn × {t0}, for some t0 ∈ R. � In the next result, we treat the case where R̃icf is not necessarily nonnegative. Theorem 4.3. Let Pnf ×ρ R1 be a weighted standard static spacetimes which is spatially f̃ -parabolic. Suppose that R̃icf ≥ −κ, for some constant κ > 0, and that ρ is a convex warping function such that 〈∇̃f, ∇̃ρ〉 ≤ 0. Let ψ : Σn ↪→ Pnf ×ρ R1 be an immersed spacelike hypersurface with constant f -mean curvature, bounded angle function Θ and such that infΣ ρ > 0. If the height function h satisfies |∇h|2 ≤ α κρ2 |A|2, (4.12) for some constant α ∈ (0, 1), then Σn is contained in a slice Pn × {t0}, for some t0 ∈ R. Proof. Noting that Hf is constant, Θ < 0 on Σn and taking into account our constraint on R̃icf , from (2.10) and (4.11) jointly with Proposition 4.1, we obtain ∆fΘ ≤ (−κρ2|∇h|2 + |A|2)Θ. (4.13) Using the hypothesis (4.12), from (4.13) we obtain ∆f (Θ) ≤ (1− α)|A|2Θ. (4.14) Hence, from (4.14) follows that Θ is a bounded f -superharmonic function on Σn. Since Proposition 3.3 guarantees that Σn is f -parabolic, Θ must be constant on Σn. So, returning to (4.14), we see that Σn is totally geodesic. Therefore, hypothesis (4.12) assures that h is constant on Σn, that is, there exists t0 ∈ R such that Σn ⊂ Pn × {t0}. � Next we study specific weight functions that will be defined in terms of the warp- ing function ρ. The following proposition give us an expression for the Laplacian of the height function h in terms of the weighted mean curvature Hlog ρ2 . Proposition 4.4. Let ψ : Σn ↪→ Pn ×ρ R1 be an immersed spacelike hypersurface and let h ∈ C∞(Σn) be the height function. Then ∆h = −nρ−2ΘHlog ρ2 , (4.15) where Θ is the angle function and Hlog ρ2 is the log ρ2-mean curvature of Σn. Proof. Let {E1, . . . , En} be an orthonormal frame defined in a neighborhood of some point of Σn. From (2.8) we note that ρ−2 div(∇h) = ρ−2 div(−ρ−2 Y >) = −ρ−2〈∇ρ−2, Y >〉 − ρ−4 div(Y >) = 〈∇ρ−2,∇h〉 − ρ−4 div(Y + ΘN) = 〈∇ρ−2,∇h〉 − ρ−4 n∑ i=1 langle∇Ei (Y + ΘN), Ei〉 EJDE-2020/83 SOLUTIONS TO MEAN CURVATURE EQUATIONS 11 = 〈∇ρ−2,∇h〉 − ρ−4 n∑ i=1 〈∇Ei (Y + ΘN), Ei〉 = 〈∇ρ−2,∇h〉 − ρ−4 n∑ i=1 〈∇Ei Y,Ei〉︸ ︷︷ ︸ 0 −ρ−4 n∑ i=1 〈∇Ei (ΘN), Ei〉 = 〈∇ρ−2,∇h〉 − ρ−4 n∑ i=1 〈Ei(Θ) 〈N,Ei〉︸ ︷︷ ︸ 0 +Θ∇Ei N,Ei〉 = 〈∇ρ−2,∇h〉+ ρ−4Θ tr(A) = 〈∇ρ−2,∇h〉 − nρ−4HΘ. Therefore, ∆h = div(∇h) = ρ2〈∇ρ−2,∇h〉 − nρ−2HΘ = 〈∇ log ρ−2,−ρ2 Y >〉 − nρ−2HΘ = −ρ−2〈∇ log ρ−2, Y >〉 − nρ−2HΘ = −ρ−2〈∇ log ρ−2, Y + ΘN〉 − nρ−2HΘ = −ρ−2 〈∇ log ρ−2, Y 〉︸ ︷︷ ︸ 0 −ρ−2〈∇ log ρ−2, N〉Θ− nρ−2HΘ = −ρ−2Θ{nH + 〈∇(log ρ−2), N〉} = −nρ−2ΘHlog ρ2 , where in the last equality we used (2.4). � In the next theorem, the weighted mean curvature Hlog ρ2 of the spacelike hy- persurface is not supposed to be constant. Indeed, we just assume a certain control on the sign of Hlog ρ2 . We recall that a slab of a standart static spacetime Pn×ρR1 is a region of the type Pn ×ρ [t1, t2] = {(t, q) ∈ Pn ×ρ R1 : t1 ≤ t ≤ t2 }. Theorem 4.5. Let Pnlog ρ2 ×ρ R1 be a weighted standard static spacetimes which is spatially log ρ̃ 2-parabolic. Let ψ : Σn ↪→ Pnlog ρ2 ×ρ R1 be an immersed spacelike hypersurface such that η is bounded. Suppose that the log ρ2-mean curvature Hlog ρ2 and the function 〈∇ρ,∇h〉 have opposite signs. If Σn lies in a slab, then Σn is contained in a slice Pn × {t0}, for some t0 ∈ R. Proof. By (2.5) and from Proposition 4.4, we have ∆log ρ2h = −nρ−2ΘHlog ρ2 − 〈∇ log ρ2,∇h〉 = −nρ−2ΘHlog ρ2 − 2 ρ 〈∇ρ,∇h〉. Taking into account that Hlog ρ2 and 〈∇ρ,∇h〉 have opposite signs, we conclude that ∆log ρ2h does not change sing. Therefore, since Proposition 3.3 guarantees the log ρ2-parabolicity of Σn, h must be constant and, consequently, Σn is contained in a slice Pn × {t0}, for some t0 ∈ R. � 12 H. F. DE LIMA, A. F. A. RAMALHO, M. A. L. VELÁSQUEZ EJDE-2020/83 We recall that a spacelike hypersurface Σn is said f -maximal if its f -mean cur- vature vanishes identically on it. In this setting, from Theorem 4.5 we also have the following result. Corollary 4.6. Let Pnlog ρ2 ×ρ R1 be a weighted standard static spacetimes which is spatially log ρ̃ 2-parabolic. Let ψ : Σn ↪→ Pn+1 be a log ρ2-maximal spacelike hypersurface, contained in a slab, such that η is bounded. If the function 〈∇ρ,∇h〉 does not change sign, then Σn is contained in a slice Pn × {t0}, for some t0 ∈ R. Proceeding as above, we obtain the following rigidity result. Theorem 4.7. Let Pnlog ρ−2 ×ρ R1 be a weighted standard static spacetimes which is spatially log ρ̃−2-parabolic. Let ψ : Σn ↪→ Pnlog ρ−2 ×ρ R1 be a maximal spacelike hypersurface such that η is bounded and infΣ ρ > 0. If Riclog ρ−2 ≥ κ, for some constant κ > 0, then Σn is contained in a slice Pn × {t0}, for some t0 ∈ R. Proof. Firstly, observe that, reasoning as in the proof of Proposition 4.4, we obtain ∆h = div(∇h) = ρ2〈∇ρ−2,∇h〉 − nρ−2HΘ = 〈∇ log ρ−2,∇h〉 − nρ−2HΘ. Therefore, using (2.5), we obtain ∆log ρ−2h = −nρ−2HΘ. (4.16) Now, from Bochner’s formula (see [32, page 378]) we have 1 2 ∆log ρ−2 |∇h|2 = |Hessh|2 + Riclog ρ−2(∇h,∇h) + 〈∇∆log ρ−2h,∇h〉. (4.17) Consequently, taking into account our restriction on Riclog ρ−2 and the assumption that Σn is maximal, from (4.16) and (4.17), we obtain 1 2 ∆log ρ−2 |∇h|2 ≥ Riclog ρ−2(∇h,∇h) ≥ κ|∇h|2 ≥ 0. (4.18) On the other hand, Proposition 3.3 guarantees that Σn is log ρ−2-parabolic. Since, from (2.10), infΣ ρ > 0 implies in the boundedness of |∇h| and, consequently, in the boundedness of |∇h|2, we conclude from log ρ−2- parabolicity of Σnthat |∇h|2 is constant, and then ∆log ρ2 |∇h|2 = 0. Returning to (4.18), we obtain that |∇h| = 0 and Σn is contained in a slice. � 5. Entire Killing graphs and the mean curvature equation in Pnf ×ρ R1 According to [10], we define the entire Killing graph Σ(z) associated with a smooth function z ∈ C∞(P) as been the hypersurface given by Σ(z) = {Ψ(x, z(x)) : x ∈ Pn} ⊂ Pn ×ρ R1. The metric induced on Pn from the Lorentzian metric (2.1) via Σ(z) is given by 〈, 〉z = 〈, 〉P − ρ2dz2. Moreover, Σ(z) is spacelike if, and only if, ρ2|Dz|2P < 1, where Dz denotes the gradient of a function z with respect to the metric 〈, 〉P of Pn. Indeed, if Σ(z) is spacelike, then 0 < 〈Dz,Dz〉z = 〈Dz,Dz〉P − ρ2〈Dz,Dz〉2P EJDE-2020/83 SOLUTIONS TO MEAN CURVATURE EQUATIONS 13 and, hence, we conclude that ρ2|Dz|2P < 1. Conversely, if ρ2|Dz|2P < 1 and X is a vector field tangent to Σ(z), we obtain, from Cauchy-Schwarz inequality, 〈X,X〉z = 〈X∗, X∗〉P − ρ2〈Dz,X∗〉2P ≥ 〈X∗, X∗〉P(1− ρ2|Dz|2P), where X∗ is the orthogonal projection of X onto TPn. Thus, 〈X,X〉z ≥ 0 and 〈X,X〉z = 0 if, and only if, X = 0. The function g : Pn × R1 → R given by g(x, t) = z(x) − t is such that Σ(z) = Ψ(g−1(0)). Thus, for each vector field X tangent to Pn ×ρ R1, we have X(g) = X∗(g)− 1 ρ2 〈X, ∂t〉∂t(g) = 〈 1 ρ2 ∂t +Dz,X〉. Hence, ∇g = 1 ρ2 ∂t +Dz is a normal vector field on g−1(0) and, consequently, N0 = Ψ∗(∇g) = 1 ρ2 Y + Ψ∗(Dz) is a normal timelike vector field on Σ(z). Since |N0| = (1− ρ2|Dz|2P)1/2 ρ , it follows that N = N0 |N0| = 1 ρ(1− ρ2|Dz|2P)1/2 (Y + ρ2Ψ∗(Dz)) (5.1) defines the future-pointing Gauss map of Σ(z) such that its angle function is Θ = 〈N,Y 〉 = − ρ (1− ρ2|Dz|2P)1/2 < 0. (5.2) Moreover, for each vector field X tangent to Pn, the shape operator A of Σ(z) with respect to N is given by AX = − ρ (1− ρ2|Dz|2P)1/2 DXDz − ρ3〈DXDz,Dz〉 (1− ρ2|Dz|2P)3/2 Dz − ρ2〈Dρ,X〉|Dz|2P (1− ρ2|Dz|2P)3/2 Dz − 〈Dρ,X〉 (1− ρ2|Dz|2P)1/2 Dz − 〈Dz,X〉 (1− ρ2|Dz|2P)1/2 Dρ, (5.3) where D denotes the Levi-Civita connections in Pn. So, it follows from (5.3) that the mean curvature Hz of a spacelike entire Killing graph Σ(z) is given by nH(z) = Div( ρDz (1 + ρ2|Dz|2P)1/2 ) + 〈Dz,Dρ〉 (1 + ρ2|Dz|2P)1/2 , where Div stands for the divergence operator on Pn with respect to the metric 〈, 〉P. A direct computation shows that the f -mean curvature is given by n(Hz)f = Divf ( ρDz (1− ρ2|Dz|2P)1/2 ) + 〈Dz,Dρ〉 (1− ρ2|Dz|2P)1/2 . 14 H. F. DE LIMA, A. F. A. RAMALHO, M. A. L. VELÁSQUEZ EJDE-2020/83 From the previous discussion, an entire Killing graph Σ(z) is spacelike with constant f -mean curvature C if, and only if, the function z ∈ C∞(P) satisfies the following elliptic partial differential equation of f -divergence form Divf ( ρDz (1− ρ2|Dz|2P)1/2 ) + 〈Dz,Dρ〉 (1− ρ2|Dz|2P)1/2 = C, in Pn ρ2|Dz|2P < 1. (5.4) In what follows, we will use the theorems obtained in the previous section, on entire Killing graph context, to obtain uniqueness results for equations of the type (5.4). We start by applying the Theorem 4.2 to get the following result. Theorem 5.1. Let Pnf ×ρ R1 be a weighted standard static spacetimes which is spatially f̃ -parabolic with convex warping function ρ, 〈∇̃f, ∇̃ρ〉 ≤ 0 and R̃icf ≥ 0. If the entire Killing graph Σ(z) associated with z ∈ C∞(P) is such that ρ|Σ(z) is bounded and R̃icf is positive at some point p0 ∈ Σ(z), then the only solutions of the problem Divf ( ρDz (1− ρ2|Dz|2P)1/2 ) + 〈Dz,Dρ〉 (1− ρ2|Dz|2P)1/2 = C, z ∈ C∞(P) sup Σ(z) (ρ2|Dz|2P) < 1, are constants. Proof. Since we are supposing that sup ρ2|Dz|2P < 1, from (5.2), the boundness of ρ|Σ(z) is equivalent to the boundness of Θ. Furthermore, we observe that the condition sup ρ2|Dz|2P < 1 also implies the boundness of η. Indeed, using (5.2) again, we have that η = 1 (1− ρ2|Dz|2P)1/2 . Hence, we can disregard the hypothesis infΣ(z) ρ > 0 in Theorem 4.2 to obtain the present result. � An important example of weighted Riemannian manifold is the so-called Gauss- ian space Gn, which corresponds to the Euclidean space Rn endowed with the Gaussian probability measure e−fdx2 = (2π)− n 2 e− |x|2 2 dx2. Concerned with the weighted product space Gn×R1, An et al extended the classi- cal Bernstein’s theorem [6] showing that the only weighted minimal graphs Σn(z) of functions z(x2, · · · , xn+1) = x1 over Gn, with supΣ(z) |Dz|G < 1, are the hyper- planes x1 = constant (see [2, Theorem 4]). Taking into account this previous discussion, from Theorem 5.1 we obtain an extension of Theorem 4 of [2]. Corollary 5.2. Consider the weighted standard static spacetime Gn ×ρ R1, where Gn is the Gaussian space and the warping function ρ is convex with 〈∇̃f, ∇̃ρ〉 ≤ 0. If the entire Killing graph Σ(z) associated with z ∈ C∞(G) is such that ρ|Σ(z) is bounded, the only solutions of the problem Divf ( ρDz (1− ρ2|Dz|2G)1/2 ) + 〈Dz,Dρ〉 (1− ρ2|Dz|2G)1/2 = C, z ∈ C∞(G) EJDE-2020/83 SOLUTIONS TO MEAN CURVATURE EQUATIONS 15 sup Σ(z) (ρ2|Dz|2G) < 1, are constants. Proof. We note that, since Volf (Gn) = 1, [18, Remark 3] guarantees that Gn is f - parabolic. Moreover, with a straightforward computation, we obtain that R̃icf = 1. Therefore, since Gn is also simply connected, the result follows from Theorem 5.1. � The next result is an application of Theorem 4.3. Theorem 5.3. Let Pnf ×ρ R1 be a weighted standard static spacetime which is spatially f̃ -parabolic with convex warping function ρ, 〈∇̃f, ∇̃ρ〉 ≤ 0 and R̃icf ≥ −κ, for some constant κ > 0. If the entire Killing graph Σ(z) associated with z is such that ρ|Σ(z) is bounded and α ∈ (0, 1) is a constant, the only solutions of the problem Divf ( ρDz (1− ρ2|Dz|2P)1/2 ) + 〈Dz,Dρ〉 (1− ρ2|Dz|2P)1/2 = C, z ∈ C∞(P) sup Σ(z) (ρ2|Dz|2P) < α|A|2 α|A|2 + κ , (5.5) are constants. Proof. From equation (5.11) we have |N∗|2P = ρ2|Dz|2P 1− ρ2|Dz|2P . (5.6) Then (2.10) and (5.6) give us the relation |∇h|2 = |Dz|2P 1− ρ2|Dz|2P . (5.7) Now, using (5.7) we conclude that the hypothesis the hypothesis (4.12) is equivalent to ρ2|Dz|2P ≤ α|A|2 α|A|2 + κ . Furthermore, since κ > 0, we have that that α|A|2 α|A|2+κ ≤ 1. Hence, the result follows from Theorem 4.3. � Reasoning as in the Corollary 5.2, we have the following result. Corollary 5.4. Consider the weighted standard static spacetime Gn ×ρ R1, where Gn is the Gaussian space and the warping function ρ is convex with 〈∇̃f, ∇̃ρ〉 ≤ 0. If the entire Killing graph Σ(z) associated with z ∈ C∞(G) is such that ρ|Σ(z) is bounded, then, for any constants k > 0 and α ∈ (0, 1), the only solutions of the problem Divf ( ρDz (1− ρ2|Dz|2G)1/2 ) + 〈Dz,Dρ〉 (1− ρ2|Dz|2G)1/2 = C, z ∈ C∞(G) sup Σ(z) (ρ2|Dz|2G) < α|A|2 α|A|2 + κ , (5.8) are constants. 16 H. F. DE LIMA, A. F. A. RAMALHO, M. A. L. VELÁSQUEZ EJDE-2020/83 From Theorem 4.5, we obtain the following result. Theorem 5.5. Let Pnlog ρ2 ×ρ R1 be a weighted standard static spacetimes which is spatially log ρ̃ 2-parabolic. If the entire Killing graph associated with z is such that 〈∇ρ,Ψ∗(Dz)〉 does not change sign, then the only bounded solutions of the problem Divlog ρ2( ρDz (1− ρ2|Dz|2P)1/2 ) + 〈Dz,Dρ〉 (1− ρ2|Dz|2P)1/2 = C, z ∈ C∞(P) sup Σ(z) (ρ2|Dz|2P) < 1. (5.9) are constants. Proof. Firstly, observe that 〈∇ρ,∇N〉 = ∇h(ρ) = − 1 ρ2 Y >(ρ) = − 1 ρ2 Y > ( (−〈Y, Y 〉)1/2 ) = − 1 ρ2 (1 2 (−〈Y, Y 〉)1/2Y >〈Y, Y 〉 ) = − 1 2ρ3 Y >〈Y, Y 〉 ) = − 1 ρ3 〈∇Y >Y, Y 〉 = − 1 ρ3 〈∇Y+ΘNY, Y 〉 = − 1 ρ3 ( 〈∇Y Y, Y 〉︸ ︷︷ ︸ 0 +〈∇ΘNY, Y 〉 ) = − 1 ρ3 〈∇ΘNY, Y 〉 = −Θ ρ3 〈∇NY, Y 〉 = − Θ 2ρ3 N〈Y, Y 〉 = − Θ 2ρ3 N(ρ2) = − Θ 2ρ3 − 2ρN∗(ρ) = Θ ρ2 〈∇ρ,N∗〉. (5.10) On the other hand, from (5.1), we have N∗ = N −N⊥ = ρΨ∗(Dz) (1− ρ2|Dz|2P)1/2 . (5.11) Hence, from (5.10) and (5.11) we obtain 〈∇ρ,∇N〉 = Θ ρ 〈∇ρ, ρΨ∗(Dz) (1− ρ2|Dz|2P)1/2 〉 = Θ ρ(1− ρ2|Dz|2P)1/2 〈∇ρ,Ψ∗(Dz)〉. Therefore, 〈∇ρ,∇N〉 do not change of sign if and only if 〈∇ρ,Ψ∗(Dz)〉 do not change of sign and the result follows from Corollary 4.6. � EJDE-2020/83 SOLUTIONS TO MEAN CURVATURE EQUATIONS 17 Taking ρ = ( e |x|2 2 +log (2π) n 2 )1/2 (5.12) in Theorem 5.5, we obtain the following consequence. Corollary 5.6. Consider the weighted standard static spacetime Gn ×ρ R1, where Gn is the Gaussian space and ρ is defined in (5.12). If the entire Killing graph associate to z is such that 〈∇ρ,Ψ∗(Dz)〉 does not change sign, then the only bounded solutions of the problem Divf ( ρDz (1− ρ2|Dz|2G)1/2 ) + 〈Dz,Dρ〉 (1− ρ2|Dz|2G)1/2 = C, z ∈ C∞(G) sup Σ(z) (ρ2|Dz|2G) < 1, are constants. Applying the Theorem 4.7 we obtain the following result. Theorem 5.7. Let Pnlog ρ−2 ×ρ R1 be a weighted standard static spacetimes which is spatially log ρ̃−2-parabolic. If the entire Killing graph associate to z is such that |Dz|2P is bounded and Riclog ρ−2 ≥ κ, for some constant κ > 0, then the only bounded solutions of the problem Div( ρDz (1− ρ2|Dz|2P)1/2 ) + 〈Dz,Dρ〉 (1− ρ2|Dz|2P)1/2 = 0, z ∈ C∞(P) sup Σ(z) (ρ2|Dz|2P) < 1, (5.13) are constants. Proof. We observe that if z ∈ C∞(P) is solution of problem (5.13), then the entire Killing graph Σ(z) is spacelike and maximal. Moreover using (5.7), we note that the boundness of |∇h|2 follows from the boundness of |Dz|2P. Then, the result follows from Theorem 4.7. � Finally, considering ρ = (e |x|2 2 +log (2π) n 2 )−1/2 (5.14) in Theorem 5.7, we have the following result. Corollary 5.8. Consider the weighted standard static spacetime Gn ×ρ R1, where Gn is the Gaussian space and ρ is defined in (5.14). If the entire Killing graph associate to z is such that |Dz|2P is bounded and Riclog ρ−2 ≥ κ, for some constant κ > 0, then the only bounded solutions of the problem Div( ρDz (1− ρ2|Dz|2G)1/2 ) + 〈Dz,Dρ〉 (1− ρ2|Dz|2G)1/2 = 0, z ∈ C∞(G) sup Σ(z) (ρ2|Dz|2G) < 1, are constants. Acknowledgements. H. F. de Lima, and M. A. L. Velásquez were partially sup- ported by CNPq, Brazil, grants 301970/2019-0 and 311224/2018-0, respectively. 18 H. F. DE LIMA, A. F. A. RAMALHO, M. A. L. VELÁSQUEZ EJDE-2020/83 References [1] A. Aledo, A. Romero, R. Rubio; The existence and uniqueness of standard static splitting, Classical Quant. Grav., 32 (2015), 105004. [2] H. V .Q. An, D. V. Cuong, N. T. M. Duyen, D. T. Hieu, T. L. Nam; On entire f-maximal graphs in the Lorentzian product Gn × R1, J. Geom. Phys., 114 (2017), 587–592. [3] J. Barbosa, M. do Carmo, J. Eschenburg; Stability of hypersurfaces with constant mean curvature, Math. Z., 197 (1988), 123–138. [4] V. Bayle; Propriétés de concavité du profil isopérimétrique et applications, Ph.D. thesis, Institut Fourier, Grenoble, 2003. [5] J. K. Beem, P. E. Ehrlich, K. L. Easley; Global Lorentzian geometry, Marcel Dekker Inc., New York, 1996. [6] S. Bernstein; Sur les surfaces d’efinies au moyen de leur courboure moyenne ou totale, Ann. Ec. Norm. Sup., 27 (1910), 233–256. [7] E. Calabi, Examples of Bernstein problems for some nonlinear equations, in: Global Analysis (Proc. Sympos. Pure Math., Vol. XV, Berkeley, CA, 1968), Amer. Math. Soc., Providence, RI, 1970, pp. 223–230. [8] J. Case; Singularity theorems and the Lorentzian splitting theorem for the Bakry-Émery-Ricci tensor, J. Geom. Phys., 60 (2010), 477–490. [9] S. Y. Cheng, S.T. Yau; Maximal space-like hypersurfaces in the Lorentzian Minkowski spaces, Ann. of Math., 104 (1976), 407–419. [10] M. Dajczer, P. Hinojosa, J. H. de Lira; Killing graphs with prescribed mean curvature, Calc. Var. Partial Diff. Eq. 33 (2008), 231–248. [11] E. L. de Lima, H. F. de Lima, F. R. dos Santos; On the stability and parabolicity of complete f-minimal hupersurfaces in weighted warped products, Results Math., 73 (2018), 1–14. [12] E. L. de Lima, H. F. de Lima, E. A. Lima J, A. A. Medeiros; Parabolicity and rigidity of spacelike hypersurfaces immersed in a Lorentzian Killling warped product, Comm. Math. Unive. Carolinae, 58 (2017), 183–196. [13] M. P. do Carmo; Riemannian Geometry, Birkhäuser Basel, New York, 1992. [14] A. Grigor’yan; Analytic and geometric background of recurrence and non-explosion of the Brownian motion on Riemannian manifolds, Bull. American Math. Soc., 36 (1999), 135– 249. [15] A. Grigor’yan; Escape rate of Brownian motion on Riemannian manifolds, Appl. Anal., 71 (1999), 63–89. [16] M. Gromov; Isoperimetry of waists and concentration of maps, Geom. Funct. Anal., 13 (2003), 178–215. [17] S. Hawking, G. Ellis; The large scale structure of space-time, Cambridge University Press, 1973. [18] D. Impera, M. Rimoldi, Stability properties and topology at infinity of f-minimal hypersur- faces, Geom. Ded., 178 (2015), 21–47. [19] M. Kanai, Rough isometries and combinatorial approximations of geometries of noncompact Riemannian manifolds, J. Math. Soc. Japan, 37 (1985), 391–413. [20] S. Kobayashi, K. Nomizu; Foundations of Differential Geometry, Vol. II, Interscience, New York, 1969. [21] J. E. Marsden, F.J Tipler; Maximal hypersurfaces and foliations of constant mean curvature in general relativity, Phys. Rep., 66 (1980), 109–139. [22] F. Morgan; Geometric Measure Theory. A Beginners Guide, Fourth ed., Elsevier/Academic Press, Amsterdam, 2009. [23] B. O’Neill; Semi-Riemannian Geometry with Applications to Relativity, Academic Press, London, 1983. [24] J. A. S. Pelegŕın, A. Romero, R. M. Rubio; Spacelike hypersurfaces in spatially parabolic standard static spacetimes and Calabi-Bernstein-type problems, Mediterr. J. Math., (2019), 16: 34. [25] A. Romero, R. Rubio, J. Salamanca; Parabolicity of spacelike hypersurfaces in generalized Robertson-Walker spacestimes: Applications to uniqueness results, Int. J. Geom. Meth. Mod. Phys. 10 (2013), 1360014. EJDE-2020/83 SOLUTIONS TO MEAN CURVATURE EQUATIONS 19 [26] A. Romero, R. Rubio, J. Salamanca; Uniqueness of complete maximal hypersurfaces in spa- tially parabolic generalized Robertson-Walker spacetimes, Class. Quantum Grav., 30 (2013), 1–13. [27] M. Sánchez; Lorentzian manifolds admitting a Killing vector field, Nonl. Anal., 30 (1997), 643–654. [28] M. Sánchez; Geodesics in static spacetimes and t-periodic trajectories, Nonl. Anal., 35 (1999), 677–686. [29] M. Sánchez; On the geometry of static spacetimes, Nonl. Anal., 63 (2005), 455–46. [30] M. Sánchez; On causality and closed geodesics of compact Lorentzian manifolds and static spacetimes, Diff. Geom. App., 24 (2006), 21–32. [31] S. M. Stumbles; Hypersurfaces of constant mean curvature. Ann. Physics, 133 (1981), 28–56. [32] G. Wei, W. Willie; Comparison geometry for the Bakry-Émery Ricci tensor, J. Diff. Geom., 83 (2009), 377–405. Henrique F. de Lima Departamento de Matemática, Universidade Federal de Campina Grande, 58.429-970 Campina Grande, Paráıba, Brazil Email address: henrique@mat.ufcg.edu.br André F. A. Ramalho Departamento de Matemática, Universidade Federal de Campina Grande, 58.429-970 Campina Grande, Paráıba, Brazil Email address: andre@mat.ufcg.edu.br M. A. L. Velásquez Departamento de Matemática, Universidade Federal de Campina Grande, 58.429-970 Campina Grande, Paráıba, Brazil Email address: marco.velasquez@mat.ufcg.edu.br 1. Introduction 2. Weighted standard static spacetimes 3. An f-parabolicity criterion for spacelike hypersurfaces in (PnR1)f 4. Rigidity results for spacelike hypersurfaces in PnfR1 5. Entire Killing graphs and the mean curvature equation in PnfR1 Acknowledgements References