� EXTRACTA MATHEMATICAE Volumen 33, Número 2, 2018 instituto de investigación de matemáticas de la universidad de extremadura EXTRACTA MATHEMATICAE Vol. 34, Num. 1 (2019), 123 – 134 doi:10.17398/2605-5686.34.1.123 Available online February 8, 2019 Characterizations of minimal hypersurfaces immersed in certain warped products Eudes L. de Lima 1, Henrique F. de Lima 2,@ , Eraldo A. Lima Jr 3, Adriano A. Medeiros 3 1 Unidade Acadêmica de Ciências Exatas e da Natureza Universidade Federal de Campina Grande, 58900–000 Cajazeiras, Paráıba, Brazil, eudes.lima@ufcg.edu.br 2 Departamento de Matemática, Universidade Federal de Campina Grande, 58.429–970 Campina Grande, Paráıba, Brazil, henrique@mat.ufcg.edu.br 3 Departamento de Matemática, Universidade Federal da Paráıba, 58.051–900 João Pessoa, Paráıba, Brazil, eraldo@mat.ufpb.br , adrianoalves@mat.ufpb.br Received October 4, 2018 Presented by Teresa Arias-Marco Accepted January 15, 2019 Abstract: Our purpose in this paper is to investigate when a complete two-sided hypersurface im- mersed with constant mean curvature in a Killing warped product Mn×ρR, whose Riemannian base Mn has sectional curvature bounded from below and such that the warping function ρ ∈ C∞(M) is supposed to be concave, is minimal (and, in particular, totally geodesic) in the ambient space. Our approach is based on the application of the well known generalized maximum principle of Omori- Yau. Key words: Killing warped product, constant mean curvature hypersurfaces, minimal hypersurfaces, totally geodesic hypersurfaces. AMS Subject Class. (2010): Primary 53C42; Secondary 53B30 and 53C50. 1. Introduction Killing vector fields are important objects which have been widely used in order to understand the geometry of submanifolds and, more particularly, of hypersurfaces immersed in Riemannian spaces. Into this branch, Aĺıas, Da- jczer and Ripoll [1] extended classical Bernstein’s theorem [4] to the context of complete minimal surfaces in Riemannian spaces of nonnegative Ricci curva- ture carrying a Killing vector field. This was done under the assumption that the sign of the angle function between a global Gauss mapping and the Killing vector field remains unchanged along the surface. Afterwards, Dajczer, Hino- josa and de Lira [10] defined a notion of Killing graph in a class of Riemannian manifolds endowed with a Killing vector field and solved the corresponding Dirichlet problem for prescribed mean curvature under hypothesis involving @ Corresponding author ISSN: 0213-8743 (print), 2605-5686 (online) https://doi.org/10.17398/2605-5686.34.1.123 mailto:eudes.lima@ufcg.edu.br mailto:henrique@mat.ufcg.edu.br mailto:eraldo@mat.ufpb.br mailto:adrianoalves@mat.ufpb.br mailto:henrique@mat.ufcg.edu.br https://www.eweb.unex.es/eweb/extracta/ https://creativecommons.org/licenses/by-nc/3.0/ 124 e.l. de lima, h.f. de lima, e.a. lima jr, a.a. medeiros domain data and the Ricci curvature of the ambient space. More recently, Dajczer and de Lira [7] showed that an entire Killing graph of constant mean curvature lying inside a slab must be a totally geodesic slice, under certain restrictions on the curvature of the ambient space. To prove their Bernstein type result, they used as main tool the generalized maximum principle of Omori [11] and Yau [15] for the Laplacian in the sense of Pigola, Rigoli and Setti given in [13] (see also [2] for a modern and accessible reference to the generalized maximum principle of Omori-Yau). When the ambient space is a Riemannian product of the type Mn × R, it was shown by Rosenberg, Schulze and Spruck [14] that if the Ricci curva- ture of the base Mn is nonnegative and its sectional curvature is bounded from below, then any entire minimal graph over Mn with nonnegative height function must be a slice. This result extends the celebrated theorem due to Bombieri, De Giorgi and Miranda [5] for entire minimal hypersurfaces in the Euclidean space. In [8], the second and third authors jointly with Parente studied complete two-sided hypersurfaces immersed in Mn × R, whose base is also supposed to have sectional curvature bounded from below. In this setting, they extended a technique developed in [9] obtaining sufficient con- ditions which assure that such a hypersurface is a slice of the ambient space, provided that its angle function has some suitable behavior. We recall that a hypersurface is said to be two-sided if its normal bundle is trivial, that is, there exists on it a globally defined unit normal vector field. These aforementioned works allow us to discuss a natural question: Question. Under what reasonable geometric restrictions on a Rieman- nian manifold M n+1 endowed with a Killing vector field must a complete two-sided hypersurface Σn immersed with constant mean curvature in M n+1 be minimal and, in particular, totally geodesic in this ambient space? As is well known, under suitable assumptions on such a Killing vector field, the ambient space M n+1 can be regarded as a Killing warped product Mn×ρR, for an appropriate n-dimensional Riemannian base Mn and a certain warping function ρ ∈ C∞(M). Assuming that the base Mn has sectional curvature bounded from below and supposing that the warping function ρ is concave on Mn, our purpose in this paper is just to present satisfactory answers for the above stated question. For this, in order to use the generalized maximum principle of Omori-Yau, first we establish sufficient conditions to guarantee that the Ricci curvature of a complete two-sided hypersurface is bounded from below (see Proposition 1). Afterwards, in Section 4 we state characterizations of minimal hypersurfaces 125 and prove our main results (see Theorems 1 and 2, and Corollaries 1 and 2). Finally, we also discuss the plausibility of the assumptions assumed in our results (see Remark 1). 2. Killing warped products Let M n+1 be an (n + 1)-dimensional Riemannian manifold endowed with a Killing vector field K. Suppose that the distribution D orthogonal to K is of constant rank and integrable. We denote by Ψ : Mn × I → Mn+1 the flow generated by K, where Mn is an arbitrarily fixed integral leaf of D labeled as t = 0, which we will suppose to be connected, and I is the maximal interval of definition. Without loss of generality, in what follows we will also consider I = R. In this setting, M n+1 can be regard as the Killing warped product Mn×ρR, that is, the product manifold Mn ×R endowed with the warping metric 〈 , 〉 = π∗M (〈 , 〉M ) + (ρ◦πM ) 2π∗R ( dt2 ) , (2.1) where πM and πR denote the canonical projections from M × R onto each factor, 〈 , 〉M is the induced Riemannian metric on the base Mn and the warping function ρ ∈ C∞ is ρ = |K| > 0, where | · | denotes the norm of a vector field on Mn+1. Throughout this work, we will deal with hypersurfaces ψ : Σn → Mn+1 immersed in a Killing warped product M n+1 = Mn ×ρ R and which are two- sided. This condition means that there exists a globally defined unit normal vector field N on Σn. Let ∇, ∇ and D denote the Levi-Civita connections in M n+1 , Σn and Mn, respectively. Then, as in [12], the curvature tensor R of the hypersurface Σn is given by R(X,Y )Z = ∇[X,Y ] − [∇X,∇Y ]Z, where [ , ] denotes the Lie bracket and X,Y,Z ∈ X(Σ). A well known fact is that the curvature tensor R of the hypersurface Σn can be described in terms of the shape operator A and of the curvature tensor R of the ambient space M n+1 = Mn ×ρ R by the Gauss equation given by R(X,Y )Z = ( R(X,Y )Z )> + 〈AX,Z〉AY −〈AY,Z〉AX, (2.2) 126 e.l. de lima, h.f. de lima, e.a. lima jr, a.a. medeiros for every tangent vector fields X,Y,Z ∈ X(Σ), where ( )> denotes the tangen- tial component of a vector field in X(M) along Σn. In this paper, we will also consider two particular smooth functions on a connected two-sided hypersurface ψ : Σn → Mn+1 immersed in a Killing warped product M n+1 = Mn ×ρ R, namely, the (vertical) height function h = πR ◦ ψ and the angle function Θ = 〈N,K〉, where we recall that N denotes the unit normal vector field globally defined on Σn. From the decomposition K = K> + ΘN, it is easy to see that ∇h = 1 ρ2 K> and |∇h|2 = ρ2 − Θ2 ρ4 . (2.3) Moreover, assuming the constancy of the mean curvature function H = 1 n trace(A), from Proposition 2.12 of [3] (see also Proposition 6 of [1] or Proposition 2.1 of [6]) we have the following formula ∆Θ = − ( Ric(N,N) + |A|2 ) Θ, (2.4) where Ric denotes the Ricci tensor of M n+1 and |A| stands for the Hilbert- Schmidt norm of the shape operator A of Σn. Finally, we also recall that it holds the following algebraic relation |A|2 = nH2 + n(n− 1)(H2 −H2), (2.5) where H2 = 2 n(n− 1) S2 is the mean value of the second elementary symmetric function S2 on the eigenvalues of A. 3. Auxiliary results In order to prove our main theorems in the next section, we will need use two auxiliary results. The first one is the well known generalized maximum principle of Omori [11] and Yau [15], which is quoted below (see also [2] for a modern and accessible reference to the generalized maximum principle of Omori-Yau). Lemma 1. Let Σn be a n-dimensional complete Riemannian manifold whose Ricci curvature is bounded from below and let u : Σn → R be a smooth function satisfying inf Σ u > −∞. Then, there exists a sequence of points {pk}⊂ Σn such that lim k u(pk) = inf Σ u, lim k |∇u(pk)| = 0 and lim inf k ∆u(pk) ≥ 0 . characterizations of minimal hypersurfaces 127 The next auxiliary result will give sufficient conditions to guarantee that the Ricci curvature of a two-sided hypersurface immersed in a Killing warped product M n+1 = Mn ×ρ R is bounded from below. In order to prove this result, we will develop some preliminaries computations. Let us consider a two-sided hypersurface ψ : Σn → Mn+1 immersed in Killing warped product M n+1 = Mn ×ρ R. For vector fields U,V,W tangent to M n+1 , we can write U = U∗ + Û, where U∗ and Û are the orthogonal projections of U onto TM and TR, re- spectively. Thus, Û = 〈U,K〉 〈K,K〉 K = 〈U,K〉 ρ2 K, where (as in the previous section) K = ∂t. Thus, with a straightforward computation it is not difficult to verify that R(U,V )W = RM (U ∗,V ∗)W∗ − 〈V,K〉 ρ2 R(K,U∗)W∗ + 〈V,K〉〈W,K〉 ρ4 R(U∗,K)K + 〈U,K〉 ρ2 R(K,V ∗)W∗ − 〈U,K〉〈W,K〉 ρ4 R(V ∗,K)K. Then, from Lemma 7.34 and Proposition 7.42 of [12] we get R(U,V )W = RM (U ∗,V ∗)W∗ − 〈V,K〉HessM ρ(U∗,W∗) ρ3 K + 〈V,K〉〈W,K〉〈K,K〉 ρ5 ∇U∗∇(ρ◦πM ) + 〈U,K〉HessM ρ(V ∗,W∗) ρ3 K − 〈U,K〉〈W,K〉〈K,K〉 ρ5 ∇V∗∇(ρ◦πM ), where HessM is the Hessian on M n. So, we have that R(U,V )W = RM (U ∗,V ∗)W∗ − 〈V,K〉HessM ρ(U∗,W∗) ρ3 K + 〈V,K〉〈W,K〉 ρ3 DU∗Dρ + 〈U,K〉HessM ρ(V ∗,W∗) ρ3 K − 〈U,K〉〈W,K〉 ρ3 DV∗Dρ. 128 e.l. de lima, h.f. de lima, e.a. lima jr, a.a. medeiros In particular, taking a local orthonormal frame {E1, . . . ,En} tangent to Σn and X a vector field tangent to Σn, we can take U = W = X and V = Ei in the last equation to obtain R(X,Ei)X = RM (X ∗,E∗i )X ∗ − 〈Ei,K〉HessM ρ(X∗,X∗) ρ3 K + 〈Ei,K〉〈X,K〉 ρ3 DX∗Dρ + 〈X,K〉HessM ρ(E∗i ,X ∗) ρ3 K − 〈X,K〉2 ρ3 DE∗i Dρ. Hence, we conclude that 〈 R(X,Ei)X,Ei 〉 = 〈RM (X∗,E∗i )X ∗,Ei〉− 〈Ei,K〉2 ρ3 HessM ρ(X ∗,X∗) + 〈Ei,K〉〈X,K〉 ρ3 〈DX∗Dρ,Ei〉− 〈X,K〉2 ρ3 〈DE∗i Dρ,Ei〉 + 〈Ei,K〉〈X,K〉 ρ3 HessM ρ(E ∗ i ,X ∗) = 〈RM (X∗,E∗i )X ∗,E∗i 〉− 〈Ei,K〉2 ρ3 HessM ρ(X ∗,X∗) + 〈Ei,K〉〈X,K〉 ρ3 HessM ρ(X ∗,E∗i ) + 〈Ei,K〉〈X,K〉 ρ3 HessM ρ(X ∗,E∗i ) − 〈X,K〉2 ρ3 HessM (E ∗ i ,E ∗ i ). Consequently, we get〈 R(X,Ei)X,Ei 〉 = KM (X ∗,E∗i ) ( 〈X∗,X∗〉〈E∗i ,E ∗ i 〉−〈X ∗,E∗i 〉 2 ) − 〈Ei,K〉2 ρ3 HessM ρ(X ∗,X∗) − 〈X,K〉2 ρ3 HessM ρ(E ∗ i ,E ∗ i ) + 2 〈Ei,K〉〈X,K〉 ρ3 HessM ρ(X ∗,E∗i ) = KM (X ∗,E∗i ) ( 〈X∗,X∗〉〈E∗i ,E ∗ i 〉−〈X ∗,E∗i 〉 2 ) − 1 ρ HessM ρ ( X̃∗i ,X̃ ∗ i ) + 2 ρ HessM ρ ( X̃∗i , Ẽ ∗ i ) − 1 ρ HessM ρ ( Ẽ∗i , Ẽ ∗ i ) , where X̃∗i = 〈Ei,K〉 ρ X∗ and Ẽ∗i = 〈X,K〉 ρ E∗i . characterizations of minimal hypersurfaces 129 Hence,〈 R(X,Ei)X,Ei 〉 = KM (X ∗,E∗i ) ( 〈X∗,X∗〉〈E∗i ,E ∗ i 〉−〈X ∗,E∗i 〉 2 ) − 1 ρ HessM ρ ( X̃∗i − Ẽ ∗ i ,X̃ ∗ i − Ẽ ∗ i ) . (3.1) Therefore, we obtain that n∑ i=1 〈 R(X,Ei)X,Ei 〉 = n∑ i=1 KM (X ∗,E∗i ) ( 〈X∗,X∗〉〈E∗i ,E ∗ i 〉−〈X ∗,E∗i 〉 2 ) − n∑ i=1 1 ρ HessM ρ ( X̃∗i − Ẽ ∗ i ,X̃ ∗ i − Ẽ ∗ i ) . (3.2) At this point, we recall that a concave function defined on a Rieman- nian manifold Mn is a smooth function ρ ∈ C∞(M) whose Hessian operator HessMρ is negative semidefinite. Now, we are in position to establish the following result: Proposition 1. Let M n+1 = Mn×ρ R be a Killing warped product with concave warping function ρ and whose base Mn has sectional curvature satis- fying KM ≥−κ, for some constant κ ≥ 0. Let ψ : Σn → M n+1 be a two-sided hypersurface with bounded mean curvature H and H2 bounded from below. Then, the Ricci curvature Ric of Σn is bounded from below. Proof. From the Gauss equation (2.2), taking a local orthonormal frame {E1, . . . ,En} tangent to Σn, we have that the Ricci curvature Ric of Σn is given by Ric(X,X) = n∑ i=1 〈R(X,Ei)X,Ei〉 + nH〈AX,X〉−〈AX,AX〉, (3.3) for all vector field X tangent to Σn. Now, we observe that, for each i = 1, . . . ,n, (2.3) implies that 〈X∗,X∗〉〈E∗i ,E ∗ i 〉 = 〈X −〈X,∇h〉K,X −〈X,∇h〉K〉〉〈Ei −〈Ei,∇h〉K,Ei −〈Ei,∇h〉K〉 = |X|2 −ρ2|X|2〈Ei,∇h〉2 −ρ2〈X,∇h〉2 + ρ4〈X,∇h〉2〈Ei,∇h〉2 and 130 e.l. de lima, h.f. de lima, e.a. lima jr, a.a. medeiros 〈X∗,E∗i 〉 2 = 〈X −〈X,∇h〉K,Ei −〈Ei,∇h〉K〉2 = 〈X,Ei〉2 − 2ρ2〈X,∇h〉〈X,Ei〉〈Ei,∇h〉 + ρ4〈X,∇h〉2〈Ei,∇h〉2. Consequently, we get n∑ i=1 〈X∗,X∗〉〈E∗i ,E ∗ i 〉−〈X ∗,E∗i 〉 2 = (n− 1)|X|2 −ρ2|X|2|∇h|2 − (n− 2)ρ2〈X,∇h〉2 ≤ (n− 1)|X|2. Hence, taking into account our constraint on the sectional curvature of Mn, we obtain n∑ i=1 KM (X ∗,E∗i ) ( 〈X∗,X∗〉〈E∗i ,E ∗ i 〉−〈X ∗,E∗i 〉 2 ) ≥−(n− 1)κ|X|2. (3.4) On the other hand, we have that nH〈AX,X〉−〈AX,AX〉≥−|A|(|nH| + |A|)|X|2 for all tangent vector field X ∈ X(Σ). Since ρ is concave, it follows from (3.2), (3.3) and (3.4) that Ric(X,X) ≥− ( (n− 1)κ + |A|(|nH| + |A|) ) |X|2. Therefore, taking into account relation (2.5), our hypothesis on H and H2 assure that the Ricci curvature Ric of Σn is bounded from below. 4. Main results In this section, we present our main results concerning the characterization of minimal (and, in particular, totally geodesic) complete two-sided hypersur- faces immersed in a Killing warped product. So, we state and prove our first one. Theorem 1. Let M n+1 = Mn ×ρ R be a Killing warped product with concave warping function ρ and whose base (not necessarily complete) Mn has nonnegative sectional curvature KM . Let ψ : Σ n → Mn+1 be a complete two-sided hypersurface with constant mean curvature H and H2 bounded from below. Suppose that the angle function Θ of Σn is bounded away from zero. Then, Σn is minimal. Moreover, if H2 is constant, then Σ n is totally geodesic. characterizations of minimal hypersurfaces 131 Proof. Firstly, since we are assuming that Θ is bounded away from zero, for an appropriated choice of N we can suppose that Θ > 0 and, consequently, inf Σ Θ > 0. Then, taking into account Proposition 1, we can apply Lemma 1 to guarantee the existence of a sequence of points {pk}⊂ Σn such that lim k Θ(pk) = inf Σ Θ and lim inf k ∆Θ(pk) ≥ 0 . On the other hand, from Corollary 7.43 of [12] we get Ric(N,N) = Ric(N∗,N∗) + Ric(N⊥,N⊥) (4.1) = RicM (N ∗,N∗) − 1 ρ HessM ρ(N ∗,N∗) −〈N⊥,N⊥〉 ∆Mρ ρ = RicM (N ∗,N∗) − 1 ρ HessM ρ(N ∗,N∗) − Θ2 ρ3 ∆Mρ, where HessM and ∆M are the Hessian and the Laplacian on M n, respectively. Thus, from (2.4) and (4.1) we obtain the following formula ∆Θ = − ( RicM (N ∗,N∗) − 1 ρ HessM ρ(N ∗,N∗) − Θ2 ρ3 ∆Mρ + |A|2 ) Θ . (4.2) Since ρ is concave, from (4.2) we have that ∆Θ ≤− ( RicM (N ∗,N∗) + |A|2 ) Θ . (4.3) So, taking into account relation (2.5) jointly with the hypothesis that H is constant and H2 is bounded from below, it follows from (4.3) that 0 ≤ lim inf k ∆Θ(pk) ≤− lim k ( RicM (N ∗,N∗) + |A|2 ) Θ(pk) ≤− lim k ( RicM (N ∗,N∗) + nH2 ) Θ(pk) ≤ 0 . (4.4) Consequently, since RicM is nonnegative and inf Σ Θ > 0, we conclude that H = 0, that is, Σn is minimal. Finally, assuming that H2 is constant, from relation (2.5) we obtain that |A| is also constant. Therefore, returning to (4.4) we get that |A| must be identically zero and, hence, Σn is totally geodesic. It is not difficult to verify that from the proof of Theorem 1 we obtain the following result: 132 e.l. de lima, h.f. de lima, e.a. lima jr, a.a. medeiros Corollary 1. Let M n+1 = Mn ×ρ R be a Killing warped product with concave warping function ρ and whose base (not necessarily complete) Mn has nonnegative sectional curvature KM . Let ψ : Σ n → Mn+1 be a complete two-sided hypersurface with constant mean curvature H and H2 ≥ 0 (not necessarily constant). Suppose that the angle function Θ of Σn is bounded away from zero. Then, Σn is totally geodesic. Proceeding, we consider the case that the sectional curvature of the Rie- mannian base of the ambient space can be negative. In order to obtain our next result we need to assume a suitable constraint on the norm of gradient of the height function. More precisely, we get the following result: Theorem 2. Let M n+1 = Mn ×ρ R be a Killing warped product with concave warping function ρ and whose base (not necessarily complete) Mn has sectional curvature satisfying KM ≥ −κ, for some constant κ > 0. Let ψ : Σn → Mn+1 be a complete two-sided hypersurface with constant mean curvature H and H2 bounded from below. Suppose that the angle function Θ of Σn is bounded away from zero. If the height function of Σn satisfies |∇h|2 ≤ α (n− 1)κρ2 |A|2, (4.5) for some constant 0 < α < 1, then Σn is minimal. Moreover, if H2 is constant, then Σn is a slice and Mn is complete. Proof. As in the the proof of Theorem 1, we can choose N such that inf Σ Θ > 0. Then, taking into account our constraint on KM , it follows from (4.3) that ∆Θ ≤ ( (n− 1)κρ2|∇h|2 −|A|2 ) Θ (4.6) Using our hypothesis (4.5), from (4.6) we have that ∆Θ ≤ (α− 1)|A|2Θ . (4.7) Thus, in a similar way of the proof of Theorem 1, it follows from (4.7) that there exists a sequence of points {pk}⊂ Σn such that 0 ≤ lim inf k ∆Θ(pk) ≤ lim k ( (α− 1)|A|2Θ ) (pk) = (α− 1) inf Σ Θ lim k |A|2(pk) ≤ 0 . (4.8) Hence, from (4.8) we obtain that limk |A|2(pk) = 0. Consequently, since H is constant and (2.5) implies that nH2 ≤ |A|2, we conclude that H = 0, characterizations of minimal hypersurfaces 133 that is, Σn is minimal. Assuming that H2 is constant, using again rela- tion (2.5) we obtain that |A| must be identically zero. Therefore, using once more hypothesis (4.5) we get that Σn is a slice and, in particular, Mn is complete. From the proof of Theorem 2 we also get the following consequence: Corollary 2. Let M n+1 = Mn ×ρ R be a Killing warped product with concave warping function ρ and whose base (not necessarily complete) Mn has sectional curvature satisfying KM ≥ −κ, for some constant κ > 0. Let ψ : Σn → Mn+1 be a complete two-sided hypersurface with constant mean curvature H and H2 ≥ 0 (not necessarily constant). Suppose that the angle function Θ of Σn is bounded away from zero. If the height function of Σn satisfies condition (4.5), then Σn is a slice and Mn is complete. We close our paper discussing the plausibility of the assumptions assumed in Theorems 1 and 2. Remark 1. We observe that Theorem 1 does not hold when the base of the ambient space has negative sectional curvature and that hypothesis (4.5) in Theorem 2 cannot be extended for α = 1. Indeed, let H2 = {(x,y) ∈ R2 : y > 0} be the 2-dimensional hyperbolic space endowed with its canonical complete metric 〈 , 〉H2 = 1 y2 ( dx2 + dy2 ) and let u : H2 → R be the smooth function given by u(x,y) = a ln y, where a ∈ R\{0}. Let us consider the entire vertical graph Σ(u) = {(x,y,u(x,y)) : y > 0}⊂ H2 ×R. According to Example 10 in [8], Σ(u) has constant mean curvature H = a 2(1 + a2)1/2 and H2 = 0. Moreover, its angle function is given by Θ = 1 (1 + a2)1/2 > 0 and its height function h satisfies |∇h|2 = |Du|2H2 1 + |Du|2H2 = a2 1 + a2 = |A|2. 134 e.l. de lima, h.f. de lima, e.a. lima jr, a.a. medeiros Acknowledgements The authors would like to thank the referee for his/her valuable sug- gestions and useful comments which improved the paper. 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Spruck, The half-space property and entire positive minimal graphs in M × R, J. Diff. Geom. 95 (2013), 321 – 336. [15] S.T. Yau, Harmonic functions on complete Riemannian manifolds, Comm. Pure Appl. Math. 28 (1975), 201 – 228. Introduction Killing warped products Auxiliary results Main results