ENTIRE SPACELIKE H -GRAPHS IN LORENTZIAN PRODUCT SPACES

: In this work we establish suﬃcient conditions to ensure that an entire spacelike graph immersed with constant mean curvature in a Lorentzian product space, whose Riemannian ﬁber has sectional curvature bounded from below, must be a trivial slice of the ambient space.


Introduction and statements of the results
The last few decades have seen a steadily growing interest in the study of the geometry of spacelike hypersurfaces immersed in a Lorentzian space.Apart from physical motivations, from the mathematical point of view this is mostly due to the fact that such hypersurfaces exhibit nice Bernstein-type properties, and one can truly say that the first remarkable results in this branch were the rigidity theorems of Calabi in [10] and Cheng and Yau in [11], who showed (the former for n ≤ 4, and the latter for general n) that the only maximal (that is, with zero mean curvature) complete noncompact spacelike hypersurfaces of the Lorentz-Minkowski space L n+1 are the spacelike hyperplanes.However, in the case that the mean curvature is a positive constant, Treibergs [20] astonishingly showed that there are many entire solutions of the corresponding constant mean curvature equation in L n+1 , which he was able to classify by their projective boundary values at infinity.
On the other hand, Xin [21] and Aiyama [1], working independently, characterized spacelike hyperplanes as the only complete constant mean curvature spacelike hypersurfaces in L n+1 whose Gauss mapping image is contained in a geodesic ball of the n-dimensional hyperbolic space.Later on, Aledo and Alías [6], among other results, showed that a complete constant mean curvature spacelike hypersurface which lies between two parallel spacelike hyperplanes of L n+1 must be, in fact, a spacelike hyperplane.
It is also natural to treat these same questions in a wide class of Lorentzian manifolds.When the ambient space is a Lorentzian product space, Salavessa [19] considered spacelike graphs in −R×M n and, under the assumption that the Cheeger constant of the fiber M n is zero and some conditions on the second fundamental form at infinity, she concluded that if the spacelike graph has parallel mean curvature then the graph must be maximal.When M n is the hyperbolic space H n , for any constant c ∈ R the author described an explicit foliation of −R × H n by hypersurfaces with constant mean curvature c.Meanwhile, Albujer [2] obtained new explicit examples of complete and non-complete entire maximal spacelike graphs in −R × H 2 .
Afterwards, Albujer and Alías [3] established Calabi-Bernstein results for maximal spacelike surfaces immersed into a Lorentzian product space −R × M 2 .In particular, when M 2 is a Riemannian surface with nonnegative Gaussian curvature, they proved that any complete maximal spacelike surface in −R × M 2 must be totally geodesic.Besides, assuming that the fiber M 2 is non-flat, the authors concluded that it must be a slice {t} × M 2 .In [14], Li and Salavessa generalized such results of [3] to higher dimension and codimension.
In [5], the first author jointly with Albujer and Camargo established uniqueness results concerning complete spacelike hypersurfaces with constant mean curvature immersed in −R × H n .Next, Albujer and Alías [4] obtained some parabolicity criteria for maximal surfaces immersed into a Lorentzian product space −R × M 2 , where M 2 is supposed to have nonnegative Gaussian curvature.As an application of their main result, they deduced that every maximal graph over a starlike domain Ω ⊂ M 2 is parabolic.This allowed them to give an alternative proof of the nonparametric version of the Calabi-Bernstein theorem for entire maximal graphs in such ambient space.Later, the first author jointly with Parente [13] obtained a lower estimate of the index of relative nullity of complete maximal spacelike hypersurfaces immersed in a so-called Robertson-Walker spacetime and, in particular, we also proved a sort of weak extension of the Calabi-Bernstein theorem in Lorentzian product spaces.More recently, the authors [12] applied some generalized maximum principles in order to establish uniqueness results concerning complete spacelike hypersurfaces with constant mean curvature in −R × M n , extending the results of [5].
Motivated by these works described above, in this article we deal with entire spacelike graphs Σ(u) = {(u(x), x); x ∈ M n } with constant mean curvature in a Lorentzian product space −R × M n , whose Riemannian fiber M n has sectional curvature bounded from below.According to the current literature, since the mean curvature H of Σ(u) is supposed to be constant, we call Σ(u) an entire spacelike H-graph.In this setting, we obtain the following Calabi-Bernstein type result: Theorem 1.Let M n+1 = −R×M n be a Lorentzian product space, such that the sectional curvature K M of its Riemannian fiber M n satisfies K M ≥ −κ, for some positive constant κ.Let Σ(u) be an entire spacelike H-graph over M n , with u bounded and H 2 bounded from below.If Here, H 2 = 2 n(n−1) S 2 is the mean value of the second elementary symmetric function S 2 on the eigenvalues of the shape operator A of Σ(u), Du stands the gradient of the smooth function u : M n → R in M n and |Du| M its norm, both with respect to the metric of M n .
In the context of Lorentzian product spaces, we note that our restriction on the sectional curvature K M of the fiber M n in Theorem 1 is a weaker restriction when compared with the so-called null (timelike) convergence condition, which means that the Ricci curvature of the ambient space is nonnegative on null or lightlike (timelike) directions (for a thorough discussion about such convergence conditions, see for example [7,8,9,15]).Furthermore, through the example described in Remark 3, we see that Theorem 1 is sharp in the sense that it does not hold when the function u is unbounded.
The proof of Theorem 1 is given in Section 3. From Theorem 1 jointly with Theorem 3.3 of [3], it is not difficult to see that we also get the following result, where 1-maximal means that H 2 vanishes identically on the graph: We observe that, when the ambient space is the Lorentz-Minkowski space L n+1 , Corollary 1 reads as follows: Corollary 2. The only bounded entire 1-maximal spacelike H-graphs over a spacelike hyperplane of L n+1 are the spacelike hyperplanes.

Preliminaries
In what follows, we deal with a spacelike hypersurface Σ n immersed into an (n + 1)-dimensional Lorentzian product space M n+1 of the form R × M n , where M n is an n-dimensional connected Riemannian manifold and M n+1 is endowed with the Lorentzian metric , where π R and π M denote the canonical projections from R × M onto each factor, and , M is the Riemannian metric on M n .
For simplicity, we will just write In this setting, for each fixed t 0 ∈ R, we say that M n t0 = {t 0 } × M n is a slice, which is a totally geodesic spacelike hypersurface of M n+1 .We recall that a smooth immersion ψ : Σ n → −R×M n of an n-dimensional connected manifold Σ n is said to be a spacelike hypersurface if the induced metric via ψ is a Riemannian metric on Σ n , which, as usual, is also denoted for , . Since , is a unitary timelike vector field globally defined on the ambient spacetime, then there exists a unique timelike unitary normal vector field N globally defined on the spacelike hypersurface Σ n which is in the same time-orientation as ∂ t .By using Cauchy-Schwarz inequality, we get N, ∂ t ≤ −1 on Σ n .We will refer to that normal vector field N as the future-pointing Gauss map of the spacelike hypersurface Σ n .
Let ∇ and ∇ denote the Levi-Civita connections in −R×M n and Σ n , respectively.Then the Gauss and Weingarten formulas for the spacelike hypersurface ψ : Σ n → −R × M n are given by for every tangent vector fields X, Y ∈ X(Σ).Here A : X(Σ) → X(Σ) stands for the shape operator (or Weingarten endomorphism) of Σ n with respect to the future-pointing Gauss map N .
As in [17], the curvature tensor R of the spacelike hypersurface Σ n is given by where [ ] denotes the Lie bracket and X, Y, Z ∈ X(Σ).Another fact well known is that the curvature tensor R of the spacelike hypersurface Σ n can be described in terms of the shape operator A and the curvature tensor R of the ambient spacetime −R × M n by the so-called Gauss equation given by for every tangent vector fields X, Y, Z ∈ X(Σ).Now, we consider two particular functions naturally attached to a spacelike hypersurface Σ n immersed into a Lorentzian product space −R × M n , namely, the (vertical) height function h = (π R )| Σ and the support function N, ∂ t , where we recall that N denotes the futurepointing Gauss map of Σ n and ∂ t is the coordinate vector field induced by the universal time on −R × M n .
Let us denote by ∇ and ∇ the gradients with respect to the metrics of −R × M n and Σ n , respectively.Then, a simple computation shows that the gradient of π R on −R × M n is given by where ( ) denotes the tangential component of a vector field in X(M n+1 ) along Σ n .Thus, we get (2.5) where | | denotes the norm of a vector field on Σ n .Since ∂ t is parallel on −R × M n , we have that for every tangent vector field X ∈ X(Σ).Writing ∂ t = −∇h − N, ∂ t N along the hypersurface Σ n and using formulas (2.1) and (2.2), from (2.4) and (2.6) we get that for every tangent vector field X ∈ X(Σ).Therefore, from (2.7) we obtain that the Laplacian on Σ n of its height function h is given by where H = − 1 n tr(A) denotes the mean curvature of Σ n with respect to its future-pointing Gauss mapping N .
Moreover, from (2.4) and (2.6) we also have that for all X ∈ X(Σ).Thus, Supposing that Σ n is a constant mean curvature spacelike hypersurface of −R × M n , as a particular case of Corollary 8.2 in [7] we also obtain that the Laplacian on Σ n of its support function N, ∂ t is given by where Ric M is the Ricci curvature of the fiber M n , N * = N + N, ∂ t ∂ t is the projection of N onto M n and |A| stands for the Hilbert-Schmidt norm of the shape operator A of Σ n .

Proof of Theorem 1
Let −R × M n be a Lorentzian product space.We recall that an entire graph over the fiber M n is determined by a smooth function u ∈ C ∞ (M ) and it is given by The metric induced on M n from the Lorentzian metric on the ambient space via Σ(u) is Remark 1.It can be easily seen from (3.1) that an entire graph Σ(u) is a spacelike hypersurface if, and only if, |Du| 2 M < 1.Note that, when the fiber M n is simply connected, every complete spacelike hypersurface in −R × M n is an entire graph in such space (see, for instance, Lemma 3.1 of [3]).However, according to the examples of non-complete entire maximal graphs in −R × H 2 due to Albujer in Section 3 of [2], we see that an entire spacelike graph in a Lorentzian product space is not necessarily complete, in the sense that the induced Riemannian metric (3.1) is not necessarily complete.
If Σ(u) is an entire graph over the fiber M n , with a straightforward computation we verify that the vector field defines the future-pointing Gauss map of Σ(u).
Let us study the shape operator A of Σ n (u) with respect its orientation given by (3.2).For any X ∈ X(Σ(u)), since X = X * − Du, X * M ∂ t , we have that (3.3), and with aid of Proposition 7.35 of [17], we verify that Du, where D denotes the Levi-Civita connection in M n with respect to the metric , M .
From (3.4) we obtain that the mean curvature of Σ(u) is given by where Div stands for the divergence operator on M n with respect to the metric , M .In order to prove Theorem 1, we will need two key lemmas.The first one gives a suitable lower estimate for the Ricci curvature of a spacelike hypersurface immersed in −R × M n .Lemma 1.Let Σ n be a spacelike hypersurface immersed in a Lorentzian product space −R × M n , whose sectional curvature K M of its fiber M n verifies K M ≥ −κ for some positive constant κ.Then, for all X ∈ X(Σ), the Ricci curvature of Σ n satisfies the following inequality Proof: let us consider X ∈ X(Σ) and a local orthonormal frame {E 1 , . . ., Moreover, we have that On the other hand, since , and ∇h = −∂ t , with a straightforward computation we see that Therefore, since we are supposing that K M ≥ −κ for some positive constant κ, we obtain which jointly with (3.7) yields (3.6).
The second auxiliary lemma is the well known generalized maximum principle due to Omori [16] and Yau [22], which is quoted below.Lemma 2. Let Σ n be an n-dimensional complete Riemannian manifold whose Ricci curvature is bounded from below and ϑ be a smooth function on Σ n which is bounded from below.Then, for each ε > 0 there exists a point Now, we are in position to present the proof of Theorem 1.
Proof of Theorem 1: Observe first that, under the assumptions of the theorem, Σ(u) is indeed a complete spacelike hypersurface.In fact, from (3.1) and the Cauchy-Schwarz inequality we get On the other hand, we have that the Hilbert-Schmidt norm of the shape operator A of Σ(u) satisfies the following algebraic identity (3.9) Thus, since H is constant and H 2 is supposed to be bounded from below, from (3.9) it holds that sup p∈Σ(u) |A p | 2 < +∞.So, from (1.1) we see that there exists a constant 0 < α < 1 such that |Du| M ≤ α.Hence, from (3.8) we get This implies that L ≥ √ cL M , where L and L M denote the length of a curve on Σ(u) with respect to the Riemannian metrics , and , M , respectively, and c = 1 − α 2 .As a consequence, since we are supposing that M n is complete, then the induced metric on Σ(u) from the metric of −R × M n is also complete.Now, let us consider on Σ(u) the functions η = 1 − e −ku , with k ∈ N, and W = 1 − |Du| 2 M .Since we are supposing that u is bounded, we have that the function ϑ = ηW is bounded from below.On the other hand, since H is constant and taking into account hypothesis (1.1) jointly with (3.12), from Lemma 1 we have that the Ricci curvature of Σ(u) is also bounded from below.Hence, we can apply Lemma 2 to the function ϑ, obtaining a sequence of points {p k,ε } in Σ(u) such that, for each fixed k > 0, Computing ∆ϑ we obtain Therefore, since W ∇η = ∇ϑ − η∇W , from (3.10) we get On the other hand, since N = − N, ∂ t ∂ t + N * where N * denotes the projection of N onto the fiber M n , from equation (2.4) it is not difficult to see that N * = − N, ∂ t ∇u and |∇u| 2 = N * , N * M .Here, we are taking into account that the height function h of Σ(u) is nothing but the function u regarded as a function on Σ(u).Thus, from (3.2) we obtain that (3.12) Consequently, from (2.5) and (3.12) we have that Hence, taking into account that we can use formula So, using Cauchy-Schwartz inequality in (3.15), we obtain that On the other hand, since we are assuming that K M ≥ −κ for some positive constant κ, we have But, from (1.1) and (3.12) it holds that .
Consequently, from (3.17) we obtain Furthermore, up to translation, we can assume u > 0 and, hence, we have that η > 0 on Σ(u).Therefore, from (3.16) ).We claim that ∇W is also bounded.Indeed, from (3.13) we have that Hence, from (2.9) and (3.23) we get Thus, letting ε → 0 in (3.22) and taking the lim sup on ε, we obtain the following estimate Since these sequences are minimizing, by Lemma 2 on an arbitrary point we have the ensuing where η * = inf Σ(u) η and u * = inf Σ(u) u.Without loss of generality, denoting u * = sup Σ(u) u, we can suppose that u * ≥ u ≥ u * > 0. Thus, Since ε does not appear in the left hand side of (3.27), we can take lim sup ε→0 on both sides of (3.27) obtaining In an analogous way, taking lim sup k→∞ on (3.28), we finally conclude that |Du| 2 M = 0 on Σ(u), that is, u ≡ t 0 for some t 0 ∈ R. Remark 2. We recall that the Cheeger constant b(M ) of a complete Riemannian manifold M n is given by b , where D ranges over all open submanifolds of M n with compact closure in M n and smooth boundary, and where V (D), A(∂D) are the volume of D resp. the area of ∂D, relative to the metric of M n .

3
(3.24) by 1 k and making k → ∞ as we take the lim sup over k we get the next (

2 M
, ν dA ≤ n (n − 1)κ HA(∂D),where ν is the outward unit normal of ∂D.Yielding the following lower estimate for the Cheeger constant of the fiber M n n(n − 1)κ ≤ b(M ).
1, entire spacelike H-graph is, in fact, a slice and therefore Ric(∂ t , ∂ t ) = 0 and |A| 2 ≡ 0. Hence, in this case, from (3.29) we see that such graph is stable.Remark 3.According to Example 4.4 of [12], taking 0 < |a| < 1, we have that the entire vertical graphΣ(u) = {(a ln y, x, y); y > 0} ⊂ −R × H 2 is such that |Du| 2 H 2 = |a| 2 and, hence, Σ(u) is a complete spacelike surface in −R × H 2 .Moreover, with a straightforward computation we verify that Σ 2 (u) has constant mean curvature H = − a Hence, we conclude that Theorem 1 does not hold when the function u is unbounded.Furthermore, since N, ∂ t is constant on Σ(u), from formula (2.10) and taking into account equations (3.12) and (3.30), we get(3.31)∆N, ∂ t = (|A| 2 − |∇h| 2 ) N, ∂ t = 0.Consequently, according to the stability criteria given in (3.29), from equation (3.31) we also conclude that Σ(u) constitutes a nontrivial example of stable surface in −R × H 2 .Therefore, concerning the context of Theorem 1, we see that the stability of the entire spacelike H-graph cannot alone guarantee the uniqueness result.