# Complete vertical graphs with constant mean curvature in semi-riemannian warped products.

1 Introduction

This paper deals with complete non-compact constant mean curvature graphs over a horosphere of theHyperbolic space, aswell as over horizontal hyperplanes (slices) in the Steady State space. In connection with our work, L.J. Alias and [[bar.M].sup.n+1]. Dajczer (cf. [2]) studied properly immersed complete surfaces of the 3--dimensional Hyperbolic space contained between two horospheres, obtaining a Bernsteintype result for the case of constant mean curvature between 11 and 1. In de Sitter space, K. Akutagawa (cf. [4]) proved that complete spacelike hypersurfaces having constant mean curvature in a specific interval of the real line are totally umbilical. Also for de Sitter space, among other interesting results S. Montiel (cf. [15]) proves that, under an appropriate restriction on their hyperbolic Gauss map, complete spacelike hypersurfaces of constant mean curvature greater than or equal to 1 must actually have mean curvature 1. More recently, A.L. Albujer and L.J. Alias (cf. [1]) have proved that if a hypersurface is bounded away from the infinity of the Steady State space [[??].sup.n+1], then its mean curvature must be identically 1. As a consequence of this result, they concluded that the only complete spacelike surfaces with constant mean curvature in [H.sup.3] which are bounded away from the infinity are the totally umbilical flat surfaces.

For the Lorentz case, our motivation to restrict attention to the Steady State space comes from the fact that there exists a natural duality between the Gauss maps of Riemannian hypersurfaces of this space and those of the Hyperbolic space, provided we model these as hyperquadrics of the Lorentz-Minkowski space (cf. Section 5). Besides, in physical context the Steady State space appears naturally as an exact solution for the Einstein equations, being a cosmological model where matter is supposed to travel along geodesics normal to horizontal hyperplanes; these, in turn, serve as the initial data for the Cauchy problem associated to those equations (cf. [7], Chapter 5).

In this work we model both our ambient spaces as semi-Riemannian warped products to obtain necessary conditions for the existence of the types of graphs mentioned in the beginning of this introduction. More precisely, under appropriate restrictions on the values of the mean curvature and the growth of the height function of these graphs, we actually prove that the mean curvature has to be identically 1 (cf. Theorem 4.1 and Theorem 5.1). We also prove (under a slightly stronger hypothesis in the hyperbolic case) that the scalar curvature of our graphs cannot be globally bounded away from zero in a certain sense. The analytical framework we use to prove the above-mentioned results consists of the generalizedmaximumprinciple of Omori and Yau. Specifically, we apply Lemma 3 of [4] on nonnegative solutions to the partial differential inequality [Delta]g [is greater than or equal to] [ag.sup.2] (a being a positive real constant) to a carefully chosen combination of functions naturally attached to our immersions.

In dimension 2, for complete surfaces of non-negative Gaussian curvature, we are able to obtain Bernstein-type theorems related to our previous general results by using the fact that those surfaces are parabolic in the sense of Riemann surfaces (cf. [8]). Indeed, if the size of the gradient of the height function of the graph is suitably bounded, then the graph has to be a horosphere in the 3--dimensional Hyperbolic space (cf. Theorem 5.2), or a horizontal plane in the 3--dimensional Steady State space (cf. Theorem 4.5).

This paper is organized in the following manner: in Section 2 we discuss general semi-Riemannian manifolds furnished with conformal vector fields, and derive a formula for the Laplacian of a support-like function associated to an oriented Riemannian hypersurface of such an ambient space. Section 3 recasts the result of the previous one in the particular context of semi-Riemannian warped products with Riemannian fiber; we also compute the Laplacian of a general height function and close the section by defining the objects of our main interest, namely, vertical graphs over fibers of such an warped product. Finally, Sections 4 and 5 are respectively devoted to applications of this general picture to the special cases of the Steady State space and the Hyperbolic space.

2 Conformal vector fields

Let [[bar.M].sup.n+1]

be a connected semi-Riemannian manifold with metric [bar.g] = h <, > of index v [is less than or equal to] 1, and semi-Riemannian connection [bar.[nabla]] For a vector field X [member of] X([[bar.M].sup.n+1]), let [epsilon](X) = <X, X>; X is said to be a unit vector field if [epsilon](X) = [+ or -]1, timelike if [epsilon](X) = -1.

A vector field V on [[bar.M].sup.n+1] is said to be conformal if [L.sub.V] < , > = 2[phi]< , > (2.1)

for some function [phi] [member of] [C.sup.[infinity]]([[bar.M].sup.n+1]), where L stands for the Lie derivative of the metric of [[bar.M].sup.n+1]. The function [phi] is called the conformal factor of V. Since [L.sub.V](X) = [V, X] for all X [member of] X([[bar.M].sup.n+1]), it follows from the tensorial character of [L.sub.V] that V [member of] X([[bar.M].sup.n+1]) is conformal if and only if

MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (2.2)

for all X,Y [member of] X([[bar.M].sup.n+1]). In particular, V is a Killing vector field relatively to [bar.g] if and only if [phi] [equivalent to] 0.

In all that follows, we consider Riemannian immersions [psi] : [[summation].sup.n] [right arrow] [[bar.M].sup.n+1], namely, immersions from a connected, n--dimensional orientable differentiable manifold [summation] into [[bar.M].sup.n+1], such that the induced metric g = [psi]*([bar.g]) turns [summation] into a Riemannian manifold (in the Lorentz case v = 1, we refer to ([summation], g) as a spacelike hypersurface of [[bar.M].sup.n+1]), with Levi-Civita connection [nabla]. We orient [summation] by the choice of a unit normal vector field N on it, let A denote the corresponding shape operator and H = [epsilon](N) tr(A)/n the corresponding mean curvature.

The following proposition appeared for the first time in [17], there in the Riemannian setting. In a joint work with A.B. Barros and A. Brasil (cf. [5]) the first author generalized it to the Lorentz setting. Here we present a unified version of it, together with a proof.

Proposition 2.1. Let [[bar.M].sup.n+1] be semi-Riemannian manifold furnished with a conformal vector field V with conformal factor [phi] : [[bar.M].sup.n+1] [right arrow] R, and [psi] : [[summation].sup.n] [right arrow] [[bar.M].sup.n+1] a Riemannian immersion. If [eta] = <V, N>, then

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (2.3)

where [epsilon] = [epsilon](N), [nabla]H the gradient of H in the metric of [summation], [bar.Ric] is the Ricci tensor of [bar.M] and |A| is the Hilbert-Schmidt norm of A.

Proof. Fix p [member of] [summation] and let {[e.sub.k]} be an orthonormal moving frame on a neighborhood of p in [summation], geodesic at p. Extend the [e.sub.k] to a neighborhood of p in [bar.M], so that ([[bar.[nabla]].sub.N][e.sub.k])(p) = 0, and let

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

Then

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

so that

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (2.4)

Now, differentiating [Ae.sub.k] = [[summation].sub.l] [h.sub.kl][e.sub.l] with respect to [e.sub.k], one gets at p

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (2.5)

Asking further that [Ae.sub.k] = [[lambda].sub.k][e.sub.k] at p (which is always possible), we have at p

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (2.6)

In order to compute the last summand of (2.4), note that the conformality of V gives

<[[bar.[nabla]].sub.N] V,, [e.sub.k]> + <N, [[bar.[nabla]].sub.ek]V> = 0

for all k. Hence, differentiating the above relation in the direction of [e.sub.k], we get

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

However, at p one has

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

and

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

so that

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] 0 (2.7)

at p. On the other hand, since

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

it follows from (2.7) that

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

and hence

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (2.8)

Finally,

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

and

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

so that

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

and it follows from (2.8) that

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (2.9)

Substituting (2.5), (2.6) and (2.9) into (2.4), one gets the desired formula (2.3).

3 Semi-Riemannian warped products Let [M.sup.n] be a connected, n-dimensional oriented Riemannian manifold, I [subset] R an interval and [phi] : I [right arrow] R a positive smooth function. In the product differentiable manifold [[bar.M].sup.n+1] = I x [M.sup.n], let [[pi].sub.I] and [[pi].sub.M] denote the projections onto the I and M factors, respectively.

A particular class of semi-Riemannian manifolds having conformal vector fields is the one obtained by furnishing [bar.M] with the metric

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

where [epsilon] = -1 or [epsilon] = 1 for all p [epsilon] [bar.M] and all v, w [member of] [T.sub.p][bar.M]. Indeed (cf. [13] and [14]), the vector field

V = ([phi] [??] [[pi].sub.I])[partial derivative]t

is conformal and closed (in the sense that its dual 1.form is closed), with conformal factor [phi] = f', where the prime denotes differentiation with respect to t [member of] I. Such a space is called a semi-Riemannian warped product, and in what follows we shall write [[bar.M].sup.n+1] = [epsilon]I [x.sub.f] [M.sup.n] to denote it.

If [psi] : [[summation].sup.n] [right arrow] [[bar.M].sup.n+1] is a Riemannian immersion, with [summation] oriented by the unit vector field N, one obviously has [epsilon] = [epsilon]([[partial derivative].sub.t]) = [epsilon](N). The following result restates Proposition 2.1 in this context, in the spirit of [3].

Proposition 3.1. Let [[bar.M].sup.n+1] = [epsilon]I [x.sub.f] [M.sup.n]. In the notations of proposition 2.1, if [summation] has constant mean curvature H, then

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (3.1)

where Ric denotes the Ricci tensor of M and [N.sup.T] = [([[pi].sub.M]).sub.*]N.

Proof. First of all, [eta] = <V, N> = f <N, [[partial derivative].sub.t]>, and it thus follows from (2.3) that

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

Now, N(f') = [epsilon]f"<N, [[partial derivative].sub.t]> = [epsilon][(f"/f).sub.[eta]]. On the other hand, since N = [N.sub.T + [epsilon]<N, [[partial derivative].sub.t]> [[partial derivative].sub.t], it follows from Corollary 7.43 of [16] that

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

where we used that <[N.sup.T], [N.sup.T]> = [epsilon](1 - <N, [[[partial derivative].sub.t]>.sup.2]) in the last equality above.

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

If [psi] : [[summation].sup.n] [right arrow] [[bar.M].sup.n+1] is a Riemannian immersion as above, we let

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

I denote the height function of [summation] with respect to the unit vector field [[partial derivative].sub.t]. As far as we know, the following proposition appeared for the first time in [3], as a special case of Lemma 4.1; here we present a direct proof of the particular case which is needed for the applications we have in mind.

Proposition 3.2. In the above notation,

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (3.2)

where H denotes the mean curvature of [summation] with respect to N.

Proof. Since [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII], one has

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

where [bar.[nabla]] denotes the gradient with respect to the metric of the ambient space, and [X.sup.T] the tangential component of a vector field X [member of] X([[bar.M].sup.n+1]) in [summation]. Now fix p [member of] M, v [member of] [T.sub.p]M and let A denote the Weingarten map with respect to N. Write v = w + [epsilon]<v, [[partial derivative].sub.t]>[[partial derivative].sub.t], so that w [member of] [T.sub.p]M is tangent to the fiber of [bar.M] passing through p. Therefore, by repeated use of the formulas of item (2) of Proposition 7.35 of [16], we get

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

so that

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

Now, fixing p [member of] [summation] and an orthonormal frame {[e.sub.i]} at [T.sub.p][summation], one gets

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

Let us consider again a semi-Riemannian warped product [[bar.M].sup.n+1] = [epsilon]I [x.sub.f] [M.sup.n. For [t.sub.0] [member of] R, we orient the fiber [[[M.sup.n].sub.t].sub.0] = {[t.sub.0]} x [M.sup.] by using the unit normal vector field [[partial derivative].sub.t]. According to Proposition 1 of [13] (see also Proposition 1 of [14]), [M.sub.[t.sub.0]] has constant mean curvature -[epsilon]f'([t.sub.0])/f([t.sub.0]). We are finally in position to define the objects of our main concern.

Definition 3.3. Let [psi] : [[summation].sup.n] [right arrow] [[bar.M].sup.n+1] be a Riemannian immersion. We say that [summation] is a vertical graph over the fiber [M.sub.[t.sub.0]] if [psi](x) = (u(x), x) for some smooth function u : [M.sub.[t.sub.0]] [right arrow] [0, + [infinity]).

Three remarks are in order. First of all, if we let h denote the height function associated to a vertical graph over the fiber [M.sub.[t.sub.0]] , with corresponding function u : [M.sub.[t.sub.0]] [right arrow] [0, + [infinity]), then one obviously has u = h [??] [psi] - [t.sub.0]. Secondly, in the Lorentz case the condition that [psi] is Riemannian in the above definition amounts to |Du| < 1, where by Du we mean the gradient of u . ? with respect to the metric of M, where i : M [right arrow] [M.sub.[t.sub.0]] is the canonical map (cf. [15], Section 4). At last, our applications in the following sections all deal with semi-Riemannian warped products with warping function f(t) = [e.sup.t]. According to the discussion preceding the above definition, in this setting all fibers have mean curvature -[epsilon] and due to this fact we will assume that our vertical graphs are those over [M.sub.0], i.e., such that u = h [??] [psi] [is greater than or equal to] 0. This agreement clarifies our exposition and does not imply in any loss of generality; indeed, changing u by u + [t.sub.0], all of the arguments to come can be easily adapted to vertical graphs over [M.sub.[t.sub.0]].

4 Vertical graphs in the Steady State space

In this section we consider a particular model of Lorentzian warped product, the Steady State space, namely, the warped product

[[??].sup.n+1] = -R [??] [R.sup.n] (4.1)

In Cosmology, this space corresponds to the steady state model of the universe proposed by Bondi, Gold and Hoyle (cf. [7], p. 126).

An alternative description of the Steady State space [[??].sup.n+1] can be given as follows (cf. [15]; see also [9]). Let h [L.sup.n+2] denote the (n + 2)-dimensional Lorentz-Minkowski space (n [is greater than or equal to] 2), that is, the real vector space [R.sup.n+2], endowed with the Lorentz metric

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

for all v, w [member of] [R.sup.n+2]. We define the (n + 1)-dimensional de Sitter space [[S.sub.1].sup.n+1] as the hyperquadric

[[[summation].sub.1]sup.n+1] = {p [member of] [L.sup.n+2]; <p, p> = 1}

of [L.sup.n+2]. From the above definition it is easy to show that the metric induced from h , i turns [[S.sub.1].sup.n+1] into a Lorentz manifold with constant sectional curvature 1. Moreover, for p [member of] [[S.sub.1].sup.n+1], we have

[T.sub.p][[S.sub.1].sup.n+1] = {v [member of] [L.sup.n+2]; <v, p> = 0}.

Let a [member of] [L.sup.n+2] be a nonzero null vector of the null conewith vertex in the origin, such that <a, [e.sub.n+2]> 0, where [e.sub.n+2] = (0, . . . , 0, 1). It can be shown that the open region

{p [member of] [[S.sub.1].sup.n+1]; <p, a> > 0}

of the de Sitter space [[S.sub.1].sup.n+1] is isometric to [[??].sup.n+1]. Therefore, as a subset of [[S.sub.1].sup.n+1] the boundary of [[??].sup.n+1] is the null hypersurface

{p [member of] [[S.sub.1].sup.n+1]; <p, a> = 0}.

Back to the warped product model of [[??].sup.n+1], if [psi] : [[summation].sup.n] [right arrow] [[??].sup.n+1] is a spacelike hypersurface oriented by the timelike unit vector field N such that <N, [[partial derivative].sub.t]> < 0, the hyperbolic angle [theta] of [psi] is the smooth function [theta] : [psi]([summation]) [right arrow] [0,+[infinity]) such that

cosh [theta] = -<N, [[partial derivative].sub.t]> [is greater than or equal to] 1. (4.2)

In the following result, the right hand side of (4.3) must be interpreted as + [infinity] when cosh [theta] = 1.

Theorem 4.1. Let [psi] : [[summation].sup.n] [right arrow] [[??].sup.n+1] be a complete spacelike vertical graph in the (n+1)-dimensional Steady State space, with constant mean curvature H [is greater than or equal to] 1. If

h [is less than or equal to] -log(cosh [theta] - 1), (4.3)

then:

(a) H = 1 on [summation].

(b) The scalar curvature R of [summation] is non-negative and not globally bounded away from zero.

Proof. Let g : [summation] [right arrow] R be defined by g = -[e.sup.h] - [eta]. It follows easily from (4.2) and the definition of h that g [is greater than or equal to] 0 on [summation]. On the other hand, our hypothesis on the growth of h assures that g [is less than or equal to] 1 on [summation].

A straightforward computation gives us [[Delta]e.sup.h] = [e.sup.h]{[|[nabla]h|.sup.2] + [Delta]h}. Moreover, since the Riemannian fiber of [[??].sup.n+1] is [R.sup.n], by computing the Laplacian of g with the aid of Propositions 3.1 and 3.2 we get

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

Now, let [S.sub.2] denote the second elementary symmetric function on the eigen-values of A, and [H.sub.2] = 2[S.sub.2]/n(n - 1) denote the mean value of [S.sub.2]. Elementary algebra gives

[|A|.sup.2] = [n.sup.2][H.sup.2] - n(n - 1)[H.sub.2],

which put into the above formula gives, after a little more algebra,

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (4.4)

where for the inequality we used that -[eta] [is greater than or equal to] [e.sup.h] [is greater than or equal to] 1.

(a) Suppose, by contradiction, that H > 1. Since 0 [is less than or equal to] g [is less than or equal to] 1 and (from the Cauchy-Schwarz inequality) [H.sup.2] - [H.sub.2] [is greater than or equal to] 0, we get [Delta]g [is greater than or equal to] n(H - 1)[g.sup.2].

Now let [Ric.sub.[summation]] denote the Ricci curvature of [summation]; by applying Gauss' equation, we get the estimate

[Ric.sub.[summation]] [is greater than or equal to] (n - 1) - [n.sup.2][H.sub.2]/4, (4.5)

so that we are in position to apply Lemma 3 of [4] to conclude that g [equivalent to] 0. Thus, [eta] = -[e.sup.h], so that <N, [[partial derivative].sub.t]> [equivalent to] -1, i.e., [psi]([summation]) is a slice of [??]. However, such a slice has constant mean curvature 1, and we arrive at a contradiction. Thus H = 1.

(b) Back to (4.4), we obtain

[Delta]g = n(n - 1)(1 - [H.sub.2]) = R [is greater than or equal to] 0,

where we used Gauss' equation once more to get the last equality, and [H.sub.2] - [H.sub.2] [is greater than or equal to] 0 to get the sign for R.

Hence, if there exists a > 0 such that R = a on [summation], from the above we could derive the inequality

[Delta]g [is greater than or equal to] [[alpha]g.sup.2],

which once more would give us g [equivalent to] 0, so that [psi]([summation]) would also be a slice. However, such a slice is isometric to [R.sup.n], thus having scalar curvature R [equivalent to] 0. We, therefore, have got another contradiction.

Remark 4.2. It is easy to see that hypothesis (4.3) on the growth of h is implied by the estimate

|[nabla]h| [is less than or equal to] [e.sup.-h/2]

for the gradient of h, which in turn is taken as a natural one in the literature (see, for instance, Corollary 16.6 of [6]).

Remark 4.3. As a consequence of Bonnet-Myers Theorem , a complete spacelike hypersurface [psi] : [[summation].sup.n] [right arrow] [[??].sup.n+1] having (not necessarily constant) mean curvature H, such that |H| [is less than or equal to] [??] < 2[square root of n - 1/n] (. constant), has to be compact; in fact, for such a bound on H, equation (4.5) would give us

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

However, since [psi]([summation]) is a graph over [R.sup.n], it cannot be compact. Therefore, since 2[square root of n - 1/n] [is less than or equal to] 1 for n [is greater than or equal to] 2, in a certain sense it is natural to restrict attention to H [is greater than or equal to] 1.

As a consequence of the previous result, we have the following Bernstein-type theorem in [[??].sup.3]:

Theorem 4.4. Let [psi] : [[summation].sub.2][right arrow] [[??].sup.3] be a complete spacelike vertical graph in the 3-dimensional Steady State space, with constant mean curvature H [is greater than or equal to] 1. If

h [is less than or equal to] -log(cosh [theta] - 1),

then [psi]([summation]) is a slice of [[H??].sup.3].

Proof. From the previous result, H = 1 on [summation]. Now apply the main Theorem 1 of [4] and the classification of umbilical hypersurfaces of the de Sitter space (cf. [12], Example 1).

We can also apply the result of Proposition 3.2 to prove yet another Bernstein-type theoremfor complete surfaces (not necessarily graphs) of the 3--dimensional Steady State space.

Theorem 4.5. Let [psi] : [[summation].sub.2] [right arrow] [[??].sup.3] be a Riemannian immersion of a complete surface of nonnegative Gaussian curvature [K.sub.[summation]], with constant mean curvature H [is greater than or equal to] 1. If

[|[nabla]h|.sup.2] [is less than or equal to] [H.sub.2] - 1, (4.6)

then [psi] ([summation]) is a slice of [[??].sup.3].

Proof. By applying the result of Proposition 3.2, we get

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

On the other hand, since [|[nabla]h|.sup.2] = [<N, [[partial derivative].sub.t]>.sup.2] - 1, hypothesis (4.6) is equivalent to

[|[nabla]h|.sup.2] + 1 + H<N, [[partial derivative].sub.t]> [is less than or equal to] 0,

so that the function [e.sup.-h] is a superharmonic positive function on [summation]. However, a classical result due to A. Huber [8] assures that complete surfaces of non-negative Gaussian curvature must be parabolic; therefore, h is constant on [summation], i.e., [psi] ([summation]) is a slice.

Remark 4.6. We observe that, in relation to our Theorem s 4.4 and 4.5, A.L. Albujer and L.J. Alias have recently found another interesting Bernstein-type results in the 3--dimensional Steady State space (cf. [1], Theorem s 3 and 5).

5 Vertical graphs in the Hyperbolic space

In this section, instead of the more commonly used half-space model for the (n + 1)--dimensional Hyperbolic space, we consider the warped product model

[H.sup.n+1] = R [??] [R.sup.n].

An explicit isometry between these two models can be found at [2], from where it can easily be seen that the fibers [M.sub.[t.sub.0]] = {[t.sub.0]} x [R.sup.n] of the warped product model are precisely the horospheres of[H.sup.n+1]. Moreover, according to the last paragraph of section 3, these have constant mean curvature 1 if we take the orientation given by the unit normal vector field N = -[[partial derivative].sub.t].

Another useful model for [H.sup.n+1] is (following the notation of the previous section) the so-called Lorentz model, obtained by furnishing the hyperquadric

{p [member of] [L.sup.n+2]; <p, p> = -1, [p.sub.n+2] > 0}

with the (Riemannian) metric induced by the Lorentz metric of [L.sup.n+2]. In this setting, if a [member of] [L.sup.n+2] denotes a fixed null vector as in the beginning of the previous section, a typical horosphere is

[L.sub.[tau]] = {p [member of] [H.sup.n+1]; <p, a> = [tau]},

where [tau] is a positive real number. A straightforward computation shows that

[[xi].sub.p] = p + 1/[tau]a [member of] [[??].sup.n+1]

is a unit normal vector field along [L.sub.[tau]], with respect to which [L.sub.[tau]] has mean curvature -1 (cf. [11]). Therefore, any isometry [Phi] between the warped product and Lorentz models of [H.sup.n+1] must carry [([[partial derivative].sub.t]).sub.q] to [[Phi].sub.*][([[partial derivative].sub.t])] = [[xi].sub.[Phi]](q).

If [psi] : [[summation].sup.n] [right arrow] [H.sup.n+1] is a vertical graph over [R.sup.n], we orient [summation] by choosing a unit normal vector field N such that [eta] = <N,V> < 0, and hence -[e.sup.h] [is less than or equal to] ? < 0. Following the discussion of the previous paragraph, it is natural to consider the Lorentz Gauss map of [summation] with respect to N as given by

[[summation].sup.n] [right arrow] [[??].sup.n+1]

p [??] -[[Phi].sub.*].([N.sub.p])

We are finally in position to state and prove, in the Hyperbolic setting, analogues of two of the results of the previous section, starting with Theorem 4.1.

Theorem 5.1. Let [summation] be a complete Riemannian manifold with Ricci curvature globally bounded from below, and [Psi] : [[summation].sup.n] [right arrow] [H.sup.n+1] be a vertical graph in the (n+1)-dimensional hyperbolic space, with constant mean curvature 0 [is less than or equal to] H [is less than or equal to] 1. If

h [is less than or equal to] -log(1 + <N, [[partial derivative].sub.t]>), (5.1)

then:

(a) H = 1 on S.

(b) If the closure of the image of the Lorentz Gauss map of [psi] with respect to N is contained in [[??].sup.n+1], then the scalar curvature R of [summation] is non-positive and not globally bounded away from zero.

Proof. Let g : [summation] [right arrow] R be defined by g = [e.sup.h] + [eta]. The definition of h, together with Cauchy-Schwarz inequality, gives us g [is greater than or equal to] 0 on [summation]; on the other hand, our hypothesis on the growth of h assures that g = 1 on S.

A straightforward computation gives us [[Delta]e.sup.h] = [e.sup.h]{|[nabla]h|2 + [Delta]h}. Moreover, since the Riemannian fiber of [H.sup.n+1] is [R.sup.n], by computing the Laplacian of g with the aid of Propositions 3.1 and 3.2 we get

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

Now, let [S.sub.2] denote the second elementary symmetric function on the eigen-values of A, and [H.sub.2] = 2[S.sub.2]/n(n - 1) denote the mean value of [S.sub.2]. Elementary algebra gives

[|A|.sup.2] = [n.sup.2][H.sub.2] . n(n - 1)[H.sub.2],

which put into the above formula gives, after a little more algebra,

[Delta]g = n(1 - H){[e.sup.h] + H[eta]} . n(n - 1)([H.sup.2] - [H.sub.2])[eta]

= n(1 - H)g - n(n - 1)([H.sup.2] - [H.sub.2])[eta]. (5.2)

(a) Suppose, by the sake of contradiction, that H < 1 on [summation]. Since 0 [is less than or equal to] g [is less than or equal to] 1, -[eta] > 0 and [H.sup.2] . [H.sub.2] [is greater than or equal to] 0 (from Cauchy-Schwarz inequality), we get

[Delta]g [is greater than or equal to] n(1 - H)[g.sup.2].

Thus, from our hypothesis on the Ricci curvature of [summation] we are in position to apply Lemma 3 of [4] to conclude that g [equivalent to] 0, which is the same as <N, [[partial derivative].sub.t]> [equivalent to] -1. Therefore, [psi]([summation]) is a horosphere of [H.sup.n+1]. However, such a horosphere has constant mean curvature 1, and we reached a contradiction.

(b) Back to (5.2), we get

[Delta]g = n(n - 1)([H.sub.2] - 1)[eta] = R[eta] [is greater than or equal to] R <N, [[partial derivative].sub.t]>,

where Gauss' equation was applied for the last equality and we used that [eta] < 0 and [H.sub.2] - [H.sup.2] [is less than or equal to] 0 for the last inequality. The condition on the Lorentz Gauss map of [summation] amounts to the existence of a real number [beta] > 0 such that <-N, [[partial derivative].sub.t]> = [beta] on [summation]. Therefore, if there existed a positive real number [alpha] such that R [is less than or equal to] -[alpha] on [summation], we would get from 0 [is less than or equal to] g [is less than or equal to] 1 that

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

so that applying Lemma 3 of [4] once more would give us g [equivalent to] 0. However, horospheres of [H.sup.n+1] are isometric to [R.sup.n], thus having scalar curvature identically 0, which is a contradiction.

We close this paper with an analogue of Theorem 4.5 for the Hyperbolic space. Theorem 5.2. Let [psi] : [[summation].sub.2] [right arrow] [H.sup.3] be a complete vertical graph with nonnegative Gaussian curvature [K.sub.[summation]] and constant mean curvature [square root of 2/2 [is less than or equal to] H [is less than or equal to] 1. If

[|[nabla]h|.sup.2] [is less than or equal to] 1. [H.sup.2], (5.3)

then [psi] ([summation]) is a horosphere of [H.sup.3].

Proof. Once more from Proposition 3.2, we have

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

On the other hand, since [|[nabla]h|.sup.2] = 1 . [<N, [[partial derivative].sub.t]>.sup.2] and <N, [[partial derivative].sub.t]> does not change sign, hypothesis (5.3) is equivalent to

[|[nabla]h|.sup.2] - 1 - H<N, [[partial derivative].sub.t]> [is less than or equal to] 0,

so that [e.sup.-h] is a superharmonic and positive on [[summation].sup.2]. Hence, as in the proof of Theorem 4.5, h is constant on [[summation].sup.2], i.e., [psi]([summation]) is a horosphere.

Remark 5.3. Since Gauss's equation gives

[K.sub.[summation]] = 2[H.sup.2] - 1 - 1/2[|A|.sup.2],

the assumption [K.sub.[summation]] [is greater than or equal to] 0 forces one to restrict attention to the case H [is greater than or equal to] [square root of 2/2] .

Remark 5.4. Under the assumption of properness for [psi], a result similar to the above can be found in [2].

Acknowledgements

This paper is part of the second author's doctoral thesis [10] at the Universidade Federal do Ceara. We would like to thank professors L.J. Alias and A.G. Colares for having shown us their preprint [3], which enabled us to set Propositions 3.1 and 3.2. The first author was partially supported by Funcap/CNPq/PPP. The second author was partially supported by CAPE[summation], Brazil.

Received by the editors January 2008. Communicated by L. Vanhecke.

References

[1] A.L. Albujer L.J. Alias. Spacelike Hypersurfaces with Constant Mean Curvature in the Steady State Space, preprint (http://arxiv.org/abs/0709.4398).

[2] L.J. Alias M. Dajczer. Uniqueness of constant mean curvature surfaces properly immersed in a slab, Comment. Math. Helv. 81, (2006) 653-663.

[3] L.J. Alias A. G. Colares. Uniqueness of spacelike hypersurfaces with constant higher order mean curvature in Generalized Robertson-Walker spacetimes, to appear in Math. Proc. Cambridge Philos. Soc.

[4] K. Akutagawa. On spacelike hypersurfaces with constant mean curvature in the de Sitter space, Math. Z. 196, (1987) 13-19.

[5] A.B. Barros, A. Brasil Jr. A. Caminha. Stability of Spacelike Hypersurfaces in Foliated Spacetimes, to appear in Diff. Geom. and its Applications.

[6] D. Gilbarg N. Trudinger. Elliptic Partial Differential Equations of Second Order, Springer-Verlag, Berlin (1983).

[7] S.W. Hawking G.F.R. Ellis. The Large Scale Structure of Spacetime, Cambridge Univ. Press, Cambridge (1973).

[8] A. Huber. On subharmonic functions and differential geometry in the large, Comment. Math. Helv. 32, (1957) 13-72.

[9] H.F. de Lima. Spacelike hypersurfaces with constant higher order mean curvature in de Sitter space, J. of Geom. and Physics 57 (2007), 967-975.

[10] H.F. de Lima. Spacelike Hypersurfaces with Constant Higher Order Mean Curvature (in portuguese), PhD Thesis, Universidade Federal do Ceara, Brazil, 2007.

[11] R. Lopez S. Montiel. Existence of constant mean curvature graphs in hyperbolic space, Calc. Var. 8, (1999) 177-190.

[12] S. Montiel. An integral inequality for compact spacelike hypersurfaces in de Sitter space and applications to the case of constant mean curvature, Indiana Univ. Math. J. 37, (1988) 909-917.

[13] S. Montiel. Unicity of Constant Mean Curvature Hypersurfaces in Some Riemannian Manifolds, Indiana Univ. Math. J. 48, (1999) 711-748.

[14] S. Montiel. Uniqueness of Spacelike Hypersurfaces of Constant Mean Curvature in foliated Spacetimes, Math. Ann. 314, (1999) 529-553.

[15] S. Montiel. Complete non-compact spacelike hypersurfaces of constant mean curvature in de Sitter Space, J. Math. Soc. Japan 55, (2003) 915-938.

[16] B. O'Neill. Semi-Riemannian Geometry with Applications to Relativity, London, Academic Press (1983).

[17] P.A. Sousa. The Laplacian of a Support-type Function and Applications (in portuguese), Master's Dissertation, Universidade Federal do Ceara, Brazil, 2004.

A. Caminha

Departamento de Matematica, Universidade Federal do Ceara, Fortaleza, Ceara, Brazil. 60455-760. Tel.: +55 085 33669313. Fax: +55 085 33669889. email:antonio.caminha@gmail.com

H. F. de Lima

Departamento de Matematica e Estatistica, Universidade Federal de Campina Grande, Campina Grande, Paraiba, Brazil. 58109-970. Tel.: +55 083 33101161. Fax: +55 083 33101030. email:henrique@dme.ufcg.edu.br

This paper deals with complete non-compact constant mean curvature graphs over a horosphere of theHyperbolic space, aswell as over horizontal hyperplanes (slices) in the Steady State space. In connection with our work, L.J. Alias and [[bar.M].sup.n+1]. Dajczer (cf. [2]) studied properly immersed complete surfaces of the 3--dimensional Hyperbolic space contained between two horospheres, obtaining a Bernsteintype result for the case of constant mean curvature between 11 and 1. In de Sitter space, K. Akutagawa (cf. [4]) proved that complete spacelike hypersurfaces having constant mean curvature in a specific interval of the real line are totally umbilical. Also for de Sitter space, among other interesting results S. Montiel (cf. [15]) proves that, under an appropriate restriction on their hyperbolic Gauss map, complete spacelike hypersurfaces of constant mean curvature greater than or equal to 1 must actually have mean curvature 1. More recently, A.L. Albujer and L.J. Alias (cf. [1]) have proved that if a hypersurface is bounded away from the infinity of the Steady State space [[??].sup.n+1], then its mean curvature must be identically 1. As a consequence of this result, they concluded that the only complete spacelike surfaces with constant mean curvature in [H.sup.3] which are bounded away from the infinity are the totally umbilical flat surfaces.

For the Lorentz case, our motivation to restrict attention to the Steady State space comes from the fact that there exists a natural duality between the Gauss maps of Riemannian hypersurfaces of this space and those of the Hyperbolic space, provided we model these as hyperquadrics of the Lorentz-Minkowski space (cf. Section 5). Besides, in physical context the Steady State space appears naturally as an exact solution for the Einstein equations, being a cosmological model where matter is supposed to travel along geodesics normal to horizontal hyperplanes; these, in turn, serve as the initial data for the Cauchy problem associated to those equations (cf. [7], Chapter 5).

In this work we model both our ambient spaces as semi-Riemannian warped products to obtain necessary conditions for the existence of the types of graphs mentioned in the beginning of this introduction. More precisely, under appropriate restrictions on the values of the mean curvature and the growth of the height function of these graphs, we actually prove that the mean curvature has to be identically 1 (cf. Theorem 4.1 and Theorem 5.1). We also prove (under a slightly stronger hypothesis in the hyperbolic case) that the scalar curvature of our graphs cannot be globally bounded away from zero in a certain sense. The analytical framework we use to prove the above-mentioned results consists of the generalizedmaximumprinciple of Omori and Yau. Specifically, we apply Lemma 3 of [4] on nonnegative solutions to the partial differential inequality [Delta]g [is greater than or equal to] [ag.sup.2] (a being a positive real constant) to a carefully chosen combination of functions naturally attached to our immersions.

In dimension 2, for complete surfaces of non-negative Gaussian curvature, we are able to obtain Bernstein-type theorems related to our previous general results by using the fact that those surfaces are parabolic in the sense of Riemann surfaces (cf. [8]). Indeed, if the size of the gradient of the height function of the graph is suitably bounded, then the graph has to be a horosphere in the 3--dimensional Hyperbolic space (cf. Theorem 5.2), or a horizontal plane in the 3--dimensional Steady State space (cf. Theorem 4.5).

This paper is organized in the following manner: in Section 2 we discuss general semi-Riemannian manifolds furnished with conformal vector fields, and derive a formula for the Laplacian of a support-like function associated to an oriented Riemannian hypersurface of such an ambient space. Section 3 recasts the result of the previous one in the particular context of semi-Riemannian warped products with Riemannian fiber; we also compute the Laplacian of a general height function and close the section by defining the objects of our main interest, namely, vertical graphs over fibers of such an warped product. Finally, Sections 4 and 5 are respectively devoted to applications of this general picture to the special cases of the Steady State space and the Hyperbolic space.

2 Conformal vector fields

Let [[bar.M].sup.n+1]

be a connected semi-Riemannian manifold with metric [bar.g] = h <, > of index v [is less than or equal to] 1, and semi-Riemannian connection [bar.[nabla]] For a vector field X [member of] X([[bar.M].sup.n+1]), let [epsilon](X) = <X, X>; X is said to be a unit vector field if [epsilon](X) = [+ or -]1, timelike if [epsilon](X) = -1.

A vector field V on [[bar.M].sup.n+1] is said to be conformal if [L.sub.V] < , > = 2[phi]< , > (2.1)

for some function [phi] [member of] [C.sup.[infinity]]([[bar.M].sup.n+1]), where L stands for the Lie derivative of the metric of [[bar.M].sup.n+1]. The function [phi] is called the conformal factor of V. Since [L.sub.V](X) = [V, X] for all X [member of] X([[bar.M].sup.n+1]), it follows from the tensorial character of [L.sub.V] that V [member of] X([[bar.M].sup.n+1]) is conformal if and only if

MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (2.2)

for all X,Y [member of] X([[bar.M].sup.n+1]). In particular, V is a Killing vector field relatively to [bar.g] if and only if [phi] [equivalent to] 0.

In all that follows, we consider Riemannian immersions [psi] : [[summation].sup.n] [right arrow] [[bar.M].sup.n+1], namely, immersions from a connected, n--dimensional orientable differentiable manifold [summation] into [[bar.M].sup.n+1], such that the induced metric g = [psi]*([bar.g]) turns [summation] into a Riemannian manifold (in the Lorentz case v = 1, we refer to ([summation], g) as a spacelike hypersurface of [[bar.M].sup.n+1]), with Levi-Civita connection [nabla]. We orient [summation] by the choice of a unit normal vector field N on it, let A denote the corresponding shape operator and H = [epsilon](N) tr(A)/n the corresponding mean curvature.

The following proposition appeared for the first time in [17], there in the Riemannian setting. In a joint work with A.B. Barros and A. Brasil (cf. [5]) the first author generalized it to the Lorentz setting. Here we present a unified version of it, together with a proof.

Proposition 2.1. Let [[bar.M].sup.n+1] be semi-Riemannian manifold furnished with a conformal vector field V with conformal factor [phi] : [[bar.M].sup.n+1] [right arrow] R, and [psi] : [[summation].sup.n] [right arrow] [[bar.M].sup.n+1] a Riemannian immersion. If [eta] = <V, N>, then

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (2.3)

where [epsilon] = [epsilon](N), [nabla]H the gradient of H in the metric of [summation], [bar.Ric] is the Ricci tensor of [bar.M] and |A| is the Hilbert-Schmidt norm of A.

Proof. Fix p [member of] [summation] and let {[e.sub.k]} be an orthonormal moving frame on a neighborhood of p in [summation], geodesic at p. Extend the [e.sub.k] to a neighborhood of p in [bar.M], so that ([[bar.[nabla]].sub.N][e.sub.k])(p) = 0, and let

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

Then

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

so that

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (2.4)

Now, differentiating [Ae.sub.k] = [[summation].sub.l] [h.sub.kl][e.sub.l] with respect to [e.sub.k], one gets at p

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (2.5)

Asking further that [Ae.sub.k] = [[lambda].sub.k][e.sub.k] at p (which is always possible), we have at p

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (2.6)

In order to compute the last summand of (2.4), note that the conformality of V gives

<[[bar.[nabla]].sub.N] V,, [e.sub.k]> + <N, [[bar.[nabla]].sub.ek]V> = 0

for all k. Hence, differentiating the above relation in the direction of [e.sub.k], we get

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

However, at p one has

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

and

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

so that

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] 0 (2.7)

at p. On the other hand, since

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

it follows from (2.7) that

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

and hence

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (2.8)

Finally,

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

and

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

so that

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

and it follows from (2.8) that

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (2.9)

Substituting (2.5), (2.6) and (2.9) into (2.4), one gets the desired formula (2.3).

3 Semi-Riemannian warped products Let [M.sup.n] be a connected, n-dimensional oriented Riemannian manifold, I [subset] R an interval and [phi] : I [right arrow] R a positive smooth function. In the product differentiable manifold [[bar.M].sup.n+1] = I x [M.sup.n], let [[pi].sub.I] and [[pi].sub.M] denote the projections onto the I and M factors, respectively.

A particular class of semi-Riemannian manifolds having conformal vector fields is the one obtained by furnishing [bar.M] with the metric

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

where [epsilon] = -1 or [epsilon] = 1 for all p [epsilon] [bar.M] and all v, w [member of] [T.sub.p][bar.M]. Indeed (cf. [13] and [14]), the vector field

V = ([phi] [??] [[pi].sub.I])[partial derivative]t

is conformal and closed (in the sense that its dual 1.form is closed), with conformal factor [phi] = f', where the prime denotes differentiation with respect to t [member of] I. Such a space is called a semi-Riemannian warped product, and in what follows we shall write [[bar.M].sup.n+1] = [epsilon]I [x.sub.f] [M.sup.n] to denote it.

If [psi] : [[summation].sup.n] [right arrow] [[bar.M].sup.n+1] is a Riemannian immersion, with [summation] oriented by the unit vector field N, one obviously has [epsilon] = [epsilon]([[partial derivative].sub.t]) = [epsilon](N). The following result restates Proposition 2.1 in this context, in the spirit of [3].

Proposition 3.1. Let [[bar.M].sup.n+1] = [epsilon]I [x.sub.f] [M.sup.n]. In the notations of proposition 2.1, if [summation] has constant mean curvature H, then

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (3.1)

where Ric denotes the Ricci tensor of M and [N.sup.T] = [([[pi].sub.M]).sub.*]N.

Proof. First of all, [eta] = <V, N> = f <N, [[partial derivative].sub.t]>, and it thus follows from (2.3) that

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

Now, N(f') = [epsilon]f"<N, [[partial derivative].sub.t]> = [epsilon][(f"/f).sub.[eta]]. On the other hand, since N = [N.sub.T + [epsilon]<N, [[partial derivative].sub.t]> [[partial derivative].sub.t], it follows from Corollary 7.43 of [16] that

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

where we used that <[N.sup.T], [N.sup.T]> = [epsilon](1 - <N, [[[partial derivative].sub.t]>.sup.2]) in the last equality above.

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

If [psi] : [[summation].sup.n] [right arrow] [[bar.M].sup.n+1] is a Riemannian immersion as above, we let

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

I denote the height function of [summation] with respect to the unit vector field [[partial derivative].sub.t]. As far as we know, the following proposition appeared for the first time in [3], as a special case of Lemma 4.1; here we present a direct proof of the particular case which is needed for the applications we have in mind.

Proposition 3.2. In the above notation,

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (3.2)

where H denotes the mean curvature of [summation] with respect to N.

Proof. Since [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII], one has

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

where [bar.[nabla]] denotes the gradient with respect to the metric of the ambient space, and [X.sup.T] the tangential component of a vector field X [member of] X([[bar.M].sup.n+1]) in [summation]. Now fix p [member of] M, v [member of] [T.sub.p]M and let A denote the Weingarten map with respect to N. Write v = w + [epsilon]<v, [[partial derivative].sub.t]>[[partial derivative].sub.t], so that w [member of] [T.sub.p]M is tangent to the fiber of [bar.M] passing through p. Therefore, by repeated use of the formulas of item (2) of Proposition 7.35 of [16], we get

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

so that

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

Now, fixing p [member of] [summation] and an orthonormal frame {[e.sub.i]} at [T.sub.p][summation], one gets

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

Let us consider again a semi-Riemannian warped product [[bar.M].sup.n+1] = [epsilon]I [x.sub.f] [M.sup.n. For [t.sub.0] [member of] R, we orient the fiber [[[M.sup.n].sub.t].sub.0] = {[t.sub.0]} x [M.sup.] by using the unit normal vector field [[partial derivative].sub.t]. According to Proposition 1 of [13] (see also Proposition 1 of [14]), [M.sub.[t.sub.0]] has constant mean curvature -[epsilon]f'([t.sub.0])/f([t.sub.0]). We are finally in position to define the objects of our main concern.

Definition 3.3. Let [psi] : [[summation].sup.n] [right arrow] [[bar.M].sup.n+1] be a Riemannian immersion. We say that [summation] is a vertical graph over the fiber [M.sub.[t.sub.0]] if [psi](x) = (u(x), x) for some smooth function u : [M.sub.[t.sub.0]] [right arrow] [0, + [infinity]).

Three remarks are in order. First of all, if we let h denote the height function associated to a vertical graph over the fiber [M.sub.[t.sub.0]] , with corresponding function u : [M.sub.[t.sub.0]] [right arrow] [0, + [infinity]), then one obviously has u = h [??] [psi] - [t.sub.0]. Secondly, in the Lorentz case the condition that [psi] is Riemannian in the above definition amounts to |Du| < 1, where by Du we mean the gradient of u . ? with respect to the metric of M, where i : M [right arrow] [M.sub.[t.sub.0]] is the canonical map (cf. [15], Section 4). At last, our applications in the following sections all deal with semi-Riemannian warped products with warping function f(t) = [e.sup.t]. According to the discussion preceding the above definition, in this setting all fibers have mean curvature -[epsilon] and due to this fact we will assume that our vertical graphs are those over [M.sub.0], i.e., such that u = h [??] [psi] [is greater than or equal to] 0. This agreement clarifies our exposition and does not imply in any loss of generality; indeed, changing u by u + [t.sub.0], all of the arguments to come can be easily adapted to vertical graphs over [M.sub.[t.sub.0]].

4 Vertical graphs in the Steady State space

In this section we consider a particular model of Lorentzian warped product, the Steady State space, namely, the warped product

[[??].sup.n+1] = -R [??] [R.sup.n] (4.1)

In Cosmology, this space corresponds to the steady state model of the universe proposed by Bondi, Gold and Hoyle (cf. [7], p. 126).

An alternative description of the Steady State space [[??].sup.n+1] can be given as follows (cf. [15]; see also [9]). Let h [L.sup.n+2] denote the (n + 2)-dimensional Lorentz-Minkowski space (n [is greater than or equal to] 2), that is, the real vector space [R.sup.n+2], endowed with the Lorentz metric

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

for all v, w [member of] [R.sup.n+2]. We define the (n + 1)-dimensional de Sitter space [[S.sub.1].sup.n+1] as the hyperquadric

[[[summation].sub.1]sup.n+1] = {p [member of] [L.sup.n+2]; <p, p> = 1}

of [L.sup.n+2]. From the above definition it is easy to show that the metric induced from h , i turns [[S.sub.1].sup.n+1] into a Lorentz manifold with constant sectional curvature 1. Moreover, for p [member of] [[S.sub.1].sup.n+1], we have

[T.sub.p][[S.sub.1].sup.n+1] = {v [member of] [L.sup.n+2]; <v, p> = 0}.

Let a [member of] [L.sup.n+2] be a nonzero null vector of the null conewith vertex in the origin, such that <a, [e.sub.n+2]> 0, where [e.sub.n+2] = (0, . . . , 0, 1). It can be shown that the open region

{p [member of] [[S.sub.1].sup.n+1]; <p, a> > 0}

of the de Sitter space [[S.sub.1].sup.n+1] is isometric to [[??].sup.n+1]. Therefore, as a subset of [[S.sub.1].sup.n+1] the boundary of [[??].sup.n+1] is the null hypersurface

{p [member of] [[S.sub.1].sup.n+1]; <p, a> = 0}.

Back to the warped product model of [[??].sup.n+1], if [psi] : [[summation].sup.n] [right arrow] [[??].sup.n+1] is a spacelike hypersurface oriented by the timelike unit vector field N such that <N, [[partial derivative].sub.t]> < 0, the hyperbolic angle [theta] of [psi] is the smooth function [theta] : [psi]([summation]) [right arrow] [0,+[infinity]) such that

cosh [theta] = -<N, [[partial derivative].sub.t]> [is greater than or equal to] 1. (4.2)

In the following result, the right hand side of (4.3) must be interpreted as + [infinity] when cosh [theta] = 1.

Theorem 4.1. Let [psi] : [[summation].sup.n] [right arrow] [[??].sup.n+1] be a complete spacelike vertical graph in the (n+1)-dimensional Steady State space, with constant mean curvature H [is greater than or equal to] 1. If

h [is less than or equal to] -log(cosh [theta] - 1), (4.3)

then:

(a) H = 1 on [summation].

(b) The scalar curvature R of [summation] is non-negative and not globally bounded away from zero.

Proof. Let g : [summation] [right arrow] R be defined by g = -[e.sup.h] - [eta]. It follows easily from (4.2) and the definition of h that g [is greater than or equal to] 0 on [summation]. On the other hand, our hypothesis on the growth of h assures that g [is less than or equal to] 1 on [summation].

A straightforward computation gives us [[Delta]e.sup.h] = [e.sup.h]{[|[nabla]h|.sup.2] + [Delta]h}. Moreover, since the Riemannian fiber of [[??].sup.n+1] is [R.sup.n], by computing the Laplacian of g with the aid of Propositions 3.1 and 3.2 we get

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

Now, let [S.sub.2] denote the second elementary symmetric function on the eigen-values of A, and [H.sub.2] = 2[S.sub.2]/n(n - 1) denote the mean value of [S.sub.2]. Elementary algebra gives

[|A|.sup.2] = [n.sup.2][H.sup.2] - n(n - 1)[H.sub.2],

which put into the above formula gives, after a little more algebra,

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (4.4)

where for the inequality we used that -[eta] [is greater than or equal to] [e.sup.h] [is greater than or equal to] 1.

(a) Suppose, by contradiction, that H > 1. Since 0 [is less than or equal to] g [is less than or equal to] 1 and (from the Cauchy-Schwarz inequality) [H.sup.2] - [H.sub.2] [is greater than or equal to] 0, we get [Delta]g [is greater than or equal to] n(H - 1)[g.sup.2].

Now let [Ric.sub.[summation]] denote the Ricci curvature of [summation]; by applying Gauss' equation, we get the estimate

[Ric.sub.[summation]] [is greater than or equal to] (n - 1) - [n.sup.2][H.sub.2]/4, (4.5)

so that we are in position to apply Lemma 3 of [4] to conclude that g [equivalent to] 0. Thus, [eta] = -[e.sup.h], so that <N, [[partial derivative].sub.t]> [equivalent to] -1, i.e., [psi]([summation]) is a slice of [??]. However, such a slice has constant mean curvature 1, and we arrive at a contradiction. Thus H = 1.

(b) Back to (4.4), we obtain

[Delta]g = n(n - 1)(1 - [H.sub.2]) = R [is greater than or equal to] 0,

where we used Gauss' equation once more to get the last equality, and [H.sub.2] - [H.sub.2] [is greater than or equal to] 0 to get the sign for R.

Hence, if there exists a > 0 such that R = a on [summation], from the above we could derive the inequality

[Delta]g [is greater than or equal to] [[alpha]g.sup.2],

which once more would give us g [equivalent to] 0, so that [psi]([summation]) would also be a slice. However, such a slice is isometric to [R.sup.n], thus having scalar curvature R [equivalent to] 0. We, therefore, have got another contradiction.

Remark 4.2. It is easy to see that hypothesis (4.3) on the growth of h is implied by the estimate

|[nabla]h| [is less than or equal to] [e.sup.-h/2]

for the gradient of h, which in turn is taken as a natural one in the literature (see, for instance, Corollary 16.6 of [6]).

Remark 4.3. As a consequence of Bonnet-Myers Theorem , a complete spacelike hypersurface [psi] : [[summation].sup.n] [right arrow] [[??].sup.n+1] having (not necessarily constant) mean curvature H, such that |H| [is less than or equal to] [??] < 2[square root of n - 1/n] (. constant), has to be compact; in fact, for such a bound on H, equation (4.5) would give us

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

However, since [psi]([summation]) is a graph over [R.sup.n], it cannot be compact. Therefore, since 2[square root of n - 1/n] [is less than or equal to] 1 for n [is greater than or equal to] 2, in a certain sense it is natural to restrict attention to H [is greater than or equal to] 1.

As a consequence of the previous result, we have the following Bernstein-type theorem in [[??].sup.3]:

Theorem 4.4. Let [psi] : [[summation].sub.2][right arrow] [[??].sup.3] be a complete spacelike vertical graph in the 3-dimensional Steady State space, with constant mean curvature H [is greater than or equal to] 1. If

h [is less than or equal to] -log(cosh [theta] - 1),

then [psi]([summation]) is a slice of [[H??].sup.3].

Proof. From the previous result, H = 1 on [summation]. Now apply the main Theorem 1 of [4] and the classification of umbilical hypersurfaces of the de Sitter space (cf. [12], Example 1).

We can also apply the result of Proposition 3.2 to prove yet another Bernstein-type theoremfor complete surfaces (not necessarily graphs) of the 3--dimensional Steady State space.

Theorem 4.5. Let [psi] : [[summation].sub.2] [right arrow] [[??].sup.3] be a Riemannian immersion of a complete surface of nonnegative Gaussian curvature [K.sub.[summation]], with constant mean curvature H [is greater than or equal to] 1. If

[|[nabla]h|.sup.2] [is less than or equal to] [H.sub.2] - 1, (4.6)

then [psi] ([summation]) is a slice of [[??].sup.3].

Proof. By applying the result of Proposition 3.2, we get

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

On the other hand, since [|[nabla]h|.sup.2] = [<N, [[partial derivative].sub.t]>.sup.2] - 1, hypothesis (4.6) is equivalent to

[|[nabla]h|.sup.2] + 1 + H<N, [[partial derivative].sub.t]> [is less than or equal to] 0,

so that the function [e.sup.-h] is a superharmonic positive function on [summation]. However, a classical result due to A. Huber [8] assures that complete surfaces of non-negative Gaussian curvature must be parabolic; therefore, h is constant on [summation], i.e., [psi] ([summation]) is a slice.

Remark 4.6. We observe that, in relation to our Theorem s 4.4 and 4.5, A.L. Albujer and L.J. Alias have recently found another interesting Bernstein-type results in the 3--dimensional Steady State space (cf. [1], Theorem s 3 and 5).

5 Vertical graphs in the Hyperbolic space

In this section, instead of the more commonly used half-space model for the (n + 1)--dimensional Hyperbolic space, we consider the warped product model

[H.sup.n+1] = R [??] [R.sup.n].

An explicit isometry between these two models can be found at [2], from where it can easily be seen that the fibers [M.sub.[t.sub.0]] = {[t.sub.0]} x [R.sup.n] of the warped product model are precisely the horospheres of[H.sup.n+1]. Moreover, according to the last paragraph of section 3, these have constant mean curvature 1 if we take the orientation given by the unit normal vector field N = -[[partial derivative].sub.t].

Another useful model for [H.sup.n+1] is (following the notation of the previous section) the so-called Lorentz model, obtained by furnishing the hyperquadric

{p [member of] [L.sup.n+2]; <p, p> = -1, [p.sub.n+2] > 0}

with the (Riemannian) metric induced by the Lorentz metric of [L.sup.n+2]. In this setting, if a [member of] [L.sup.n+2] denotes a fixed null vector as in the beginning of the previous section, a typical horosphere is

[L.sub.[tau]] = {p [member of] [H.sup.n+1]; <p, a> = [tau]},

where [tau] is a positive real number. A straightforward computation shows that

[[xi].sub.p] = p + 1/[tau]a [member of] [[??].sup.n+1]

is a unit normal vector field along [L.sub.[tau]], with respect to which [L.sub.[tau]] has mean curvature -1 (cf. [11]). Therefore, any isometry [Phi] between the warped product and Lorentz models of [H.sup.n+1] must carry [([[partial derivative].sub.t]).sub.q] to [[Phi].sub.*][([[partial derivative].sub.t])] = [[xi].sub.[Phi]](q).

If [psi] : [[summation].sup.n] [right arrow] [H.sup.n+1] is a vertical graph over [R.sup.n], we orient [summation] by choosing a unit normal vector field N such that [eta] = <N,V> < 0, and hence -[e.sup.h] [is less than or equal to] ? < 0. Following the discussion of the previous paragraph, it is natural to consider the Lorentz Gauss map of [summation] with respect to N as given by

[[summation].sup.n] [right arrow] [[??].sup.n+1]

p [??] -[[Phi].sub.*].([N.sub.p])

We are finally in position to state and prove, in the Hyperbolic setting, analogues of two of the results of the previous section, starting with Theorem 4.1.

Theorem 5.1. Let [summation] be a complete Riemannian manifold with Ricci curvature globally bounded from below, and [Psi] : [[summation].sup.n] [right arrow] [H.sup.n+1] be a vertical graph in the (n+1)-dimensional hyperbolic space, with constant mean curvature 0 [is less than or equal to] H [is less than or equal to] 1. If

h [is less than or equal to] -log(1 + <N, [[partial derivative].sub.t]>), (5.1)

then:

(a) H = 1 on S.

(b) If the closure of the image of the Lorentz Gauss map of [psi] with respect to N is contained in [[??].sup.n+1], then the scalar curvature R of [summation] is non-positive and not globally bounded away from zero.

Proof. Let g : [summation] [right arrow] R be defined by g = [e.sup.h] + [eta]. The definition of h, together with Cauchy-Schwarz inequality, gives us g [is greater than or equal to] 0 on [summation]; on the other hand, our hypothesis on the growth of h assures that g = 1 on S.

A straightforward computation gives us [[Delta]e.sup.h] = [e.sup.h]{|[nabla]h|2 + [Delta]h}. Moreover, since the Riemannian fiber of [H.sup.n+1] is [R.sup.n], by computing the Laplacian of g with the aid of Propositions 3.1 and 3.2 we get

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

Now, let [S.sub.2] denote the second elementary symmetric function on the eigen-values of A, and [H.sub.2] = 2[S.sub.2]/n(n - 1) denote the mean value of [S.sub.2]. Elementary algebra gives

[|A|.sup.2] = [n.sup.2][H.sub.2] . n(n - 1)[H.sub.2],

which put into the above formula gives, after a little more algebra,

[Delta]g = n(1 - H){[e.sup.h] + H[eta]} . n(n - 1)([H.sup.2] - [H.sub.2])[eta]

= n(1 - H)g - n(n - 1)([H.sup.2] - [H.sub.2])[eta]. (5.2)

(a) Suppose, by the sake of contradiction, that H < 1 on [summation]. Since 0 [is less than or equal to] g [is less than or equal to] 1, -[eta] > 0 and [H.sup.2] . [H.sub.2] [is greater than or equal to] 0 (from Cauchy-Schwarz inequality), we get

[Delta]g [is greater than or equal to] n(1 - H)[g.sup.2].

Thus, from our hypothesis on the Ricci curvature of [summation] we are in position to apply Lemma 3 of [4] to conclude that g [equivalent to] 0, which is the same as <N, [[partial derivative].sub.t]> [equivalent to] -1. Therefore, [psi]([summation]) is a horosphere of [H.sup.n+1]. However, such a horosphere has constant mean curvature 1, and we reached a contradiction.

(b) Back to (5.2), we get

[Delta]g = n(n - 1)([H.sub.2] - 1)[eta] = R[eta] [is greater than or equal to] R <N, [[partial derivative].sub.t]>,

where Gauss' equation was applied for the last equality and we used that [eta] < 0 and [H.sub.2] - [H.sup.2] [is less than or equal to] 0 for the last inequality. The condition on the Lorentz Gauss map of [summation] amounts to the existence of a real number [beta] > 0 such that <-N, [[partial derivative].sub.t]> = [beta] on [summation]. Therefore, if there existed a positive real number [alpha] such that R [is less than or equal to] -[alpha] on [summation], we would get from 0 [is less than or equal to] g [is less than or equal to] 1 that

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

so that applying Lemma 3 of [4] once more would give us g [equivalent to] 0. However, horospheres of [H.sup.n+1] are isometric to [R.sup.n], thus having scalar curvature identically 0, which is a contradiction.

We close this paper with an analogue of Theorem 4.5 for the Hyperbolic space. Theorem 5.2. Let [psi] : [[summation].sub.2] [right arrow] [H.sup.3] be a complete vertical graph with nonnegative Gaussian curvature [K.sub.[summation]] and constant mean curvature [square root of 2/2 [is less than or equal to] H [is less than or equal to] 1. If

[|[nabla]h|.sup.2] [is less than or equal to] 1. [H.sup.2], (5.3)

then [psi] ([summation]) is a horosphere of [H.sup.3].

Proof. Once more from Proposition 3.2, we have

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

On the other hand, since [|[nabla]h|.sup.2] = 1 . [<N, [[partial derivative].sub.t]>.sup.2] and <N, [[partial derivative].sub.t]> does not change sign, hypothesis (5.3) is equivalent to

[|[nabla]h|.sup.2] - 1 - H<N, [[partial derivative].sub.t]> [is less than or equal to] 0,

so that [e.sup.-h] is a superharmonic and positive on [[summation].sup.2]. Hence, as in the proof of Theorem 4.5, h is constant on [[summation].sup.2], i.e., [psi]([summation]) is a horosphere.

Remark 5.3. Since Gauss's equation gives

[K.sub.[summation]] = 2[H.sup.2] - 1 - 1/2[|A|.sup.2],

the assumption [K.sub.[summation]] [is greater than or equal to] 0 forces one to restrict attention to the case H [is greater than or equal to] [square root of 2/2] .

Remark 5.4. Under the assumption of properness for [psi], a result similar to the above can be found in [2].

Acknowledgements

This paper is part of the second author's doctoral thesis [10] at the Universidade Federal do Ceara. We would like to thank professors L.J. Alias and A.G. Colares for having shown us their preprint [3], which enabled us to set Propositions 3.1 and 3.2. The first author was partially supported by Funcap/CNPq/PPP. The second author was partially supported by CAPE[summation], Brazil.

Received by the editors January 2008. Communicated by L. Vanhecke.

References

[1] A.L. Albujer L.J. Alias. Spacelike Hypersurfaces with Constant Mean Curvature in the Steady State Space, preprint (http://arxiv.org/abs/0709.4398).

[2] L.J. Alias M. Dajczer. Uniqueness of constant mean curvature surfaces properly immersed in a slab, Comment. Math. Helv. 81, (2006) 653-663.

[3] L.J. Alias A. G. Colares. Uniqueness of spacelike hypersurfaces with constant higher order mean curvature in Generalized Robertson-Walker spacetimes, to appear in Math. Proc. Cambridge Philos. Soc.

[4] K. Akutagawa. On spacelike hypersurfaces with constant mean curvature in the de Sitter space, Math. Z. 196, (1987) 13-19.

[5] A.B. Barros, A. Brasil Jr. A. Caminha. Stability of Spacelike Hypersurfaces in Foliated Spacetimes, to appear in Diff. Geom. and its Applications.

[6] D. Gilbarg N. Trudinger. Elliptic Partial Differential Equations of Second Order, Springer-Verlag, Berlin (1983).

[7] S.W. Hawking G.F.R. Ellis. The Large Scale Structure of Spacetime, Cambridge Univ. Press, Cambridge (1973).

[8] A. Huber. On subharmonic functions and differential geometry in the large, Comment. Math. Helv. 32, (1957) 13-72.

[9] H.F. de Lima. Spacelike hypersurfaces with constant higher order mean curvature in de Sitter space, J. of Geom. and Physics 57 (2007), 967-975.

[10] H.F. de Lima. Spacelike Hypersurfaces with Constant Higher Order Mean Curvature (in portuguese), PhD Thesis, Universidade Federal do Ceara, Brazil, 2007.

[11] R. Lopez S. Montiel. Existence of constant mean curvature graphs in hyperbolic space, Calc. Var. 8, (1999) 177-190.

[12] S. Montiel. An integral inequality for compact spacelike hypersurfaces in de Sitter space and applications to the case of constant mean curvature, Indiana Univ. Math. J. 37, (1988) 909-917.

[13] S. Montiel. Unicity of Constant Mean Curvature Hypersurfaces in Some Riemannian Manifolds, Indiana Univ. Math. J. 48, (1999) 711-748.

[14] S. Montiel. Uniqueness of Spacelike Hypersurfaces of Constant Mean Curvature in foliated Spacetimes, Math. Ann. 314, (1999) 529-553.

[15] S. Montiel. Complete non-compact spacelike hypersurfaces of constant mean curvature in de Sitter Space, J. Math. Soc. Japan 55, (2003) 915-938.

[16] B. O'Neill. Semi-Riemannian Geometry with Applications to Relativity, London, Academic Press (1983).

[17] P.A. Sousa. The Laplacian of a Support-type Function and Applications (in portuguese), Master's Dissertation, Universidade Federal do Ceara, Brazil, 2004.

A. Caminha

Departamento de Matematica, Universidade Federal do Ceara, Fortaleza, Ceara, Brazil. 60455-760. Tel.: +55 085 33669313. Fax: +55 085 33669889. email:antonio.caminha@gmail.com

H. F. de Lima

Departamento de Matematica e Estatistica, Universidade Federal de Campina Grande, Campina Grande, Paraiba, Brazil. 58109-970. Tel.: +55 083 33101161. Fax: +55 083 33101030. email:henrique@dme.ufcg.edu.br

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Author: | Caminha, A.; de Lima, H.F. |
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Publication: | Bulletin of the Belgian Mathematical Society - Simon Stevin |

Geographic Code: | 3BRAZ |

Date: | Jan 1, 2009 |

Words: | 5905 |

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