International Journal of Mathematics and Mathematical Sciences

International Journal of Mathematics and Mathematical Sciences / 2009 / Article

Research Article | Open Access

Volume 2009 |Article ID 673508 | https://doi.org/10.1155/2009/673508

Henrique F. de Lima, Joseilson R. de Lima, "Compact Spacelike Hypersurfaces with Constant Mean Curvature in the Antide Sitter Space", International Journal of Mathematics and Mathematical Sciences, vol. 2009, Article ID 673508, 12 pages, 2009. https://doi.org/10.1155/2009/673508

Compact Spacelike Hypersurfaces with Constant Mean Curvature in the Antide Sitter Space

Academic Editor: Christian Corda
Received08 Jul 2009
Accepted06 Oct 2009
Published25 Oct 2009

Abstract

We obtain a height estimate concerning to a compact spacelike hypersurface immersed with constant mean curvature in the anti-de Sitter space , when its boundary is contained into an umbilical spacelike hypersurface of this spacetime which is isometric to the hyperbolic space . Our estimate depends only on the value of and on the geometry of As applications of our estimate, we obtain a characterization of hyperbolic domains of and nonexistence results in connection with such types of hypersurfaces.

1. Introduction

Interest in the study of spacelike hypersurfaces in Lorentzian manifolds has increased very much in recent years, from both the physical and mathematical points of view. For example, it was pointed out by J. Marsden and F. Tipler in [1] and S. Stumbles in [2] that spacelike hypersurfaces with constant mean curvature in arbitrary spacetimes play an important part in the relativity theory. They are convenient as initial hypersurfaces for the Cauchy problem in arbitrary spacetime and for studying the propagation of gravitational radiation. From a mathematical point of view, that interest is also motivated by the fact that these hypersurfaces exhibit nice Bernstein-type properties. Actually, E. Calabi in [3], for , and Cheng and Yau in [4], for arbitrary , showed that the only complete immersed spacelike hypersurfaces of the -dimensional Lorentz-Minkowski space with zero mean curvature are the spacelike hyperplanes.

Related with the compact case, Alías and Malacarne in [5] showed that the only compact spacelike hypersurfaces having constant higher-order mean curvature and spherical boundary in are the hyperplanar balls with zero higher-order mean curvature, and the hyperbolic caps with nonzero constant higher-order mean curvature (cf. [6] for the case of constant mean curvature and [7] for the case of constant scalar curvature; see also [8] for the case of -dimensional surfaces in ). Also considering the compact case, R. López obtained a sharp estimate for the height of compact spacelike surfaces immersed into the -dimensional Lorentz-Minkowski space with constant mean curvature (cf. [9], Theorem ). For the case of constant higher-order mean curvature, by applying the techniques used by D. Hoffman et al. in [10], the first author obtained another sharp height estimate for compact spacelike hypersurfaces immersed in the -dimensional Lorentz-Minkowski space (cf. [11], Theorem ).

Also recently, the first author obtained some geometric estimates concerning to a spacelike hypersurface immersed with some constant higher-order mean curvature in de Sitter space (cf. [12]), also, in a Lorentzian product space with Colares (cf. [13]) and in a conformally stationary Lorentz manifold with A. Caminha (cf. [14]). We note that, in each one of these papers, the authors have used their geometric estimates to study the existence of certain types of spacelike hypersurfaces in such spacetimes.

In [15] the first author and A. Caminha have studied complete vertical graphs of constant mean curvature in the hyperbolic and steady state spaces. Under appropriate restrictions on the values of the mean curvature and the growth of the height function, they obtained necessary conditions for the existence of such a graph. In the -dimensional case they applied their analytical framework to prove Bernstein-type results in each of these ambient spaces.

We note that Albujer and Alías have also recently considered in [16] complete spacelike hypersurfaces with constant mean curvature in the steady state space. They proved that if the hypersurface is bounded away from the infinity of the ambient space, then the mean curvature must be . In the -dimensional case they concluded that the only complete spacelike surfaces with constant mean curvature which are bounded away from the infinity are the totally umbilical flat surfaces. Moreover, considering the generalized Robertson-Walker spacetime model of the steady state space, they extended their results to a wider family of spacetimes.

In this paper we deal with a compact spacelike hypersurface immersed with constant mean curvature in the antide Sitter space , which is a particular model of Robertson-Walker spacetime given by , where denotes the -dimensional hyperbolic space (cf. Section 3). In this setting, by supposing its boundary contained into some slice of , we obtain an estimate for its vertical height function . We prove the following result (cf. Theorem 3.2):

Let be a compact spacelike hypersurface whose boundary is contained in some slice . Suppose that the mean curvature is constant.

(i)If and is contained into the chronological past with respect to , then the height of satisfies (ii)If and is contained into the chronological future with respect to , then the height of satisfies

Here and is the hyperbolic angle between the Gauss map of and .

Suitable formulae for the Laplacians of the height function and a support-like function naturally attached to a spacelike hypersurface immersed in constitute the analytical tools that we use to get our estimate (cf. Lemma 2.1).

It is important to point out that our estimate depends only on the value of the mean curvature and on the geometry of the boundary of the hypersurface. On the other hand, we recall that an integral curve of the unit time-like vector field is called a comoving observer and, when is a point of a spacelike hypersurface immersed into a Robertson-Walker spacetime , is called an instantaneous comoving observer. In this setting, among the instantaneous observers at , and appear naturally. From the orthogonal decomposition where denotes the canonical projection from onto the Riemannian fiber , we have that corresponds to the energy that measures for the normal observer . Furthermore, the speed of the Newtonian velocity that measures for satisfies the equation . So, a physical consequence of the boundedness of the hyperbolic angle between the Gauss map of the spacelike hypersurface and is that the speed of the Newtonian velocity that the instantaneous comoving observer measures for the normal observer does not approach the speed of light 1 on (see [17], Sections and , and [18]; see also [19], Chapter ).

As an application of our height estimate, we obtain an characterization of hyperbolic domains of (cf. Corollary 4.3). Furthermore, we establish nonexistence results in connection with such types of hypersurfaces (cf. Corollaries 4.4 and 4.5). For example, we prove the following.

There is no compact spacelike hypersurface with constant mean curvature and tangent to the slice along its boundary.

Finally, we observe that an interesting feature of the four-dimensional antide Sitter space is that, as a cosmological model, this spacetime is a maximally symmetric universe with constant negative curvature, which is conformally related to half of the Einstein static universe. Consequently, represents a (locally) unique solution to Einstein's equation in the absence of any ordinary matter or gravitational radiation. In this setting, this spacetime may be thought of as a ground state of general relativity (cf. [20], Chapter ; see also [21], Chapter , and [22], Chapter ).

2. Preliminaries

In what follows, if is a connected semi-Riemannian manifold with metric , we let denote the ring of smooth functions and the algebra of smooth vector fields on . We also write for the Levi-Civita connection of .

Let be a connected, -dimensional () oriented Riemannian manifold, a -dimensional manifold (either a circle or an open interval of ), and a positive smooth function. In the product differentiable manifold , let and denote the projections onto the factors and , respectively.

A particular class of Lorentzian manifolds (spacetimes) is the one obtained by furnishing with the metric

for all and all . Such a space is called (following the terminology introduced in [23]) a Generalized Robertson-Walker (GRW) spacetime, and in what follows we shall write to denote it. In particular, when has constant sectional curvature, then is classically called a Robertson-Walker (RW) spacetime (cf. [19]). It is not difficult to see that a GRW spacetime has constant sectional curvature if, and only if, the Riemannian fiber has constant sectional curvature (i.e., is in fact a RW spacetime) and the warping function satisfies the following differential equations:

(see, for instance, [24], Corollary ).

We recall that a tangent vector field on a spacetime is said to be conformal if the Lie derivative with respect to of the metric of satisfies

for a certain smooth function . Since for all , it follows from the tensorial character of that is conformal if and only if

for all . In particular, is a Killing vector field relatively to the metric if and only if .

We observe that when is a GRW spacetime, the vector field

is conformal and closed (in the sense that its dual form is closed), with conformal factor , where the prime denotes differentiation with respect to (cf. [25]).

A smooth immersion of an -dimensional connected manifold is said to be a spacelike hypersurface if the induced metric via is a Riemannian metric on , which, as usual, is also denoted for . In that case, since

is a unitary time-like vector field globally defined on the ambient GRW spacetime, then there exists a unique time-like unitary normal field globally defined on the spacelike hypersurface which is in the same time-orientation as , so that

We will refer to that normal field as the future-pointing Gauss map of the spacelike hypersurface . Its opposite will be referred as the past-pointing Gauss map of .

In this setting, let stand for the shape operator (or Weingarten endomorphism) of with respect to either the future or the past-pointing Gauss map . It is well known that defines a self-adjoint linear operator on each tangent space , and its eigenvalues are the principal curvatures of at . Associated to the shape operator there are algebraic invariants given by

where is the elementary symmetric function in given by

Observe that the characteristic polynomial of can be written in terms of the as

where by definition. The -mean curvature of the spacelike hypersurface is then defined by

In particular, when ,

is the mean curvature of , which is the main extrinsic curvature of the hypersurface. The choice of the sign in our definition of is motivated by the fact that in that case the mean curvature vector is given by . Therefore, at a point if and only if is in the same time-orientation as (in the sense that ).

When , defines a geometric quantity which is related to the (intrinsic) scalar curvature of the hypersurface. For instance, when the ambient spacetime has constant sectional curvature , it follows from the Gauss equation that

Moreover, in the case of a -dimensional surface, denoting by the Gaussian curvature of the spacelike surface , we have that

As before, let be a GRW. For a fixed , we say that is a slice of . It was proved by L.J. Alías et al. in [23] that each slice is an umbilical spacelike hypersurface with constant -mean curvature, equal to with respect to (see also Example in [26]). Whenever we talk about the mean curvature of the slices of a GRW, we shall assume that it is computed with respect to . Also, if the (vertical) height function of , given by , is such that () for some , then we say that is a spacelike hypersurface contained into the chronological past (chronological future) with respect to the slice .

To close this section, we present the analytical framework that we will use to obtain our estimates. The formulae collected in the following lemma are particular cases of ones obtained by L.J. Alías and A.G. Colares (cf. [27], Lemma and Corollary ).

Lemma 2.1. Let be a spacelike hypersurface immersed into a GRW spacetime, with Gauss map and denote for the height function of . Then Moreover, by supposing an RW spacetime with Riemannian fiber of constant sectional curvature ,

Remark 2.2. For alternative proofs of the previous lemma, we suggest [11, 15, 18, 28].

3. Height Estimate for Spacelike Hypersurfaces in

In what follows we consider a particular model of RW spacetime, the antide Sitter space, namely where denotes the -dimensional hyperbolic space (see [29], Chapter ).

Remark 3.1. The spacetime can also be regarded as the hyperquadric in the indefinite index two flat space . For any timelike unit vector , we have that the closed and conformal vector field given by is timelike on the open set consisting of the points such that . This open set has two connect components and the distribution on orthogonal to provides a foliation in this spacetime by means of the umbilical spacelike hypersurfaces , , which are isometric to two copies of hyperbolic spaces with constant sectional curvature . Consequently, each of these two components can be described as the Lorentzian warped product (see [25], Example and Proposition ).

Now, we present our main result.

Theorem 3.2. (Height Estimate) Let be a compact spacelike hypersurface whose boundary is contained in some slice . Suppose that the mean curvature is constant.(i)If and is contained into the chronological past with respect to , then the height of satisfies (ii)If and is contained into the chronological future with respect to , then the height of satisfies Here and is the hyperbolic angle between the Gauss map of and .

Proof. Suppose initially that and is contained into the chronological past with respect to . From Lemma 2.1, we have Then, since , as a consequence of the maximum principle we must have for some point . Consequently, taking into account that we are supposing , we conclude that the Gauss map of is future-pointing, that is, on . Now, in order to get our estimate, we define on the function where is a negative constant to be determined. By computing the Laplacian of with the aid of Lemma 2.1, we get where we have used the fact that the Riemannian fiber of has constant sectional curvature . Moreover, again as a consequence of the maximum principle, if , then on , and We claim that it is possible to choose such that . In fact, for all constant , it yields Putting this together with the Cauchy-Schwarz inequality into the above expression of , we obtain Thus, since the Gauss map of is future-pointing, by taking we get that . Therefore,
Now, suppose that and that is contained into the chronological future with respect to . In this case (again as a consequence of the maximum principle applied to the height function ), we have that Gauss map of is past-pointing, that is, on . Thus, we define on the function where is a positive constant to be determined. From this point, by taking and working in a similar way as in the previous case we conclude that

Remark 3.3. Related to our previous theorem, it is important to observe the following facts.
We note that, while in the Riemannian case (from the Cauchy-Schwarz inequality) the support function of is always bounded, in the Lorentzian setting this boundedness occurs in a natural manner only when the spacelike hypersurface is compact. Consequently, in this last case, it is plausible that for an estimate of the vertical height must appear a term that depends on the geometry of the spacelike hypersurface. For example, the estimate of López for the height of a compact spacelike surface immersed with constant mean curvature into the -dimensional Lorentz-Minkowski space and whose boundary is included in a plane depends on the value of the mean curvature and on the area of the region of above the plane (cf. [9], Theorem ). On the other hand, from Theorem 3.2, we see that our estimate depends on the value of the mean curvature and on the geometry of the boundary .
Geometrically, observing that , we see that the boundedness of the hyperbolic angle means that (at each point ) the normal direction remains far from the light cone corresponding to . So, a physical consequence of this fact is that the speed of the Newtonian velocity that the instantaneous comoving observer measures for the normal observer does not approach the speed of light on (see [17], Sections and , and [18]; see also [19], Chapter ).

4. Hyperbolic Domains of

When a compact spacelike hypersurface is entirely contained into some slice , it is called a hyperbolic domain of . As applications of Theorem 3.2, we obtain the following results.

Proposition 4.1. Let be a compact spacelike hypersurface whose boundary is contained in some slice . Suppose that is not a hyperbolic domain, i.e. mean curvature is constant, and that one of the following conditions is satisfied. (i) and is contained into the chronological past with respect to . (ii) and is contained into the chronological future with respect to . Then where , is the hyperbolic angle between the Gauss map of and , and .

In what follows, we say that is tangent to along its boundary if is contained into , and the restriction of the Gauss map of to is equal to or (that is, the hyperbolic angle between and is identically zero along ).

Proposition 4.2. Let be a compact spacelike hypersurface, which is tangent to some slice along its boundary. Suppose that is not a hyperbolic domain, that its mean curvature is constant and that one of the following conditions is satisfied. (i) and is contained into the chronological past with respect to . (ii) and is contained into the chronological future with respect to . Then

Proof. Initially, we observe, from Proposition 4.1 and from our assumption, that is tangent to along its boundary: for all such that . Therefore, taking in the previous inequality the limit , we conclude that

As a consequence of the previous result, we get the following characterization of hyperbolic domains of .

Corollary 4.3. Let be a compact spacelike hypersurface, which is tangent to some slice along its boundary. Suppose that one of the following conditions is satisfied. (i) and is contained into the chronological past with respect to . (ii) and is contained into the chronological future with respect to . If its mean curvature is constant, then is a hyperbolic domain.

Finally, we obtain the following nonexistence results.

Corollary 4.4. There is no compact spacelike hypersurface tangent to some slice along its boundary, with constant mean curvature and satisfying one of the following conditions. (i) is contained into the chronological past with respect to , with. (ii) is contained into the chronological future with respect to , with .

Corollary 4.5. There is no compact spacelike hypersurface with constant mean curvature and tangent to the slice along its boundary.

Acknowledgments

The second author wants to thank Professor Antonio Gervasio Colares for his guidance. The first author is partially supported by FAPESQ/CNPq/PPP. The second author was partially supported by CAPES, Brazil. The authors wish to thank the referees for the useful and constructive suggestions and comments.

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Copyright © 2009 Henrique F. de Lima and Joseilson R. de Lima. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.


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