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

Authors: Henrique F. de Lima; Joseilson R. de Lima
Journal: International Journal of Mathematics and Mathematical Sciences
 (2009)

---

## Abstract

We obtain a height estimate concerning to a compact spacelike hypersurfaceΣn
immersed with constant mean curvature H
in the anti-de Sitter space ℍ1n+1, when its boundary ∂Σ
is contained into an umbilical spacelike hypersurface
of this spacetime which is isometric to the hyperbolic space ℍn. Our estimate depends only on the value of H
and on the geometry
of ∂Σ.
As applications of our estimate, we obtain a characterization of hyperbolic domains of ℍ1n+1
and nonexistence results in connection with such types of hypersurfaces.


---

## Body

## 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 n≤4, and Cheng and Yau in [4], for arbitrary n, showed that the only complete immersed spacelike hypersurfaces of the (n+1)-dimensional Lorentz-Minkowski space 𝕃n+1 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 𝕃n+1 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 2-dimensional surfaces in 𝕃3). Also considering the compact case, R. López obtained a sharp estimate for the height of compact spacelike surfaces Σ2 immersed into the 3-dimensional Lorentz-Minkowski space 𝕃3 with constant mean curvature (cf. [9], Theorem 1). 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 (n+1)-dimensional Lorentz-Minkowski space 𝕃n+1 (cf. [11], Theorem 4.2).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 2-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 H=1. In the 2-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 hypersurfaceΣn immersed with constant mean curvature H in the antide Sitter spaceℍ1n+1, which is a particular model of Robertson-Walker spacetime given by ℍ1n+1=-(-π/2,π/2)×costℍn, where ℍn denotes the n-dimensional hyperbolic space (cf. Section 3). In this setting, by supposing its boundary ∂Σ contained into some slice of ℍ1n+1, we obtain an estimate for its vertical height function h. We prove the following result (cf. Theorem 3.2):Letψ:Σn→ℍ1n+1be a compact spacelike hypersurface whose boundary∂Σis contained in some sliceℍt0n={t0}×ℍn. Suppose that the mean curvatureH>1is constant.(i)
If-π/2<t0≤0 and Σn is contained into the chronological past with respect to ℍt0n, then the height h of Σn satisfies
(1.1)t0≥h≥t0-1H-1(Ccost0-cosh).(ii)
If0≤t0<π/2 and Σn is contained into the chronological future with respect to ℍt0n, then the height h of Σn satisfies
(1.2)t0≤h≤t0+1H-1(Ccost0-cosh).HereC=max∂Σ(coshθ) and θ is the hyperbolic angle between the Gauss map N of Σ and ∂t.Suitable formulae for the Laplacians of the height function and a support-like function naturally attached to a spacelike hypersurfaceΣn immersed in ℍ1n+1 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∂t is called a comoving observer and, when p is a point of a spacelike hypersurface Σn immersed into a Robertson-Walker spacetime -ℝ×fMn, ∂t(p) is called an instantaneous comoving observer. In this setting, among the instantaneous observers at p, ∂t(p) and N(p) appear naturally. From the orthogonal decomposition N(p)=-〈N,∂t〉p∂t(p)+(πM)*N(p) where πM denotes the canonical projection from -ℝ×fMn onto the Riemannian fiber Mn, we have that coshθ(p) corresponds to the energy e(p) that ∂t(p) measures for the normal observer N(p). Furthermore, the speed |υ(p)| of the Newtonian velocityυ(p):=(1/e(p))(πM)*N(p) that ∂t(p) measures for N(p) satisfies the equation |υ(p)|2=tanhθ(p). So, a physical consequence of the boundedness of the hyperbolic angle θ between the Gauss map N of the spacelike hypersurface Σn and ∂t 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 Σn (see [17], Sections 2.1 and 3.1, and [18]; see also [19], Chapter 12).As an application of our height estimate, we obtain an characterization of hyperbolic domains ofℍ1n+1 (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 hypersurfaceψ:Σn→ℍ1n+1with constant mean curvatureH≥2and tangent to the sliceℍ0nalong its boundary.Finally, we observe that an interesting feature of the four-dimensional antide Sitter spaceℍ14 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, ℍ14 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 8; see also [21], Chapter 6, and [22], Chapter 14).


## 2. Preliminaries

In what follows, ifM¯n+1 is a connected semi-Riemannian manifold with metric g¯=〈,〉, we let 𝒟(M¯) denote the ring of smooth functions ϕ:M¯n+1→ℝ and 𝔛(M¯) the algebra of smooth vector fields on M¯n+1. We also write ∇¯ for the Levi-Civita connection of M¯n+1.LetMn be a connected, n-dimensional (n≥2) oriented Riemannian manifold, I a 1-dimensional manifold (either a circle or an open interval of ℝ), and f:I→ℝ a positive smooth function. In the product differentiable manifold M¯n+1=I×Mn, let πI and πM denote the projections onto the factors I and Mn, respectively.A particular class of Lorentzian manifolds (spacetimes) is the one obtained by furnishing M¯n+1 with the metric(2.1)〈v,w〉p=-〈(πI)*v,(πI)*w〉+(f∘πI)(p)2〈(πM)*v,(πM)*w〉
for all p∈M¯n+1 and all v,w∈TpM¯. Such a space is called (following the terminology introduced in [23]) a Generalized Robertson-Walker (GRW) spacetime, and in what follows we shall write M¯n+1=-I×fMn to denote it. In particular, when Mn has constant sectional curvature, then -I×fMn is classically called a Robertson-Walker (RW) spacetime (cf. [19]). It is not difficult to see that a GRW spacetime -I×fMn has constant sectional curvature κ¯ if, and only if, the Riemannian fiber Mn has constant sectional curvature κ (i.e., -I×fMn is in fact a RW spacetime) and the warping function f satisfies the following differential equations:(2.2)f″f=κ¯=(f′)2+κf2
(see, for instance, [24], Corollary 9.107).We recall that a tangent vector fieldK on a spacetime M¯n+1 is said to be conformal if the Lie derivative with respect to K of the metric 〈,〉 of M¯n+1 satisfies(2.3)ℒK〈,〉=2ϕ〈,〉
for a certain smooth function ϕ∈𝒟(M¯n+1). Since ℒK(X)=[K,X] for all X∈𝔛(M¯), it follows from the tensorial character of ℒK that K∈𝔛(M¯) is conformal if and only if(2.4)〈∇¯XK,Y〉+〈X,∇¯YK〉=2ϕ〈X,Y〉
for all X,Y∈𝔛(M¯). In particular, K is a Killing vector field relatively to the metric 〈,〉 if and only if ϕ≡0.We observe that whenM¯n+1=-I×fMn is a GRW spacetime, the vector field(2.5)K=f∂t=(f∘πI)∂t
is conformal and closed (in the sense that its dual 1-form is closed), with conformal factor ϕ=f′, where the prime denotes differentiation with respect to t∈I (cf. [25]).A smooth immersionψ:Σn→M¯n+1 of an n-dimensional connected manifold Σn is said to be a spacelike hypersurface if the induced metric via ψ is a Riemannian metric on Σn, which, as usual, is also denoted for 〈,〉. In that case, since(2.6)∂t=(∂∂t)(t,x),(t,x)∈-I×fMn
is a unitary time-like vector field globally defined on the ambient GRW spacetime, then there exists a unique time-like unitary normal field N globally defined on the spacelike hypersurface Σn which is in the same time-orientation as ∂t, so that(2.7)〈N,∂t〉≤-1<0onΣn.
We will refer to that normal field N as the future-pointing Gauss map of the spacelike hypersurface Σn. Its opposite will be referred as the past-pointing Gauss map of Σn.In this setting, letA:𝔛(Σ)→𝔛(Σ) stand for the shape operator (or Weingarten endomorphism) of Σn with respect to either the future or the past-pointing Gauss map N. It is well known that A defines a self-adjoint linear operator on each tangent space TpΣ, and its eigenvalues κ1(p),…,κn(p) are the principal curvatures of Σn at p. Associated to the shape operator A there are n algebraic invariants given by(2.8)Sr(p)=σr(κ1(p),…,κn(p)),1≤r≤n,
where σr:ℝn→ℝ is the elementary symmetric function in ℝn given by(2.9)σr(x1,…,xn)=∑i1<⋯<irxi1⋯xir.
Observe that the characteristic polynomial of A can be written in terms of the Sr′s as(2.10)det(tI-A)=∑r=0n(-1)rSrtn-r,
where S0=1 by definition. The r-mean curvature Hr of the spacelike hypersurface Σn is then defined by(2.11)(nr)Hr=(-1)rSr(κ1,…,κn)=Sr(-κ1,…,-κn).
In particular, when r=1,(2.12)H1=-1n∑i=1nκi=-1ntr(A)=H
is the mean curvature of Σn, which is the main extrinsic curvature of the hypersurface. The choice of the sign (-1)r in our definition of Hr is motivated by the fact that in that case the mean curvature vector is given by H⃗=HN. Therefore, H(p)>0 at a point p∈Σn if and only if H⃗(p) is in the same time-orientation as N(p) (in the sense that 〈H⃗,N〉p<0).Whenr=2, H2 defines a geometric quantity which is related to the (intrinsic) scalar curvature R of the hypersurface. For instance, when the ambient spacetime M¯n+1 has constant sectional curvature κ¯, it follows from the Gauss equation that(2.13)R=n(n-1)(κ¯-H2).
Moreover, in the case of a 2-dimensional surface, denoting by KΣ the Gaussian curvature of the spacelike surface ψ:Σ2→M¯3, we have that(2.14)KΣ=κ¯-H2.As before, letM¯n+1=-I×fMn be a GRW. For a fixed t0∈I, we say that Mt0n={t0}×Mn is a slice of M¯n+1. It was proved by L.J. Alías et al. in [23] that each slice Mt0n is an umbilical spacelike hypersurface with constant r-mean curvature, equal to (f′(t0)/f(t0))r with respect to ∂t (see also Example 5.6 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 ∂t. Also, if the (vertical) height function h:Σn→I of Σn, given by h=πI∘ψ, is such that h≤t0 (h≥t0) for some t0∈I, then we say that Σn is a spacelike hypersurface contained into the chronological past (chronological future) with respect to the slice Mt0n.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 4.1 and Corollary 8.4).Lemma 2.1.
Letψ:Σn→-I×fMn be a spacelike hypersurface immersed into a GRW spacetime, with Gauss map N and denote for h=πI∘ψ the height function of Σn. Then
(2.15)Δh=-(lnf)′(h)(n+|∇h|2)-nH〈N,∂t〉.
Moreover, by supposing M¯n+1 an RW spacetime with Riemannian fiber Mn of constant sectional curvature κ,
(2.16)Δ(f(h)〈N,∂t〉)=nf(h)〈∇H,∂t〉+nHf′(h)+nf(h)〈N,∂t〉(nH2-(n-1)H2)+(n-1)f(h)〈N,∂t〉(κf2(h)-(lnf)″(h))|∇h|2.Remark 2.2.
For alternative proofs of the previous lemma, we suggest [11, 15, 18, 28].


## 3. Height Estimate for Spacelike Hypersurfaces inℍ1n+1

In what follows we consider a particular model of RW spacetime, theantide Sitter space, namely
(3.1)ℍ1n+1=-(-π2,π2)×costℍn,
where ℍn denotes the n-dimensional hyperbolic space (see [29], Chapter 5).Remark 3.1.
The spacetimeℍ1n+1 can also be regarded as the hyperquadric
(3.2)ℍ1n+1={p∈ℝ2n+2;〈p,p〉=-1},
in the indefinite index two flat space ℝ2n+2. For any timelike unit vector a∈ℝ2n+2, we have that the closed and conformal vector field K given by
(3.3)K(p)=a+〈a,p〉p,p∈ℍ1n+1,
is timelike on the open set consisting of the points p∈ℍ1n+1 such that 〈a,p〉2<1. This open set has two connect components and the distribution on ℍ1n+1 orthogonal to K provides a foliation ℱ(K) in this spacetime by means of the umbilical spacelike hypersurfaces 〈p,a〉=τ, -1<τ<1, which are isometric to two copies of hyperbolic spaces ℍn with constant sectional curvature -1/(1+τ2). Consequently, each of these two components can be described as the Lorentzian warped product -(-π/2,π/2)×costℍn (see [25], Example 3 and Proposition 1).Now, we present our main result.Theorem 3.2.
(Height Estimate) Letψ:Σn→ℍ1n+1 be a compact spacelike hypersurface whose boundary ∂Σ is contained in some slice ℍt0n={t0}×ℍn. Suppose that the mean curvature H>1 is constant.(i)
If-π/2<t0≤0 and Σn is contained into the chronological past with respect to ℍt0n, then the height h of Σn satisfies
(3.4)t0≥h≥t0-1H-1(Ccost0-cosh).(ii)
If0≤t0<π2 and Σn is contained into the chronological future with respect to ℍt0n, then the height h of Σn satisfies
(3.5)t0≤h≤t0+1H-1(Ccost0-cosh).  Here C=max∂Σ(coshθ) and θ is the hyperbolic angle between the Gauss map N of Σ and ∂t.Proof.
Suppose initially that-π/2<t0≤0 and Σn is contained into the chronological past with respect to ℍt0n. From Lemma 2.1, we have
(3.6)Δh=tanh(n+|∇h|2)-nH〈N,∂t〉.
Then, since h≤t0≤0, as a consequence of the maximum principle we must have Δh(p)≥0 for some point p∈Σn. Consequently, taking into account that we are supposing H>1, we conclude that the Gauss map N of Σn is future-pointing, that is,
(3.7)〈N,∂t〉≤-1
on Σn. Now, in order to get our estimate, we define on Σn the function
(3.8)φ=c(h-t0)-cosh〈N,∂t〉,
where c is a negative constant to be determined. By computing the Laplacian of φ with the aid of Lemma 2.1, we get
(3.9)Δφ=Ctanh(n+|∇h|2)-cnH〈N,∂t〉+nHsinh-cosh〈N,∂t〉(n2H2-n(n-1)H2),
where we have used the fact that the Riemannian fiber of ℍ1n+1 has constant sectional curvature κ=-1. Moreover, again as a consequence of the maximum principle, if Δφ≥0, then
(3.10)φ≤φ|∂Σ=(-〈N,∂t〉|∂Σ)cost0=(coshθ|∂Σ)cost0≤max∂Σ(coshθ)cost0=Ccost0
on Σn, and
(3.11)0≥t0≥h≥t0+1c(Ccost0-cosh).
We claim that it is possible to choose c such that Δφ≥0. In fact, for all constant c<0, it yields
(3.12)Ctanh(n+|∇h|2)≥0.
Putting this together with the Cauchy-Schwarz inequality H2-H2≥0 into the above expression of Δφ, we obtain
(3.13)Δφ≥nH(-〈N,∂t〉(Hcosh+c)-1).
Thus, since the Gauss map N of Σn is future-pointing, by taking
(3.14)c=1-H
we get that Δφ≥0. Therefore,
(3.15)t0≥h≥t0-1H-1(Ccost0-cosh).
Now, suppose that0≤t0<π/2 and that Σ is contained into the chronological future with respect to ℍt0n. In this case (again as a consequence of the maximum principle applied to the height function h), we have that Gauss map N of Σn is past-pointing, that is,
(3.16)〈N,∂t〉≥1
on Σn. Thus, we define on Σ the function
(3.17)φ=c(h-t0)+cosh〈N,∂t〉,
where c is a positive constant to be determined. From this point, by taking
(3.18)c=H-1
and working in a similar way as in the previous case we conclude that
(3.19)t0≤h≤t0+1H-1(Ccost0-cosh).Remark 3.3.
Related to our previous theorem, it is important to observe the following facts.
(a) We note that, while in the Riemannian case (from the Cauchy-Schwarz inequality) the support function 〈N,∂t〉 of Σn is always bounded, in the Lorentzian setting this boundedness occurs in a natural manner only when the spacelike hypersurface Σn is compact. Consequently, in this last case, it is plausible that for an estimate of the vertical height h 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 Σ2 immersed with constant mean curvature into the 3-dimensional Lorentz-Minkowski space 𝕃3 and whose boundary ∂Σ is included in a plane Π depends on the value of the mean curvature and on the area of the region of Σ2 above the plane Π (cf. [9], Theorem 1). 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 ∂Σ.
(b) Geometrically, observing that |〈N,∂t〉|=coshθ, we see that the boundedness of the hyperbolic angle θ means that (at each point p∈Σn) the normal direction N(p) remains far from the light cone corresponding to ∂t(p). So, a physical consequence of this fact is that the speed of the Newtonian velocity that the instantaneous comoving observer ∂t(p) measures for the normal observer N(p) does not approach the speed of light on Σn (see [17], Sections 2.1 and 3.1, and [18]; see also [19], Chapter 12).


## 4. Hyperbolic Domains ofℍ1n+1

When a compact spacelike hypersurfaceψ:Σn→ℍ1n+1 is entirely contained into some slice ℍt0n={t0}×ℍn, it is called a hyperbolic domain of ℍ1n+1. As applications of Theorem 3.2, we obtain the following results.Proposition 4.1.
Letψ:Σn→ℍ1n+1 be a compact spacelike hypersurface whose boundary ∂Σ is contained in some slice ℍt0n. Suppose that Σn is not a hyperbolic domain, i.e. mean curvature H>1 is constant, and that one of the following conditions is satisfied.  (i)
-π/2<t0≤0 and Σn is contained into the chronological past with respect to ℍt0n.(ii)
0≤t0<π/2 and Σn is contained into the chronological future with respect to ℍt0n. Then
(4.1)H≤1+Ccost0-cosh*|h*-t0|,
where C=max∂Σ(coshθ), θ is the hyperbolic angle between the Gauss map N of Σ and ∂t, and h*=maxΣh.In what follows, we say thatΣn is tangent toℍt0nalong its boundary∂Σ if ∂Σ is contained into ℍt0n, and the restriction of the Gauss map N of Σn to ∂Σ is equal to (∂t)t=t0 or -(∂t)t=t0 (that is, the hyperbolic angle between N and ∂t is identically zero along ∂Σ).Proposition 4.2.
Letψ:Σn→ℍ1n+1 be a compact spacelike hypersurface, which is tangent to some slice ℍt0n along its boundary. Suppose that Σn is not a hyperbolic domain, that its mean curvature H>1 is constant and that one of the following conditions is satisfied.  (i)
-π/2<t0≤0 and Σn is contained into the chronological past with respect to ℍt0n.(ii)
0≤t0<π/2 and Σn is contained into the chronological future with respect to ℍt0n. Then
(4.2)1<H≤1+|sint0|<2.Proof.
Initially, we observe, from Proposition4.1 and from our assumption, that Σn is tangent to ℍt0n along its boundary:
(4.3)H≤1+cost0-cosh(p)|h(p)-t0|
for all p∈Σn such that h(p)≠t0. Therefore, taking in the previous inequality the limit h(p)→t0, we conclude that
(4.4)H≤1+|sint0|<2.As a consequence of the previous result, we get the following characterization of hyperbolic domains ofℍ1n+1.Corollary 4.3.
Letψ:Σn→ℍ1n+1 be a compact spacelike hypersurface, which is tangent to some slice ℍt0n along its boundary. Suppose that one of the following conditions is satisfied.  (i)
-π/2<t0≤-arctan2 and Σn is contained into the chronological past with respect to ℍt0n.(ii)
arctan2≤t0<π/2 and Σn is contained into the chronological future with respect to ℍt0n.If its mean curvature H≥2 is constant, then Σn is a hyperbolic domain.Finally, we obtain the following nonexistence results.Corollary 4.4.
There is no compact spacelike hypersurfaceψ:Σn→ℍ1n+1 tangent to some slice ℍt0n along its boundary, with constant mean curvature H≥2 and satisfying one of the following conditions. (i)
Σn is contained into the chronological past with respect to ℍt0n, with—arctan2<t0≤0.(ii)
Σn is contained into the chronological future with respect to ℍt0n, with 0≤t0<arctan2.Corollary 4.5.
There is no compact spacelike hypersurfaceψ:Σn→ℍ1n+1 with constant mean curvature H≥2 and tangent to the slice ℍ0n along its boundary.



---

*Source: 673508-2009-10-25.xml*