DOI 10.1007/s10714-008-0621-9 R E S E A R C H A RT I C L E
Space-like hypersurfaces with positive constant
r
-mean
curvature in Lorentzian product spaces
A. Gervasio Colares · Henrique F. de Lima
Received: 23 May 2007 / Accepted: 1 February 2008 / Published online: 21 February 2008 © Springer Science+Business Media, LLC 2008
Abstract In this paper we obtain a height estimate concerning compact space-like hypersurfacesn immersed with some positive constantr-mean curvature into an (n+1)-dimensional Lorentzian product space−R×Mn, and whose boundary is
contained into a slice{t} ×Mn. By considering the hyperbolic caps of the Lorentz– Minkowski spaceLn+1, we show that our estimate is sharp. Furthermore, we apply this estimate to study the complete space-like hypersurfaces immersed with some positive constantr-mean curvature into a Lorentzian product space. For instance, when the ambient space–time is spatially closed, we show that such hypersurfaces must satisfy the topological property of having more than one end which constitutes a necessary condition for their existence.
Keywords Lorentzian products spaces·Space-like hypersurfaces·Higher order mean curvatures·Height estimates
1 Introduction
In the last years, the study of space-like hypersurfaces in space–times has been of substantial interest from both the physical and mathematical aspects. For example, it was pointed out by Marsdan and Tipler [18] and Stumbles [22] that space-like hypersurfaces with constant mean curvature in arbitrary space–time play an important
A. G. Colares
Departamento de Matemática, Universidade Federal do Ceará, Fortaleza, Ceará 60455-760, Brazil e-mail: gcolares@mat.ufc.br
H. F. de Lima (
B
)Departamento de Matemática e Estatística, Universidade Federal de Campina Grande, Campina Grande, Paraíba 58109-970, Brazil
part in the relativity theory. They are convenient as initial hypersurfaces for the Cauchy problem in arbitrary space–time and for studying the propagation of gravitational radiation (for more details see [13], Chapter 5).
From a mathematical point of view, that interest is also motivated by the fact that these hypersurfaces exhibit nice Bernstein-type properties. Actually, Calabi [7], for
n≤4, and Cheng and Yau [9], for arbitraryn, showed that the only complete immersed space-like hypersurfaces of the(n+1)-dimensional Lorentz–Minkowski spaceLn+1 with zero mean curvature are the space-like hyperplanes. More recently, Aiyama [1] and Xin [23] simultaneous and independently characterized the space-like hyperplanes as the only complete constant mean curvature space-like hypersurfaces immersed in Ln+1 and having the image of its Gauss map contained into a geodesic ball of the
n-dimensional hyperbolic spaceHn.
Related to the compact space-like hypersurfaces, López [17] obtained a sharp esti-mate for the height of compact space-like surfaces2immersed with constant mean curvature into the three-dimensional Lorentz–Minkowski spaceL3. For the case of constant higher order mean curvature, the second author in [16] obtained another sharp height estimate for compact immersed space-like hypersurfaces of the(n+ 1)-dimensional Lorentz–Minkowski spaceLn+1. As an application of this estimate, he obtained a version of the theorems of Aiyama and Xin for the case of nonzero constant
r-mean curvature.
In this paper we deal with compact immersed space-like hypersurfacesnof an (n+1)-dimensional Lorentzian product space−R×Mn, having some positive constant
r-mean curvatureHrand boundary∂contained into some slice{t}×Mof−R×Mn.
In this setting, by working with the support functionη = N, ∂t, whereN denotes
the Gauss map of such a hypersurfacenand∂tis the coordinate vector field induced
by the universal time on−R×Mn, we obtain the result below (cf. Theorem3.3):
Letn be a compact immersed space-like hypersurface of a Lorentzian product space −R×Mn whose Riemannian fiber Mn has nonnegative constant sectional curvatureκM. Suppose thatnhas positive constant r -mean curvature Hr, for some
1≤r ≤n, and that its boundary∂nis contained in the slice{0} ×Mn. Then, the vertical height h ofnsatisfies the inequality
|h| ≤ C−1
Hr1/r
,
where C=max∂|η|. Moreover, in the case r =1one can replace the condition on
the sectional curvature of Mnby that of the Ricci curvature of Mnbeing nonnegative.
When the ambient space–time is the Lorentz–Minkowski spaceLn+1, we have that max
∂ |η| ≤cosh̺,
Letnbe a compact immersed space-like hypersurface of the Lorentz–Minkowski spaceLn+1. Suppose thatnhas positive constant r -mean curvature H
r, for some
1≤r≤n, and that its boundary∂is contained in some hyperplanet = {t} ×Rn. If the image of the Gauss map ofnis contained in a geodesic ball B(e1, ̺)ofHn, then the height h ofnwith respectt satisfies
|h| ≤ cosh̺−1
Hr1/r
.
To establish our estimate, we apply the technique used by Cheng and Rosenberg to prove Theorem 4.1 in [10]. There, they obtain such an estimate concerning a compact vertical graphnimmersed with positive constantr-mean curvature into a Riemannian productR×Mnand whose boundary∂ is contained into the slice{0} ×Mn (we note that Rosenberg [20] already obtained this type of estimate concerning such graphs in the Riemannian space forms; see also Hoffman et al. [14] for the case when the ambient is a productR×M2, whereM2is a Riemannian surface).
Examining the result of Cheng and Rosenberg, we see that in fact they have showed that, considering the orientation N ofn in the opposite direction with respect the coordinate vector field∂t, the vertical heighthofnsatisfies
h ≤ max∂(η)+max(−η)
Hr1/r
= 1
Hr1/r
.
Moreover, the hemisphere of the unitary round sphereSninRn+1shows that the above estimate is sharp.
We want to point out that there exists a duality between our Theorem3.3and the estimate of Cheng and Rosenberg in the following sense: from our result, considering the case when the Gauss mapN ofnis in the opposite time-orientation of∂t, we
have that the vertical heighthofnsatisfies
h≤ max∂(η)+max(−η)
Hr1/r
= C−1
Hr1/r
.
Furthermore, we note that the hyperbolic caps ofLn+1show that our estimate is also sharp (cf. Remark4.3).
We also apply our estimate to study space-like hypersurfaces immersed with some positive constantr-mean curvature into Lorentzian product spaces. For example, in the Lorentz–Minkowski space we obtain the following result (cf. Corollary4.4):
Letnbe a complete immersed space-like hypersurface of the Lorentz–Minkowski spaceLn+1, with one end. Suppose thatnhas positive constant r -mean curvature Hr, for some1 ≤r ≤ n. If the image of the Gauss map ofn is contained into a geodesic ball ofHn, then its end is not divergent.
constitutes a necessary condition for their existence: the property of having more than one end. More precisely (cf. Theorem5.1):
Letnbe a complete properly immersed space-like hypersurface of a spatially clo-sed Lorentzian product space−R×Mn. Suppose that one of the following conditions is satisfied:
(a) The Riemannian fiber Mn has nonnegative Ricci curvatureRicM andn has positive constant mean curvature H .
(b) The Riemannian fiber Mnhas nonnegative constant sectional curvatureκM and
nhas positive constant r -mean curvature Hr, for some1≤r≤n.
If the support functionηofnis bounded, then the number of ends ofnis not one.
Comparing with the corresponding Riemannian case (cf. [10, Theorem 4.3]), we see that in the Lorentzian setting we need the additional hypothesis of the boundedness of the support functionηassociated to the complete space-like hypersurfacen. Geo-metrically, this boundedness means that, at each point p ∈n, the normal direction
N(p)remains far from the light cone corresponding to∂t(p).
On the other hand, since in the hyperbolic space Hn a geodesic ball B(a, ̺)of radius̺ >0 and centered at a pointa∈Hnis characterized as
B(a, ̺)= {p∈Hn; −cosh̺≤ p,a ≤ −1},
we see that in the Lorentz–Minkowski space Ln+1 the hypothesis of the image of the Gauss map ofn be contained into a geodesic ball ofHn is equivalent to the boundedness of its support functionη. In this sense, in an arbitrary Lorentzian product space−R×Mnthe hypothesis of the boundedness ofηis the natural substitute to the boundedness of the image of the Gauss mapN of the space-like hypersurfacen.
Recall that an integral curve of the unit timelike vector field∂tis called acomoving observerand, whenpis a point of a space-like hypersurfacenimmersed into a space– time−R×Mn,∂
t(p)is called aninstantaneous comoving observer. In this setting,
among the instantaneous observers at p,∂t(p)andN(p)appear naturally. From the
orthogonal decompositionN(p)= −η(p)∂t(p)+(πM)∗N(p)whereπMdenotes the
canonical projection from−R×Monto the Riemannian fiberMn, we have that|η(p)|
corresponds to theenergy e(p)that∂t(p)measures for the normal observer N(p).
Furthermore, the speed|υ(p)|of theNewtonian velocityυ(p) := e(1p)(πM)∗N(p)
that∂t(p)measures forN(p)satisfies the equation|υ(p)|2 =tanh(cosh−1|η(p)|).
So, a physical consequence of the boundedness of the support functionηof the space-like hypersurfacenis that the speed of the Newtonian velocity that the instantaneous comoving observer measures for the normal observer do not approach the speed of light 1 onn(cf. [21, Sects. 2.1, 3.1], and [15]; see also [19, Chap. 12]).
2 Preliminaries
if the induced metric viaψ is a Riemannian metric onn, which, as usual, is also denoted by,. In this setting,∇and∇stand for the Levi-Civita connection ofMn+1
andn, respectively.
LetAbe the shape operator ofninMn+1associated to the choice of an orientation
Nofn. Associated with the shape operatorAthere arenalgebraic invariants, which are the elementary symmetric functionsSrof its principal curvaturesκ1, . . . , κn, given
by
Sr =Sr(κ1, . . . , κn)=
i1<···<ir
κi1· · ·κir, 1≤r≤n.
Ther-mean curvatureHr of the space-like hypersurfacenis then defined by
n
r
Hr =(−1)rSr(κ1, . . . , κn)=Sr(−κ1, . . . ,−κn).
In particular, whenr =1,
H1= − 1
n n
i=1 κi = −
1
ntr(A)=H
is the mean curvature ofn, which is the main extrinsic curvature of the hypersurface. The choice of the sign(−1)r in our definition ofHr is motivated by the fact that in
that case the mean curvature vector is given by−→H =H N. Therefore,H(p) >0 at a pointp∈if and only if−→H(p)is in the same time-orientation asN(p)(in the sense that−→H,Np<0).
Whenr = 2, H2defines a geometric quantity which is related to the (intrinsic) scalar curvature R of the hypersurface. For instance, when the ambient space–time
Mn+1has constant sectional curvatureκ, it follows from the Gauss’ equation that
R=n(n−1)(κ−H2).
Moreover, in the three-dimensional case, denoting byKthe Gaussian curvature of
the space-like surfaceψ:2→ M3, we have that
K=κ−H2.
It is a classical fact that the higher order mean curvatures satisfy a very useful set of inequalities, usually alluded as Newton’s inequalities. For future reference, we collect them here. A proof can be found in [12, page 52] (see also [8, Proposition 1]). Lemma 2.1 For each1≤r≤n, if H1,H2, . . . ,Hr are nonnegative, then:
(a) Hr−1Hr+1≤ Hr2;
(b) H1≥ H 1 2 2 ≥ H
1 3
3 ≥ · · · ≥ H 1
Now, we introduce the corresponding Newton transformations
Pr :X()→X() , 0≤r≤n,
which arise from the shape operator A. According to our definition of ther-mean curvatures, the Newton transformations are given by
Pr =
n
r
HrI+
n
r−1
Hr−1A+ · · · +
n
1
H1Ar−1+Ar,
whereI denotes the identity inX(), or, inductively,
P0=I and Pr =
n
r
HrI+A◦ Pr−1.
Observe that the characteristic polynomial ofAcan be written in terms of theHr as
det(t I−A)=
n
r=0
n
r
Hrtn−r,
where H0=1. By the Cayley–Hamilton theorem, this implies thatPn =0. Besides, we have the following properties ofPr (cf. [3]).
(i) If{e1, . . . ,en}is a local orthonormal frame onwhich diagonalizes A, i.e., Aei =κiei,i =1, . . . ,n, then it also diagonalizes eachPr, and Prei =λi,rei
with
λi,r =(−1)r
i1<···<ir,ij=i
κi1· · ·κir =
i1<···<ir,ij=i
(−κi1)· · ·(−κir).
(ii) For each 1≤r ≤n−1,
tr(Pr)=(r+1)
n
r+1
Hr,
tr(A◦Pr)= −(r+1)
n r+1
Hr+1
and
trA2◦Pr
=
n
r+1
(n H1Hr+1−(n−r−1)Hr+2).
(iii) For everyV ∈X()and for each 1≤r≤n−1,
tr(Pr ◦ ∇VA)= −
n
r+1
Associated to each Newton transformation Pr one has the second order linear
differential operatorLr :D()→D(), given by
Lr(f)=tr(PrHess f).
Given a local coordinate frame∂∂xiofnat a point p, by a direct computation we obtain the local expression of the linear operatorLr:
Lr(f)(p)=
i,j,k,l
gi ktklgl j ∂
2f ∂xi∂xj
−
i,j,k,l,s
gi ktklgl jŴi js ∂f ∂xs
,
where
gi j =
∂f
∂xi,
∂f
∂xj
, G=gi j
, G−1=gi j, ti j = Pr
∂f
∂xi,
∂f
∂xj
,
andŴi js are the connection coefficients of∇.
From the above local expression, we know that the linear operator Lr is elliptic
if, and only if, Pr is positive definite. Clearly, L0 = △is always elliptic. To obtain our estimates, it will be useful to have some geometric conditions which guarantee the ellipticity of Lr when r ≥ 1. Forr = 1 we have the following lemma (see
[4, Lemma 3.2]; see also [11, Lemma 3.10] and [10, Proposition 3.1]).
Lemma 2.2 Letnbe an immersed space-like hypersurface of a space–time Mn+1. If H2>0onn, then the operator L1is elliptic.
Proof Observe that by Cauchy–Schwarz inequality we haveH2 ≥ H2 >0, and H does not vanish on. By choosing the appropriate Gauss map N, we may assume thatH >0. Recall thatH2does not depend on the chosenN. Since
n2H2=
n
i=1
κi2+n(n−1)H2> κ2j,
for every j =1, . . . ,n, thenλj,1=n H+κj >0 for all j and, consequently,P1is
positive definite. ⊓⊔
When r ≥ 2, the next lemma give us sufficient conditions to guarantee the ellipticity of Lr. The proof follows from that of Proposition 3.2 in [10], taking into account our sign convention in the definition of ther-mean curvatures (see also [6, [Proposition 3.2], and [4, Lemma 3.3]).
3 Height estimate in Lorentzian products spaces
In what follows, we deal with a space-like hypersurface n immersed into a (n+1)-dimensional Lorentzian product space Mn+1 of the form R×Mn, where
Mnis an-dimensional connect Riemannian manifold andMn+1is endowed with the Lorentzian metric
, = −πR∗(dt2)+πM∗(,M),
whereπRandπM denote the canonical projections fromR×Mnonto each factor,
and,M is the Riemannian metric onMn. For simplicity, we will just writeM n+1
= −R×Mn. In particular, when Mn =Rnis then-dimensional flat Euclidean space thenMn+1=Ln+1is the(n+1)-dimensional Lorentz–Minkowski space.
In this setting, we consider two particular functions naturally attached to a space-like hypersurfacenimmersed into a Lorentzian product space−R×Mn: the vertical
height functionh =(πR)|and the support functionη= N, ∂t, where Ndenotes
the Gauss map ofn and∂t is the coordinate vector field induced by the universal
time on−R×Mn. The following proposition corresponds to the analytical framework that we will use to obtain our main result. For a detailed proof of it see [4, Lemma 4.1, Corollaries 8.2, 8.4].
Proposition 3.1 Letnbe an immersed space-like hypersurface of a Lorentzian pro-duct space−R×Mn, with Gauss map N . For every r=0, . . . ,n−1we have:
(a) Lrh = −(r+1)
n
r+1
Hr+1η; (b) △η=n∇H, ∂t +
|A|2+RicM((πM)∗N, (πM)∗N)
η,
whereRicM denotes the Ricci tensor of Mn. Moreover, if the Riemannian fiber Mnhas constant sectional curvatureκM, then
Lrη=
n r+1
∇Hr+1, ∂t +tr(A2◦Pr)η
+κM
(r+1)
n r+1
Hr|∇h|2− Pr∇h,∇h
η.
Remark 3.2 The formulae collected in the above lemma are the Lorentzian versions of the ones obtained by Cheng and Rosenberg (cf. [10, Lemmas 4.1, 4.2]). We also note that Alías jointly with the first author obtained a generalization of these formulae in the context of the Generalized Robertson–Walker space–times (cf. [4, Lemma 4.1, Corollary 8.5]). Moreover, Albujer and Alías obtained in [2] the corresponding for-mulae for the Laplacian of the height and support functions of a space-like surface immersed in a three-dimensional Lorentzian product space.
Theorem 3.3 Let n be a compact immersed space-like hypersurface of a Lorentzian product space−R×Mn whose Riemannian fiber Mn has nonnegative constant sectional curvatureκM. Suppose thatnhas positive constant r -mean cur-vature Hr, for some1 ≤r ≤n, and that its boundary∂nis contained in the slice
{0} ×Mn. Then, the vertical height h ofnsatisfies the inequality
|h| ≤ C−1
Hr1/r
,
where C=max∂|η|. Moreover, in the case r =1one can replace the condition on
the sectional curvature of Mnby that of the Ricci curvature of Mnbeing nonnegative.
Proof Suppose, for example, thatNis in the same time-orientation of∂t(i.e.,N, ∂t ≤
−1). At a lowest point, all the principal curvatures have the same sign. Since we are assume thatHr >0, we know that at this point all the principal curvatures are negative and hence we can apply Lemmas2.2and2.3to obtain thatLr−1is elliptic andHjare
positive, 1≤ j ≤r−1.
Now, we define onthe function
ϕ=c h−η,
wherecis a negative constant. We have thatϕ|∂ ≤C, whereC=max∂|η|.
On the other hand, sincePr−1is positive definite, we have that
Pr−1∇h,∇h ≤tr(Pr−1)|∇h|2=r
n
r
Hr−1|∇h|2.
Thus, from Proposition3.1and using the assumption that the Riemannian fiberMn
has nonnegative constant sectional curvatureκM, we obtain that
Lr−1ϕ = −N, ∂t
r
n
r
c Hr +tr(A2◦ Pr−1)
−κM
r
n
r
Hr−1|∇h|2− Pr−1∇h,∇h
η
≥ −N, ∂t
r
n r
c Hr +tr(A2◦ Pr−1)
.
Furthermore, we have in the caser =1 (by the Cauchy–Schwarz inequality)
H Hr −Hr+1=H2−H2≥0. In the caser >1, we know from Lemma2.1that
and also that
H ≥Hr−1/(1r−1).
Moreover, also from Lemma2.1, we have thatHr2−Hr−1Hr+1≥0. Thus,
Hr+1≤
Hr2 Hr−1
.
Then, from these above inequalities, we obtain that
H Hr −Hr+1 ≥
Hr Hr−1
(H Hr−1−Hr)≥ Hr Hr−1
H Hr−1−Hrr−/(1r−1)
= HrH−Hr1−/(1r−1)≥0. Therefore,
tr(A2◦Pr−1)=
n
r
(n H Hr −(n−r)Hr+1)≥r
n
r
Hr(r+1)/r.
Consequently, by taking
c= −Hr1/r
in the definition of the functionϕ, we get thatLr−1ϕ ≥0 on. Then, we conclude
from the maximum principle thatϕ ≤Con. Therefore,
0≥h≥ 1−C
Hr1/r
.
In the case of N is in the opposite time-orientation of∂t (i.e.,N, ∂t ≥ 1), the
proof carries in a similar way and we conclude that
0≤h≤ C−1
Hr1/r
.
Finally, since
△η=n∇H, ∂t +|A|2+RicM((πM)∗N, (πM)∗N)η,
we note that in the caser =1 one can replace the condition on the sectional curvature ofMnby that of the Ricci curvature ofMnbeing nonnegative. ⊓⊔
also considered that the boundary∂ is contained into the slice{0} ×Mn, and that the fiberMnof the ambient space has nonnegative constant sectional curvature.
Examining their proof, we see that in fact they have showed that, considering the orientationN is in the opposite direction with respect∂t, the vertical heighth ofn
satisfies
h ≤ max∂(η)+max(−η)
Hr1/r
= 1
Hr1/r
.
The hemisphere of the unitary round sphereSninRn+1shows that the above estimate is sharp.
In this setting, we observe that there exists a duality between our Theorem3.3and the estimate of Cheng and Rosenberg in the sense that from our result, considering the case when the Gauss mapN ofnis in the opposite time-orientation of∂t, we have
that the vertical heighthofnsatisfies
h≤ max∂(η)+max(−η)
Hr1/r
= C−1
Hr1/r
.
Besides, we note that the hyperbolic caps ofLn+1show that our estimate is also sharp (cf. Remark4.3).
Moreover, we note that, while in the Riemannian case (from the Cauchy–Schwarz inequality) the support functionηis always bounded, in the Lorentzian setting this boundedness occurs in a natural manner only when the space-like hypersurfacen 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 space-like hypersurface. For example, the estimate of López for the height of a compact space-like surface2 immersed with constant mean curvature into the three-dimensional Lorentz–Minkowski space L3 and whose boundary ∂ is included in a plane depends on the area of the region of2 above the plane (cf. [17, Theorem 1]). On the other hand, from Theorem3.3, we see that our estimate depends only on the geometry of the boundary∂.
Corollary 3.5 Let 2 be a compact immersed space-like surface immersed of a three-dimensional Lorentzian product space−R×M2, where M2is a Riemannian
surface with nonnegative constant sectional curvatureκM. Suppose that the boundary of2is contained in the slice{0} ×M2. If the Gaussian curvature Kis a constant
such that K< κM, then the vertical height h of2satisfies the inequality
|h| ≤ C−1
(κM −K)1/2
,
where C=max∂|η|.
Observing that the vertical translationt0 by an arbitrary real parametert0, t0 : {t} ×M
n→ {t+t
Corollary 3.6 Let n be a compact immersed space-like hypersurface of a Lorentzian product space−R×Mn whose Riemannian fiber Mn has nonnegative constant sectional curvatureκM. Suppose thatnhas positive constant r -mean cur-vature Hr, for some1 ≤r ≤n, and that its boundary∂nis contained in the slice
{t} ×Mn. Then,
n⊂ [t,t+α] ×Mn,
or
n⊂ [t−α,t] ×Mn,
whereα=(max∂|η| −1)Hr−1/r. Moreover, in the case r =1one can replace the condition on the sectional curvature of M by that of the Ricci curvature of M being nonnegative.
For what follows, we observe that a complete immersed space-like hypersurface nof a Lorentzian product space−R×Mn+1, with one end, can be regarded as
n=tn∪Cn,
where tn is a compact hypersurface with boundary contained into a slice
Mt = {t} ×MnandCnis diffeomorphic to the cylinder[t,∞)×Sn−1.
Given a complete space-like hypersurface with one end,n =tn∪Cn, we say
that the end ofn isdivergentif, consideringCnwith cylindrical coordinates p =
(s,q)∈ [t,∞)×Sn−1, we have that lim
s→∞h(p)= ∞,
wherehdenotes the vertical height function ofCn.
Theorem 3.7 Let n be a complete immersed space-like hypersurface of a Lorentzian product space−R×Mn, with one end. Suppose that one of the follo-wing conditions is satisfied:
(a) The Riemannian fiber Mn has nonnegative Ricci curvatureRicM andn has positive constant mean curvature H .
(b) The Riemannian fiber Mnhas nonnegative constant sectional curvatureκM and
nhas positive constant r -mean curvature Hr, for some1≤r≤n.
If the support functionηofnis bounded, then its end is not divergent.
Proof Suppose, by contradiction, that the end ofn =tn∪Cnis divergent. Then,
sincetnis a compact hypersurface with boundary contained into a sliceMt = {t} × Mn, from Corollary3.6we have for example that
where α = (sup|η| −1)Hr−1/r. Now, using the assumption that the end ofn
is divergent, we can intersect n by the slice Mt+α = {t +α} × Mn and obtain
a compact hypersurface t+n α with constant r-mean curvature Hr, with boundary
contained into the sliceMt+α, and whose height is strictly greater thanα. Therefore,
we get a contradiction with respect our estimates for the height function of a compact
space-like hypersurface. ⊓⊔
4 Applications to the Lorentz–Minkowski space
We note that when n is an immersed space-like hypersurface of the Lorentz– Minkowski spaceLn+1= −R×Rnthe timelike unit normal vector fieldN ∈X⊥()
can be regarded as a mapN :n→Hn, whereHndenotes then-dimensional hyper-bolic space, that is
Hn=
x∈Ln+1; x,x = −1, x1≥1.
In this setting, the imageN()will be called thehyperbolic imageofn. Furthermore, given a geodesic ballB(a, ̺)inHnof radius̺ >0 centered at a pointa ∈Hn, we recall thatB(a, ̺)is characterized as the following
B(a, ̺)= {p∈Hn; −cosh̺≤ p,a ≤ −1};
so, if the hyperbolic image ofnis contained into someB(a, ̺), then
1≤ |N,a| ≤cosh̺.
Therefore, given a compact immersed space-like hypersurfacenofLn+1, we have that
max
∂ |η| ≤cosh̺,
where̺ >0 is the radius of a geodesic ball of centere1=(1,0, . . . ,0)∈Hnwhich contains the hyperbolic image of. Consequently, from Theorem3.3, we obtain the following result.
Corollary 4.1 [16, Theorem 4.2]Letnbe a compact immersed space-like hyper-surface of the Lorentz–Minkowski spaceLn+1. Suppose thatnhas positive constant r -mean curvature Hr, for some1 ≤ r ≤ n, and that its boundary∂ is contained in some hyperplanet = {t} ×Rn. If the hyperbolic image ofn is contained in a geodesic ball B(e1, ̺)ofHn, then the height h ofnwith respectt satisfies
|h| ≤ cosh̺−1
Hr1/r
.
Corollary 4.2 [16, Corollary 4.3]Let2be a compact immersed space-like surface of the three-dimensional Lorentz–Minkowski spaceL3. Suppose that2has negative
constant Gaussian curvature K and that its boundary∂n is contained in some
hyperplanet = {t} ×R2. If the hyperbolic image of2is contained in a geodesic ball B(e1, ̺)ofH2, then the height h of2with respect
t is such that
|h| ≤ cosh̺−1
(−K)1/2
.
Remark 4.3 Fixed a positive constantλ, we easily verify that the hyperbolic cap
λn=x∈Ln+1; x,x = −λ2, λ≤x1≤
1+λ2
is an example of space-like hypersurface of the Lorentz–Minkowski spaceLn+1which has positive constantr-mean curvature
Hr = 1
λr,
for each 1≤r≤n(if we choose the Gauss mapNin the same time-orientation ofe1, for the caserodd). We also easily verify that the hyperbolic image ofλnis contained in the geodesic ball of centere1∈Hnand radius
̺=cosh−1
1+ 1
λ2. Therefore, since the vertical height of such hyperbolic cap is
h =1+λ2−λ= cosh̺−1
Hr1/r
,
we conclude that our estimate for the vertical height function is sharp.
Moreover, we observe that hyperbolic caps also show that if we fix therth mean curvatureHr, there exist space-like graphs immersed in the Lorentz–Minkowski space
with constant r-mean curvature Hr and with arbitrary height. This not occurs in
Euclidean setting.
In the Lorentz–Minkowski space, taking into account that the boundedness of the support functionηof a space-like hypersurface n is equivalent to the hyperbolic image ofnbe contained into a geodesic ball ofHn, we obtain the following corollary
of Theorem3.7. We observe that it is a version of the theorems of Aiyama [1] and Xin [23] for the case of nonzero constant higher order mean curvature.
Remark 4.5 Let us observe that, given a positive real constantλ,
n =x∈Ln+1; x,x = −λ2, xn+1≥λ
is an example of complete space-like hypersurface with positive constantr-mean curvature and with one end, which is divergent. On the other hand, the hyperbolic image ofnis exactlyHn.
5 An application to the spatially closed space–times
We recall that a Lorentzian product space−R×Mn is said to bespatially closed
when its Riemannian fiberMnis compact (for a thorough discussion about this class of space–times, see for example [4,5]). In the next result, we detect a topological property which constitutes an obstruction for the existence of complete space-like hypersurfaces properly immersed with positive constant r-mean curvature into a spatially closed Lorentzian product space: the property of having only one end.
Theorem 5.1 Letnbe a complete properly immersed space-like hypersurface of a spatially closed Lorentzian product space−R×Mn. Suppose that one of the following conditions is satisfied:
(a) The Riemannian fiber Mn has nonnegative Ricci curvatureRicM andn has positive constant mean curvature H .
(b) The Riemannian fiber Mnhas nonnegative constant sectional curvatureκM and
nhas positive constant r -mean curvature Hr, for some1≤r≤n.
If the support functionηofnis bounded, then the number of ends ofnis not one.
Proof Suppose on the contrary, thatnhas exactly one endCn. Sincenis a
space-like hypersurface properly immersed in−R×Mn and since the Riemannian fiber Mnis compact,Cnmust go up or down, but not both. Assume, for example, thatCn
goes down; thennhas a highest point. Since the vertical translation is an isometry of −R× Mn, as well as reflection through each slice Mt = {t} × Mn, we can
apply Alexandrov reflection coming down from abovenwith slicesMt. Sincen
is noncompact, it is not invariant by symmetry in any slice Mt; so, there not exists a
first point of contact of the symmetry of the part ofnaboveMt, with the part ofn
belowMt.
Consequently, since we are supposing that the support functionηofnis bounded, we can obtain a compact space-like hypersurfacetnwith boundary into a sliceMt,
having positive constantr-mean curvature Hr and whose vertical height is greater
thansup|η|−1
Hr1/r
. This contradicts the estimate of Theorem3.3. ⊓⊔
Geometrically, the boundedness of the support functionηmeans that, at each point
p ∈ n, 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)do not approach the speed of light onn (cf. [21, Sects. 2.1, 3.1], and [15]; see also [19, Chap. 12]).
Ackowledgments The first author is partially supported by FUNCAP, Brazil. The second author thanks the hospitality of the Departamento de Matemática of the Universidade Federal do Ceará during his post-doctoral work in 2007. We want to thank A. Caminha, J.H. de Lira and the referee for their valuable suggestions and comments.
References
1. Aiyama, R.: On the Gauss map of complete space-like hypersurfaces of constant mean curvature in Minkowski space. Tsukuba J. Math.16, 353–361 (1992)
2. Albujer, A.L., Alías L.J.: Calabi–Bernstein results for maximal surfaces in Lorentzian products spaces. http://arxiv.org/abs/0709.4363(preprint)
3. Alías, L.J., Brasil, A., Colares, A.G. Jr..: Integral formulae for spacelike hypersurfaces in conformally stationary spacetimes and applications. Proc. Edinb. Math. Soc.46, 465–488 (2003)
4. Alías, L.J., Colares, A.G.: Uniqueness of spacelike hypersurfaces with constant higher order mean curvature in Generalized Robertson–Walker spacetimes. Math. Proc. Camb. Philos. Soc.143, 703–729 (2007)
5. Alías, L.J., Romero, A., Sánchez, M.: Uniqueness of complete spacelike hypersurfaces of constant mean curvature in Generalized Robertson–Walker spacetimes. Gen. Relat. Gravit.27, 71–84 (1995) 6. Barbosa, J.L., Colares, A.G.: Stability of hypersurfaces with constantr-mean curvature. Ann. Glob.
Anal. Geom.15, 277–297 (1997)
7. Calabi, E.: Examples of Bernstein problems for some nonlinear equations. Proc. Symp. Pure Math.15, 223–230 (1970)
8. Caminha, A.: A rigidity theorem for complete CMC hypersurfaces in Lorentz manifolds. Differ. Geom. Appl.24, 652–659 (2006)
9. Cheng, S.Y., Yau, S.T.: Maximal spacelike hypersurfaces in the Lorentz–Minkowski space. Ann. Math.104, 407–419 (1976)
10. Cheng, X., Rosenberg, H.: Embedded positive constantr-mean curvature hypersurfaces inMm×R. An. Acad. Bras. Cienc.77(2), 183–199 (2005)
11. Elbert, M.F.: Constant positive 2-mean curvature hypersurfaces. Ill. J. Math.46, 247–267 (2002) 12. Hardy, G., Littlewood, J.E., Pólya, G.: Inequalities, 2nd edn. Cambridge Mathematical Library,
Cambridge (1989)
13. Hawking, S.W., Ellis, G.R.: The large scale structure of spacetime. Cambridge University Press, Cambridge (1973)
14. Hoffman, D., de Lira, J.H., Rosenberg, H.: Constant mean curvature surfaces inM2×R. Trans. Am. Math. Soc.358(2), 491–507 (2006)
15. Latorre, J.M., Romero, A.: Uniqueness of noncompact spacelike hypersurfaces of constant mean curvature in generalized Robertson—Walker spacetimes. Geom. Dedic.93, 1–10 (2002)
16. de Lima, H.F.: A sharp height estimate for compact spacelike hypersurfaces with constantr-mean curvature in the Lorentz–Minkowski space and application. Differ. Geom. Appl. (in press, 2008) 17. López, R.: Area monotonicity for spacelike surfaces with constant mean curvature. J. Geom. Phys.
52, 353–363 (2004)
18. Marsdan, J., Tipler, F.: Maximal hypersurfaces and foliations of constant mean curvature in general relativity. Bull. Am. Phys. Soc.23, 84 (1978)
19. O’Neill, B.: Semi-Riemannian Geometry with Applications to Relativity. Academic Press, London (1983)
21. Sachs, R.K., Wu, H.: General relativity for mathematicians. In: Graduate Texts in Mathematics. Springer, New York (1977)
22. Stumbles, S.: Hypersurfaces of constant mean extrinsic curvature. Ann. Phys.133, 28–56 (1980) 23. Xin, Y.L.: On the Gauss image of a spacelike hypersurface with constant mean curvature in Minkowski
