Uniqueness of Noncompact Spacelike

Hypersurfaces of Constant Mean Curvature

in Generalized Robertson–Walker Spacetimes

Dedicated to the memory of Professor Andre Lichnerowicz

JOSE M. LATORRE and ALFONSO ROMERODepartamento de Geometrıa y Topologıa, Facultad de Ciencias, Universidad de Granada,18071-Granada, Spain. e-mail: aromero@goliat.ugr.es

(Received: 12 July 2000; accepted in final form: 27 June 2001)

Abstract. On any spacelike hypersurface of constant mean curvature of a GeneralizedRobertson–Walker spacetime, the hyperbolic angle y between the future-pointing unit normalvector field and the universal time axis is considered. It is assumed that y has a local maximum.A physical consequence of this fact is that relative speeds between normal and comoving

observers do not approach the speed of light near the maximum point. By using a developmentinspired from Bochner’s well-known technique, a uniqueness result for spacelike hypersurfacesof constant mean curvature under this assumption on y, and also assuming certain matter

energy conditions hold just at this point, is proved.

Mathematics Subject Classifications (2000). Primary 53C42; Secondary 53C50, 53C80.

Key words. Bochner–Lichnerowicz’s formula, constant mean curvature, GRW spacetime,

spacelike hypersurface.

1. Introduction

Spacelike hypersurfaces of constant mean curvature in a spacetime are critical points

of the area functional under a certain volume constraint [5] (see also [4]). Such hyper-

surfaces play an important part in Relativity since it was noted that they can be used

as initial hypersurfaces where the constraint equations can be split into a linear sys-

tem and a nonlinear elliptic equation [8, 12, 14]. A summary of other reasons justi-

fying the study of these hypersurfaces in Relativity can be found in [13]. In this

paper, we will consider spacelike hypersurfaces of constant mean curvature in the

family of cosmological models called generalized Robertson–Walker (GRW) space-

times. GRW spacetimes are warped products of a (negatively definite) universal time

as base and an arbitrary Riemannian manifold as fiber (see Section 2). This notion

was introduced in [1–3] (see also [17, 19] for a systematic study of the geometry of

such Lorentzian manifolds). Thus our ambient spacetimes widely extend to those

which are classically called Robertson–Walker spacetimes. GRW spacetimes include,

for instance, the Einstein–de Sitter spacetime, the Friedmann cosmological models

Geometriae Dedicata 93: 1–10, 2002. 1# 2002 Kluwer Academic Publishers. Printed in the Netherlands.

and the static Einstein spacetime, as well as other relevant geometric models such as

the De Sitter spacetime. On the other hand, small deformations of the metric on the

fiber of classical Robertson–Walker spacetimes fit into the class of GRW spacetimes

and also conformal changes of the metric of a GRW spacetime with a t-dependent

conformal factor produce new GRW ones. Note that a GRW spacetime is not neces-

sarily spatially homogeneous, as in the classical cosmological models case. The spa-

tial homogeneity is, of course, appropriate to consider the universe in the large.

However, to consider a more accurate scale this assumption could be not realistic

and GRW spacetimes could be suitable spacetimes to model universes with inhomo-

geneous spacelike geometries [16].

In previous papers [2–4], compact spacelike hypersurfaces of constant mean cur-

vature were studied in these ambient spacetimes. The compacteness assumption is

natural if we consider spatially closed GRW spacetimes as cosmological models.

Note that the existence of a compact spacelike hypersurface in a GRW spacetime

implies that it is spatially closed (see Section 2). Thus, there is no compact spacelike

hypersurface in a GRW spacetime not spatially closed (or open, in classical terminol-

ogy). On the other hand, in the above quoted references the main tool are several

Minkowski-type integral formulas which work in the compact case. In this paper,

we will study spacelike hypersurfaces of constant mean curvature in (not necessarily

spatially closed) GRW spacetimes. As we will comment later, completeness on the

hypersurface is not related to our physical setting. So we will adopt a local viewpoint

to set up our main results.

Our approach uses a distinguished function on the spacelike hypersurface M as a

fundamental tool. Namely, the hyperbolic angle y between the future-pointing unit

normal vector field N (see the next section for the definition) and a natural unit time-

like vector field on the GRW spacetime �M: the coordinate vector field induced by the

universal time on �M, @t (which defines the time-orientation of �M). In a GRW space-

time �M the integral curves of @t are called comoving observers and @tð pÞ, p 2 �M, is

called an instantaneous comoving observer [18, p. 43]. If p is a point of a spacelike

hypersurface M in �M, among the instantaneous observers at p, @tð pÞ and Np appear

naturally. So, from the orthogonal decomposition Np ¼ eð pÞ@tð pÞ þNFp , we have

that cosh yð pÞ coincides with the energy eð pÞ ¼ �hNp; @tð pÞi that @tð pÞ meausures

for Np. On the other hand, the speed kvð pÞk of the velocity vð pÞ :¼ ð1=eð pÞÞNFp that

@tð pÞ meausures for Np satisfies kvð pÞk2 ¼ tanh2 yð pÞ, [18, pp. 45, 67].In the special but important case when M is a spacelike slice, i.e. a t ¼ constant

hypersurface in a GRW spacetime, we have kvk ¼ 0, i.e y 0. In fact, it is easily seen

that this property characterizes such a family of spacelike hypersurfaces.

Now consider a compact spacelike hypersurface M in a (necessarily spatially

closed) GRW spacetime, then kvk, as a function on M, attains a global maximum,

so that kvk do not approach to light speed 1 onM. In this direction, the natural gen-

eralization of the compacteness of M would be to assume that sup kvk<1 holds on

allM. However, this natural assumption is much too weak. In order to support this

assertion, note that a closed (and, hence, inextendible) spacelike hypersurface does

2 JOSE M. LATORRE AND ALFONSO ROMERO

not satisfy this condition, in general. For instance, the graph in L2 :¼ ðR2;

dx2 � dy2Þ of a smooth function f : R�!R such that

f 0ðxÞ < 1 if jx j< 1 and f 0ðxÞ ¼ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi1� expð� jx jÞ

pif jx j 5 1

is clearly a closed subset of L2, and it is easily seen that sup kvk ¼ 1. Note that this

spacelike graph is not complete because its length is finite [10]. On the other hand,

the same comment is also true if M is assumed to be geodesically complete. For

instance, on the spacelike hyperboloid

H2 :¼ fðx; y; zÞ 2 L3 : x2 þ y2 � z2 ¼ �1; z > 0g

in L3 :¼ ðR3; dx2 þ dy2 � dz2Þ we also have sup kvk ¼ 1. A way to control the rela-

tive speeds (locally) on M would be to assume that y has a local maximum at some

point p0 2M, which is equivalent to saying that the relative speeds satisfy kvð pÞk

4 kvð p0Þk < 1 for p near p0. This will be the assumption we will impose on the space-

like hypersurfaces of constant mean curvature of GRW spacetimes in this paper.

Recall that, in any GRW spacetime, every spacelike slice t ¼ t0 is totally umbilic

with constant mean curvature H ¼ f 0ðt0Þ=fðt0Þ (see the next section for more details).

So, under that assumption on y and under certain matter energy conditions at this

point, which have a clear physicalmeaning, we prove a uniqueness result, Theorem4.2,

which roughly says that under these assumptions a spacelike hypersurface of con-

stant mean curvature is a spacelike slice near the local maximum. In particular, in

Corollary 4.4 we generalize, the uniqueness results given by Corollaries 5.3 and

5.4 in [2]. Our technique is completly different from the one in [2]. In fact, our

approach works in the noncompact case and, of course, also in the compact one, giv-

ing, in particular, new proofs of Corollaries 5.3 and 5.4 in [2]. Our philosophy here

follows in spirit the well-known arguments of the so-called Bochner technique (see,

for instance, [21]). The fundamental fact in this paper is a differential inequality,

Proposition 3.1, which gives a lower estimate of the Laplacian of sinh2 y on the

spacelike hypersurface of constant mean curvature M. This inequality is derived

from the classical Bochner–Lichnerowicz formula (see Section 3) and takes a good

look at the critical points of y, Lemma 4.1. In fact, this is the key result in obtaining

our main goal, Theorem 4.2.

2. Preliminaries

Let ðF; gÞ be an n-dimensional (connected) Riemannian manifold and let I be an open

interval in R endowed with the metric �dt2. Throughout this paper we will denote by�M the ðnþ 1Þ-dimensional product manifold I� F with the Lorentzian metric

h;i ¼ �p�I ðdt2Þ þ f 2ðpIÞp�FðgÞ; ð1Þ

where f > 0 is a smooth function on I, and pI and pF denote the projections onto I

and F, respectively. That is, �M is a Lorentzian warped product, in the sense of [6] and

UNIQUENESS OF NONCOMPACT SPACELIKE HYPERSURFACES 3

[15], with base I, fiber F and warping function f. As introduced in [2], we will refer to�M as a Generalized Robertson–Walker (GRW ) spacetime.

Given an n-dimensional manifold M, an immersion x :M�! �M is said to be

spacelike if the Lorentzian metric on �M given by (1) induces, via x, a Riemannian

metric on M. As usual, the induced metric will be also denoted by h ; i and we will

refer to x :M�! �M as a spacelike hypersurface M.

The unitary timelike vector field @t :¼ @=@t 2 Xð �MÞ determines a time-orientation

on the Lorentzian manifold �M. Then the time-orientatibily of �M allows us to con-

sider, for each spacelike hypersurface M in �M, N 2 X?ðMÞ as the only globally

defined unitary timelike vector field normal to M in the same time-orientation of @t.

From the wrong way Cauchy–Schwarz inequality (see, for instance, [15, Prop. 5.30]),

we have hN; @ti4 �1 and hN; @ti ¼ �1 at a point p if and only if Nð pÞ ¼ @tð pÞ.

In fact, hN; @ti ¼ � cosh y, where y is the hyperbolic angle, at each point, between

the unit timelike vectors N and @t. In what follows, we will refer to y as the hyper-

bolic angle between M and @t.

For any spacelike hypersurface M, we put t :¼ pI � x :M�! I its projection on I

and �x :¼ pF � x :M�!F its projection on F. The map �x is not only smooth. In fact,

it is easy to see that

gðd �xðXÞ; d �xðXÞÞ51

fðtÞ2hX;Xi; ð2Þ

where fðtÞ :¼ f � pI � x 2 C1ðMÞ for any tangent vector X to M and, hence, �x is

a local diffeomorphism; that is, the spacelike hypersurface and the fiber are

locally diffeomorphic. If M is assumed to be geodesically complete and the func-

tion fðtÞ is bounded away from 0 on M, then from [9, Lemma 7.3.3] we conclude

that �x is a covering map and, in particular, an open map. As a consequence,

there is no compact spacelike hypersurface in a GRW spacetime with noncompact

fiber.

It should be noted that �x can be also forced to be a covering map from different

assumptions. In fact, ifM is just edgelessly immersed (e.g. x is a proper map) and if f

does not grow too quickly (specifically,RI 1=f is infinite at both ends of the interval

I), then �x is a covering map [11, Theor. 4.4]. On the other hand, if t ¼ pI � x is a

constant t0 onM, then this covering map is a local homothety and the equality holds

in (2) at t ¼ t0. As has been used before (see [2]), a spacelike hypersurface such

that pI � x is a constant t0, i.e. such that xðMÞ is contained in ft0g � F, will be called

a spacelike slice.

Now let us introduce on �M the timelike vector field K 2 Xð �MÞ given by

K ¼ fðpIÞ@t. This vector field has a nice geometrical property that says, in particular,

that it is conformal and this will be the key to obtaining the main tools which follow.

In order to decide when the hypersurface is a spacelike slice, we have to see when the

hypersurface is orthogonal to @t or, equivalently, orthogonal to K. From the rela-

tionship between the Levi-Civita connections of �M and those of the base and the fiber

4 JOSE M. LATORRE AND ALFONSO ROMERO

[15, Prop. 7.35], or [6, Chap. 3]), it is not difficult to get

�HZK ¼ f 0ðpIÞZ; ð3Þ

for any vector field Z on �M, where �H is the Levi-Civita connection of �M.

From the Gauss and Weingarten formulas we have the relationship between the

Levi-Civita connection of �M and the spacelike hypersurface one, denoted by H,

�HXY ¼ HXY� hAX;Y iN; ð4Þ

where X;Y 2 XðMÞ and A is the shape operator associated to N, which is defined by

AX :¼ � �HXN: ð5Þ

Recall that the mean curvature function corresponding to N is defined to be

H :¼ �ð1=nÞtrðAÞ.

Taking tangential components in (3), we get

HYKT þ fðtÞhN; @tiAY ¼ f 0ðtÞY; ð6Þ

where

KT ¼ fðtÞ@Tt ¼ Kþ hK;NiN

is the tangential component of K. Note, as an easy consequence of (6), that the shape

operator A of the spacelike slice t ¼ t0 satisfies A ¼ �f 0ðt0Þ=fðt0Þ I, where I is the

identity transformation. Therefore, it is totally umbilic with constant mean curvature

H ¼ f 0ðt0Þ=fðt0Þ.

Contracting (6), we get

fðtÞdivð@Tt Þ þ hHfðtÞ; @Tt i þ fðtÞhN; @titrðAÞ ¼ nf0ðtÞ; ð7Þ

where div denotes the divergence on M. Now, taking into account

Ht ¼ �@Tt ð8Þ

and (7), we get

Dt ¼ �f 0ðtÞ

fðtÞfnþ j Ht j2g � nHhN; @ti; ð9Þ

where Dt :¼ divðHtÞ is the Laplacian of t.

The curvature tensor R of M and the curvature tensor �R of �M are related by the

so-called Gauss equation

hRðX;Y ÞV;Wi

¼ h �RðX;YÞV;Wi þ hAY;WihAX;Vi � hAY;VihAX;Wi; ð10Þ

for any tangent vector fields X;Y;V;W 2 XðMÞ. From (10), we derive the relation

between the Ricci tensors Ric of M and Ric of �M:

RicðX;XÞ

¼ RicðX;XÞ þ h �RðX;NÞN;Xi þ hA2X;Xi þ nHhAX;Xi ð11Þ

for all X 2 XðMÞ.

UNIQUENESS OF NONCOMPACT SPACELIKE HYPERSURFACES 5

Now, let Z be a tangent vector to �M and write

Z ¼ �hZ; @ti@t þ ZF; ð12Þ

where ZF denotes its projection on the fiber. In particular, when Z ¼ N, from (12) we

obtain j NF j2¼j Ht j2¼ sinh2 y, where y is the hyperbolic angle betweenM and @t. On

the other hand, �M being a warped product (with a one-dimensional base), it is possi-

ble to express its curvature in terms of the warping function f and the curvature of

the fiber F [15, Props. 7.42, 7.43]. So, we have$

h �Rð@t;NÞN; @ti ¼ h �Rð@t;NFÞNF; @ti ¼ �

f 00ðpIÞfðpIÞ

j Ht j2 ð13Þ

and for the Ricci tensor Ric of �M, we get

RicðZ;ZÞ ¼ RicFðZF;ZFÞ þf 00ðtÞ

fðtÞþ ðn� 1Þ

f 0ðtÞ2

fðtÞ2

� �hZF;ZFi�

� nf 00ðtÞ

fðtÞhZ; @ti

2; ð14Þ

where RicF denotes the Ricci tensor of the fiber.

We end this section recalling important energy conditions which are usually

assumed on the spacetime. Given a spacetime �M, we will say that �M obeys the time-

like convergence condition (TCC) if RicðZ;ZÞ5 0, for all timelike vector Z. It is nor-

mally argued that TCC is the mathematical translation that gravity, on average,

attracts. On the other hand, if �M satisfies the Einstein equations (in the terminology

of [18], and, in particular, with zero cosmological constant), then it obeys TCC.

From (14) it is easily seen that a GRW spacetime �M obeys TCC if and only if its

warping function satisfies f 00 4 0 and RicF5 ðn� 1Þ ff 00 � f 02� �

g.

A spacetime �M is said to have nonvanishing matter fields, or obeys the ubiquitous

energy condition [20], if RicðZ;ZÞ > 0, for all timelike vector Z. Note that this energy

condition is stronger than TCC and if a GRW spacetime �M has nonvanishing matter

fields, then f 00 < 0.

3. An Inequality Arising from Bochner–Lichnerowicz’s Formula

Recall the well-known Bochner–Lichnerowiz formula [7, p. 83]

12DðjHu j

2Þ ¼jHess u j2 þRicðHu;HuÞ þhHu;HDui

valid for any u 2 C1ðMÞ,M a Riemannian manifold with metric h;i and Ricci tensor

Ric, and where Hu denotes the gradient of u, Hess u :¼ H2u is the Hessian tensor of u,

jHess u j2 is its square algebraic trace-norm (i.e. jHess u j2:¼ trðHu �HuÞ where Hudenotes the operator defined by hHuðXÞ;Yi :¼ HessðuÞðX;YÞ, for all X;Y 2 XðMÞ),

and Du is the Laplacian of u.

$ Note in this paper we are using the notation �RðX;YÞZ ¼ �HX �HYZ� �HY �HXZ� �H½X;Y�Z for the curvature

tensor of �M.

6 JOSE M. LATORRE AND ALFONSO ROMERO

Now we will apply this formula to the function u ¼ t on the spacelike hypersurface

M. By using (6) and (8) we get

HXHt ¼ �f 0ðtÞ

fðtÞXþ hN; @tiAX � Xðlog fðtÞÞHt ð15Þ

for all tangent vector field X. From the last formula we achieve

j Hess t j2 ¼ nf 0ðtÞ2

fðtÞ2þ hN; @ti

2trðA2Þ þf 0ðtÞ2

fðtÞ2j Ht j4 þ2

f 0ðtÞ2

fðtÞ2j Ht j2 þ

þ 2nHf 0ðtÞ

fðtÞhN; @ti � 2

f 0ðtÞ

fðtÞhN; @tihAHt;Hti: ð16Þ

On the other hand, from (11), (13) and (14) we obtain

RicðHt;HtÞ ¼ �ðn� 1ÞðlogfÞ00 j Ht j4 þðn� 1Þf 0ðtÞ2

fðtÞ2j Ht j2 þ

þ hN; @ti2RicFðNF;NFÞ þ nHhAHt;Hti þ hA2Ht;Hti: ð17Þ

A direct computation from (3) gives

HhN; @ti ¼ AHt�f 0ðtÞ

fðtÞhN; @tiHt: ð18Þ

Now, we assume that H is constant. Under this assumption, from (9) and (18)

we get

hHt;HDti ¼ �ðlog fÞ00fnþ j Ht j2g j Ht j2 �2f 0ðtÞ

fðtÞHessðtÞðHt;HtÞ þ

þ nHf 0ðtÞ

fðtÞhN; @ti j Ht j2 �nHhAHt;Hti ð19Þ

Now recall that Schwarz’s inequality gives nH2 4 trðA2Þ, and equality holds if and

only if A is, pointwise, a multiple of the identity transformation, i.e. M is totally

umbilic in �M. Using this fact, from (16), (17) and (19) and taking into account

j Ht j ¼ sinh y, we obtain from Bochner–Lichnerowicz’s formula:

PROPOSITION 3.1. Let �M be a GRW spacetime and let M be a spacelike hyper-

surface of constant mean curvature H in �M. The hyperbolic angle y between M and @tsatisfies the differential inequality

12D sinh2 y5 n

f 0ðtÞ2

fðtÞ2þ hA2Ht;Hti � 2

f 0ðtÞ

fðtÞHessðtÞðHt;HtÞ þ

þ nH2 cosh2 yþ cosh2 yRicFðNF;NFÞ � 2nHf 0ðtÞ

fðtÞcosh y þ

þ 2f 0ðtÞ

fðtÞcosh yhAHt;Hti � nH

f 0ðtÞ

fðtÞcosh y sinh2 y þ

UNIQUENESS OF NONCOMPACT SPACELIKE HYPERSURFACES 7

þ ð2nþ 1Þf 0ðtÞ2

fðtÞ2sinh2 y� n

f 00ðtÞ

fðtÞsinh2 y þ

þ ðnþ 1Þf 0ðtÞ2

fðtÞ2sinh4 y� n

f 00ðtÞ

fðtÞsinh4 y

and the equality holds if and only if M is totally umbilic in �M.

4. Main Results

It should be noticed that the right-hand side of differential inequation obtained in

Proposition 3.1 is very complicated and not signed, in general. However, if we

restrict our attention only on the critical points of y we get:

LEMMA 4.1. Let �M be a GRW spacetime and let M be a spacelike hypersurface of

constant mean curvature H in �M. If the hyperbolic angle y between M and @t has a

critical point p0, then

12D sinh2 yð p0Þ5 n 1þ 1

2 sinh2 yð p0Þ

� �H�

f 0ðt0Þ

fðt0Þcosh yð p0Þ

� �2

�

� nf 00ðt0Þ

fðt0Þsinh2 yð p0Þ cosh

2 yð p0Þ þ cosh2 yð p0ÞRicFq0ðNF;NFÞ;

where t0 :¼ tð p0Þ and q0 :¼ �xð p0Þ.

Proof. From (18), we have at the critical point p0,

AHtp0 ¼ �f 0ðt0Þ

fðt0Þcosh yð p0ÞHtp0 ð20Þ

On the other hand, we also have at p0

HessðtÞðHt;HtÞp0 ¼ 0: ð21Þ

Now, to verify the inequality we use (20) and (21) in the inequality of Proposi-

tion 3.1, and throw away some nonnegative terms. &

THEOREM 4.2. Let �M be a GRW spacetime and let x :M�! �M be a spacelike

hypersurface of constant mean curvature. Assume that the hyperbolic angle between M

and @t attains a local maximun at some point p0 2M. If either

ð1Þ f 00ðtð p0ÞÞ < 0,

ð2Þ The Ricci tensor of the fiber of �M is positive semi-definite at �xð p0Þ.

or

ð10Þ f 00ðtð p0ÞÞ4 0,

ð20Þ The Ricci tensor of the fiber of �M is positive definite at �xð p0Þ,

then, there exists an open neighborhood U of p0 in M which is a spacelike slice.

8 JOSE M. LATORRE AND ALFONSO ROMERO

Proof. From Lemma 4.1, we have

12D sinh2 yð p0Þ5 � n

f 00ðt0Þ

fðt0Þsinh2 yð p0Þ cosh

2 yð p0Þ þ

þ cosh2 yð p0ÞRicFq0 ðNF;NFÞ ð22Þ

and now we know

D sinh2 yð p0Þ4 0: ð23Þ

Consider now the first couple of assumptions. Using RicFq0ðNF;NFÞ5 0 in (22) and

taking into account (23), we obtain

f 00ðt0Þ sinh2 yð p0Þ ¼ 0;

but we are also assuming f 00ðt0Þ < 0, which gives yð p0Þ ¼ 0. Finally, we know that

yð pÞ4yð p0Þ on some open neighborhood U of p0. Therefore, yð pÞ ¼ 0, for all p

in U, which means that U is a spacelike slice.

Next, consider the second couple of assumptions. From f 00ðt0Þ4 0 we get

RicFq0ðNF;NFÞ ¼ 0;

but now we know that RicFq0 is positive definite, and therefore NF ¼ 0 at this point.

We end the proof recalling that jNF j2¼ sinh2y, which implies again yð p0Þ ¼ 0.

As immediate consequences we get the following corollaries:

COROLLARY 4.3. The only analytical spacelike hypersurfaces of constant mean

curvature in a GRW spacetime, whose hyperbolic angle with @t attains a local max-

imum, where one of the two couple of assumptions of Theorem 4:2 are fulfilled, are open

subsets of spacelike slices.

COROLLARY 4.4. The only spacelike hypersurfaces of constant mean curvature in a

GRW spacetime, whose hyperbolic angle with @t attains a global maximum, where one

of the two couple of assumptions of Theorem 4:2 are fulfilled, are open subsets of

spacelike slices.

Note that this result widely extends and give a new proof of [2, Cors. 5.3, 5.4].

To end this paper, we come back to the Physical motivation in Section 1. So, we

have obtained:

In a GRW spacetime obeying the ubiquitous energy condition, the onlyspacelike hypersurfaces of constant mean curvature possessing a globalmaximun of the observed speed by the comoving observers are open sub-sets of spacelike slices.

Acknowledgements

The authors would like to thank Miguel Sanchez for his clarifying comments on the

physical interpretation of geometric assumptions in this paper. We also thank the

referee for his criticism and valuable suggestions. Partially supported by a MCYT-

FEDER Grant BFM2001-2871-C04-01.

UNIQUENESS OF NONCOMPACT SPACELIKE HYPERSURFACES 9

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10 JOSE M. LATORRE AND ALFONSO ROMERO