Spacelike Hypersurfaces of Constant Mean Curvature in Spacetimes with Symmetries

Luis J. Al´ıas1, Alfonso Romero2 and Miguel Sanchez´ 2

1 Departamento de Matem´aticas, Universidad de Murcia, 30100-Espinardo, Murcia, Spain

2 Departamento de Geometr´ıay Topolog´ıa,Universidad de Granada, 18071-Granada, Spain.

Abstract The role of some symmetries in General Relativity is discussed, espe- cially those involving a timelike conformal vector field. The importance of spacelike hypersurfaces of constant mean curvature in the study of the Einstein equation is explained. Uniqueness results of compact spacelike hypersurfaces of constant mean curvature in spacetimes which admits a timelike conformal vector field are given by means of several integral formulas. As an application, several uniqueness consequences for the spacelike constant mean curvature differential equation are obtained.

1 Introduction

Our purpose is to explain some uniqueness results of spacelike constant mean curvature hypersurfaces in certain ambient spacetimes; these hypersurfaces become relevant for the study of the Einstein equation. Our ambient spacetimes are not so restrictive as, for example, Lorentzian space forms, which have been widely studied from a purely mathematical point of view. Nevertheless, many of them contain some general symmetries such that the hypersurfaces can be seen as solutions to a partial differential equation. In the remainder of this section we explain the role of symmetries in General Relativity and define our ambient spacetimes; in Section 2 we see how spacelike hypersurfaces of constant mean curvature are useful to simplify Einstein equation; in Section 3 the associated differential equation and variational problem is introduced; in Section 4 we write some integral formulas which are our basic tool; in Section 5 the main results, essentially contained in [2], [3], [4], are explained. The concept of symmetry is basic to Physics. In General Relativity, symmetry is usually based on a local one-parameter group of isometries generated 2 Spacelike Hypersurfaces of Constant Mean Curvature by a Killing or, more generally, conformal, vector field. In fact, the main simplification for the search of exact solutions to the Einstein equation, is to assume, a priori, the existence of such symmetries [10], [12]. Even more, the use of affine and affine conformal vector fields has been useful to obtain new exact solutions [11]. We remark that a completely general approach to symmetries in General Relativity has been developed in [31]. In the above mentioned references, the causal character of the Killing or conformal vector field is not always prefixed. However, it is natural to assume that this vector field is timelike. This is supported by very well-known examples of exact solutions. At the same time, under this assumption, the integral curves of such vector field provide a privileged class of observers (in the sense of [24]) or test particles in spacetime. A Lorentzian manifold which admits a timelike Killing vector field K is called a stationary spacetime. Locally, a set of coordinates (t, x1, . . . , xn) can be chosen such that K = ∂/∂t, each ∂/∂xi is spacelike and the metric com- ponents are independent of the coordinate t. Such vector fields have shown to be useful in a purely mathematical area, in fact, they are used to classify Lorentzian manifolds of constant sectional curvature. So, in [15], a Lorentzian space form is called standard if its time-orientable double cover is stationary. In [25] the geometry of stationary spacetimes is analysed from several viewpoints, some of them of physical interest. When K is irrotational (its orthogonal distribution is involutive) the stationary spacetime is called static.A standard static spacetime is a warped product, in the sense of [21] (see also [7]), M¯ =M ×I, with base a Riemannian manifold (M, g), fiber (I, −dt2) and warping function h ∈ C∞(M), h > 0, so that the Lorentzian metric on M¯ is writen ∗ 2 ∗ 2 as πM g − h(πM ) πI (dt ). If we add in this last metric cross terms which are independent of t, the spacetime is standard stationary. Locally, every static or stationary spacetime looks like the corresponding standard one. The simplest stationary spacetimes are the Lorentzian products of a definite negative real line (I, −dt2) and a Riemannian manifold (M, g). Under several geometric assumptions a stationary spacetime is such a Lorentzian product. In fact, it was proven in [6] that a geodesically complete simply connected stationary spacetime with non-negative timelike sectional curvatures is the Lorentzian product of (R, −dt2) and a geodesically complete Riemannian manifold. Nev- ertheless, recall that, in this result, the assumed inequality for the curvature implies an inequality for the Ricci tensor which is opposite to the physically relevant timelike convergent condition (see Section 5). More generally, we will deal with Lorentzian manifolds which admit a timelike conformal vector field. As in [4], we call them conformally stationary spacetimes (CS spacetimes). If (M,¯ g¯) is an (n + 1)-dimensional CS spacetime, Luis J. Al´ıas, Alfonso Romero and Miguel Sanchez´ 3 then K will denote a timelike conformal vector field on M¯ , and ρ the smooth function on M¯ given by LK g¯ = 2ρg¯. The concept of CS spacetime is motivated in part by the following simple fact, which can be easily shown from the last equality: if a Lorentzian metricg ¯ admits a timelike conformal vector field K, then K is a timelike Killing vector field for a new Lorentzian metric pointwise conformally related tog ¯, concretely (−1/g¯(K,K))¯g. If K is a conformal vector field and its corresponding ρ is constant on all the spacetime, then K is said to be homothetical. From a physical viewpoint, for any timelike vector field K on a spacetime, the normalized vector field Z := √ 1 K is a reference frame; −g¯(K,K) i.e. a vector field each of whose integral curves is an observer [24, Definition 2.3.1]. If K is a timelike conformal vector field for (M,¯ g¯), a direct computation gives div(Z) = √ n−1 ρ, where div is the divergence with respect to the −g¯(K,K) Lorentz metricg ¯. Thus, if ρ > 0 on an open set U, then div(Z) > 0 on U, which indicates that the observers in Z are on average spreading apart in U. Similarly, ρ < 0 on U indicates that the observers come together [24, p. 121]. An interesting subclass of CS spacetimes is the family of Generalized Robertson-Walker spacetimes (GRW spacetimes). For a GRW spacetime we mean a product manifold M¯ = I × M with the Lorentzian metricg ¯ = ∗ 2 2 ∗ −πI (dt ) + f(πI ) πM (g) where πI and πM denote the projections onto I and M, respectively and g is a Riemannian metric on M. Then (M,¯ g¯) is a warped product, with base (I, −dt2), fiber (M, g) and warping function f. The notion of GRW spacetime is taken from [2],[3] (see also [1], [13], [23], [26], [27] for a systematic study of the geometry of such Lorentzian manifolds). Let us remark that, in this case, the timelike conformal vector field K may be 0 chosen as f(πI )∂/∂t, and so ρ = f (πI ). Note that, in our definition of GRW spacetime the fiber (M, g) is not assumed to be of constant sectional curvature, in general. When this holds and n = 3 the GRW spacetime is a (classical) Robertson-Walker spacetime. Thus, GRW spacetimes widely extend to Robertson-Walker spacetimes and include, for instance, Einstein-de Sitter spacetime, Friedmann cosmological models, the static Einstein spacetime and the de Sitter spacetime. Note that conformal changes of the metric of a GRW spacetime, with a conformal factor which only depends on t, produce new GRW spacetimes. Moreover, small deformations of the metric on the fiber of Robertson-Walker spacetimes also fit into the class of GRW spacetimes. Thus, a GRW spacetime is not necessarily spatially homogeneous, as in the classical cosmological models. Recall that spatial homogeneity seems appropriate just as a rough approach to consider the universe in the large. However, in order to consider it in a more accurate scale, this assumption could not be realistic and GRW spacetimes could be suitable spacetimes to model universes with inhomogeneous spacelike geometry [22]. On the other hand, the family of CS 4 Spacelike Hypersurfaces of Constant Mean Curvature spacetimes is general enough to include GRW spacetime as well as stationary spacetimes. Moreover, recall that a globally conformal Lorentzian manifold to a CS spacetime is itself a CS spacetime. Therefore, CS spacetimes include Lorentzian manifolds which are globally conformal to GRW spacetimes as well as to stationary spacetimes. We will show how the timelike conformal vector field can be used as a tool to study compact spacelike hypersurfaces of constant mean curvature in CS spacetimes (note that if a GRW spacetime admits a compact spacelike hypersurface then its fiber is also compact [2]). The topological requirement compact is assumed, on one hand, because we will use a simple but powerful tool: the classical divergence theorem; on the other, to include in our study spatially closed spacetimes.

2 Spacelike hypersurfaces as initial hypersurfaces in Relativity

Next, we will explain how spacelike hypersurfaces of constant mean curvature are interesting to study the Einstein equation. Let (M,¯ g¯) be a 4-dimensional spacetime. Let us denote by Ric and R(¯g) its Ricci tensor and its scalar curvature, respectively. Consider a stress-energy tensor ﬁeld T on M¯ , that is, a 2–covariant symmetric tensor which is assumed to satisfy some reasonable conditions from a physical viewpoint (say, T (v, v) ≥ 0 for any timelike vector v, see for example [24, Section 3.3]). It is said that (M,¯ g¯) obeys the Einstein equation with source T (and with zero cosmological constant) if 1 Ric − R(¯g)¯g = T. (1) 2 When T = 0 this equation is called the Einstein vacuum equation, and it is equivalent to: Ric = 0. (2) If (1) holds, then the following constraint equations are satisﬁed on each spacelike hypersurface S in M¯ :

R(g) − tr(A2) + (trA)2 = ϕ (3)

div(A) − ∇tr(A) = X, (4) where g is the Riemannian metric on S induced byg ¯, R(g) its scalar curvature, A is the shape operator (or Weingarten endomorphism) relative to a unit normal vector field, and ϕ ∈ C∞(S) and X ∈ X (S) depend on the stress energy tensor T , in such way that T = 0 implies ϕ = 0 and X = 0. We Luis J. Al´ıas, Alfonso Romero and Miguel Sanchez´ 5 remark that equations (3) and (4) are respectively obtained by classical Gauss and Codazzi equations for the spacelike hypersurface S of M¯ . Moreover, they can be seen as differential equations with unknows g and A. In fact, consider the easiest case (2). As in [8], [9], we set the following definitions: An initial data set for the vacuum equation (2) is a triple (S, g, A) where S is a 3-dimensional manifold, g is a Riemannian metric on S and A is a (1, 1)-tensor field self-adjoint with respect to g, which satisfy the constraint equations (3) and (4) (with ϕ = 0 and X = 0). A solution of the Cauchy problem for (2) corresponding to the initial data set (S, g, A), is a spacetime (M,¯ g¯) such thatg ¯ is Ricci-flat and there exists a spacelike embedding j : S −→ M¯ such that j∗g¯ = g and A is the shape operator with respect to a chosen unit timelike normal vector field. Thus, the Cauchy problem for the Einstein equation requires to solve the constraint equations (3) and (4) previously. We recall, following [8], the conformal method initiated by A. Lichnerowicz [17], for the vacuum Einstein equation. We choose an arbitrary Riemannian metric g0 on S and we set

1 g := φ4g , φ ∈ C∞(S), φ > 0 and B := φ6(A − tr(A)I) 0 3 and denote τ := tr(A). Then, the constraint equations (3) and (4) become

R(g ) tr(B2) 1 1 ∆0φ − 0 φ + − τ 2φ5 = 0, (5) 8 8 φ7 12

2 div0(B) − φ6∇0τ = 0. (6) 3 where ∆0, div0 and ∇0 respectively denote the Laplacian, the divergence and the gradient with respect to g0. Equation (5) is known as the Lichnerowicz equation. As in [8] and [20], assume B is a solution of the linear system

div0(B) = 0, tr(B) = 0,

4 and φ > 0 is a solution of the elliptic equation (5). If we put g := φ g0 1 1 and A := φ6 B + 3 τI, τ ∈ R, then (S, g, A) is an initial data set for the vacuum Einstein equation. Note that S is a spacelike hypersurface of constant −1 ¯ mean curvature H = 3 τ of the solution (M, g¯) of the corresponding Cauchy problem. 6 Spacelike Hypersurfaces of Constant Mean Curvature

3 Variational approach to spacelike hypersurfaces of constant mean curvature

The following variational problem [4] is interesting in Lorentzian geometry. Let (M, g) be a (connected) compact Riemannian manifold, with dimension n ≥ 2 and let f be a positive smooth function deﬁned on an open interval I of R. Consider the class of smooth real valued functions u on M such that u(M) ⊂ I and | ∇u |< f(u) on all M, where ∇u is the gradient of u with respect to g. Over this class consider the functional Z p A(u) := f(u)n−1 f(u)2− | ∇u |2dV (7) M where dV is the canonical measure given by g. For critical points of this functional, under the constraint

Z Z u ( f(t)ndt)dV = constant, (8) M t0 the Euler-Lagrange equation is written as

0 2 div( √ ∇u ) = nH − √ f (u) (n + |∇u| ) f(u) f(u)2−|∇u|2 f(u)2−|∇u|2 f(u)2 (9) | ∇u |< f(u) where H is a real number. This variational problem arises naturally from Lorentzian geometry. Consider the GRW spacetime (M,¯ g¯), where M¯ = I ×M ∗ 2 2 ∗ andg ¯ = −πI (dt ) + f(πI ) πM (g). The metric gu on M induced fromg ¯ via the graph Mu := {(u(p), p): p ∈ M} ⊂ M¯ is written as 2 2 gu := −du + f(u) g (10) which is positive deﬁnite if and only if | ∇u |< f(u) everywhere on M. In this case, A(u), given in (7), is the area of (M, gu) and (8) is a volume constraint for the graph. Moreover u is a critical point of (7) under the constraint (8) if and only if Mu has constant mean curvature, and so, (9) is the constant mean curvature equation for spacelike graphs in M¯ . In a more general setting, consider an (n + 1)-dimensional Lorentzian manifold (M,¯ g¯) and a spacelike immersion x : S −→ M¯ of an n-dimensional manifold S into M¯ ; i.e. an immersion such that the induced metric x∗g¯ on S is Riemannian. As usual, we will refer to x as the spacelike hypersurface S. Given −1 a unit normal vector ﬁeld N on S, then the mean curvature H = n tr(A), where A is the shape operator associated to N, is a function on the spacelike Luis J. Al´ıas, Alfonso Romero and Miguel Sanchez´ 7 hypersurface S. As it is seen in [5], S is of constant mean curvature (i.e. H is constant) if and only if x is a critical point of the area functional under a suitable constraint volume.

4 Integral formulas

Given a spacelike hypersurface x : S −→ M¯ , we put N ∈ X ⊥(S) as the (globally deﬁned) unitary timelike vector ﬁeld normal to S and in the same time-orientation of K, so thatg ¯(K,N) < 0 holds everywhere on S. We put KT := K +g ¯(K,N)N along x, so KT ∈ X (S) is the tangential component of K. A direct computation using Gauss and Weingarten formulas for S gives

div(KT ) = nρ + ng¯(K,N)H (11) where div denotes the divergence with respect to the induced Riemannian metric on S. Now using Codazzi equation we get

div(AKT ) = −ng¯(∇H,K) − Ric(KT ,N) − nρH − g¯(K,N)tr(A2) (12) where ∇H denotes the gradient of H and Ric is the Ricci tensor ofg ¯. Finally we have T div((∇¯ N K) ) = ng¯(∇¯ ρ, N) + Ric(K,N) (13)

T where (∇¯ N K) denotes, as above, tangential component, ∇¯ the Levi-Civita connection ofg ¯, and ∇¯ ρ is the gradient of ρ with respect tog ¯. Next suppose that S is compact. Integrating (11) over S and using the divergence theorem we get Z {ρ +g ¯(K,N)H}dV = 0 (14) S Now, from (11) and (12) we obtain Z {(n − 1)¯g(∇H,K) + Ric(KT ,N) +g ¯(K,N)(tr(A2) − nH2)}dV = 0 (15) S

It is relevant in formula (15) thatg ¯(K,N) < 0 and tr(A2) − nH2 ≥ 0 with equality if and only if S is totally umbilical; i.e. A = µI for some µ ∈ C∞(S). Finally, (13) directly gives Z {ng¯(∇¯ ρ, N) + Ric(K,N)}dV = 0 (16) S 8 Spacelike Hypersurfaces of Constant Mean Curvature

5 Consequences of the integral formulas

As a ﬁrst application we begin by observing how the existence of a compact hypersurface of constant mean curvature imposes some geometric restriction to the ambient spacetime M¯ . By a look at (14) we have [4]

PROPOSITION 5.1 Let M¯ be a CS spacetime. If M¯ admits a compact spacelike hypersurface with H = constant, then there exists some point p0 ∈ M¯ such that ρ(p0) > 0 (resp. ρ(p0) < 0, ρ(p0) = 0) whenever H > 0 (resp. H < 0, H = 0).

A direct consequence of (15) is the following result [4]

THEOREM 5.2 If a CS spacetime M¯ is Einstein, then every compact spacelike hypersurface of constant mean curvature in M¯ is totally umbilical.

This result can be seen as a Lorentzian extension of the main result by Y. Katsurada in [16]. In particular, Theorem 5.2 applies to the case in which M¯ is a GRW spacetime I×M with warping function f. In this case, it is not difficult to see that Ric =c ¯g¯,c ¯ ∈ R, if and only if: (1) (M, g) is of constant Ricci 00 c¯ 02 c¯ 2 c curvature c and (2) f satisfies f − n f = 0 and f − n f + n−1 = 0. Moreover, M¯ has constant sectional curvatureκ ¯ if and only if M has constant sectional curvature κ and f satisfies these equations with c = (n − 1)κ andc ¯ = nκ¯. The positive solutions of these differential equations can be easily collected (see [3] for details) obtaining a wide family of Einstein GRW spacetimes. In particular, Theorem 5.2 applies to the case of Einstein GRW spacetimes, extending also the case of De Sitter spacetime previously studied in [19]. Of course, in Theorem 5.2 we cannot expect, in general, that the hypersurface must be orthogonal to the timelike conformal vector field K. However, this holds under some extra assumption as shows the following result [4]

THEOREM 5.3 Let us consider a compact spacelike hypersurface x : S −→ M¯ , in an Einstein CS spacetime M¯ , with constant mean curvature and non- positive deﬁnite Ricci tensor. If S is orthogonal to K at some point, then it is orthogonal to K everywhere.

The proof follows from the fact that S is totally umbilical by Theorem 5.2, and so, KT is a conformal vector ﬁeld on S. Next from a well-known theorem by K. Yano [29] we get that KT is parallel and, therefore, using our last assumption, identically zero. Given a spacelike hypersurface x : S −→ M¯ in a GRW spacetime (M,¯ g¯), we will say that it is a spacelike slice when πI ◦ x= constant; i.e. when Luis J. Al´ıas, Alfonso Romero and Miguel Sanchez´ 9 it is orthogonal to ∂/∂t. This notion includes to the embedded spacelike slices t=constant, which are geometric realizations of the trivial solutions u=constant to the equation (9). Observe that, in any GRW spacetime, every spacelike slice t = t0 is totally umbilical with constant mean curvature 0 H = f (t0)/f(t0). So, the converse appears as a natural problem, i.e. to decide under what assumptions a spacelike hypersurface of constant mean curvature is totally umbilical and a spacelike slice. We remark, [3], that in T the case that M¯ is a GRW spacetime then K = −f(πI ◦ x)∇(πI ◦ x) and, therefore, a consequence of Theorem 5.3 is, [3], every compact spacelike hypersurface in an Einstein GRW spacetime with constant mean curvature and non-positive deﬁnite Ricci tensor must be a spacelike slice. In particular, if a GRW spacetime has constant sectional curvature κ¯, then the only compact spacelike hypersurfaces with constant mean curvature H such that H2 ≥ κ¯ are the spacelike slices (de Sitter spacetime shows that there exist compact spacelike hypersurfaces of constant mean curvature with H2 < κ¯ which are not spacelike slices). This result gives the following uniqueness theorem for equation (9), [3]

THEOREM 5.4 Let (M, g) be a compact Riemannian manifold with zero curvature (resp. with sectional curvature −1). Let H be a real number and let f : I −→ R+ be one of the functions f(t) = a, a > 0 or f(t) = a exp (bt), a > 0, b 6= 0 with b2 ≤ H2 (resp. f(t) = t + a, f(t) = a exp bt − (1/4ab2) exp (−bt), 2 2 a 6= 0, b > 0 with b ≤ H , or f(t) = a1 cos (bt) + a2 sin (bt), b > 0, 2 2 2 a1 + a2 = 1/b ). Then, the only solutions to the constant mean curvature equation (9) are the constant functions.

Next, we will change the assumption to be Einstein previously imposed to the ambient spacetimes by others with a more physical meaning. Given a spacetime M¯ , we will say that M¯ obeys the timelike convergence condition (TCC) if Ric(Z,Z) ≥ 0, for all timelike tangent vector Z. It is normally argued that TCC is the mathematical translation that gravity, on average, attracts, [24]. On the other hand, if M¯ satisfies the Einstein equation (1) then it obeys TCC for physically reasonable T [24, Ex. 4.3.7]. A weaker energy condition is the null convergence condition (NCC) which reads Ric(Z,Z) ≥ 0 for all null tangent vector Z. So, this energy condition only applies to light particles. Clearly, a continuity argument shows that TCC implies NCC. A spacetime M¯ is said to have non-vanishing matter fields, or obeys the ubiquitous energy condition [28], if Ric(Z,Z) > 0, for all timelike tangent vector Z. This last energy condition is stronger than TCC and roughly means a real presence of matter at any point of the spacetime. It is easily seen that a GRW spacetime M¯ obeys TCC if and only if its warping function satisfies f 00 ≤ 0 and the Ricci 10 Spacelike Hypersurfaces of Constant Mean Curvature

¡ ¢ tensor of the fiber Ric satisfies Ric ≥ (n−1) ff 00 − f 02 g. On the other hand, if a GRW spacetime M¯ has non-vanishing matter fields then f 00 < 0. Now we consider the case in which the ambient spacetime admits certain homothetic symmetries. As an inmediate consequence of integral formulas (15) and (16), using TCC, we have [4]

THEOREM 5.5 Let M¯ be a CS spacetime with ρ = constant. If M¯ obeys TCC then every compact spacelike hypersurface of constant mean curvature in M¯ is totally umbilical.

Obviously, an extra conclusion on the Ricci tensor of the ambient spacetime is obtained in Theorem 5.5, namely Ric(N,N) = 0. Thus, this result imposes a geometric restriction to the existence of compact spacelike hypersurfaces of constant mean curvature. We remark that Theorem 5.5 points out in the same direction that a previous one stated for Riemannian ambient spaces by K. Yano in [30]. Next, from (14) and the previous result we can state [4]

THEOREM 5.6 Let M¯ be a CS spacetime with ρ = 0 (i.e. M¯ is stationary). If M¯ obeys TCC then every compact spacelike hypersurface with signed mean curvature function H ≥ 0 or H ≤ 0 in M¯ is totally geodesic.

Recall that when M¯ is the Lorentzian product of (I, −dt2) and (M, g), without any curvature assumption, the following fact holds [2]: the only compact spacelike hypersurfaces with signed mean curvature function are the spacelike slices, which are totally geodesic. Now we pay attention to a more general situation [4]

THEOREM 5.7 Let M¯ be a CS spacetime and let x : S −→ M¯ be a compact spacelike hypersurface in M¯ . Assume that at any point p in S the fol- ¯ lowing conditions hold: (a) Ric(Kx(p), v) = 0 for all v ∈ Tx(p)M such that ¯ g¯(Kx(p), v) = 0, and (b) Ric(w, w) ≥ 0 for all null vector in Tx(p)M. If x has constant mean curvature, then it is totally umbilical.

It is not diﬃcult to see that Ric(KT ,N) ≤ 0 is obtained from the assumptions in Theorem 5.7, and now the conclusion follows using this inequality in formula (15). We would like to note that assumption (a) in Theorem 5.7 is automatically satisﬁed in the case of a GRW spacetime. On the other hand, assumption (b) holds whenever the spacetime obeys NCC. Thus, as a relevant case of Theorem 5.7, we have [2], [4] Luis J. Al´ıas, Alfonso Romero and Miguel Sanchez´ 11

COROLLARY 5.8 If a GRW spacetime obeys NCC, then every compact spacelike hypersurface of constant mean curvature is totally umbilical.

In order to go further, observe that in Theorem 5.7 we have also obtained the extra conclusion Ric(KT ,N) = 0. In the case that M¯ is a GRW spacetime, this gives

Ric(N ∗,N ∗) = (n − 1)(f(t).f 00(t) − f 0(t)2)g(N ∗,N ∗) in terms of the Ricci tensor, Ric, of the ﬁber, (M, g), where t = πI ◦ x and ∗ N = dπM (Nˆ). Under a bit stronger assumption than TCC (see the comments below Theorem 5.4) we get [2]

THEOREM 5.9 If a GRW spacetime M¯ satisﬁes f 00 ≤ 0 and the Ricci curvature of the ﬁber is greater than (n − 1) sup (f.f 00 − f 02) then, the only compact spacelike hypersurfaces of constant mean curvature in M¯ are the spacelike slices.

In particular, this result includes the following [2]

COROLLARY 5.10 In a GRW spacetime which obeys TCC and whose ﬁber is positively Ricci curved, the only compact spacelike hypersurfaces of constant mean curvature are the spacelike slices.

Note that this assumption on the Ricci tensor of the fiber is natural from a mathematical point of view. From a more physical eye, we can impose the existence of non-vanishing matter fields. Then the curvature assumption on the fiber can be weakened to show [2]

COROLLARY 5.11 In a GRW spacetime with non-vanishing matter ﬁelds and non-negative Ricci curved ﬁber, the only compact spacelike hypersurfaces of constant mean curvature are the spacelike slices.

Finally, we can state the following uniqueness result [2]

THEOREM 5.12 Let (M, g) be a compact Riemannian manifold. Let H be a 00 real number and let f : I −→ R+ be a smooth function. Assume either f ≤ 0 and (M, g) is positively Ricci curved or f 00 < 0 and (M, g) is non-negatively Ricci curved. Then, the only solutions to the constant mean curvature equation (9) are the constant functions. 12 Spacelike Hypersurfaces of Constant Mean Curvature

Acknowledgments

This work has been partially supported by DGICYT Grant PB-97-0784-C03. The first-named author is also supported by the Consejer´ıade Educacióny Cultura CARM Grant No. PB/5/FS/97, Programa Séneca(PRIDTYC).

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