Maximal hypersurfaces and foliations of constant mean curvature in general relativity

MAXIMAL HYPERSURFACES AND FOLIATIONS OF CONSTANT MEAN CURVATURE IN GENERAL RELATIVITY

Jerrold E. MARSDEN Department of Mathematics, University of California, Berkeley, California 94720, U.S.A.

and Frank J. TIPLER Department of Mathematics, University of California, Berkeley, California 94720, U.S.A.

and Center for Theoretical Physics, University of Texas, Austin, Texas 78712, U.S.A.

NORTH-HOLLAND PUBLISHING COMPANY -AMSTERDAM

PHYSICS REPORTS (Review Section of Physics Letters) 66. No.

3(1980)109—139. North-Holland Publishing Company

MAXIMAL HYPERSURFACES AND FOLIATIONS OF CONSTANT MEAN CURVATURE IN GENERAL RELATIVITY Jerrold E. MARSDEN Department of Mathematics. University of California. Berkeley, California 94720. U.S.A. and

Frank J. TIPLER Department of Mathematics. University of California. Berkeley, California 94720. U.S.A. and Center for Theoretical Physics. University of Texas. Austin. Texas 78712. U.S.A. Received May 1980

Contents Introduction Notation 1. Uniqueness 2. Variational methods for existence 3. The method of continuity

111 112 114 122 125

4. Sufficient conditions for avoidance of singularities and turning null Conclusions References

131 137 137

Abstract: We prove theorems on existence, uniqueness and smoothness of maximal and constant mean curvature compact spacelike hypersurfaces in globally hyperbolic spacetimes. The uniqueness theorem for maximal hypersurfaces of Brill and Flaherty, which assumed matter everywhere, is extended to spacetimes that are vacuum and non-fiat or that satisfy a generic-type condition. In this connection we show that under general hypotheses, a spatially closed universe with a maximal hypersurface must be Wheeler universe; i.e. be closed in time as well. The existence of Lipschitz achronal maximal volume hypersurfaces under the hypothesis that candidate hypersurfaces are bounded away from the singularity is proved. This hypothesis is shown to be valid in two cases of interest: when the singularities are of strong curvature type, and when the singularity is a single ideal point. Some properties of these maximal volume hypersurfaces and difficulties with Avez’ original arguments are discussed. The difficulties involve the possibility that the maximal volume hypersurface can be null on certain portions; we present an incomplete argument which suggests that these hypersurfaces are always smooth, but prove that an a priori bound on the second fundamental form does imply smoothness. An extension of the perturbation theorem of Choquet-Bruhat, Fischer and Marsden is given and conditions under which local foliations by constant mean curvature hypersurfaces can be extended to global ones is obtained.

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J.E. Marsden and Fl Tipler, Maximal hypersurfaces and foliations of constant mean curvature

111

Introduction The importance of maximal and constant mean curvature hypersurfaces in general relativity was clearly brought out in the classic paper of Lichnerowicz [43].Since then these hypersurfaces have played a key role in the dynamic aspects of general relativity (see the survey articles of Choquet-Bruhat and York [181and Fischer and Marsden [23,24] and references therein). Related studies of the structure of the singularities in the space of solutions of Einstein’s equations assume the existence of constant mean curvature hypersurfaces (see Fischer, Marsden and Moncrief [251).Numerical computation of asymptotically flat spacetimes use maximal hypersurfaces to integrate forward in time (see Smarr et al. [58] and S. Shapiro and S. Teukoisky [57]).Finally, in the proof of positivity of gravitational mass, maximal hypersurfaces play an important role: cf. Choquet-Bruhat, Fischer and Marsden [17] and Schoen and Yau [53—55]. This paper extends the existence and uniqueness theorems known for spatially closed universes; the asymptotically flat case, which may be treated by similar techniques, is also discussed. Since the open Friedmann universes contain no maximal hypersurfaces, the asymptotically flat and closed cases cover many physically realistic spacetimes which might be expected to have maximal hypersurf aces. The two main papers with uniqueness theorems are Choquet-Bruhat [16] and Brill and Flaherty [7]. Previous to this, the main result is the proof of the Bernstein conjecture for Minkowski space by Calabi [11] i.e. any closed maximal hypersurface in Minkowski space is a plane. This was extended by Cheng and Yau [15] to arbitrary dimension. The Brill and Flaherty result replaces Minkowski space by a spatially closed universe and proves uniqueness in the large, assuming the ubiquitous energy condition: RabU~5Ub= Ric(u, u)> 0 for timelike vectors u. (This terminology follows Tipler [60,61].) Our first main result relaxes this energy condition to include non-flat vacuum spacetimes or spacetimes satisfying a generic condition of the type occurring in the singularity theorems of Hawking and Penrose. Moreover, we show that there must be singularities both to the future and to the past of the maximal hypersurface and that every inextendable timelike curve begins at the past singularity and ends at the future singularity. The principal papers dealing with existence are Avez [2] (see also Goddard [33]) which uses variational methods, and Choquet-Bruhat [161,Cantor et at. [13], and Choquet-Bruhat, Fischer and Marsden [17] which use the implicit function theorem in Banach spaces. Avez [2] gave an erroneous proof that the maximal hypersurfaces constructed are actually smooth and spacelike. This error is propagated in Geroch [27,28] and in Hawking and Ellis [37]. In fact, without further hypotheses, it is not known whether or not Geroch’s singularity theorem (see Hawking and Ellis [37] pp. 274—275) is true. Roughly speaking, the variational method cannot as yet deal with the possibility that the maximal hypersurface might be nonsmooth and (hence be null) in places. We give a summary of what is known to us using this method, including some new results, and a plausibility argument of smoothness in hopes that the methods we give can be generalized. The variational method of Goddard [33] and Brill and Flaherty [8] enables one to give similar results for the existence of constant mean curvature hypersurfaces. The local results of Choquet-Bruhat [16], Cantor et at. [13], Choquet-Bruhat, Fischer and Marsden [17] and the present paper give a complete perturbation analysis of the existence and uniqueness question. It is shown that maximal (or constant mean curvature) hypersurfaces are stable under perturbation of the spacetime under generic hypotheses. It is suggested by this approach that the continuity method may yield global results. (The global part of the paper of Cantor et at. has incomplete proofs.) The continuity method is also discussed in this paper. Again one has to deal with the possibility

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J.E. Marsden and Fl Tipler, Maximal hypersurfaces and foliations of constant mean curvature

that the hypersurfaces can turn null, as in the variational method. For maximal hypersurfaces, variations of the spacetime are considered and the maximal hypersurfaces are followed; as we move farther away from a known spacetime along a curve of spacetimes, we must prevent the maximal hypersurface from turning null and from hitting the spacetime singularities. We give a technically incomplete, but plausible argument that the maximal hypersurfaces never can turn null. We also demonstrate how to bound maximal (or constant mean curvature) hypersurfaces away from the singularity in terms of a classification of singularities into strong curvature type (Tipler [62,63]) and crushing type (Eardley and Smarr [20]).There is evidence, both numerical and theoretical that this avoidance of singularities will occur in physically realistic spacetimes (see the papers of Eardley and Smarr [20], Smarr and York [59], and Shapiro and Teukolsky [57];the Eardley—Smarr counterexamples, which occur in Tolman—Bondi universes, are thought to be artifacts of the fact that the matter is dust). The method of continuity also can be used to prove the existence of a foliation of spacetime by Cauchy hypersurfaces of constant mean curvature K. Here the spacetime is fixed and we extend our foliation as far as we can from a given maximal hypersurface as the starting point using the mean curvature as a real parameter. We show that the only things that can prevent such a foliation from existing is that the hypersurfaces run into the singularity or turn null. Our arguments suggest the latter can never happen; we also prove that it never happens if one has an a priori bound on the length of the second fundamental form. Thus, under an avoidance condition (implied by strong curvature or crushing type assumptions) and a “null condition”, one gets a foliation of the entire spacetime with the mean curvature K running from +~ at the initial singularity to at the final singularity (with the sign conventions on K from Hawking and Ellis [37]).Thus the York time r = —4K/3 has the “Misner time property” (Misner [46]; see also York [69] and Mime [45]) which means that the singularities are at temporal infinity. Uniqueness of the foliation by constant mean curvature Cauchy hypersurfaces is covered by our general uniqueness theorem. The hypotheses involved are general and hold in most cases of interest. Uniqueness under different hypotheses was shown by Goddard [34]. The authors wish to acknowledge the helpful comments provided by S.Y. Cheng, Y. ChoquetBruhat, D. Eardley, A. Fischer, V. Moncrief, M. Protter, L. Smarr, A. Tromba, A. Weinstein, J.A. Wolf and S.T. Yau at various stages of the present work. —~

Notation This section recalls a number of definitions that will be employed in our main results. Throughout the paper, (M, g) will denote a spacetime i.e. a smooth four-dimensional Hausdorif, boundaryless, connected, oriented and time-oriented Lorentz mainfold. We refer to Hawking and Ellis [37] for any unexplained terminology. For us, “smooth” means C~,r 2, but we shall write C~for simplicity and C’ will mean Lipschitz. We shall depart from Hawking and Ellis’ notation in one respect. For us, the term “Cauchy surface” means an achronal set S without edge such that D(S) = M, whereas Hawking and Ellis use acausal in place of achronal. The main results will explicitly assume that (M, g) contains a compact C~spacelike Cauchy hypersurface S. (All hypersurfaces are understood to be boundaryless.) By Geroch [30] (see also Hawking and Ellis [37] pp. 211—212), if M(g) admits a C°° spacelike Cauchy surface S, then (M, g) is globally hyperbolic with M homeomorphic to S x R. By Budic et al. [9], if (M, g) admits a compact C” spacelike Cauchy surface, then any compact C1 spacelike hypersurface will be a Cauchy surface.

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(Compactness of S is often called the “cosmological case”. However, many of our results carry over to the asymptotically flat case as well. One obtains the asymptotically flat theorems by replacing the Sobolev spaces* used in the closed universe case by the weighted Sobolev spaces of Nirenberg, Walker and Cantor and then exploiting the fact that the hypersurfaces sought are asymptotically flat themselves.t We shall state the results for the asymptotically flat case parenthetically following each of the main theorems.) We recall that (M, g) is said to satisfy the timelike convergence condition if Ric(v, v) R0bV” v~’ 0 for all timelike vectors v E Th4. According to the definition of Hawking and Ellis [37] p. 95, if the Einstein equations without cosmological constant hold, then the timelike convergence condition is equivalent to the strong energy condition. If S C M is a smooth spacelike hypersurface, we will let k (or k~if there is an ambiguity as to which hypersurface is being referred to) denote its second fundamental form and we let #ctrace k=kaa, (supa=1,2,3) denote its mean curvature. Our sign conventions in the definitions of k follow Hawking and Ellis [37]. Namely if z” (a = 1, 2, 3, 4), is the unit normal to S then ka~=Zaj~,

a,/3 = 1,2,3,

where “;“ denotes the spacetime covariant derivative. The covariant derivative in a spacelike hypersurface is denoted T~1~ etc. Misner, Thorne and Wheeler [47]and ADM use the convention ka~= Za ;~and switch the roles of greek and latin indices. These convention changes are responsible for various sign changes in cited formulas throughout the paper. A smooth spacelike hypersurface S is called maximal if K = 0. (“Stationary” would be a better term, but maximal is justified by Theorem 1 below.) If K is a constant, we say S 1hasachronal constant hypersurface mean curvature. we If the of S isvolume. larger than or equal to that of any Other compact C say S isvolume of maximal The first variation formula states that if S(A) is a one parameter family of spacelike hypersurfaces with S(0) = S and if N is the forward pointing normal component of the infinitesimal deformation vector X of the family, then ~-{vo1S(A)}I=JNKd3V

(1)

(the proof is standard; see Choquet-Bruhat, Fischer and Marsden [17] for instance, noting changes in conventions). From (1) it follows that if S has maximal volume then f~ NK d3V = 0 for all N and so K = 0 i.e. S is maximal. This well known remark is valid only for smooth S. If S is only achronal and Lipschitz, but nevertheless has maximal volume, then K need not be defined (although K = 0 in a distributional or weak sense). As is explained in [2], the volume function extends uniquely from the space of smooth compact spacelike hypersurfaces to an upper semi-continuous function on the space of compact See, for example Hawking and Ellis [37]p. 234. t See Cantor 1121 and Choquet-Bruhat, Fischer and Marsden [17]for the definitions and basic properties. *

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I.E. Marsden and FJ. Tipler, Maximal hypersurfaces and foliations of constant mean curvature

achronal Lipschitz hypersurfaces. (We will further extend the volume function to the space of noncompact achronal Lipschitz hypersurfaces in section 1.) The second variation formula states that if K =0 on 5, then ~p{vol S(A)}f

where k k = ~ .

=

—J

2(k

{N

k +R±±)+(VN)2}d3V

(2)

R 11 =

Ric(Z, Z) =

RaZ~*ZI*,and Z

is the unit forward pointing normal to S.

1. Uniqueness The + R timelike convergence condition implies that the second variation of volume is non-positive. If 11 >0, it follows that S locally strictly maximizes volume, and S is locally unique. This much is well-known, going back at least to Komar [40]. Brill and Flaherty [7] prove global uniqueness if R~1> 0, and they prove a local maximizing property in the degenerate case k =0, R~1= 0 for non-flat vacuum spacetimes; our Theorem 1 improves this further by proving global uniqueness and maximization of volume of all non-flat vacuum spacetimes, and in any spacetime that satisfies a generic-type condition. It was shown in Choquet-Bruhat, Fischer and Marsden [17] that hypersurfaces of constant mean curvature K 0 are locally unique. The techniques of Bril and Flaherty also apply here to give a global result, as in Theorem 1 below. Maximal hypersurfaces in non-flat spacetimes need not be unique nor be isolated without some assumptions on the spacetime. For example, the Einstein static universe (Hawking and Ellis [37]p. 139) 3, namely the hypersurfaces of homois foliated by maximal Cauchy hypersurfaces with topology S geneity and isotropy. In this example one has 2

k

Ric(v, v) = 2((V2Za)2 +

V’~Va)~

0

where Z is the unit normal to a maximal hypersurface. This spacetime has compact C” spacelike Cauchy surfaces and satisfies the timelike convergence condition, but has non-unique and non-isolated maximal hypersurlaces. Our extra assumptions that will ensure uniqueness in a non-flat spacetime is as follows: Definition. We shall say (M, g) satisfies the hypersurface generic condition provided that for each smooth compact spacelike hypersurface S on which ~ =0 and R~=0, at least one of the following holds: (a) the Ricci tensor Rab vanishes at each point of S or (b) at some point of 5, Z”Z”Z[aRbJcdEeZfl

0

where Z is the unit forward pointing normal to S. This condition is automatically satisfied for vacuum spacetimes, or for spacetimes that satisfy

Ric(v, v)> 0 for all timelike v (the Brill—Flaherty ubiquitous energy assumption).

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Notice that (a) is more general than the energy condition: R~ = R~

implies

R

0b

0

that is often assumed at every point of spacetime. It is also more general than the “mixed energy condition”: ~

~

=

0 implies

Rab =

0

(see Choquet-Bruhat, Fischer and Marsden [17]Theorem 4.1). The next assumption is of a technical nature and is often assumed implicitly. It will hold if the spacetime is generated by coupled Einstein-matter equations that are well-posed; in particular, it is automatic for vacuum spacetimes. See Hawking and Ellis [37] Ch. 7 and Fischer and Marsden [27] section 4.5. Definition. We say (M, g) satisfies the Development Uniqueness Assumption when following 4~Rabcd = 0, thenthe S has a flat condition Cauchy holds: Ifdevelopment S is a partialD(S). Cauchy which ka~=equations 0 and ~ are such that the maximal maximal Thissurface holds on if the matter development of the initial data on S is unique. The existence and uniqueness of hypersurfaces of constant mean curvature is related to the spacetime singularities. We shall need a number of global concepts relevant to the problem. The first is the notion of a Wheeler universe. This concept is believed to be generic for closed universes. Definition. A Wheeler Universe is a spacetime with a compact C” spacelike Cauchy surface and which satisfies d(M, M) < c~• See Hawking and Ellis [37] p. 215 for the definition of d(M, M); the condition d(M, M) < x is equivalent to the condition that there be a constant L >0 such that the lengths of all timelike curves in M be bounded above by L. This definition is slightly different from one given in Tipler [62], but we believe it is physically the same. We will prove that compact maximal hypersurfaces have maximal value amongst all achronal Lipschitzhypersurfaces in Wheeler universes. To do this, we extend the volume function to non-compact hypersurfaces. First of all, we note that, by Avez’ [2] arguments, the volume of compact achronal Lipschitz hypersurfaces with Lipschitz boundary is defined. We define the volume of a general Lipschitz achronal hypersurface S by vol(S) = sup vol(S’) s,

where the sup is over compact subsets of S that are closures of open subsets of S with Lipschitz boundary. In the proof of Theorem 1 we shall see that in Wheeler universes satisfying the timelike convergence condition, the volume function is upper semicontinuous. (Without the timelike convergence condition, vol(S) could be +c~.) Theorem 1. Let (M, g) be a spacetime with a compact C” spacelike Cauchy surface and let (M, g) satisfy the timelike convergence condition.

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I.E. Marsden and Fl. Tipler, Maximal hypersurfaces and foliations of constant mean curvature

(A) If S C M is a smooth compact spacelike maximal hypersurface, then S has maximal volume

(amongst all achronal Lipschitz hypersurfaces). (B) IfS C M is a smooth compact spacelike hypersurface of constant mean curvature

0, then there is no other smooth compact spacelike hypersurface in M with the same constant mean curvature. (C) Assume the hypersurface generic condition holds and suppose (M, g) is not identically flat. Let (M, g) contain a smooth compact spacelike maximal hypersurface S. Then (a) (M,g) is a Wheeler universe and (b) if development uniqueness also holds, then S is the only smooth spacelike maximal hypersurface in (M,g). i~

The uniqueness assertion in (CXb) within the class of achronal Lipschitz maximal volume hypersur-

laces is not known. There is some partial information however. For example, one can show that if S is a compact achronal Lipschitz maximal volume hypersurface and S~is a compact smooth spacelike hypersurface with mean curvature KS~< 0, then fl 5) < vol(1(S) fl S4). vol(L4~(S~) {In the globally hyperbolic, asymptotically flat case, we abandon (A), (B) and (C)(a). In (C) part (b), we have uniqueness amongst any two maximal hypersurfaces that are asymptotic (again in a suitable function space) to the same t = constant hypersurface at spatial infinity. The proof of Theorem 1(CXb) can be modified to cover this case. The hypersurface generic condition and development uniqueness are no longer needed, and flat space is no longer exceptional.} Now we turn to the proof of Theorem 1. We begin by recalling the following results from Hawking and Ellis [37]p. 217. Lemma 1. Let (M,g) be a spacetime, S a C2 partial Cauchy surface, D~(S)the future Cauchy

development of S and q E D~(S). Then there is a future-directed timelike geodesic orthogonal to S of length d(q, 5) that contains no point conjugate to S between S and q.

~q

from S to q

We shall also require the following: Lemma 2. (M, g) contain a smooth compact Cauchy surface and satisfy the timelike convergence

condition. Let S~be a compact smooth spacelike hypersurface with i~+<0 and let S be a smooth spacelike compact maximal hypersurface. Then (a) SflI~(S~)ø and (b) if S’ is any achronal Lipschitz hypersurface, then vol(S’)

vol(S).

Proof of Lemma 2. First of all, we observe that S and S~are both Cauchy surfaces by Budic et al. [9]. (a) Let T~= S flI~(S~), an open set in S, and assume T1 0. See fig. 1. For p E T~,let y~be the geodesic of length d(p, S~)from S~to p guaranteed by Lemma 1. By construction, the past endpoint q of y,, lies in T, so T = S~nL(S) ~ 0. Let d V(t) denote the 3-volume element determined by three Jacobi fields along 7p that are

I.E. Marsden and F) Tipler, Maximal hypersurfaces and foliations of constant mean curvature

1

117

(Si

:~:~<: Fig. 1. Figure for the proof of Lemma 2.

orthonormal tangent vectors to S~at q at t = 0. Now, d V(t) satisfies the following equations*

with

~2(d V)113 + ~ (R0bV’~v”+ 2u2Xd V)1~’3=0

(3a)

3~4j(d V)”3

(3b)

= Ks+(d V)”3 <0

where v = y,, o• is the shear of the Jacobi fields (see Tipler [60]and Hawking and Ellis [37]p. 97) and d/dt denotes covariant differentiation. Since v,~has no points conjugate to S~between p and q, d V is well defined along y~(but d V may vanish at p). By the timelike convergence condition and i~÷<0, we have dV(,p)< dV(q). Let d,s~(p)be the volume element on the spacelike hyperplane v1, the Lorentz orthogonal complement of v. Let Z be the unit normal to S at p and let d~z(p)denote the volume element on Z~= T~S.We have, by construction, vol(T~)=

Let I’,, v .

-~

J

dp~z(p).

T~Sbe

the Lorentz orthogonal projection of T~Monto T~S,restricted to v It is an ~.

exercise in Lorentz geometry to show that P,, transforms volume elements as follows: P~:dj.s~(p)=—(v Z)d .

4az(p).

Equation (3a)

is

essentially the Raychaudhuri equation; see Hawking and Ellis [37]p. 97. The relationship between the expansion 0 and the

volume element dV along a timelike geodesic is given by (1/dV)d(dV)/dt = 8; see Penrose [50]Prop. 7.14 on p. 61.

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I.E. Marsden and F.J. Tipler, Maximal hypersurfaces and foliations of constant mean curvature

Note that —(v Z) 1, with equality only if v and Z are equal. The result dV(,p)< dV(q) above means the following, by the geometric interpretation of Jacobi propagation: the map q ~p where 4 near q is mapped to the intersection of the normal geodesic at 4 to p on the hyperplane v at p, has Jacobian determinant at p given by d V(p)Id V(q) which is less than 1 i.e. d V(q) d V(p) is how geodesics transform volume elements. The Jacobian of the map 4l’-~’fiE T~is therefore .

~-*

dV(p) 1 dV(q)xJac0l~1a~b0f P~ dV(q) —(v Z)< 1. dV(p)

.

.

Thus, the map p ~-÷q from T~to T has Jacobian strictly greater than 1. Thus, by the change of variables formula, volume (T~)< volume (T). However, we may reverse time and make the same construction starting at S and propagating to S~to conclude volume T
since in eq. (3), the initial condition

,~+

<0 is replaced by

Ks

=0. Reversing time, we

get volume (5’ fl I(S)) s volume (S fl I~(S’)). Since S and 5’ are both Cauchy surfaces, M = I~(S)uI(S)uS= I~(S’)U1(S)

US’

so addition of the two inequalities and the trivial equality volume (S’ flS) = volume (S’ flS’) yields

volume [S’fl (I~(S)U 1(S) US)] volume [S fl (I~(S’)U F(S’) US’)] i.e. volume (S’) volume (5). If S’ is a compact achronal Lipschitz hypersurface, then S’ is the Lipschitz limit of compact smooth spacelike hypersurfaces. Since in this case the volume function for Lipschitz achronal hypersurfaces is upper semi-continuous, the same inequality holds.

J.E. Marsden and FJ. Tipler, Maximal hypersurfaces and foliations of constantmean curvature

119

The argument above proving that volume (5’) volume (S) works for any compact smooth spacelike hypersurface S’ with boundary, since 5’ can be extended to a Cauchy surface. Thus, it also holds for

compact Lipschitz achronal hypersurfaces with Lipschitz boundary by upper semi-continuity. The inequality then extends to the non-compact case by definition of vol(S’). The above argument and the timelike convergence condition shows that vol(S’) is uniformly bounded by vol(S). I We now continue the proof of Theorem 1. We have proved (A) in part (b) of Lemma 2. The proof of (B) will use the following assertion: if S is a smootk spacelike hypersurface of constant mean curvature # 0, then in any neighborhood U of S there exist smooth spacelike hypersurfaces S_ and S+ with mean curvatures K~ and K+ satisfying K~
In fact, this follows from the evolution equation for K in Gaussian normal coordinates; using u for the time parameter, we have 3KJL9U

=—(k k+R11)<0 .

(<0 since k k 1(2>0). Thus, K is strictly decreasing, so our assertion follows. The proof of (B) is based on an idea due to Frankel [26]and Brill and Flaherty [7].Suppose on the contrary that there existed another compact spacelike hypersurface 5’ with constant mean curvature = # 0. By Lemma 1 and compactness of S and S’, there exists a timelike geodesic y(t) from S to S’ of length d(S, S’); it follows that y(t) must be normal to both S and S’. The second variation formula for curve length is .

~

(4)

~0)=JRabvavbdt+Ks_i~

1~3 0

where e,~(j.L = 1,2, 3) are vector fields along y(t) obtained by parallel translating an orthonormal basis along y(t) from S to 5’. This formula follows from the first equation in the proof of Theorem 3.3 of Brifi and Flaherty [7].It should be noted that (4) is not time invariant; it is assumed that y nS’ C I~(S).St~ ince #0 y(t) maximizes proper length between S’, wesince must ‘~ haveKS~ ~i If ~ If RabVaV somewhere along “ thetime right-hand side of (4)S isand positive Ra~V’V”=0 0. along y(t), then we use the fact that in a sufficiently small neighborhood U of S there is a compact spacelike hypersurf ace S+ with mean curvature ~ < i~.As before, there exists a timelike geodesic j~from S+ to 5’ such that j~is locally maximal and 5’ flS’ C I~(S÷)(for S÷sufficiently close to S). Equation (4) applies, giving

1~~3 0 ~ ~~(o)

JRabVaVbdt+KS._KS+>O.

This contradicts the maximality of

‘~

and completes the proof of (B) of Theorem 1.

Remark. This argument may be used to prove a generalization of Lemma 2(a). Namely if (M, g) has a

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I.E. Marsden and FJ. Tipler, Maximal hypersurfaces and foliations of constant mean curvature

compact Cauchy surface and satisfies the timelike convergence condition and furthermore S~and S2 are smooth compact spacelike hypersurfaces with mean curvatures K1 and K2 that satisfy sup ici SI

S2

then S~flI~(S2)= 0 i.e. Si C J(S2). In other words, “as ic decreases, the hypersurfaces move to the future”. This result will be used in Theorem 6 and in our discussion of foliations in Theorem 3. To prove Theorem 1(C) we will need the following fact: in any neighborhood of a maximal hypersurface S, there exists smooth compact spacelike hypersurfaces S~and S such that Kg~<0< K~.

The proof of this result can be given along the lines of Appendix II of Geroch [28].In Lemmas 4 and 5 below we shall prove a sharper version of this result that is needed in Theorem 4, but for now the above will suffice. (Fhe result quoted depends crucially on the hypersurface generic condition and development uniqueness; the reader is cautioned that if Geroch’s argument is to be used, the part dealing with case (b) of the hypersurface generic condition is inadequately described in Hawking and Ellis [37]p. 273.)

The existence of S~and S can be used to give a proof of part (b) of (C) using the method of Brill and Flaherty discussed in the proof of (B). We shall give a different proof below. The proof of (C) part (a) relies on the following lemma proved in Hawking and Ellis [37]p. 199. Lemma 3. Assume the existence of a compact Cauchy surface and the timelike convergence condition. Let

S~be a (compact) spacelike hypersurface with KS~< —a <0, and y(s) a geodesic normal to S~.There is a point conjugate to S~along y(s) within a distance 3/a from S~,provided y(s) can be extended that far.

Proof of Theorem 1(C) part (a). Let U be a neighborhood of S with d(U, U) = e 0 and K5~ —a <0 for constants a, b > 0, which is possible as S~,S are compact. By Lemma 3, every timelike geodesic normal to S~has a point conjugate to S÷ within a distance 3/a, provided the geodesic can be extended that far. Therefore, by Lemma 1, no causal curve can be extended in D~(S~) to the future of S~for a length greater than 3/a. Similarly, no causal

curve can be extended in D(S) for a length greater than 3/b. Thus d(M,M)_
I

One of Avez’ original goals in his 1963 paper was to prove what we would view today as a singularity theorem; in particular, he attempted to prove that any vacuum time-periodic spacetime is fiat. This

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conjecture remains open as far as we know. However, if a smooth maximal spacelike hypersurface exists in a time-periodic spacetime then the spacetime is flat; i.e. the conjecture is true in this case. This is stated in the following consequence of Theorem 1, C(b). Corollary 1. Let (M, g) be a spacetime satisfying (a) of the hypersurface generic condition, the timelike convergence condition and development uniqueness. Suppose (M, g) contains a maximal Cauchy hypersurface and is time-periodic (i.e., there is a timelike action of S1 on M by isometries). Then (M, g) is flat and equals Minkowski space with identifications. Remark. In Tipler [65,66] it is shown that spacetimes cannot be time-periodic if they satisfy the timelike convergence condition and the standard generic condition. (The Einstein static universe is time periodic, satisfies the timelike convergence condition but does not satisfy the standard generic condition.) Proof of Corollary 1. Since (M, g) is time-periodic, it must contain a countable number of distinct smooth maximal spacelike hypersurfaces. By Theorem 1(C), (b), (M,g) is flat. It follows from eq. (2) and the locally Minkowski nature of (M, g) that a maximal hypersurface S is a surface of time symmetry (k 5 = 0), and so, by the Gauss-~Codazziequations, S is flat. By Cartan—Ambrose—Hicks [68] 3, the ö), where & is the Eucidean theorem metric. Thus p. 42, S is covered by standard, complete Euclidean space (R (M, g) is covered by standard, complete Minkowski space (R4, ~). I Corollary 2. Let (M,g) satisfy the timelike convergence condition and contain a C compact Cauchy hypersurface S of constant mean curvature and suppose the conditions of (B) or (C) of Theorem 1 hold. Let X be a Killing vector field of M. Then X is tangent to S. Proof. If X were not tangent to 5, its flow would map S to another hypersurface with the same constant mean curvature, contradicting uniqueness. I In particular, there are no stationary spacetimes satisfying condition (C). This is a slight strengthening of a well-known theorem of Lichnerowicz [42]. One can also study the question of uniqueness in open universes. A simple application of the techniques used earlier in this section gives: Proposition 1: Let S~and S

2 be two spacelike constant meancurvature Cauchy surfaces with KS, = the timelike convergence condition holds, then d(S1, S2) is bounded.

K~2

0. If

This proposition indicates that if non-compact constant mean curvature hypersurfaces are Cauchy surfaces, they are either unique or close together. In both the compact and non-compact case, the Cauchy surface condition is essential in the proof of uniqueness. For example, in both Minkowski space and in the open Friedman models there are constant mean curvature hypersurfaces which hit scri and hence are not Cauchy surfaces. Stumbles [75] has shown that these surfaces are not unique; in fact shehas shown that a large family of such surfaces exists which is indexed by BMS supertranslations. The techniques used by Stumbles are based again on those in Choquet-Bruhat, Fischer and Marsden [17]. (This paper by Choquet-Bruhat, Fischer and Marsden also makes some brief conjectural remarks and supplies some references on how the geometrodynamics relative to a BMS-slicing might go.) —

—

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However, if some reasonable conditions are imposed on the behavior of constant mean curvature

hypersurfaces at spatial infinity, one can prove they are unique if they are Cauchy surfaces. Proposition 2: Let S~and S2 be as in Proposition 1. Ifthe timelikeconvergence condition holds, and there are compact sets K1 and K2 with K1 C S~and K2 C S2 such that d(q, p) < d(S1, S2) for all q C S1\K1 and pES2\K2, then S1=S2.

Proof: The existence of the sets K1 and K2 would ensure that there exists a timelike geodesic of length d(S1, S2) between S1 and 52. One then applies the Brill-Flaherty argument [see the proof of Theorem 1(B), above] to get uniqueness. 2. Variational methods for existence This section gives sufficient conditions for the existence of Lipschitz achronal maximal volume hypersurfaces. The major obstacle in this method is keeping the candidate hypersurfaces away from the singularity. We phrase the condition we need abstractly in the following definition. The verification of

this condition under realistic hypotheses is the major subject of section 4. Definition. We say that large volume hypersurtaces in (M, g) avoid singularities if there exist two compact smooth spacelike hypersurfaces SF and S with S~C J~(S) such that for any achronal Lipschitz hypersurface S in M, there exists a compact achronal Lipschitz hypersurface 5’ such that S’ C J~(S)nJ(S~) and vol(S’) vol(S). Theorem 2. Let (M, g) be a spacetime with a compact C’°spacelike Cauchy surface and suppose that large volume hypersurfaces avoid singularities. Then there exists a compact achronal Lipschitz Cauchy surface S of maximal volume (amongst all achronal Lipschitz hypersurfaces). Furthermore (a) The subset S0 C S at which S is (strictly) spacelike,* is open in S, is a smooth hypersurface and is maximal. (b) Ifp, q ES and there is a null curve yjoiningp to q, then y is a null geodesic joining p to q and y lies in S. (That is, S is ruled by null geodesics on its null portion.) {In the asymptotically flat case, the implicit function theorem argument in Choquet-Bruhat, Fischer and Marsden [17]can be used to construct a portion of a maximal hypersurface asymptotic to the t =0 hypersurface at infinity. One can then consider all Lipschitz achronal extensions of this hypersurface

and maximize the volume of the extension by the techniques of Theorem 2. By elliptic theory one can show that the hypersurface is smooth in a neighborhood of infinity and smooth across the joining surface between the piece at infinity produced by the implicit function theorem and the piece produced by maximizing volume. In the definition of large volume hypersurfaces avoiding singularities, we require i.e. at any point p ES,,

S

lies exterior to an enlarged light cone at p in a neighborhood of p.

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in the asymptotically flat case that S~,S~and S be compact with boundary a fixed two sphere of large radius in the t = 0 hypersurface relative to the background Minkowski metric.} Proof of Theorem 2. By hypothesis, candidate maximal volume hypersurfaces may be chosen in J~(S)fl J(S~).We consider the set ~ of all compact smooth spacelike hypersurfaces in J~(S)flJ(S~).By Budic et al. [9]they are all Cauchy surfaces. By the arguments in Avez [2], there exists a compact achronal Lipschitz hypersurface S of maximal volume that is the Lipschitz limit of a sequence S~in This limit S is also a (achronal) Cauchy surface. Indeed, every timelike curve in M intersects SN at least once, say at By compactness, has a convergent subsequence that converges to p which lies in S and on the given timelike curve. If the timelike curve intersected S more than once, it would also intersect 5N more than once for N sufficiently large. The assertion (a) follows from the fact that the equations describing S are elliptic at Po if S is strictly spacelike at Po. The smoothness then follows by elliptic regularity (see Morrey [48]for example). It remains to prove that S is ruled by null geodesics on its null portion. Let p, q C S and y be a null curve from p to q. If y did not lie entirely in S, there is a point r C y that is in I~(S)or in 1(S), since M = I~(S)U I~(S)US, as S is a Cauchy surface. Suppose r C 1(S). Thus there is a w E S such that r 4 w. Since p < r we have p 4 w, by Penrose [50] Prop. 2.18. Since p and w are both in 5, this contradicts the achronality of S. Similarly r E I~(S)contradicts achronality of S. Thus y C S. Finally, y is a null geodesic by the proof of Proposition 2.20 of Penrose [50]. I ‘~.

PN.

PN

There is an analogous theorem for surfaces of constant mean curvature; they are obtained by maximizing the function vol(S) + ic vol(S, S~) where ic is a fixed real number, S

0 is a fixed compact C~spacelike Cauchy surface and vol(S, S~)is the 4-volume of the region between S and S~.In this case one needs hypersurfaces with large values of vol(S) + K vol(S, S0) to avoid singularities. An extremum that is smooth is of constant mean curvature, as with maximal hypersurfaces. See Goddard [33]for more information. Conjecture. The Lipschitz achronal hypersurface produced in Theorem 2 is actually C~and spacelike.

(Similarly for constant mean curvature hypersurfaces.) An incorrect proof of this conjecture was given by Avez [2].The flaw is that the volume function fails to satisfy the required ellipticity conditions for a hypersurface which is null in places. The resolution of the conjecture may rest on the methods of Cheng and Yau [15]or on a careful study of degenerate elliptic equations using the results of Krylov [41] or a combination of these. For constant mean curvature hypersurfaces in the positive definite case, see Gilbarg and Trudinger [32] Ch. 15. Our theorem below shows that it is enough to get an a priori bound on the length of the second fundamental form of the hypersurface. Plausibility argumentfor the conjecture Suppose that S is not spacelike at p E S. We want to derive a contradiction. From assertion (b) in Theorem 2 we can assume p is chosen so that no point in S to the future of p is null-related to p.

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Consider the open future set I~i,p)of p; the argument will be local so we can restrict attention to a neighborhood of p. Let C be the future light cone of p. so C is tangent to I~(p)at p. Using a reference background Riemannian metric, enlarge C to a cone C(O) by opening the vertex angle by an amount 0 >0. Correspondingly, enlarge I~(,p)to I~(p,0). Since S is not spacelike, we can assume that S intersects I~(p,0) (there is a similar argument for I1,p, 0)). Since we chose p to be the maximal future point common to S and a null geodesic, S flI~(p,0) shrinks to a point as 0 -~0+.Replace 5 nJ~(~~ 0) by the corresponding portion S~of 3I~(p,0). This produces a new Lipschitz achronal hypersurface S9. We claim that for 0 sufficiently small, vol(S9)> vol(S) which will give a contradiction. 1 future pointing curve -y(t) with y(O) = p and ‘~“(0+) Unless S is very will containinaaCneighborhood of y, by basic Lorentz geometry. Thus null. Moreover, S flpathological, I~ip,0) will itconcentrate the volume element on S flI~ip,0) will be bounded above as follows: li-s

C~6

for a constant C 1. On the other hand, a straightforward computation in Minkowski geometry shows that the volume element on âI (p, 0) is bounded below as follows: lLI~(p,o)

C2Vë.

These two inequalities imply that for 0 small enough, vol(Se)> vol(S fl ~

0))

and hence vol(S9)> vol(S).

I

The authors are not able to construct a sufficiently pathological candidate S that invalidates this argument which leads us to believe the conjecture is true. In the caseof one space and one time dimension (1 + I dimensions), the above plausibility argument is in fact a proof of the conjecture. Thus we have: Proposition 3: If the dimension ofspacetime is (1 + 1), then the Lipschitz achronal hypersurface produced in

Theorem 2 is actually C~,spacelike, and is generated by a geodesic. Furthermore, in (1+1) dimensions constant mean curvature hypersurfaces don’t turn null (see Theorem 3 below).

This problem of keeping constant mean curvature hypersurfaces bounded away from the light cone is one of the most important unsolved problemsin initial value theory, because it tends to appear in one form or another in most initial value problems. (Cantor and York call it the “problem of bounding interior gradients”.) Audonetand Bancel [71,72] have recently set up a nice formalism for this problem, and proved that the places where maximal hypersurfaces go null form at most a set of measure zero. This work by Audonet and Bancel [71, 72] and also Lacaze [73] has been concerned with the problem of maximal hypersurfaces with boundary.

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3. The method of continuity We shall first deal with the question of finding foliations on a fixed spacetime given a maximal hypersurface. We need an avoidance condition analogous to that in the previous section, as follows: Definition. Let (M, g) be a spacetime with a compact CC spacelike Cauchy surface. We say that constant mean curvature hypersurfaces avoid singularities if for any number R >0 there exist two spacelike CC compact hypersurfaces S~ and S~ with S~C I~(S~) such that if S is a compact smooth spacelike hypersurface of constant mean curvature ic where IKI R, then S C J~(S~)flJ~(S~). Note that from Hawking and Ellis [37]p. 207, J~(S~) flJ~(S~) is a compact set. The next section will give conditions under which one can verify that constant mean curvature hypersurfaces avoid singularities. The turning null condition is formalized in the following definition. (See the above conjecture and the next section for a discussion.) Definition. Let (M, g) be a spacetime with a compact C’°spacelike Cauchy surface. We say that constant mean curvature hypersurface foliations don’t turn null if for any number R >0 there is a number ~ >0 such that ifS is a spacelike CC compact hypersurface of constant mean curvature K, 1K
Remarks. 1. Theorem 1 already established uniqueness of individual hypersurfaces. However for uniqueness of the whole foliation only Assumptions 1 and 2 are needed. 2. Our earlier conjecture can be stated as follows: the assumption that constant mean curvature hypersurfaces don’t turn null is superfluous. In the previous section, this conjecture was proved in the (1 + 1)-dimensional case. Many problems of physical interest can be reduced to this case. For example, the problem of finding the constant mean curvature foliation in a closed Friedmann universe with a single spherically symmetric black hole can be so reduced, as pointed out by Qadir and Wheeler 1741. 3. The theorem is easily modified to cover the case where only the future (or past) of the maximal hypersurface is to be foliated. Also, if Condition 5 is dropped, one can still obtain a maximal foliation including the maximal hypersurface. In this case, however, the maximum foliation need not fill the spacetime. 4. By Theorem 1, (M, g) in Theorem 3 is a Wheeler universe.

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5. Since the constant mean curvature foliation is unique, it defines an absolute time function. York [69] was the first to suggest using constant mean curvature foliation to define a time function. (See also Misner, Thorne and Wheeler [47] section 21.11.) York proposed measuring time intervals in extrinsic time r = —4,c/3. Under the conditions of Theorem 3, the past and future singularities are at “temporal infinity” as measured in York extrinsic time. Misner [46] (see also Mime [45]) has argued that the appropriate “physical” time to be used near a singularity is not proper time, but a time coordinate in which it takes infinite time to reach the singularity. The York time produced by Theorem 3 thus has this “Misner time property”. {In the asymptotically flat case one attempts to foliate (M, g) by maximal hypersurfaces asymptotic to t = constant hypersurfaces at spatial infinity. Assumptions 2 and 6 are retained, 1, 3 and 4 are dropped and 5 is retained with the avoidance of singularities part of 5 modified as follows: Given any number R > 0, there exist two smooth spacelike hypersurfaces S~and S~ with S~C I~(S~) and asymptotic to t = R and t = —R hypersurfaces in Minkowski space respectively, such that if S is a smooth space/ike maximal hypersurface asymptotic to a t = c hypersurface, id
condition. Let S be a smooth spacelike hypersurface of constant mean curvature ic 0. Then there is an e >0 and a neighborhood U of S such that for Ia ic
Proof (based on Choquet-Bruhat, Fischer and Marsden [17]; an outline of this idea of proof was first given by York [70]).Let V be a neighborhood of S obtained using Gaussian normal coordinates. Let g~’ denote the set of all Hs(= W~in the notation of Hawking and Ellis) real valued functions N on S and s 3. Let 5N be the hypersurface that is the graph of N over S in Gaussian normal coordinates. If N is sufficiently small, SN is well defined. Define P: ~‘ ~ ~ P(N) = mean curvature of SN~Then P is a smooth mapping with derivative at zero given by

dP(0).N=—(k . k+R

11+4)N

For example maximally extended Schwarzchild spacetime cannot be foliated by maximal Cauchy surfaces; see Smarr and York discussion.

(591

or a

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where k is the second fundamental form of S and 4 = —V2 is the positive Laplacian. (See ChoquetBruhat, Fischer and Marsden [17] and note that our k differs in sign from theirs.) Since ic ~ 0, k . k >0. By assumption, R 11 0 so k k + R11 >0. It follows from elliptic theory (see, for instance Morrey [48]or Hormander [39]Ch. X), that dP(0) is an isomorphism. Thus P is a local diffeomorphism by the implicit function theorem, so the result follows. This proves the HS result. For the CC result one needs to employ a regularity argument as in Ebin and Marsden [21]for instance. Lemma 5. Let (M, g) satisfy Conditions 1, 2, 3 and 4.of Theorem 3 and not be identically flat. Let S be a smooth compact spacelike maximal hypersurface. Then there is a neighborhood U of S that is foliated by hypersurfaces 5(K) of constant mean curvature whose mean curvatures ic comprise a neighborhood of zero and S(0)= S. If ic >0, S(K)C1(S)flUand if ic <0 then S(ic)CI~(S)flU. 4~RJ..L is not identically zero, the result Proof. Let second fundamental formtheof non-trivial S. If k or case ~ follows fromk be the the proof of Lemma 4. Thus is when k and ~4~R 1J. both vanish identically. If Condition 4 of Theorem 3 and part (a) of the hypersurface generic condition hold, it is well known that this implies that S and hence (M, g) are fiat. (See, e.g., Geroch [28]Appendix II.) Thus, assume part (1,) of the hypersurface generic condition 2 defined in the holds. previous proof. Again by elliptic theory we know again of the4map P: g~’ —* gu of the constant functions. Thus the map P is transversal to the thatConsider the cokernel consists exactly

constant functions. It follows by transversality theory that the embeddings giving rise to hypersurfaces of constant mean curvature form a smooth infinite dimensional manifold (see, e.g. Abraham and Marsden [1] p. 50). In fact, this manifold is diffeomorphic to the solutions of P (restricted to ker DP(io) and projected to the cokernel) equal to a constant. That is, hypersurfaces of constant mean curvature o are in one to one correspondence with real numbers a which are solutions of the equation

F(a) =

J

=

iç

where Ka is the mean curvature of a t = a = constant hypersurface in Gaussian normal coordinates. One computes that

4!~ da

=f(_lK~_2~2+4R~d3S \

a0

3

/

=0

a0

S

and similarly d2P T~

ua

a~O

0.

However, Raape,Z”Z”

(see Hawking and Ellis Raa~ZaZb

—

o~~

[371eq.

—

~O~CPK + ~Sa~~

+20.2)

(4.25), Tipler [64] eq. (7)). Now

~RaapbZ”Zt’

(5)

ZCZdZEaRbIC

4EeZIJ ~ 0 is equivalent to 0, so &.Ta~/~9ais not identically zero (Hawking and Penrose [38] p. 540).

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It follows that d~PJda~Ja =~<0and so by calculus P is equivalent to a cubic (i.e. equals a cubic after a change of coordinates). Thus for any small constant /3, P(a) = /3 has a unique small solution a. The lemma now follows. (Note that Lemma 2 only gave S(K) C J(S) if ic > 0, but the proof here shows that S(ic)C1(S).)

The proof of this lemma is useful in its own right. For example, it shows that if (M, g) is flat, any perturbation of it has a nearby hypersurface of constant mean curvature. The lemma can be regarded as a strengthening of the local isolation of maximal hypersurfaces in vacuum spacetimes obtained by Brill and Flaherty [7] Theorem 4.3. The lemma is also useful in studying the space of solutions of Einstein’s equations; see Fischer, Marsden and Moncrief [25]. In order not to be tied to Gaussian normal coordinates, it is important to now enlarge ~ to the space of all (H5) embeddings i :S -+ M that are spacelike. By standard arguments in manifolds of mappings (Palais [49]Ch. IV and Ebin—Marsden [21])~ is a smooth manifold. Now define P : Scalar functions on S by mapping i C to the mean curvature of i(S). Let c 0 E R be a given constant. We seek i C g such that P(i) = ci,. Let A denote the set of all constants c E [—icol,coil such that there is a curve of embeddings i(A) with i(0) the given embedding of S onto a maximal (hypersurface with P(i(A)) constant, A lies in the interval [0,Al say, and P(i(A)) = c, the given constant. By Lemma 5, A is an open set relative to [—icol, Icol] since any constant mean curvature hypersurface is part of a (regular) foliation. By connectedness of R, we will have hypersurfaces of all possible constant mean curvatures if we can show that A is closed. (This ‘method of continuity’ is suggested in Choquet-Bruhat, Fischer and Marsden [17].) We proceed in several steps. ~

~‘

Lemma 6. Suppose Conditions 1 and 5a hold. Let 0< A <°o and i(A) for A E [0, A), be a C curve in ~ such that P(i(A)) is a constant for each A; i.e. i(A) is a curve of spacelike, compact, constant mean curvature hypersurfaces in M. Then there is a compact set C C M such that i(A )(S) C C for all A E [0,A). Proof. This follows from Assumption 5a of Theorem 3 and compactness of J~(S~) flJ~(S~).

I

With these lemmas in hand we can now prove that A is closed. Let i(A) be a curve as above, and = i(AXS). We need to show that this curve can be extended to include A. However as A —IA, the operators DP(i(A)) are uniformly elliptic by the proof of Lemma 4 and by Assumption Sb. Since A 0, it follows by elliptic estimates that DP(i(A)) has a bounded right inverse and that its bound is uniform as A =+A. By the inverse function theorem, the radius of the ball around 1(A) in which P(i) constant is solvable is bounded away from zero as A A. Therefore i(A) can be extended as a spacelike embedding to include i(A). This proves the existence of a foliation ~ = a set of hypersurfaces of constant mean curvatures running from to ~ that is regular. The proof given above shows that U~= {p C M~pbelongs to a leaf in ~} is an open subset of M. Since M is connected, to show that U~= M, it suffices to show that U~is closed in M. To do this, let p~lie on a hypersurface S~C .9 of constant mean curvature i~ and assume p~ p C M as n We claim p C U~.If ic~contains a bounded subsequence, this follows by extracting a convergent subsequence of the 5,,. (Use Condition 5a and the proof ofTheorem 2.)Thuswe can assume that K,, say. Since p CM there is a point q to the future of p; say d(p, q) = I > 0. Choose n so large that —*

—~

~

—~

~.

—~ —~,

l>—3/K,,

and qEI~(S~)

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ic,, ~ Pn -*p and S,, is a Cauchy surface. By Lemma 1 there is a future directed geodesic -y,, from S,, to q of length d(q, 5,,) that contains no conjugate point to 5, between S,,

which can be done since

-*

and q. By 1> —3/ic,,, the length of y,, satisfies length y,,

>

—3/ic,,.

Since y,, is normal to S,, we have a contradiction to Lemma 3. Thus the case ic~ with p,, —~pEM does not occur. Similarly we get a contradiction to ic,, +00 by looking to the past of p. Thus U~is closed and so U~= M. This completes the proof of Theorem 3. I —~ —~

-~

The method of continuity may also be used to prove a global existence theorem for maximal hypersurfaces under avoidance and null hypotheses similar to those in Theorem 3. Definition. We say that (M, g) satisfies the continuation conditions if there is a smooth curve of Lorentz metrics g(A), say OA 1, on Msuch that 1. each (M, g(A)) satisfies Conditions 1, 2, 3, 4 of Theorem 3 and is not flat, 2. there exist compact smooth spacelike hypersurfaces S~and S~varying continuously with A such that S~C Jf(5_) and such that any compact smooth spacelike maximal hypersurface S~of (M, g(A)) lies in J~(S~) flJ(S~)(we refer to this by saying that maximal hypersurfaces avoid singularities), 3. any maxima! hypersurface in (M, g(A)) avoids turning null uniformly in A, in the sense of the

definition preceding Theorem 3, 4. (M, g(0)) has a smooth spacelike maxima! hypersurface S0,

5. g(1) = g, the given Lorentz metric. Theorem 4. If (M, g) satisfies the continuation conditions then it has a unique smooth compact maximal

hypersurface S (by Theorem 1, S has maximal volume and (M, g) is a Wheeler universe). {In the asymptotically flat case, omit Conditions 3 and 4 of Theorem 3 and the non-flat assumption. In Condition 2 of the preceding definition the hypersurf aces are to be asymptotic to the t =0 hypersurface in the Minkowski background. Then Theorem 4 carries over. The statements in parenthesis in Theorem 4 of course do not apply to the asymptotically flat case.} The proof of Theorem 4 proceeds along lines similar to the proof of Theorem 3. This time we consider a curve i(A) of spacelike embeddings of S to (M, g(A)) such that i(AXS) is maximal and i(OXS) = S~is the given maximal hypersurface in Assumption 4. This curve is defined in a neighborhood of A =0 by the perturbation argument in Lemma 5. In case b of the generic hypersurface condition we 3P/da3(,, =~<0 are both use the factbythata small transversality is an ofopen it and the maintained perturbation the condition, backgroundsometric see condition the proof dof Lemma 5. To extend this curve from A = 0 to A = 1 we again need to show that if it is defined for A C [0,A), then it can be extended to (and hence by openness, beyond) A. This can be done since DP(i(A)) be uniformly elliptic and hence the neighborhood of solvability of P(i) = 0 is uniform as A —s’A. This uniform ellipticity is guaranteed by Conditions 2 and 3 in the preceding definition, as in the proof of Theorem 3. I —

The proof shows that as one deforms away from a known solution (such as Friedmann) with a maximal hypersurface, and if the spacetimes through which the deformations take place satisfy

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Conditions 1, 2, 3 and 4 of Theorem 3 and are non-flat, then the maximal hypersurface will move as

well, and will remain smooth and spacelike until such a point that it turns null or collides with a singularity. In other words, collision with a singularity, turning null or an inability to maintain the energy condition and non-flatness along the path of deformation are the only ways of maximal hypersurface can disappear. As in Theorem 2 we conjecture that the turning null condition is superfluous. We refer to Eardley and Smarr [20]forsome Bondi—Tolman type models where collision with the singularity occurs. Theorems 3 and 4 prove the existence of maximal and constant mean curvature hypersurfaces by making global hypotheses on the spacetime. Ideally, one might like to draw such conclusions from hypotheses on the Cauchy data. This is an important and difficult problem. For purposes of numerical computation our results may be useful since they guarantee that under reasonable conditions, maximal or constant mean curvature hypersurfaces can fail to exist only by colliding with a singularity or turning null; in particular, if one is numerically constructing the foliation, successively marching forward by inverting the relevant elliptic equation (see Lemma 4) one is guaranteed that the hypersurfaces will not spontaneously disappear. They will be lost only by hitting a singularity or turning null, and numerical computation can warn one of this possibility. As hinted at by Goddard and others, it may be that the spatial topology will play an important role in the existence of a maximal Cauchy hypersurface. For example, any non-trivial (i.e., non-flat) perturbation of the flat spacetime T3 x R has a hypersurface of non-zero constant mean curvature, but not a maximal hypersurface. (This follows from Fischer and Marsden [22]p. 257, Brill [5]and the proofs of Theorems 1 and 4 above.) Brill [51and Fischer and Marsden [22] p. 257 showed that maximal hypersurfaces cease to exist locally when flat T3 x R space is perturbed. The global non-existence of maximal hypersurfaces in non-flat T3 x R spacetimes follows from a theorem of Schoen and Yau [53,56].They showed that if g is a Riemannian metric on T3 with R(g) 0 then g is fiat. It follows that if the timelike convergence condition and development uniqueness hold, and there exists a maximal, smooth spacelike Cauchy surface with topology T3 then the entire spacetime is flat. The global structure of a non-trivial perturbation of flat T3 x R space is not known, but its maximal development must have an all-encompassing singularity in one time direction (this follows from the proof of Theorem 1). Brill [5] conjectured that such a spacetime would expand forever in the other time direction. That is, if singularities occur in the other time direction, they would be hidden inside black holes and not be all-encompassing. This behavior may be general for T3 x R spacetimes satisfying the timelike convergence condition.* For example, let (M, g) be the open Friedmann solution with identifications on the surfaces of homogeneity and isotropy to convert them to T3. This spacetime begins with an all-encompassing singularity, and expands forever. It has no maximal hypersurface, but it is foliated by Cauchy hypersurfaces of constant mean curvature. On the other hand, the globally hyperbolic inhomogeneous vacuum 53 Gowdy [35,36]metrics are Wheeler and have a maximal hypersurface, as do the known spatially S3 homogeneous spacetimes (Ryan and Shepley [52],Collins and Hawking [191).It may not be outlandish to conjecture that any inextendible spacetime which has an S3 Cauchy surface and which satisfies the timelike convergence condition is Wheeler and contains a smooth compact spacelike maximal hypersurface. Our results show that for non-flat spacetimes, the existence of a maximal hypersurface and the maximal development being Wheeler are open conditions. For instance, any spacetime near the closed (53) Friedmann universe (near in the sense of C3 close initial data) has a maximal hypersurface and its maximal development is a Wheeler universe. * The results of Fischer, Marsden and Moncrief [25]which show that vacuum solutions near flat T3 xR space can be completely characterized by second order perturbation theory and the vanishing of Taub’s conserved quantities, may be useful for this problem.

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4. Sufficient conditions for avoidance of singularities and turning null

Theorems 2, 3 and 4 made assumptions that require further investigation. These are the avoidance of singularities assumption and the null condition. We have already given evidence that the null condition is always superfluous. In this section we will relate it to an a priori estimate on the length of the second fundamental form. This section also discusses various realistic hypotheses on the spacetime singularities which guarantee avoidance of singularities. Theorem 5. Let (M, g) have a C°° spacelike compact Cauchy surface and satisfy the timelike convergence condition. Let SK foliate an open set U C M, where SK has constant mean curvature ic, and ~< ,c
where A1 0, A2 0. Assume that the second fundamental forms of S~are uniformly bounded in length. Then the metrics on SK relative to the slicing {S~}with zero shift are uniformly positive definite. Remarks. 1. This conclusion is not the same as the definition of uniformly spacelike that was used

earlier, but it is sufficient for the proof of Theorem 3 to go through. 2. The results of Cheng and Yau [15]suggest that the assumed estimate on the length of the second fundamental form is automatically satisfied. This is consistent with our earlier conjecture. 3. There is a similar theorem for a family of maximal hypersurfaces relative to a curve of Lorentz metrics on M. 4. The theorem can be readily modified to cover the asymptotically flat case.

The proof of Theorem 5 relies on the maximum principle, which we now recall. Lemma 7 (Maximum Principle). Let L be a strongly elliptic scalar second order differential operator on a

compact smooth manifold Q without boundary; i.e. in coordinates, Lu =

~

a0(x) êx’êx’ + ~ bi(x)f!_

where ~

a1~~j

for some >0. Suppose h(x) is a smooth function such that h(x) 0 on 0, h is not identically zero, and u is a smooth function satisfying (L + h)u 0. Then u(x)sO fora!! xEQ.

Proof. Since u is smooth and 0 is compact, u attains a maximum at some point of 0. By the interior maximum principle (Protter and Weinberger [51] Theorem 6, p. 64), this maximum cannot be positive. I Proof of Theorem 5. We parametrize the constant mean curvature hypersurfaces by their mean curvatures and recall that K increases as SK moves to the future. (The argument will be similar for the

past directions.) Relative to this slicing of spacetime, the lapse function N obeys the equation

I.E. Marsden and Fl Tipler, Maximal hypersurfaces and foliations of constant mean curvature

132

1=(k.k+~4~R±1+,i)N,N>0 (see Choquet-Bruhat, Fischer andchoose Marsden [19],noting convention differences regarding the sign of t~~RI.L 0 and a constant C k). Let a = k k + 1 such that C1

sup (1/a). C

This is possible because for A near A 0, k~k +

t4~R

1J.is bounded away from zero. It follows that

(—A —aXN—Ci)=~—1+aCi0. Thus by Lemma 7, with L = —A a and u = N —

—

C1,

we get

0
In other words, N is uniformly bounded above. In the same way, we prove that 0
by choosing u = C2 N in Lemma 7. By hypothesis, C2 is bounded away from zero. —

The spacetime metric relative to this slicing is given by 2 dt2 + gij dx1 dx’ gab dx~ dx” = —N in which the unit normal has the form z” = (0, 1/N);

Za

= (0,

N)

so that the components of Z are uniformly bounded in ic, in both covariant and contravariant form. Since g~has signature (+ + +

—),

these bounds on N imply that g

0 is uniformly positive definite.

I

Next we turn to a study of the avoidance of singularities. The following concept will play a basic role. Definition (Tipler [62,631). A causal geodesic A (t) will be said to terminate in a strong curvature singularity at affine parameter value t0 if the following holds: Let jz (t) be the three-form on the normal space to A ‘(t) determined by any three linearly independent vorticity-free Jacobi fields along A (t). [IfA (t) is null, j~(t)is a two-form.] If p(t) vanishes for at most finitely many tin some neighborhood [t1, t0) of t0, then we require limit5.,0 js(t) = 0. If the timelike convergence condition holds and if

~i(t)

vanishes only for finitely many

t

in a

I.E. Marsden and FJ. Tipler, Maximal hypersurfaces and foliations of constant mean curvature

neighborhood of

t

0,

133

it can be shown that limit~..50~t(t) exists (I’ipler [63]).The above definition demands

that this limit be zero. The basic equation satisfied by ~a(t)is 2)js(t)1° =0 (3a) ~i? ,z(t)U3 + (Rai,V” + 2o~ where v = A ‘(t) and °ab is the shear of the Gaussian congruence along A (t); (for null geodesics replace 1/3 by 1/2; see Tipler [61]). The assumption that all singularities in a spacetime of the strong curvature type is an inextendibiity assumption. However, some inextendibility assumption must be made to insure the existence of a maximal hypersurface. For example, if we remove J~(S)in a closed (S3) Friedmann universe, where S is a surface of homogeneity and isotropy with KS >0, the resulting spacetime is Wheeler but it contains no maximal hypersurface. Tipler [63]has shown that requiring all singularities to be of the strong curvature type is a physically reasonable inextendibility assumption. Tipler, Clarke and Ellis [671have argued that requiring all singularities to be of the strong curvature type is an essential assumption in a correct formulation of cosmic censorship. In fact, they contend that these are the only physically reasonable singularities. If this is accepted, then Theorem 6 shows that maximal hypersurfaces and constant mean curvature foliation necessarily avoid all physically reasonable singularities. Theorem 6C below shows that the volume of any achronal spacelike hypersurface with or without boundary, must vanish as the hypersurface approaches a strong curvature singularity. Any physical object can be thought of as an achronal spacelike hypersurface with the surface of the object as the boundary. Thus in Wheeler universes in which all singularities are strong curvature singularities, “all physical objects are crushed to zero volume at the singularity, no matter what the properties of the physical object”. However, maximal hypersurfaces and even foliations of spacetimes by Cauchy hypersurfaces of constant mean curvature can exist in spacetimes which contain singularities but no strong curvature singularities. For example, Taub space is foliated by constant mean curvature Cauchy hypersurfaces, and has a maximal hypersurface as well. Although Taub space is the maximal Cauchy development of each of these Cauchy surfaces, it can be analytically extended into NUT space, and so it contains no strong curvature singularities. Taub space taken without the NUT part is a Wheeler universe, and the past and future singularities are of the “crushing” type. ~b

Definition (Eardley and Smarr [20]). Let (M, g) have a smooth compact Cauchy surface and suppose that all timelike geodesics are future incomplete. The future singularity is called a crushing singularity if the future of some smooth compact Cauchy surface in (M, g) can be foliated by compact Cauchy surfaces S(t) whose mean curvatures Ks(t) satisfy max

Ks(t)(J))-~’—~

as S(t) approaches the future singularity. There is a similar definition for the past singularity. Actually, for our results, we need only a weaker notion. The weakness is only apparent because under hypotheses 1 to 4 of Theorem 3, the proof of Theorem 4 shows that the preceding and following definitions are equivalent.

134

I.E. Marsden and FJ. Tipler, Maximal hypersurfaces and foliations of constant mean curvature

Definition. Let (M, g) have a smooth compact Cauchy surface and suppose that all timelike geodesic are future incomplete. The future singularity is called a crushing singularity if there exists a sequence S~of compact smooth Cauchy surfaces whose mean curvatures K~satisfy maxK~(p)-~’—ct~ as n-~. ,, mS,, For another example of a crushing singularity, consider the region of Minkowski space defined by x4 <0 and (x’)2 + (x2)2 + (x3)2 (x4)2 <0; i.e., the chronological past of the origin. The hypersurface (x~+ (x2~+ (x3? (x4)2 = —1, x4 <0, defines a Cauchy surface S for this region with mean curvature —3 (see Hawking and Ellis [37]p. 119). If we identify points of S under a discrete group G of isometries of Minkowski space such that S/G is compact, then the spacetime D(SIG) (with the induced metric from Minkowski space) is globally hyperbolic and is foliated by constant mean curvature compact Cauchy surfaces generated by timelike geodesics normal to S/G, and the mean curvature of these surfaces tends to as the hypersurfaces approach the singularity at x4 0, (x~ + (x2)2 + (x3? = (x4)2. (See Löbell [44]and Hawking and Ellis [37]p. 274.) Although this singularity is crushing, it is not a strong curvature singularity. In fact, the spacetime is locally extendible. Not all scalar polynominal curvature singularities (as defined in [37]p. 260) are crushing singularities. Eardley and Smarr [20] (see also Smarr and York [59])have constructed an asymptotically flat Tolman—Bondi spacetime which has a scalar polynomial curvature singularity that is not crushing and not of the strong curvature type. In this example the constant mean curvature hypersurfaces do not foliate the spacetime near the singularity. —

—

—~

Theorem 6. Let (M, g) be a spacetime with a C°’compact spacelike Cauchy surface and satisfying the

timelike convergence condition. (A) Suppose there are two smooth compact spacelike hypersurfaces S~and S with KS~<0 and Ks~->0. Then large volume and maximal hypersurfaces avoid singularities. (B) Suppose (M, g) is a Wheeler universe and that the future and past singularities are crushing singularities. Then large volume, maximal and constant mean curvature hypersurfaces avoid singularities. (C) Let (M,g) be a Wheeler universe in which either the future or the past singularity (or both) is a strong curvature singularity. Then that singularity is crushing and so (B) applies. (D) Let (M, g) be a Wheeler universe in which eitherthe past or the future singularity (or both) consists of a single c-boundary point.* Then that singularity is crushing and so (B) applies. Proof of Theorem 6. (A) Suppose that S flI~(S~) By the proof of Lemma 2, 0.

vo1(S~11 I(S))

vol(S

fl J+(5f))

with a similar inequality for S~.Thus we can choose S’ = [S nJ~(S)nJ(S~)J U [S~n 1(S)J U [S

n I~(S)]

* The c-boundary of a spacetime is one way of defining the singularity. See Geroch, Kronheimer and Penrose [31]and Hawking and Ellis [37] section 6.8. The significance of having a singularity consists of a single point is discussed by Budic and Sachs [10].

I.E. Marsden and F.J. Tipler, Maximal hypersurfaces and foliations of constant mean curvature

135

which will have the desired properties for the definition of large volume hypersurfaces to avoid singularities. The same thing for maximal hypersurfaces follows from Lemma 2. (B) The hypothesis of crushing singularities is stronger than the hypothesis in (A) and so the same conclusions hold. Taking S~to a member of the sequence hypothesized in the definition of crushing singularity with —R mean curvature of S at each point of S~,and S~similarly defined, the remark following the proof of Theorem 1(B) shows that constant mean curvature hypersurfaces avoid singularities. (C) The method is to show that one can construct a compact smooth hypersurface S near the future singularity, with I(~as negative as one likes (but not necessarily constant). A similar argument applies to the past singularity. Let S be a given arbitrary smooth spacelike hypersurface and 0 a given (large negative) number. Lemma 8. There exists another compact smooth hypersurface S9 in J~(~) such that for each maximal geodesic y normal to 5, the expansion 0.~along y (with 0, = K on 5) is less than 0 when y reaches S9. Proof. By the definition of strong curvature singularity, d V 0 as the singularity or a conjugate point is approached. The argument on p. 97letof xHawking andsoEllis the timelike condition 3/dt-/~0. Indeed, = (dV)113, (3a) [37]and holds. Thus, letting tconvergence = 0 be the parameter shows that d(dV)” value corresponding to a point where dx/dt is negative (since x -+0 and x 0 on y as the singularity is approached or a conjugate point is reached, dxldt must be negative at some point of y), we have from (3a) -~

=

_~J

(Rai~Z)’~V” +2cr2)x

dt+~(.

We have dx/dtI 2)x dt 0. Thus, <0 andaway by the timelike condition ~~fO’(RabV”V” + 2crd(dV)hI’s/dt, 0, dx/dt remains 5.~o bounded from zero. convergence Since 8,, = (lid V) d(dV)/dt = (31(d V)~’3) —cc as the future singularity or a conjugate point is approached. By Proposition 7.24 of Penrose L50], 0, varies continuously with y up to the first conjugate point. Using compactness of S we then get S 9. • ..+

The argument in Lemma 1 above gives the following. Lemma 9. There is a constant ic <8 and a compact achronalLipschitz hypersurfaceS to thefuture of~and ~e made up oft = constant cross-sections of the geodesic congruence from Son each piece of which S has mean curvature K. If S were smooth the proof would be complete. However S could have singularities of two types: 1. conjugate points, where the synchronous coordinate system has broken down, 2. points where two different t = constant hypersurfaces intersect. We can, however, deform S to a smooth compact hypersurfacewith ic < 0 everywhere, as follows. By modifying S in the vicinity of each singular point p E S, we claim to be able to deform S to a new smooth compact hypersurface S” lying to the future of S. To do this, we keep S unchanged except in a neighborhood U with compact closure of the singular points, and in U flJ~(S)we deform S through a one-parameter familyofsmooth hypersurfaces S(a),where a increasesto the future. We require U to be of

fE. Marsden and Fl. Tipler, Maximal hypersurfaces and foliations of Constant mean curvature

136

the form JY( V), where V C S is some closed neighborhood ofthe singular points of S, and V hasa smooth boundary. Now by Lemma 1 the geodesic normal coordinate system from the arbitrary Cauchy surface S will cover each S(a) almost everywhere, and since the expansion of this coordinate system is everywhere negative, there exists a family S(a) such that the normals to the S(a)have increasing angle /3 with respect to the geodesic normal coordinate system from S as a increases. Furthermore, the deformation S(a) can be continued with increasing /3 until the S(a) are arbitrarily close to either the singularity orthe light cone. We claim that fora family of S(a)which issufficiently close to the singularity or the light cone, wehave icS(a) < 8. To see this, note that forincreasing a, S(a)becomes ever closer to the light cone (since /3 is increasing), and since d(d V)/dt is monotone decreasing along each geodesic of the S cordinate system, we have d(d VS(a ))/dt~S(a) d(d V)/dt(s(~ d(dV)Idtjo along each y at each point of each S(a), where d Vs(,,~is the proper volume element on S(a), d V is the volume element of the geodesic normal coordinate system from S, t measures proper time along the geodesics of this coordinate system, and the parameter t’ is the proper time deformation parameter of the S(a); ~/ât’ is normal to S(a). Note that d(dV)/dt is bounded away from zero and infinity at each point of S(a), even though the coordinate system from S has broken down at some points ofS(a). This follows from an argument on p.97of Hawkingand Ellis [37].The volume element d V of the S coordinate system will vary in U fl S from some value d Vmax to zero at the points where the coordinate system breaks down, and thus wedo not have d Vs(,,) d V. However, by increasing a so that the S(a) are sufficiently close to either the singularity or the light cone, we can obtain ,-~

d VS(a) d Vmax

(

inf [d(dV)/dt])/[d(d V)/dt] the point where dV=~dV,,~~

pEUflS

Thus sufficiently close to the singularity or the light cone, we will have 1

=

d(dVs(~))< 1 A

U

~S(a)

ut

I

d(dV)

..157 ~‘max

A ut

1

d(dV)

dVmax

dt

U

< -

uns

[d(dV) L

ut

11~ d(dV)~ ~i rnf ~

dVma~JLUnS

ut


(Recall that all first derivatives of volume elements in the above inequalities are negative.) Let S” = S(a) with a sufficiently large so the above inequality holds. The hypersurface 5” is a smooth spacelike compact hypersurface with mean curvature

ic

<0. The singularity is therefore crushing.

(D) Suppose it is the future singularity which is a single c-boundary point. We first show that it is impossible for a spacelike achronal edgeless hypersurface to intersect the singularity. That is, any achronal spacelike edgeless hypersurface S in I~(S’)for some Cauchy surface S’ must be a compact Cauchy surface. Suppose S is non-Cauchy (and hence non-compact). Then there is a future-endless timelike curve y in M which does not intersect S; i.e. y flIt(S) is empty. Now 1(y)flS is also empty, since if there were a point p C 1(y) fl S, then there would be a future-directed timelike curve from p to y and hence y flIt(S) would not be empty. However, there is some future-endless timelike curve a C 14(S), so 1(a) fl S is non-empty. Thus L(a) I~(y),which contradicts the assumption that the

future c-boundary is a single point.

Now every smooth continuous foliation of Cauchy surfaces must have leaves whose volume vanishes

as the future singularity is approached. If this were not the case, then the

light

rays outgoing from a

I.E. Marsden and FJ. Tipler, Maximal hypersurfaces and foliations of constantmean curvature

137

point p would not have time to generate a compact r9J4~p)before the light rays intersect the singularity (recall that (M, g) is a Wheeler universe). However, Budic and Sachs [101have shown that for globally hyperbolic spacetimes whose future c-boundaries which are single points, 87(p) must be compact for any p in M We can thus use a deformation argument similar to that used in (C) to construct a smooth spacelike Cauchy surface with K arbitrarily small by getting sufficiently close to the singularity. This implies the singularity is crushing. U Remark. By combining the argument of Lemma 2 and the fact that the volume elements from any Cauchy surface must decrease to zero sufficiently close to a strong curvature singularity, we see that the volume of any achronal spacelike hypersurface with or without boundary must vanish as the hypersurface approaches the singularity. Conclusions

In this paper we have studied the questions of the existence and uniqueness of both maximal hypersurfaces and foliations of spacetimes by constant mean curvature hypersurfaces. In the closed universe case — the Wheeler universe case we have completely solved the uniqueness question: we have generalized the work of Brill and Flaherty, and proven the most general uniqueness theorem for maximal hypersurfaces and constant mean curvature hypersurfaces. We have also shown that compact maximal hypersurfaces have a volume which is greater than or equal to every other achronal Lipschitz hypersurface, in the spacetime, whether compact or non-compact. This generalizes a previous result of Brill and Flaherty, who showed that a maximal hypersurface locally maximizes volume. We have also shown that under certain conditions which are not quite as general as we would like, Wheeler universes contain maximal hypersurfaces and can be uniquely foliated by spacelike Cauchy surfaces of constant mean curvature. The chief weakness of our existence theorem is our assumption that such hypersurfaces stay uniformly bounded away from the light cone. We suspect this assumption can be proven, but we have been unable to do so. We present plausability arguments in support of this assumption. We also had to assume that such hypersurfaces avoid singularities. We have proven that they avoid crushing singularities, strong curvature singularities, and singularities whose c-boundary is a single point. We present reasons why physically realistic singularities must be one of the above types. The questions of existence and uniqueness of maximal hypersurfaces in asymptotically flat spacetimes are also discussed. —

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