ABSTRACT. We prove new estimates for the volume of a Lorentzian manifold and show especially that cosmological spacetimes with crushing singularities have finite volume.

Date: April 18, 2002.

2000 Mathematics Subject Classification. 35J60, 53C21, 53C44, 53C50, 58J05.

Key words and phrases. Lorentzian manifold, volume estimates, cosmological spacetime, general relativity, constant mean curvature, CMC hypersurface.

Let N be a (n + 1)-dimensional Lorentzian manifold and suppose that N can be decomposed in the form

where N_{0} has finite volume and N_{-} resp. N_{+} represent the critical past
resp. future Cauchy developments with not necessarily a priori bounded
volume. We assume that N_{+} is the future Cauchy development of a
Cauchy hypersurface M_{1}, and N_{-} the past Cauchy development of a
hypersurface M_{2}, or, more precisely, we assume the existence of a time
function x^{0}, such that

and that the Lorentz metric can be expressed as

where x = (x^{i}) are local coordinates for the space-like hypersurface M_{
1}
if N_{+} is considered resp. M_{2} in case of N_{-}.

The coordinate system (x^{})_{
0<<n} is supposed to be future directed,
i.e. the past directed unit normal (^{}) of the level sets

is of the form

If we assume the mean curvature of the slices M(t) with respect to the past directed normal--cf. [4] for a more detailed explanation of our conventions--is strictly bounded away from zero, then, the following volume estimates can be proved

Theorem 0.1. Suppose there exists a positive constant _{0} such
that

then

These estimates also hold locally, i.e. if E_{i} M(t_{i}), i = 1, 2, are
measurable subsets and E_{1}^{+},E_{
2}^{-} the corresponding future resp. past
directed cylinders, then,

and

In the following we shall only prove the estimate for N_{+}, since the
other case N_{-} can easily be considered as a future development by
reversing the time direction.

Let x = x() be an embedding of a space-like hypersurface
and (^{}) be the past directed normal. Then, we have the Gauß
formula

where (h_{ij}) is the second fundamental form, and the Weingarten
equation

We emphasize that covariant derivatives, indicated simply by indices, are always full tensors.

The slices M(t) can be viewed as special embeddings of the form

where (x^{i}) are coordinates of the initial slice M(t_{
1}). Hence, the
slices M(t) can be considered as the solution of the evolution
problem

with initial hypersurface M(t_{1}), in view of ( 0.5).

From the equation ( 1.4) we can immediately derive evolution
equations for the geometric quantities g_{ij},h_{ij},, and H = g^{ij}h_{
ij} of
M(t), cf. e.g. [3, Section 4], where the corresponding evolution
equations are derived in Riemannian space.

For our purpose, we are only interested in the evolution equation for the metric, and we deduce

in view of the Weingarten equation.

Let g = det(g_{ij}), then,

and thus, the volume of M(t),|M(t)|, evolves according to

where we shall assume without loss of generality that |M(t_{1}| is finite,
otherwise, we replace M(t_{1}) by an arbitrary measurable subset of
M(t_{1}) with finite volume.

Now, let T [t_{1},T_{+}) be arbitrary and denote by Q(t_{1},T) the
cylinder

then,

where we omit the volume elements, and where, M = M(x^{0}).

By assumption, the mean curvature H of the slices is bounded
from below by _{0}, and we conclude further, with the help of
( 1.7),

Letting T tend to T_{+} gives the estimate for |N_{+}|.

To prove the estimate ( 0.10), we simply replace M(t_{1}) by
E_{1}.

If we relax the conditions ( 0.6) and ( 0.7) to include the case _{0} = 0,
a volume estimate is still possible.

Theorem 1.1. If the assumptions of Theorem 0.1 are valid with
_{0} = 0, and if in addition the length of any future directed curve
starting from M(t_{1}) is bounded by a constant _{1} and the length of any
past directed curve starting from M(t_{2}) is bounded by a constant _{2},
then,

Proof. As before, we only consider the estimate for N_{+}.

From ( 1.6) we infer that the volume element of the slices M(t) is decreasing in t, and hence,

Furthermore, for fixed x M(t_{1}) and t > t_{1}

because the left-hand side is the length of the future directed curve

Let us now look at the cylinder Q(t_{1},T) as in ( 1.8) and ( 1.9). We
have

by applying Fubini’s theorem and the estimates ( 1.13) and ( 1.14).

A cosmological spacetime is a globally hyperbolic Lorentzian manifold N with compact Cauchy hypersurface S0, that satisfies the timelike convergence condition, i.e.

If there exist crushing singularities, see [1] or [2] for a definition, then, we proved in [2] that N can be foliated by spacelike hypersurfaces M() of constant mean curvature , - < < ,

where _{0} consists either of a single maximal slice or of a whole
continuum of maximal slices in which case the metric is stationary in
_{0}. But in any case _{0} is a compact subset of N.

In the complement of _{0} the mean curvature function is a regular
function with non-vanishing gradient that can be used as a new time
function, cf. [4] for a simple proof.

Thus, the Lorentz metric can be expressed in Gaussian coordinates
(x^{}) with x^{0} = as in ( 0.3). We choose arbitrary _{
2} < 0 < _{1} and
define

Then, N_{0} is compact, and the volumes of N_{-},N_{+} can be estimated
by

Hence, we have proved

Remark 2.2. Let N be a spacetime with compact Cauchy
hypersurface and suppose that a subset N_{-} N is foliated by
constant mean curvature slices M() such that

and suppose furthermore, that x^{0} = is a time function--which will be
the case if the timelike convergence condition is satisfied--so that the
metric can be represented in Gaussian coordinates (x^{}) with
x^{0} = .

Consider the cylinder Q(,_{2}) = { __<__ x^{0} __<__ _{
2}} for some fixed .
Then,

and we obtain in view of ( 1.7)

and conclude further

i.e.

Suppose that N is a Riemannian manifold that is decomposed as in ( 0.1) with metric

The Gauß formula and the Weingarten equation for a hypersurface now have the form

As default normal vector--if such a choice is possible--we
choose the outward normal, which, in case of the coordinate slices
M(t) = {x^{0} = t} is given by

Thus, the coordinate slices are solutions of the evolution problem

and, therefore,

i.e. we have the opposite sign compared to the Lorentzian case leading to

The arguments in Section 1 now yield

Theorem 3.1. (i) Suppose there exists a positive constant _{0} such
that the mean curvature H(t) of the slices M(t) is estimated by

then

(ii) On the other hand, if the mean curvature H is negative in N_{+}
and positive in N_{-}, then, we obtain the same estimates as Theorem
0.1, namely,

[1] D. Eardley & L. Smarr, Time functions in numerical relativity: marginally bound dust collapse, Phys. Rev. D 19 (1979) 2239-2259.

[2] C. Gerhardt, H-surfaces in Lorentzian manifolds, Commun. Math. Phys. 89 (1983) 523-553.

[3] _________ , Hypersurfaces of prescribed Weingarten curvature, Math. Z. 224 (1997) 167-194. gerhardt/preprints

[4] _________ , On the foliation of space-time by constant mean curvature hypersurfaces, e-print, 7 pages, math.DG/0304423

[5] _________ , Hypersurfaces of prescribed curvature in Lorentzian manifolds, Indiana Univ. Math. J. 49 (2000) 1125-1153. gerhardt/preprints

[6] S. W. Hawking & G. F. R. Ellis, The large scale structure of space-time, Cambridge University Press, Cambridge, 1973.

RUPRECHT-KARLS-UNIVERSITÄT, INSTITUT FÜR ANGEWANDTE MATHEMATIK, IM NEUENHEIMER FELD 294, 69120 HEIDELBERG, GERMANY

E-mail address: gerhardt@math.uni-heidelberg.de