ON LONG-TIME EVOLUTION IN GENERAL RELATIVITY AND GEOMETRIZATION OF 3-MANIFOLDS

MICHAEL T. ANDERSON

Contents

0. Introduction. 1 1. Background and Preliminary Results. 7 2. Curvature Estimates on CMC Surfaces. 12 3. Asymptotics of Future Complete Space-Times. 17 4. The Future Boundary of M_{F} in M. 28 5. Remarks on the Curvature Assumption. 32 References 33

Abstract. This paper introduces relations between the long-time asymptotic behavior of Einstein vacuum space-times and the geometrization of 3-manifolds envisioned by Thurston. The relations are obtained by analysing the asymptotic behavior of a CMC foliation by compact Cauchy surfaces and the induced curve of 3-manifold geometries. The Cheeger-Gromov theory is introduced in this context, and a number of open problems are considered from this viewpoint.

0. Introduction.

In this paper, we describe certain relations between the vacuum Einstein evolution equations in general relativity and the geometrization of 3-manifolds. In its simplest terms, these relations arise by analysing the long-time asymptotic behavior of natural space-like hypersurfaces Σ_{τ} , diffeomorphic to a fixed Σ, in a vacuum space-time. In the best circumstances, the induced asymptotic geometry on Σ_{τ} would implement the geometrization of the 3-manifold Σ.

We present several results illustrating this relationship, some of which require however rather strong hypotheses. Thus, besides proving these results, a second purpose of this paper is to clarify and make precise what some of the major difficulties are in carrying this program out to a deeper level. In this setting, we thus discuss a number of open problems and conjectures, some of which are well-known, which are of interest both in general relativity and in 3-manifold geometry. A third purpose is to introduce and use the Cheeger-Gromov theory of convergence and collapse of Riemannian manifolds in this setting; this theory, which has hardly been used to date in general relativity, should be a versatile and useful tool for a number of issues.

The idea of geometrizing 3-manifolds, and its proof in many important cases, is due to Thurston. We refer to [39] for an introduction from the point of view of hyperbolic geometry. A survey of geometrization from the point of view of general Riemannian geometries is given in [1], where some early relations with issues in general relativity were also presented. We also refer to [6] and [35] for surveys on topics in general relativity related to this paper.

Let (M, g) be a vacuum space-time, i.e. a 4-manifold M with smooth (C^{∞}) Lorentz metric g, of signature (−, +, +, +), satisfying the Einstein vacuum equations

Ric_{g} = 0.  (0.1)

Partially supported by NSF Grant DMS 9802722.

We assume that (M, g) is a cosmological space-time in the sense that M admits a compact Cauchy surface Σ. In particular, (M, g) is then globally hyperbolic and M is diffeomorphic to Σ × ℝ. The space-time (M, g) is the maximal C^{∞} vacuum Cauchy development of initial data on Σ, c.f. [30, Ch.7],[19].

By a result of [14] any compact space-like hypersurface is then a Cauchy surface for (M, g). It is thus natural to seek a preferred Cauchy surface with respect to which to view the evolution of the space-time. We assume throughout the paper that (M, g) has a compact Cauchy surface Σ with constant mean curvature

H = trK = const < 0.  (0.2)

The sign of H is chosen so that H < 0 corresponds to expansion of Σ in the future direction.

It is a well-known conjecture that any vacuum cosmological space-time admits a CMC Cauchy surface, and this is certainly the case for all known examples; c.f. [11] for further discussion. We will call a cosmological space-time satisfying (0.2) a CMC cosmological space-time.

One of the reasons for preferring compact CMC surfaces is that any such surface Σ, with a given value of H, is unique, c.f. [32]. Further, any such surface embeds at least locally, to the past and future, in a (unique) smooth foliation F = {Σ_{τ} , τ ∈ I} of a domain in (M, g) by CMC Cauchy surfaces. Each leaf Σ_{τ} is diffeomorphic to Σ and the function H : F → ℝ is monotone decreasing w.r.t. future proper time, c.f. again [32]. Let

M_{F} ⊂ M  (0.3)

be the maximal domain in M foliated in this way. Thus, the function

τ = H  (0.4)

defines a natural time function on M_{F}. We will use τ to denote the time parameter and H to denote the mean curvature. The range of τ : M_{F} → ℝ is a connected open interval I ⊂ ℝ, c.f. [25], and correspondingly, the subset M_{F} ⊂ M is a domain, i.e. a connected open subset, of M.

The Hawking-Penrose singularity theorem, c.f. [30, Ch.8.2], implies that if (M, g) is a CMC cosmological space-time, then there is a singularity in the finite proper time past of Σ. For the purposes of this Introduction, a singularity to the past (or future) of Σ merely means that (M, g) is not time-like geodesically complete to the past (or future) of Σ. Unfortunately, comparatively little is known in general about the structure of such an ”initial” singularity or singularities, c.f. [30], [40].

In this paper, we mainly, (although not only), consider issues concerning the future development of (M, g) from the Cauchy surface Σ.

Perhaps the first natural issue to consider is to characterize the situation when (M, g) is future geodesically complete, i.e. time-like geodesically complete to the future of Σ. This depends, at the very minimum, on topological properties of Σ. The basic reason derives from one of the constraint equations for the geometry of CMC surfaces in (M, g); namely on any leaf (Σ_{τ} , g_{τ}) ⊂ M_{F}, one has

s = |K|^{2} − H^{2},  (0.5)

where s is the scalar curvature of (Σ_{τ} , g_{τ}) and K is the second fundamental form of (Σ_{τ} , g_{τ}) ⊂ (M, g). Now the scalar curvature plays an important role in understanding the geometry and topology of 3-manifolds, just as it does in dimension 2. Thus, define the Sigma constant σ(Σ) of a closed oriented 3-manifold Σ to be the supremum of the scalar curvatures of unit volume Yamabe metrics on Σ, c.f. [1], [37]. This is a topological invariant, which divides the family of closed 3-manifolds naturally into three classes according to

σ(Σ) < 0, σ(Σ) = 0, σ(Σ) > 0.  (0.6)

By the solution to the Yamabe problem, σ(Σ) > 0 if and only if Σ admits a metric of positive scalar curvature. In fact, σ(Σ) > 0 if Σ admits a non-flat metric with s ≥ 0, c.f. [13].

If M_{F} contains a (non-flat) leaf Σ_{o} with H = 0, then (0.5) implies that s ≥ 0, so that σ(Σ_{o}) > 0. Since M_{F} is open, (M, g) admits CMC Cauchy surfaces with H > 0 and H < 0 and hence the Hawking-Penrose singularity theorem implies that there exist singularities to the finite proper time future and past of Σ_{o}.

This observation, together with the behavior in certain model cases, namely the space homogeneous Bianchi geometries on S^{3} and Kantowski-Sachs geometries on S^{2} × S^{1}, have led to following conjecture, c.f. [9],[35].

Recollapse Conjecture. If σ(Σ) > 0, then

M = M_{F},

and the leaves Σ_{τ} of F have mean curvature taking on all values monotonically in (−∞, +∞). The space-time (M, g) recollapses in finite proper time, i.e. there is a singularity to the finite proper time future of Σ.

This conjecture basically says that if Σ ⊂ M topologically can admit a metric of positive scalar curvature, then under the CMC time evolution, it will evolve in finite proper time to a maximal surface (Σ_{o}, g_{o}) with H = 0 where it realizes positive scalar curvature, and then recollapses to the future in finite time. Perhaps the strongest general result on this conjecture is the work in [25], which resolves it in case (M, g) has a past and a future crushing singularity [21], i.e. there exists a sequence of compact space-like hypersurfaces P_{i} with H(P_{i}) → −∞ uniformly, and similarly a such a sequence F_{i} with H(F_{i}) → +∞ uniformly.

We mention that the only known examples of 3-manifolds Σ with σ(Σ) > 0 are spherical space forms S^{3}/Γ, S^{2} × S^{1}, and connected sums of such manifolds.

The vast majority of 3-manifolds satisfy σ(Σ) ≤ 0, i.e. admit no metric of positive scalar curvature. This is the case for instance if Σ has a K(π, 1) factor in its prime or sphere decomposition, c.f. [27]. For such Cauchy surfaces, the space-time (M, g) admits no compact maximal hypersurfaces, and so the CMC time evolution is restricted to a subset of the half-line (−∞, 0). For much of the paper, we assume then that

σ(Σ) ≤ 0.

A natural generalization of the recollapse conjecture to all values of σ(Σ) is the following, raised in [33], c.f. also [21].

Global existence problem in CMC time. The CMC foliation F = {Σ_{τ} } in (M, g) exists for all allowable CMC time τ, i.e. for all τ ∈ (−∞, 0) if σ(Σ) ≤ 0 and for all τ ∈ (−∞, +∞) if σ(Σ) > 0.

This problem arises from a general belief that a CMC foliation should avoid any singularities at the boundary of (M, g), c.f. also [32].

The following result is of some relevance to this problem.

Theorem 0.1. Suppose σ(Σ) ≤ 0. Then either the CMC foliation F exists for all allowable CMC time or there exists a sequence of points x_{i} ∈ Σ_{τ_{i}}, with τ_{i} → τ_{o} < 0, such that {x_{i}} approaches a point of ∂M. Further, this is the case only if the curvature R of (M, g) blows up on {x_{i}}, in the sense that

|R|(x_{i}) → ∞.

Here |R| is measured in terms of the electric-magnetic decomposition of R with respect to the future unit normal T of the leaves Σ_{τ} of F, i.e.

|R|^{2} = |E|^{2} + |B|^{2},  (0.7)

where E, B are the symmetric bilinear forms E(X, Y) = ⟨R(X, T)T, Y ⟩, B(X, Y) = ⟨(∗R)(X, T)T, Y ⟩, c.f. [18,§0].

Theorem 0.1 implies that global existence in CMC time fails, (when σ(Σ) ≤ 0), only when a sequence of leaves Σ_{τ_{i}} of F approaches ∂M somewhere. This behavior is special to CMC foliations, and is not shared by other geometrically defined foliations, i.e. 3+1 decompositions of M. For instance the Gauss or proper time-equidistant foliation from a given space-like hypersurface will typically break down before reaching ∂M.

The assumption σ(Σ) ≤ 0 in Theorem 0.1 is not necessary and an analogous result is valid for any value of σ(Σ), c.f. Remark 2.6. In this context, Theorem 0.1 gives some information on the ’initial’ singularity or singularities to the finite past of Σ. Namely, it implies that either the curvature in the sense of (0.7) blows-up in the finite proper-time past of Σ, or the initial singularity is a crushing singularity.

In contrast to the positive case, when σ(Σ) ≤ 0 one does not expect, even if one has global existence in CMC time, that in general,

M_{F} = M,

i.e. the space-time (M, g) may well extend past the CMC foliated region. In this context, assuming global CMC time existence, we prove in Theorem 4.1 that the future boundary of M_{F} in M, i.e. the boundary

∂_{o}M_{F} ⊂ M,

to the future of the initial surface Σ in (0.2), is a countable union of smooth, connected, complete maximal hypersurfaces Σ_{o} ⊂ M. Each component Σ_{o} of ∂_{o}M_{F} is non-compact, and so not diffeomorphic to Σ. However, any smooth compact domain in Σ_{o} does embed in Σ. The hypersurface ∂_{o}M_{F} is a Cauchy surface for the region

M^{+} = M \ M_{F}

to the future of M_{F} in M. Of course any causal curve starting in M_{F} which enters M^{+} can never reenter M_{F}.

In analogy to the Hawking-Penrose singularity theorem, (which only applies to compact CMC hypersurfaces), we have:

Singularity Formation Conjecture. If ∂_{o}M_{F} ≠ ∅ in (M, g), then (M, g) possesses a singularity to the finite proper-time future in each component of M^{+}.

It is natural to conjecture that M^{+} itself has a foliation M_{F}^{+} by non-compact CMC surfaces limiting on crushing singularities to the finite future, c.f. [21]; this is discussed in more detail in §4. The region M^{+}, when non-empty, may be expected to be related to the formation of black holes in (M, g).

The global existence problem in CMC time and the singularity formation conjecture thus imply that (M, g) is future geodesically complete if and only if σ(Σ) ≤ 0 and M = M_{F}.

Next, suppose (M, g) is a space-time which is future geodesically complete. It is then natural to consider the possible asymptotic behavior of the space-time as the proper time t approaches infinity. This brings us to the relations with the geometrization of the 3-manifold Σ.

Definition 0.2. Let Σ be a closed, oriented and connected 3-manifold, with σ(Σ) ≤ 0. A weak geometrization of Σ is a decomposition of Σ,

Σ = H ∪ G,  (0.8)

where H is a finite collection of complete connected hyperbolic manifolds of finite volume embedded in Σ and G is a finite collection of connected graph manifolds embedded in Σ. The union is along a finite collection of embedded tori T = ∪T_{i}, T = ∂H = ∂G.

A strong geometrization of Σ is a weak geometrization as above, for which each torus T_{i} ∈ T is incompressible in Σ, i.e the inclusion of T_{i} into Σ induces an injection of fundamental groups.

Of course, it is possible that the collection T of tori dividing H and G in (0.8) is empty, in which case weak and strong geometrizations are the same. In such a situation, Σ is then either a closed hyperbolic manifold or a closed graph manifold.

For a strong geometrization, the decomposition (0.8) is unique up to isotopy, c.f. [2], [31], but this is far from being the case for a weak geometrization. Any 3-manifold, for instance S^{3} or T^{3}, has numerous knots or links whose complement admits a complete hyperbolic metric of finite volume, and hence gives rise to a weak geometrization.

We recall that a graph manifold is a union of S^{1} fibrations over surfaces, i.e. Seifert fibered spaces, glued together along toral boundary components. More precisely, a graph manifold G has a decomposition into Seifert fibered spaces S_{j}, with ∂S_{j} given by a union of tori {T_{k}}. The manifold G is then assembled by glueing (some of) the boundary tori together by toral automorphisms. In case ∂G = ∅, or ∂G consists of incompressible tori, there exists such a decomposition into Seifert fibered pieces along tori {T_{k}}, each of which is incompressible in G. Such an incompressible decomposition, called the JSJ decomposition, is unique up to isotopy, except in a few, well understood special cases. A typical exceptional case are the Sol manifolds, i.e. T^{2} bundles over S^{1} or over an interval or finite quotients of such bundles. The topology of graph manifolds is completely classified, c.f. [41].

The Seifert fibered spaces above each admit a geometric structure in the sense of Thurston, i.e. a complete locally homogeneous Riemannian metric, c.f. [39], [38]. Thus if Σ has a strong geometrization as above, then Σ admits a further decomposition by incompressible tori into domains, each of which has a complete geometric structure. Such a structure is called the geometrization of Σ in the sense of Thurston.

Not all 3-manifolds Σ have such a geometric decomposition however. The essential 2-spheres S^{2} ⊂ Σ are obstructions. This may be related to the fact that M_{F} ≠ M in general, and is discussed further in Remark 4.3.

Let t: M → ℝ be the proper time to a fixed (initial) CMC Cauchy surface Σ as in (0.2). Thus, t(x) is the maximal length of time-like curves from x to Σ. The maximum is achieved by a time-like geodesic, since (M, g) is globally hyperbolic. Similarly, let t_{τ} be the proper time of the CMC surface Σ_{τ} ∈ M_{F} to an initial surface Σ, i.e.

t_{τ} = t_{max}(τ) = max{t(x) : x ∈ Σ_{τ} }.  (0.9)

Thus t_{τ} is the usual Lorentzian distance t_{τ} = dist_{g}(Σ_{τ} , Σ). If (M, g) is future geodesically complete, t_{τ} → ∞ as τ → 0. In this situation, the volume of the slices (Σ_{τ} , g_{τ}) typically diverges to +∞, and the metrics g_{τ} typically become more and more flat, as t_{τ} → ∞. Hence it is natural to consider the rescaled metrics

ḡ_{τ} = t_{τ}^{−2} · g_{τ} ,  (0.10)

which at least give Σ_{τ} uniformly bounded volume, c.f. (3.16).

We raise the following:

Weak Global Asymptotics Problem.

Suppose Σ is closed, oriented, connected and σ(Σ) ≤ 0, as above. Suppose further that (M, g) is future geodesically complete and M_{F} = M. Then for any sequence τ_{i} → 0, the slices (Σ_{τ_{i}}, ḡ_{τ_{i}}) have a subsequence asymptotic to a weak geometrization of Σ.

More precisely, the problem states on the region H ⊂ Σ from (0.8), the metrics ḡ_{τ_{i}} in (0.10) converge to the complete hyperbolic metric of finite volume on H while on the region G ⊂ Σ, the metrics ḡ_{τ_{i}} collapse the graph manifold G with bounded curvature in the sense of Cheeger-Gromov to a lower dimensional space.

We note that although (0.10) is a fixed time rescaling of the metric g_{τ} for each τ = τ_{i}, i.e. the scale factor is independent of any base point in Σ_{τ} , the spatial behavior of the metrics ḡ_{τ_{i}} depends strongly on the choice of base points. Some sequences of base points x_{i} ∈ Σ_{τ_{i}} give rise to hyperbolic limits, while others give rise to a collapse along a graph manifold structure.

Of course, one may also formulate the corresponding strong problem, that the slices (Σ_{τ} , ḡ_{τ}) are asymptotic to a strong geometrization, or even the geometrization of Σ, as τ → 0.

Further discussion on more detailed aspects of this problem will be given below; see §1 and §3 for example for details on the collapse behavior. The conclusion of the strong version of this problem is essentially the same as the conclusions of Conjectures I and II of [1], [2] on the behavior of maximizing sequences of Yamabe metrics or minimizing sequences for the L^{2} norm of scalar curvature on a 3-manifold Σ.

This problem, as well as the previous conjectures and problems, still appear to be very difficult to resolve. We provide some evidence for its validity in the following result. Suppose (M, g) is a CMC cosmological space-time satisfying the following curvature assumption: there is a constant C < ∞ such that for x ∈ (M, g)

|R|(x) + t(x)|∇R|(x) ≤ \frac{C}{t^{2}(x)}.  (0.11)

Here, in analogy to (0.7), |∇R|^{2} = |∇E|^{2} +|∇B|^{2}. Observe that the bound (0.11) is scale-invariant. We then have the following concerning the asymptotic behavior of Σ_{τ} as τ → 0, (t_{τ} → ∞).

Theorem 0.3. Let (M, g) be a cosmological CMC space-time, with σ(Σ) ≤ 0. Suppose that the curvature assumption (0.11) holds, and

M_{F} = M,

to the future of an initial CMC surface Σ.

Then (M, g) is future geodesically complete and, for any sequence τ_{i} → 0, the slices (Σ_{τ_{i}}, ḡ_{τ_{i}}) have a subsequence asymptotic to a weak geometrization of Σ.

Theorem 0.3 is proved in §3, where we also describe the asymptotic behavior of the space-time (M, g), c.f. §3.1 and §3.3.

It is an interesting open problem whether any such weak geometrization is in fact a strong geometrization. This issue is discussed further in §3.3 where it is shown to be closely related to the recollapse conjecture. Note that since the weak decomposition (0.8) is not unique, different sequences τ_{i} → 0 may possibly give rise to distinct decompositions (0.8).

The bound on the derivative of the curvature R in (0.11) is needed for purely technical reasons, related to the use of the stability theorem for the Cauchy initial value problem. Of course one would like to remove this dependence on ∇R. In Theorem 3.1, we prove a version of Theorem 0.3 without the bound on ∇R, but requiring an extra hypothesis on the decay of the mean curvature H as t_{τ} → ∞. A more difficult problem is whether the decay assumption on R in (0.11) is really necessary, see §5 for further discussion.

Of course the basic reason that one may expect that hyperbolic manifolds arise from the long-time behavior of Σ_{τ} is the simple fact that the Lorentzian cone on a hyperbolic 3-manifold, i.e.

g_{o} = −dt^{2} + t^{2}g_{−1}  (0.12)

is a flat Lorentzian space-time, and so in particular a vacuum space-time; here g_{−1} is any complete metric of curvature −1 on a 3-manifold Σ, so that Σ = H^{3}(−1)/Γ, where Γ is a properly discontinuous subgroup of Isom(H^{3}(−1)). In fact, the flat metric g_{o} in (0.12) is just the quotient of the future light cone about a point {0} in flat Minkowski space (ℝ^{4}, η) by the action of Γ extended to the isometry group of (ℝ^{4}, η), i.e. a Löbell space-time, c.f. [30, p.274].

In this respect, it has recently been proved by Andersson-Moncrief [8] that if Cauchy data on a given compact CMC surface Σ are a sufficiently small perturbation of hyperbolic data as in (0.12), and the hyperbolic metric on Σ is rigid among conformally flat metrics on Σ, then the resulting space-time (M, g) is future geodesically complete, and asymptotic to the flat Lorentzian metric (0.12) as τ → 0. This is a compact analogue of the result of Christodoulou-Klainerman [18] on the global stability of Minkowski space.

We also refer to the interesting work of Fischer-Moncrief [24], where the long-term behavior of a naturally defined Hamiltonian for the space-time evolution is related to the Sigma constant (0.6) of Σ.

I would like to thank Lars Andersson and Alan Rendall for interesting and informative discussions on topics related to the paper.

1. Background and Preliminary Results.

Let (M, g) be a CMC cosmological space-time, with M_{F} ⊂ M as in (0.3), and with time function τ = H on M_{F}. The 4-metric g may then be decomposed into a 3+1 split as

g = −α^{2}dτ^{2} + g_{τ} ,  (1.1)

where g_{τ} is a Riemannian metric on Σ_{τ} . Hence g_{τ} may be viewed as a curve of metrics on a fixed slice Σ, by means of diffeomorphisms ψ_{τ} : Σ → Σ_{τ} . The function α is called the lapse function, and measures the (infinitesimal) time-like spread of the leaves Σ_{τ} , since α^{2} = −g(∂_{τ} , ∂_{τ}) > 0.

The vacuum Einstein equations Ric = 0 imply, via the Gauss and Gauss-Codazzi equations, constraints on the geometry of (Σ_{τ} , g_{τ}) ⊂ (M, g). The (CMC) constraint equations are:

s = |K|^{2} − H^{2},  (1.2)
δK = 0.  (1.3)

Here K is the 2^{nd} fundamental form or extrinsic curvature, given by K(X, Y) = g(∇_{X}Y, T) where T is the future unit normal, H = tr_{g}K, and δ is the divergence operator. The sign convention on K agrees with the convention following (0.2). The vacuum Einstein evolution equations are

L_{∂_{τ}} g = −2αK,  (1.4)
L_{∂_{τ}} K = −D^{2}α + α(Ric + H · K − 2K^{2}),  (1.5)

where L denotes the Lie derivative, D^{2} the Hessian and K^{2}(X, Y) = ⟨K(X), K(Y)⟩.

The choice of time function τ = H determines the lapse α, which satisfies the following lapse equation, derived from the 2^{nd} variational formula for volume:

− Δα + |A|^{2}α = \frac{dH}{dτ} = 1,  (1.6)

where Δ = trD^{2}. The equations (1.2)-(1.6) on the surfaces (Σ_{τ} , g_{τ}) are equivalent to the vacuum Einstein equations on (M, g). Finally, the Gauss-Codazzi equations, (in general), give

dK = R^{T} ,  (1.7)

i.e. dK(X, Y, Z) = ∇_{X}K(Y, Z) − ∇_{Y} K(X, Z) = R(T, X, Y, Z). The operator d in (1.7) is the exterior derivative with respect to the Levi-Civita connection ∇ of g_{τ} , when K is viewed as a 1-form with values in T Σ_{τ} .

The lapse equation (1.6) gives rise to the following elementary and well-known, but important estimates.

Lemma 1.1. On any leaf Σ_{τ} , the lapse α satisfies the following bounds:

supα ≤ \frac{1}{inf|K|^{2}} ≤ \frac{3}{H^{2}} ,  (1.8)

and

infα ≥ \frac{1}{sup|K|^{2}} > 0.  (1.9)

Proof: This is immediate consequence of the maximum principle applied to the elliptic equation (1.6).

Note that α is not scale-invariant - it scales as the square of the distance, i.e. as the metric and so inversely to the norm of the curvature. Namely, if g′ = λ^{−2}g is a rescaling of g, then (1.1) becomes

g′ = −α^{2}λ^{−2}dτ^{2} + λ^{−2}g_{τ} .  (1.10)

The space metric g′_{τ} = λ^{−2} · g_{τ} is of course just a rescaling of g_{τ} . Now τ is the mean curvature in a given metric and so τ′ = λτ. Thus, (1.10) becomes

g′ = −(λ^{−2}α)^{2}d(τ′)^{2} + g′_{τ} ,  (1.11)

so that

α′ = λ^{−2}α.  (1.12)

In particular, under this rescaling, the lapse equation is scale-invariant, as are the other equations (1.2)-(1.7), since the vacuum Einstein equations are scale-invariant.

It will be useful to also represent the metric w.r.t. the parameter t_{τ} defined as in (0.9) in place of τ. Thus, we have

g = −α^{2}(\frac{dτ}{dt_{τ}})^{2}dt_{τ}^{2} + g_{τ} .  (1.13)

Here, the lapse factor α\frac{dτ}{dt_{τ}} is scale-invariant. In (1.13), we are thus parametrizing the slices Σ_{τ} by the distance function t_{τ}; since t_{τ} is the maximal proper time between Σ_{τ} and Σ_{τ_{o}}, we have

α\frac{dτ}{dt_{τ}} ≤ 1,  (1.14)

everywhere on Σ_{τ} .

Next we prove a natural comparison result for cosmological CMC space-times which will be important in the proof of Theorem 0.3. It is essentially a standard argument in Lorentzian comparison geometry.

Proposition 1.2. Let Σ_{τ_{o}} be any initial compact CMC surface, with H_{o} = τ_{o} < 0 in (M, g). Then

− H(Σ_{τ}) · t_{τ} ≤ 3(1 − \frac{τ}{τ_{o}}),  (1.15)

where t_{τ} = dist_{g}(Σ_{τ} , Σ_{τ_{o}}) ≥ 0. Equality holds in (1.15) for some τ = τ_{1} > τ_{o} if and only if the domain M_{[τ_{o},τ_{1}]} = ∪_{τ∈[τ_{o},τ_{1}]}Σ_{τ} ⊂ (M, g) is flat and is isometric to a time-annulus in a flat Lorentzian cone as in (0.12). In particular, all the leaves Σ_{τ} , τ ∈ [τ_{o}, τ_{1}] are of constant negative curvature.

Proof: Given the initial surface Σ_{τ_{o}}, let S(r) = {x ∈ M : t(x, Σ_{τ_{o}}) = r} = t^{−1}(r), where t is the Lorentzian distance as preceding (0.9). Let θ be the mean curvature of S(r), with the sign convention as following (0.2) or (1.3). Since the space-time is vacuum, the Raychaudhuri equation, c.f. [30, §4.4], gives

− θ′ = −\frac{1}{3}θ^{2} − 2σ^{2} ≤ −\frac{1}{3}θ^{2},  (1.16)

where σ is the shear, the trace-free part of 2^{nd} fundamental form of S(r), along this geodesic congruence and θ′ = ∇_{N}θ, where N is the future directed unit tangent vector to the geodesics γ normal to {S(r)}. Thus by integration, one immediately obtains, as long as θ ≤ 0,

− \frac{1}{θ(s)} − \frac{1}{3}s ↑,  (1.17)

i.e. this function is monotone increasing in s along any future directed geodesic γ. Hence, for s ≥ 0,

− \frac{1}{θ(s)} − \frac{1}{3}s ≥ −\frac{1}{H(Σ_{τ_{o}})}.  (1.18)

Now on Σ_{τ} , there is a point x_{τ} realizing t_{τ} , the maximal value of t on Σ_{τ} . Hence S(t_{τ}) lies to the future of Σ_{τ} , but touches Σ_{τ} at the point x_{τ} . By a standard geometric maximum principle, c.f. [7, Thm. 3.6] for example, we then have

− θ(x_{τ}) ≥ −H(Σ_{τ}),  (1.19)

again with the sign convention above understood. Substituting this in (1.18), and using the fact that s = t_{τ} at x_{τ} gives (1.15). It is obvious that (1.15) holds in any region where H = τ ≥ 0.

If equality holds in (1.15) at some τ_{1} > τ_{o}, then equality holds in (1.18) for all τ ∈ [τ_{o}, τ_{1}]. Hence by (1.16), σ = 0 everywhere in the time annulus t^{−1}[τ_{o}, τ_{1}] and so the geodesic spheres S(r) are everywhere umbilic. Further, the inequality in (1.19) must also be an equality, and hence the maximum principle again implies that Σ_{τ} = S(t_{τ}), so that the surfaces Σ_{τ} are all equidistants. These facts imply that (M_{[τ_{o},τ_{1}]}, g) is a Lorentzian cone of the form g = −dt^{2} + t^{2}g_{τ_{o}}, where g_{τ_{o}} is the metric on Σ_{τ_{o}}, up to rescaling. The vacuum equations (0.1) then imply that g must be flat and the metrics g_{τ} on Σ_{τ} are of constant curvature.

Observe that (1.15) holds if (M, g) satisfies the strong energy condition Ric_{g} ≥ 0 in place of the vacuum equations (0.1). However, the rigidity statement in the second part of Proposition 1.2 requires the vacuum equations.

Proposition 1.2 of course immediately implies

limsup_{t_{τ} →∞}t_{τ} (−H) ≤ 3,

when M is geodesically complete to the future of Σ_{τ_{o}}.

Since H is the mean curvature, the first variational formula for volume gives

\frac{d}{ds}volΣ_{s} = − ∫_{Σ_{s}} H · fdV,

where s is any parametrization for the leaves Σ_{τ} and f > 0 is the length of the associated variation field. For example, if s = τ, then f = α as in (1.1). Consider the family Σ_{τ} parametrized w.r.t. t_{τ} , as in (1.13). Then f = α\frac{dτ}{dt_{τ}} and hence by (1.14),

\frac{d}{dt_{τ}} volΣ_{τ} ≤ − ∫_{Σ_{τ}} HdV = −HvolΣ_{τ} ≤ \frac{3}{t_{τ}} volΣ_{τ} .  (1.20)

This discussion gives the following monotonicity result on the volume growth of the leaves Σ_{τ} .

Corollary 1.3. The volume of the leaves Σ_{τ} in (M, g) satisfies

\frac{volΣ_{τ}}{t_{τ}^{3}} ↓,  (1.21)

i.e. the ratio is monotone non-increasing in t_{τ} or τ. The volume ratio is constant in [τ_{o}, τ_{1}] if and only if the corresponding time-annulus in (M, g) is a flat Lorentzian cone.

Proof: This follows immediately from (1.20) by integration and the corresponding rigidity statement in Proposition 1.2.

We will need several results from the Cheeger-Gromov theory on convergence and collapse of Riemannian manifolds, c.f. [15], [16], [4]. For simplicity, we consider only the 3-dimensional case, where the Ricci curvature determines the full curvature tensor.

Proposition 1.4. Let (D_{i}, g_{i}, x_{i}) be a sequence of Riemannian metrics on 3-manifolds D_{i}, satisfying

|Ric_{g_{i}}| ≤ Λ, diam_{g_{i}}D_{i} ≤ D, dist_{g_{i}}(x_{i}, ∂D_{i}) ≥ d_{o},  (1.22)

and

vol_{g_{i}}B_{x_{i}}(1) ≥ v_{o},  (1.23)

for some constants Λ, D < ∞, and v_{o}, d_{o} > 0. Then for any ϵ > 0 with ϵ < d_{o}, there are domains U_{i} = U_{i}(ϵ) ⊂ D_{i}, with dist_{g_{i}}(∂D_{i}, ∂U_{i}) < ϵ, and diffeomorphisms ψ_{i} of U_{i}, such that the metrics ψ_{i}^{∗}g_{i} have a subsequence converging in the C^{1,β′} topology to a limit C^{1,β} Riemannian manifold (U_{∞}, g_{∞}, x_{∞}), x_{∞} = lim x_{i}, for any β′ < β < 1. In particular, U_{∞} is diffeomorphic to U_{i}, for i sufficiently large.

Proposition 1.5. Let (D_{i}, g_{i}, x_{i}) be a sequence of Riemannian metrics on 3-manifolds D_{i}, satisfying

|Ric_{g_{i}}| ≤ Λ, diam_{g_{i}}D_{i} ≤ D, dist_{g_{i}}(x_{i}, ∂D_{i}) ≥ d_{o},  (1.24)

and

vol_{g_{i}}B_{x_{i}}(1) → 0,  (1.25)

for some constants Λ, D < ∞ and d_{o} > 0. Then there are domains U_{i} ⊂ D_{i} as above and diffeomorphisms ψ_{i} of D_{i}, such that U_{i} is either a Seifert fibered space or a torus bundle over an interval. In both cases, the g_{i} diameter of any fiber F, (necessarily a circle S^{1} or torus T^{2}), goes to 0 as i → ∞.

Further, suppose U_{i} is not diffeomorphic to a closed spherical space-form S^{3}/Γ. Then for any i < ∞ sufficiently large, π_{1}(F) injects in π_{1}(U_{i}) and there is a finite cover Ū_{i} of U_{i} such that the sequence (Ū_{i}, g_{i}, x_{i}) does not collapse, i.e. satisfies (1.23), and hence there is a subsequence converging in the C^{1,β′} topology as above to a C^{1,β} Riemannian manifold (Ū_{∞}, g_{∞}, x_{∞}). The limit (Ū_{∞}, g_{∞}) admits a locally free isometric action by one of the following Lie groups: S^{1}, S^{1} × S^{1}, T^{3}, Nil.

Both of these results essentially remain valid if diam_{g_{i}}D_{i} → ∞ as i → ∞, but now both behaviors convergence/collapse are possible depending on the choice of base points x_{i}. Thus, suppose (Σ, g_{i}) are complete Riemannian 3-manifolds with

|Ric_{g_{i}}| ≤ Λ,  (1.26)

for some constant Λ < ∞. For a given sequence of base points x_{i} ∈ Σ, suppose (1.23) holds. Then a subsequence of (Σ_{i}, g_{i}, x_{i}) converges, modulo diffeomorphisms of Σ_{i}, to a complete C^{1,β} limit Riemannian manifold (Σ_{∞}, g_{∞}, x_{∞}). The limit Σ_{∞} embeds weakly in Σ, denoted as

Σ_{∞} ⊂⊂ Σ,  (1.27)

in the sense that any domain with smooth and compact closure in Σ_{∞} embeds smoothly in Σ. (Such a result also holds if Σ = Σ_{i} varies with i, but we will not need this situation).

On the other hand, if (1.25) holds at x_{i} ∈ Σ, then the sequence collapses uniformly on compact sets, in the sense that vol_{g_{i}}B_{y_{i}}(1) → 0, for all y_{i} ∈ B_{x_{i}}(R), for any fixed R < ∞. In this case, the collapse may be unwrapped as above and one obtains a complete limit C^{1,β} Riemannian manifold (Σ̄_{∞}, g_{∞}, x_{∞}), which admits a locally free isometric action by one of the groups S^{1}, S^{1} ×S^{1}, T^{3}, Nil.

The convergence in Propositions 1.4 and 1.5 above is actually in the weak L^{2,p} topology, for any p < ∞ and the limits are L^{2,p} smooth. By Sobolev embedding, convergence in this topology implies convergence in the C^{1,β} topology, for any β < 1.

Remark 1.6. If the bound |Ric_{g_{i}}| ≤ Λ in (1.26) is strengthened to a bound on the derivatives of the curvature, i.e.

|∇^{j}Ric_{g_{i}}| ≤ Λ_{j} < ∞,  (1.28)

then one obtains convergence to a L^{j+2,p} limit in the weak L^{j+2,p} topology, p < ∞. In particular, if (1.28) holds for all j, then one has C^{∞} convergence to a C^{∞} limit.

(ii). By standard results in Riemannian comparison geometry, a lower volume bound as in (1.23) under the curvature bound (1.22) is equivalent to a lower bound on the injectivity radius at x_{i}, c.f. [34].

The collapse situation in Proposition 1.5 corresponds to the formation of families of (arbitrarily) short geodesic loops in (D_{i}, g_{i}) within bounded distance to x_{i} and the unwrapping of the collapse corresponds to the unwrapping of these short loops to loops of length about 1 in covering spaces.

In the single exceptional case of S^{3}/Γ, the collapse is along inessential loops, as for instance in the Berger collapse, c.f. [15], of S^{3} to S^{2} by shrinking the S^{1} fibers of the Hopf fibration. (Exactly this behavior occurs at the Cauchy horizon of the Taub-NUT metric, c.f. [30, §5.8]). In all other cases, the short loops are essential in bounded domains about x_{i}, (with diameter depending on the degree of collapse in (1.25)), although they are not necessarily essential globally, i.e. in all of D_{i}.

The dimension of the groups S^{1}, S^{1} × S^{1}, T^{3} or Nil corresponds to the dimension of this family of short loops. For instance, in the latter two cases of T^{3} or Nil, one has a 3-dimensional family of short loops, and so all of (D_{i}, g_{i}) is collapsing, and in fact

diam_{g_{i}}D_{i} → 0;

c.f. §3.3(I) for an example. Of course, by (1.24), this is possible only if D_{i} is closed. In the context of Proposition 1.5, this means that the base spaces of the S^{1} or S^{1} × S^{1} bundle are collapsing, so that the finite covers also unwrap the base space collapse.

(iii). Define the C^{1,β} harmonic radius r_{h}(x) = r_{h}^{1,β}(x) of a Riemannian 3-manifold (Σ, g) at x to be the largest radius of the geodesic ball about x on which there exists a harmonic coordinate chart in which the metric components are controlled in the C^{1,β} norm, i.e. as matrices,

C^{−1}δ_{ij} ≤ g_{ij} ≤ Cδ_{ij},  (1.29)

and, for p given by β = 1 −\frac{3}{p},

r_{h}(x)^{p}||∂g_{ij}||_{C^{β}(B_{x}(r_{h}(x)))} ≤ C.  (1.30)

The power of r_{h} in (1.30) is chosen so that r_{h} scales as a radius, i.e. as a distance. The assumptions (1.22)-(1.23) imply a lower bound on r_{h}, r_{h}(x_{i}) ≥ r_{o}, where r_{o} depends only on Λ, v_{o}, (and d_{o}), c.f.

[4] and references therein. Similar results hold for the C^{j+1,β} harmonic radius, defined analogously to (1.29)-(1.30) when a curvature bound of the form (1.28) is used.

2. Curvature Estimates on CMC Surfaces.

In this section, we prove Theorem 0.1 and several related results. To begin, we claim that for any given compact subset Ω ⊂ M, there is a constant Λ_{o} = Λ_{o}(Ω) < ∞ such that, for all x ∈ Ω,

|R|(x) ≤ Λ_{o},  (2.1)

where |R| is measured as in (0.7). Since (M, g) is smooth, (2.1) holds provided the future unit normal T to the leaves Σ_{τ} of F remains a uniformly bounded distance away from the null-cones in Ω, (so that T remains in a compact set in the tangent bundle T Ω). Since (M, g) is globally hyperbolic and Ω is compact, this follows directly from the apriori estimate of Bartnik, [10, Thm.3.1].

Of course, on approach to ∂M, the bound (2.1) may no longer hold. In the results to follow, we use this bound to control the intrinsic and extrinsic geometry of the leaves Σ_{τ} of M_{F}.

First, consider the symmetric bilinear form K as a 1-form on Σ_{τ} with values in T Σ_{τ} , as in (1.7). We have the following standard Weitzenbock formula, c.f. [12] for example:

δdK + dδK = 2D^{∗}DK + R(K),  (2.2)

on (Σ_{τ} , g_{τ}), where the curvature term is given by

R(K) = Ric ∘ K + K ∘ Ric − 2R ∘ K,  (2.3)

Here R ∘ K is the action of the curvature tensor R on symmetric bilinear forms given by: (R ∘ K)(X, Y) = ∑⟨R(X, e_{i})K(e_{i}), Y ⟩, {e_{i}} a local orthonormal framing for Σ_{τ} . The sign of the curvature tensor is such that R_{ijji} denotes the sectional curvature in the (ij) direction.

By the constraint equation (1.3), δK = 0 while the Gauss-Codazzi equation gives dK = R^{T} . Pairing (2.2) with K thus implies

Δ|K|^{2} = 2|DK|^{2} − ⟨δR^{T} , K⟩ + ⟨R(K), K⟩,  (2.4)

The key point is to prove that the algebraic curvature term in (2.4) is positive, on the order of |K|^{4}.

Lemma 2.1. On a leaf Σ_{τ} , with mean curvature τ = H, the curvature term in (2.4) satisfies

⟨R(K), K⟩ ≥ |K|^{4} − c(|H||K|^{3} + Λ_{o}|K|^{2} + H^{2}|K|^{2}),  (2.5)

where c is a numerical constant, independent of (M, g) and Σ_{τ} .

Proof: Let λ_{i}, i = 1, 2, 3, be the eigenvalues of K. The Gauss equation gives, for j ≠ k,

R_{jkkj} = R_{jkkj} − λ_{j}λ_{k}.  (2.6)

The minus sign in (2.6), which is crucial in the following, uses the fact that (M, g) is Lorentzian; in the case of Riemannian geometry, one has a plus sign, (which would give the wrong sign to the dominant term in (2.5)). We also have, since Σ is a 3-manifold, Ric_{ii} = \frac{s}{2} − R_{jkkj}, where (i, j, k) are distinct. Hence

⟨Ric ∘ K, K⟩ = ⟨K ∘ Ric, K⟩ = Ric_{ii} · λ_{i}^{2} = \frac{s}{2} ∑ λ_{i}^{2} − ∑ R_{jkkj}λ_{i}^{2}

= \frac{s}{2}|K|^{2} + ∑ λ_{i}^{2}λ_{j}λ_{k} − ∑ λ_{i}^{2}R_{jkkj}.

Observe that ∑ λ_{i}^{2}λ_{j}λ_{k} = Hλ_{1}λ_{2}λ_{3} = HdetK ≤ c · |H||K|^{3} and s = |K|^{2} − H^{2} by the constraint equation (1.2).

Similarly, from the definition, one computes

⟨R ∘ K, K⟩ = ∑_{i≠j} λ_{i}λ_{j}R_{ijji} = ∑_{i≠j} λ_{i}λ_{j}(R_{ijji} − λ_{i}λ_{j}) = ∑_{i≠j} λ_{i}λ_{j}R_{ijji} − ∑_{i≠j} λ_{i}^{2}λ_{j}^{2}.

Combining these estimates and using the bound (2.1) gives

⟨R(K), K⟩ ≥ |K|^{4} − c(|H||K|^{3} + Λ_{o}|K|^{2} + H^{2}|K|^{2}) + 2 ∑_{i≠j} λ_{i}^{2}λ_{j}^{2},

which implies (2.5).

Proposition 2.2. Suppose (2.1) holds and let Σ_{τ} be any compact leaf of M_{F}, Σ_{τ} ⊂ Ω, with

|H| ≤ H_{o} < ∞,  (2.7)

for some constant H_{o} > 0. Then there is a constant Λ_{1} = Λ_{1}(Λ_{o}, H_{o}) such that, on (Σ_{τ} , g_{τ}),

|K|_{L^{∞}} ≤ Λ_{1}, |Ric|_{L^{∞}} ≤ Λ_{1}.  (2.8)

Proof: By the Gauss equation (2.6) and the bound (2.1), the two estimates in (2.8) are equivalent, so it suffices to prove the first estimate.

Since Σ_{τ} is compact, we may choose a point x ∈ Σ_{τ} realizing the maximal value of |K| on Σ_{τ} , i.e.

|K|(y) ≤ |K|(x) ≡ K̄,  (2.9)

for all y ∈ Σ_{τ} . We will prove (2.8) by contradiction, although with some further work it is possible to give a direct argument, (without passing to the limit below), in which case the dependence of Λ_{1} on (Λ_{o}, H_{o}) is more explicit.

Thus, if (2.8) is false, there exist compact CMC surfaces (Σ_{i}, g_{i}) in space-times (M_{i}, g_{i}) satisfying (2.1) and (2.7), but for which K̄_{i} = |K_{i}|(x_{i}) = sup_{Σ_{i}}|K_{i}| → ∞.

Consider the rescaled metric

g̃_{i} = (K̄_{i})^{2} · g_{i}  (2.10)

on Σ_{i}, and the corresponding space-time (M_{i}, g̃_{i}). By the scaling properties of the extrinsic curvature K, (2.9) becomes

|K̃_{i}|(y_{i}) ≤ |K̃_{i}|(x_{i}) = 1  (2.11)

on (Σ_{i}, g̃_{i}). Similarly, the scaling properties of the curvature R and mean curvature H imply

|R̃_{i}| ≤ Λ_{o} · (K̄_{i})^{−2} → 0, as i → ∞,  (2.12)
|H̃_{Σ_{i}}| ≤ H_{o}(K̄_{i})^{−1} → 0, as i → ∞,  (2.13)

on (M_{i}, g̃_{i}) and (Σ_{i}, g̃_{i}) respectively. Hence, by the Gauss equation (2.6), we obtain

|Ricg̃_{i}|(x_{i}) ≥ \frac{1}{2}, while |Ricg̃_{i}|(y_{i}) ≤ 2,  (2.14)

for all y_{i} ∈ (Σ_{i}, g̃_{i}).

Thus, the manifold (Σ_{i}, g̃_{i}) has uniformly bounded curvature. By Proposition 1.5, if the geodesic ball (B̃_{x_{i}}(10), g̃_{i}) in Σ_{i} is sufficiently collapsed, we may then unwrap the collapse by passing to a sufficiently large finite cover, so that vol B̃_{x_{i}}(1) ≥ 10^{−1}. We assume this has been done, without change of the notation.

It follows from Proposition 1.4 and Remark 1.6(iii) that the local C^{1,β} geometry of (Σ_{i}, g̃_{i}) is uniformly controlled, (independent of (Σ_{i}, g̃_{i})), in the ball (B̃_{x_{i}}(10), g̃_{i}). Thus, there are harmonic coordinate charts on balls B̃(r_{o}) of a uniform size r_{o} > 0, within B̃_{x_{i}}(10), in which the metric components are uniformly controlled in C^{1,β} norm.

Next, the Gauss-Codazzi equation (1.7) and the constraint equation (1.3) give

dK̃ = R̃^{T}, δK̃ = 0  (2.15)

on (Σ_{i}, g̃_{i}), where K̃ = K̃_{i}. The system (2.15) is first order uniformly elliptic system for K̃. In local harmonic coordinates, the coefficients of this system are uniformly bounded in C^{1,β}. Hence, from the bound (2.11) and (2.12), standard elliptic interior regularity estimates imply a uniform bound

|K̃|_{L^{1,p}} ≤ C,  (2.16)

where p < ∞, C depends only on p, and the L^{1,p} norm is taken over any ball B̃(r_{o}) in B̃_{x_{i}}(5). In particular, choosing p > 3 and using Sobolev embedding, (which is applicable again by the uniform C^{1,β} local control on the metric), we may find a uniform r_{1} > 0 such that

osc_{B̃(r_{1})}|K̃| ≤ \frac{1}{4},  (2.17)

on all balls B̃(r_{1}) ⊂ B̃_{x_{i}}(5).

By Proposition 1.4, the manifolds (Σ_{i}, g̃_{i}, x_{i}) have a subsequence converging in the C^{1,β′} topology to a C^{1,β} limit (Σ_{∞}, g̃_{∞}, x_{∞}). The equation (2.2) holds on each Σ_{i} and so it holds weakly on the limit, i.e. when paired by integration with any L^{1,p} test form of compact support. Hence, since R̃_{i} → 0 in L^{∞} by (2.12), on the limit (Σ_{∞}g̃_{∞}), the limit form K = K̃_{∞} satisfies

2D^{∗}DK = R(K)  (2.18)

weakly in L^{1,p}. This is an elliptic equation with C^{1,β} coefficients, so elliptic regularity, c.f. [26, Ch. 9.6], implies that K is L^{3,p} smooth, for any p < ∞, and hence by Sobolev embedding, K is C^{2,β} smooth. In particular, as in (2.4), we then have

Δ|K|^{2} = 2|DK|^{2}+ < R(K), K > .  (2.19)

However, at the limit base point x_{∞}, |K| is maximal, so Δ|K|^{2} ≤ 0. Further, the estimates (2.12)- (2.13) together with (2.5) imply that the curvature term in (2.19) is non-negative.

Hence, at x_{∞}, we must have |K| = 0. However, this contradicts the estimates (2.11) and (2.17), which pass to the limit. This contradiction thus establishes (2.8).

By Remark 1.6(iii), the estimate (2.8) on the Ricci curvature gives apriori local C^{1,β} control on the metric g_{τ} in harmonic coordinates, i.e. an apriori lower bound on the C^{1.β} harmonic radius at any x ∈ Σ_{τ} ,

r_{h}(x) ≥ r_{o} = r_{o}(Λ_{o}, H_{o}) > 0,  (2.20)

provided one has a lower bound on the volume vol_{g_{τ}} B_{x}(1).

This will not be the case when vol_{g_{τ}} B_{x}(1) is too small, but in that case B_{x}(1) may be unwrapped by passing to covering spaces as in Proposition 1.5, (assuming B_{x}(1) ≠ S^{3}/Γ). Thus, one obtains such a lower bound on r_{h}(x) in the covering.

Given such a lower bound on r_{h}, one may apply standard elliptic estimates, c.f. [26, §8.8], to the lapse equation (1.6) to control the behavior of α, since the coefficients of Δ are controlled in C^{1,β} in harmonic coordinates on B_{x}(r_{h}(x)). (Note that the lapse equation is invariant under coverings). Thus, together with the uniform bound on |K| in (2.8), it follows that

\frac{supα}{infα} ≤ A_{o} < ∞,  (2.21)

where the sup and inf are taken over any B_{x}(r_{o}) ⊂ Σ_{τ} , and A_{o} depends only on Λ_{o}, H_{o}. Similarly, given any fixed x ∈ Σ_{τ} , if α is renormalized so that ᾱ = α/α(x), then elliptic regularity [26, §9.5] applied to the lapse equation (1.6) divided by α(x) gives a uniform bound

|D^{2}ᾱ|_{L^{p}(B_{x}(r_{o}))} ≤ A_{1},  (2.22)

where A_{1} depends only on Λ_{o}, H_{o}; the estimate (2.22) is assumed to be taken over the local unwrapping covering in case Σ_{τ} is sufficiently collapsed at x. In particular, by Sobolev embedding, this gives

|∇ᾱ|_{L^{∞}} ≤ A_{2},  (2.23)

with A_{2} = A_{2}(Λ_{o}, H_{o}).

Similarly, consider the equation (2.2) again, i.e.

2D^{∗}DK = δR^{T} + R(K).

This is a second order elliptic system in K, and the terms R^{T} and R(K) are bounded in L^{∞} by (2.1) and (2.8). Thus, consider the elliptic operator D^{∗}D as a mapping

D^{∗}D : L^{1,p} → L^{−1,p}.

In local harmonic coordinates, i.e. within the harmonic radius, the coefficients of D^{∗}D are controlled in C^{1,β}. It follows from elliptic regularity and the bounds above that K is controlled in L^{1,p}, for any p < ∞, i.e.

||K||_{L^{1,p}(B(r_{o}))} ≤ K_{1},  (2.24)

where K_{1} depends only on Λ_{o}, H_{o}.

There is of course no general apriori lower bound on the volumes vol B_{x}(1), (c.f. also the collapse discussion in §3). However, if some initial CMC surface Σ_{τ_{o}} as in (0.2) is given which has a fixed lower bound on vol B_{p}(1), ∀p ∈ Σ_{τ_{o}}, then the following result shows that one has a lower bound on vol B_{x}(1) at all points x ∈ Σ_{τ} within bounded proper distance to Σ_{τ_{o}}.

Lemma 2.3. Let Σ_{τ_{o}} be a CMC surface in the space-time (M, g) satisfying (2.1), and suppose

volB_{p}(1) ≥ ν_{o} > 0,  (2.25)

for all p ∈ Σ_{τ_{o}}. Let Σ_{τ} be any other compact CMC surface in (M, g) with mean curvature τ = H and suppose, for x ∈ Σ_{τ} , t(x) = dist_{g}(x, Σ_{τ_{o}}) ≤ T_{o}. Then there is a constant ν_{1} = ν_{1}(H, H_{o}, Λ_{o}, ν_{o}, T_{o}) > 0, such that

volB_{x}(1) ≥ ν_{1}.  (2.26)

In particular, there is then a constant r_{1} = r_{1}(H, H_{o}, Λ_{o}, ν_{o}, T_{o}) > 0 such that

r_{h}^{1,β}(x) ≥ r_{1} > 0.  (2.27)

Proof: Given x ∈ Σ_{τ} , we consider neighborhoods U of x in Σ_{τ} as graphs over Σ_{τ_{o}} via the timelike normal exponential map from Σ_{τ_{o}}. Thus, if p ∈ Σ_{τ_{o}} is such that dist_{g}(x, p) = dist_{g}(x, Σ_{τ_{o}}), let U be the set of points y in Σ_{τ} for which there exists a maximal geodesic from y to B_{p}(1) ⊂ Σ_{τ_{o}} realizing dist_{g}(y, Σ_{τ_{o}}). Hence the normal exponential map to Σ_{τ_{o}} induces a continuous map F : B_{p}(1) → U, F (q) = exp(ϕ(q) · T), where T is the time-like unit normal to Σ_{τ_{o}} and ϕ is a positive (or negative) function.

We claim that F is a quasi-isometry, with distortion factor depending only on the bounds (H_{o}, Λ_{o}, T_{o}). To see this, consider the 1-parameter interpolation F_{s}(q) = exp(sϕ(q) · T). The apriori bounds on |K| from (2.8), which controls the infinitesimal distortion of g_{τ} in the unit normal direction, and α from (2.21), imply that for s small, F_{s} has metric distortion at most C(H_{o}, Λ_{o}) · s onto its image. Iterating this control inductively to s = 1 then gives the claim.

Since F is a quasi-isometry, it is clear that the estimate (2.25) implies (2.26). The bound (2.27) then follows from the arguments preceding the Lemma.

Observe that if H = τ is bounded away from 0, i.e. |τ| ≥ τ̄ > 0, then the estimate (1.8) implies that α is bounded above, depending only on τ̄. Hence, one has a global estimate for the proper time

t(x) ≤ T_{o}(τ̄),  (2.28)

∀x ∈ Σ_{τ} in this case.

We need one further apriori estimate for the proof of Theorem 0.1.

Lemma 2.4. For Σ_{τ} ⊂ Ω, as in (2.1), there is a constant Λ_{1} < ∞, depending only on H_{o}, Λ_{o}, vol Σ_{τ} and an initial surface Σ_{τ_{o}} as in (0.2) such that

||∇Ric_{Σ_{τ}} ||_{L^{2}} ≤ Λ_{1}, and ||∇^{2}K_{Σ_{τ}} ||_{L^{2}} ≤ Λ_{1}.  (2.29)

Proof: Given the estimates above from Proposition 2.2 through Lemma 2.3, (2.29) follows from estimates for the 1^{st} order Bel-Robinson energy and we refer to [18], [6], [8] for some further details.

Thus, let Q^{1} be the 1^{st} order Bel-Robinson tensor associated to the Weyl-type field W^{1} = ∇_{T} W, where W is the Weyl tensor of (M, g) and T is the unit normal to the foliation F. Then there are numerical constants c, C such that, for

Q^{1}(τ) = ∫_{Σ_{τ}} Q^{1}(T, T, T, T)  (2.30)

one has

cQ^{1}(τ) ≤ ||∇Ric_{Σ_{τ}} ||_{L^{2}}^{2} + ||∇^{2}K_{Σ_{τ}} ||_{L^{2}}^{2} ≤ CQ^{1}(τ).  (2.31)

Further, Q^{1} obeys the following differential inequality:

|\frac{d}{dτ} Q^{1}(τ)| ≤ M(1 + Q^{1}(τ) + F (τ)),  (2.32)

where M depends only on the L^{∞} norm of K and |∇logα| on Σ_{τ} and

F (τ) = ∫_{Σ_{τ}} (|Ric|^{4} + |DK|^{4}).

By (2.8) and (2.23), M is uniformly bounded in terms of Λ_{o}, H_{o} while (2.8) and (2.1) give, via the Hölder inequality, a uniform bound on F (τ) depending only on Λ_{o}, H_{o} and an upper bound for volΣ_{τ} . This gives uniform control on M and F (τ) on the right side of (2.32), and hence (2.29) follows by integration w.r.t. τ.

Remark 2.5. In §4, we will also use the analogous 0^{th} order Bel-Robinson estimate. Thus, the Bel-Robinson energy Q^{o}(τ), given by

Q^{o}(τ) = ∫_{Σ_{τ}} |Ric|^{2} + |dK|^{2},  (2.33)

satisfies

|\frac{d}{dτ} Q^{o}(τ)| ≤ c · MQ^{o}(τ),  (2.34)

where M = |K|_{L^{∞}} + |∇logα|_{L^{∞}} on Σ_{τ} , and c is a numerical constant.

This discussion now easily leads to the proof of Theorem 0.1.

Proof of Theorem 0.1.

Suppose σ(Σ) ≤ 0 and let Σ_{τ_{i}} be a sequence of compact leaves in M_{F}, with τ_{i} → τ̄ < 0. We may assume that {τ_{i}} is either increasing or decreasing. Suppose Σ_{τ_{i}} does not approach any point of ∂M, so that by the discussion at the beginning of §2, the estimate (2.1) holds, for some constant Λ_{o}, uniformly on {Σ_{τ_{i}}}.

Since τ̄ < 0, the estimate (2.28) implies that the proper time function t is uniformly bounded above on {Σ_{τ_{i}}}. Thus, the proof of Lemma 2.3 implies that the surfaces {Σ_{τ_{i}}} are all uniformly quasi-isometric. Together with the bound on |Ric| from (2.8), we see that the hypotheses (1.22)- (1.23) of Proposition 1.4 hold on {Σ_{τ_{i}}}.

Proposition 1.4, Remark 1.6(i) and Lemma 2.4 thus imply that a subsequence of {Σ_{τ_{i}}} converges in the weak L^{3,2} topology to a limit L^{3,2} CMC surface Στ̄ ⊂ (M, g) with L^{2,2} extrinsic curvature Kτ̄. Similarly, the extrinsic curvatures K_{i} of Σ_{τ_{i}} converge in weak L^{2,2} to the limit Kτ̄ on Στ̄. Now the Cauchy problem on Στ̄ with data (gτ̄, Kτ̄) in (L^{3,2}, L^{2,2}) is uniquely locally solvable, c.f. [23], and hence there exist unique local developments of this data for a short time into the past and future of Στ̄. By the uniqueness, such a development must lie in a region of the C^{∞} smooth space-time (M, g). Hence, by elliptic regularity applied to the equation (2.2), it follows that Στ̄ is C^{∞} smooth, with C^{∞} extrinsic curvature. The limit CMC surface Στ̄ is the unique surface in (M, g) with mean curvature τ̄, [32], and hence the sequence {Σ_{τ_{i}}}, (and not just a subsequence), converges smoothly to the limit Στ̄. Thus, the smooth foliation M_{F} extends a definite amount past Στ̄.

This completes the proof.

Remark 2.6. The only place the assumption σ(Σ) ≤ 0 was used in the proof of Theorem 0.1 was to obtain a uniform upper bound on the lapse function α from (1.8), which in turn was used only to obtain an upper bound on the proper time to {Σ_{τ_{i}}} via (2.28). Hence, Theorem 0.1 remains true for any value of σ(Σ) whenever there is an upper bound for t_{τ} on {Σ_{τ_{i}}}. More generally, since Lemma 2.3 and the results preceeding it are all local, one needs only an upper bound on t(x) on domains in {Σ_{τ_{i}}}. This will be discussed further in §4.

Remark 2.7. As remarked in the Introduction, Theorem 0.1 may be applied to the past of the initial surface Σ, for any value of σ(Σ) by Remark 2.6. It implies that either the curvature in the sense of (0.7) blows up in the finite proper and CMC past of Σ, or one has global CMC time existence to the past, and in the limit τ → −∞, the space-time (M, g) approaches a crushing singularity.

Observe that the globally hyperbolic space-time (M, g) does not extend, as a globally hyperbolic space-time, to the past of a (compact) crushing singularity. To see this, suppose (M′, g′) is such a globally hyperbolic extension of (M, g), and choose a point x ∈ M′ \ M. Since M′ is globally hyperbolic, there is a maximal time-like geodesic from x to any leaf Σ_{τ} , which realizes the maximal distance of x to Σ_{τ} . Hence γ has no focal points. But the standard argument in the Hawking singularity theorem via the Raychaudhuri equation (1.16), implies that γ must have a focal point if τ is chosen sufficiently large and negative.

It follows that in any extension (M′, g′) of (M, g), the boundary ∂_{P} M_{F} of M_{F} to the past of Σ_{τ_{o}} is the past Cauchy horizon of Σ. Examples with this behavior of course do occur, most notably in the Taub-NUT metric, c.f. [30, Ch.5.8].

3. Asymptotics of Future Complete Space-Times.

This section is mainly concerned with the proof of Theorem 0.3. In §3.1 we prove the result under an extra hypothesis concerning the asymptotic behavior of the mean curvature H = τ, but without the bound on ∇R in (0.11). This hypothesis is then removed, i.e. proved to always hold, in §3.2. In §3.3 we make a number of remarks on the collapse situation.

First however we note that a space-time satisfying the assumptions of Theorem 0.3 must be geodesically future complete. In fact, the much weaker assumption of a uniform curvature bound

|R| ≤ Λ_{o} < ∞  (3.1)

to the future of an initial surface Σ_{τ_{o}} implies by Theorem 0.1 that the CMC foliation F exists for all CMC time τ ∈ [τ_{o}, 0). Hence, since by the assumption in Theorem 0.3 that ∂_{o}M_{F} = ∅ in M, it follows that (M, g) is future geodesically complete.

The remainder of the proof is thus concerned only with the weak geometrization of Σ.

§3.1. As mentioned above, we first prove Theorem 0.3 in a special case. Thus, in contrast to (0.9), define

t_{min}(τ) = min{t(x) : x ∈ Σ_{τ} }.  (3.2)

As in (1.15), the product H · t_{min} is scale invariant.

Theorem 3.1. Suppose (M, g) is a CMC cosmological space-time with M = M_{F}, (to the future of Σ_{τ_{o}}), and that (M, g) satisfies the curvature assumption

|R|(x) ≤ \frac{C}{t^{2}(x)},  (3.3)

for some C < ∞, compare with (0.11). Further, given a sequence {τ_{i}} → 0, suppose there exists a constant δ > 0 such that

|H_{Σ_{τ_{i}}} | ≥ \frac{δ}{t_{min}(τ_{i})}.  (3.4)

Then there is a subsequence of {τ_{i}}, denoted also τ_{i}, such that the rescaled metrics (Σ_{τ_{i}}, ḡ_{τ_{i}}) as in (0.10) converge to a weak geometrization of Σ.

Proof: Given a sequence of surfaces Σ_{τ_{i}}, τ_{i} → 0, to simplify notation, we let Σ_{i} = Σ_{τ_{i}}. Then combining the assumption (3.4) with the general bound (1.15) gives

\frac{δ}{t_{min}(τ_{i})} ≤ |H_{Σ_{i}}| ≤ \frac{3}{t_{max}(τ_{i})} = \frac{3}{t_{τ_{i}}} ,  (3.5)

and hence on each Σ_{i},

t_{τ_{i}} = t_{max}(τ_{i}) ≤ \frac{3}{δ}t_{min}(τ_{i}).  (3.6)

This means that the surfaces Σ_{i} lie in time-annuli of uniformly bounded ratios, i.e.

Σ_{i} ⊂ A(\frac{δ}{3}t_{τ_{i}}, t_{τ_{i}}),  (3.7)

where A(r, s) = t_{τ}^{−1}(r, s).

Consider then the rescaled space-time metric

ḡ_{i} = t_{τ_{i}}^{−2} · g,  (3.8)

and the corresponding 3 + 1 decomposition

ḡ_{i} = −ᾱ_{i}^{2}dτ̄_{i}^{2} + ḡτ̄_{i}.  (3.9)

Note that the vacuum Einstein equations are of course invariant under rescaling. Here, the mean curvature parameter τ is replaced by τ̄_{i} = τ ·t_{τ_{i}} and ᾱ_{i} = α/t_{τ_{i}}. Hence the surfaces Σ_{i}, now denoted by Σ̄_{i} in this rescaled metric, have mean curvature H̄_{i} = H̄(Σ̄_{i}) satisfying

δ ≤ |H̄_{i}| ≤ 3.  (3.10)

Let t̄_{i} be the proper time function w.r.t. ḡ_{i}, so that t̄_{i} = t/t_{τ_{i}}. Thus, (3.7) translates into the statement that

Σ̄_{i} ⊂ Ā(\frac{δ}{3}, 1),  (3.11)

where Ā(r, s) = t̄_{i}^{−1}(r, s) so that the surfaces Σ̄_{i} lie in time-annuli of uniformly bounded inner and outer radii. Of course, the leaves Σ_{τ} of the foliation F are now considered as leaves (Σ̄τ̄_{i}, ḡτ̄_{i}) of the foliation F̄_{i} of the rescaled space-time (M̄_{i}, ḡ_{i}).

By scaling properties of curvature and the curvature assumption (3.3), we have

|R̄_{i}|(x) = t_{τ_{i}}^{2}|R|(x) ≤ C · t_{τ_{i}}^{2} · t(x)^{−2} = C · t̄_{i}(x)^{−2},  (3.12)

and hence, for any given κ > 0 we have

|R̄_{i}| ≤ C = C(κ),  (3.13)

within any time-annulus Ā(κ, κ^{−1}). Thus, by Proposition 2.2 the leaves (Σ̄τ̄_{i}, ḡτ̄_{i}) of F̄_{i} within Ā(κ, κ^{−1}), c.f. (3.11), have uniformly bounded intrinsic curvature and extrinsic curvature K̄. Let H̄ also denote the mean curvature of the leaves Σ̄τ̄_{i}, and recall that H̄ is monotone. Since |K̄|^{2} ≥ \frac{1}{3}|H̄|^{2} and |H̄| is bounded away from 0 on Σ̄_{i} by (3.10), we thus obtain

c ≤ |K̄| ≤ c^{−1}  (3.14)

for some constant c = c(κ) > 0 in Ā(κ, κ^{−1}).

The bound (3.14) applied to the lapse estimates (1.8)-(1.9) in the ḡ_{i} scale immediately gives uniform bounds for the lapse function ᾱ_{i} on the leaves Σ̄τ̄_{i} within Ā(κ, κ^{−1}), i.e.

1 ≤ \frac{supᾱ_{i}}{infᾱ_{i}} ≤ C,  (3.15)

where sup and inf are taken over Σ̄τ̄_{i} and C = C(κ). At least when (3.4) holds for all τ ≥ −1, (3.15) implies that the original lapse function α on (M, g) satisfies α ∼ t^{2}, i.e. the ratio α/t^{2} is bounded above and below away from 0 within A(κt_{τ_{i}}, κ^{−1}t_{τ_{i}}) as t_{τ_{i}} → ∞.

The volume estimate (1.21) is also scale-invariant, and hence we also have a uniform upper bound

volḡ_{τ_{i}} Σ̄_{i} ≤ V_{o},  (3.16)

for some V_{o} < ∞. Note however that the diameter of (Σ̄_{i}, ḡ_{τ_{i}}) may or may not be uniformly bounded. The bounds (3.15) on the lapse function do not control the size of the diameter of (Σ_{i}, ḡ_{τ_{i}}). Given the lapse bounds (3.15), the growth of the diameter is determined by the extrinsic curvature, in particular by the growth of its eigenvalues, which are not controlled by bounds on the lapse function. (We remark this is in contrast to the Riemannian situation, where one can obtain apriori diameter bounds, in addition to volume bounds, from (0.1), or just the assumption of non-negative Ricci curvature).

Thus, in passing to limits, one must choose base points and the form of the limits will then depend on the choice of the base points. Again however note that all base points on Σ̄_{i} lie in fixed time annuli as in (3.11).

Thus, let x_{i} ∈ Σ̄_{i} be any sequence of base points and consider the behavior of the pointed sequence (Σ̄_{i}, ḡ_{τ_{i}}, x_{i}). By Propositions 1.4 and 1.5, we have two possible cases.

Case A. (Non-Collapse)

Suppose the sequence (Σ̄_{i}, ḡ_{τ_{i}}, x_{i}) is non-collapsing, in that (1.23) holds with ḡ_{τ_{i}} in place of g_{i}. This lower volume bound implies that there exists a constant ν_{o} > 0 such that

volḡ_{τ_{i}} Σ̄_{i} = \frac{volΣ_{i}}{t_{τ_{i}}^{3}} ≥ ν_{o},  (3.17)

for t_{τ_{i}} → ∞; the first equality follows from the scale invariance of the ratio. By the monotonicity property (1.21), it follows that (3.17) holds for all t_{τ} , and we may assume that ν_{o} is the limiting minimal value.

As noted above, all leaves Σ̄τ̄_{i} in annuli Ā(κ, κ^{−1}) have uniform L^{∞} bounds on the intrinsic curvature Ric and hence the spatial metrics ḡτ̄_{i} are uniformly controlled locally in L_{x}^{2,p}, where L_{x}^{2,p} denotes the spatial Sobolev space, along the leaves Σ̄τ̄_{i}. By (2.24), there is a uniform local L_{x}^{1,p} bound on the extrinsic curvature K while by (2.22), the lapse function ᾱ_{i} also is uniformly bounded locally in L_{x}^{2,p} in Ā(κ, κ^{−1}).

Similarly, we claim that the time derivative ∂_{τ}^{2}ḡ_{i} is uniformly bounded locally in L^{p} in Ā(κ, κ^{−1}), where τ = τ̄_{i} is the time parameter. For by (1.4) and the estimates above, ∂_{τ} ḡ_{i} = −2ᾱ_{i}K̄_{i} is uniformly bounded in L_{x}^{1,p}. We have ∂_{τ}^{2}ḡ_{i} = −2(∂_{τ} ᾱ_{i})K̄_{i} − 2ᾱ_{i}∂_{τ} K̄_{i}. The second term is uniformly bounded locally in L^{p} by the estimates above applied to the evolution equation (1.5). Further, by differentiating the lapse equation (1.6) w.r.t. τ and using standard formulas for the derivative of Δ, (c.f. [13, 1.184]), a straightforward calculation shows (−Δ + |K|^{2})∂_{τ} ᾱ_{i} is uniformly bounded locally in L_{x}^{p} and hence by elliptic regularity, ∂_{τ} ᾱ_{i} is uniformly bounded locally in L_{x}^{2,p}. This proves the claim.

It follows that the sequence of Lorentzian vacuum space-times in (3.9) has a subsequence converging in the weak L^{2,p} topology to a maximal limit C^{1,β} ∩ L^{2,p} Lorenztian vacuum space-time (M̄_{∞}, ḡ_{∞}), of the form

ḡ_{∞} = −ᾱ_{∞}^{2}dτ̄_{∞}^{2} + ḡτ̄_{∞}.  (3.18)

(More precisely, one should first take a limit within the time-annuli Ā(κ, κ^{−1}) and then let κ = κ_{j}, j = j(i) → 0 in a suitable diagonal subsequence). Here τ̄_{∞} is the mean curvature parametrization given by the limit of the parametrizations τ̄_{i} from (3.9) and ᾱ_{∞} is the limit of the lapse functions ᾱ_{i}. The metric (3.18) is a weak solution of the vacuum equations (0.1).

Now by the volume monotonicity (1.21) used above, this limit space-time is a ’volume cone’ in the sense that

volΣ̄τ̄_{∞}/(t̄_{∞})^{3} ≡ ν_{o} > 0,  (3.19)

where t̄_{∞} is the limit of the renormalized proper time functions t̄_{i} following (3.10). By Corollary 1.3, it thus follows that ḡ_{∞} is flat, and ḡτ̄_{∞} is complete and hyperbolic, (i.e. of constant negative curvature). Thus, the limit is a flat Lorentzian cone on a hyperbolic manifold, as in (0.12).

Note that the limit surface Σ̄_{∞} = lim Σ̄_{i} may be compact, in which case it is diffeomorphic to Σ, or non-compact, corresponding to the possibility diamḡ_{i} Σ̄_{i} → ∞. The limit lapse function ᾱ_{∞} ≡ 1, so that the leaves Σ̄τ̄_{∞} are level sets of t̄_{∞}, while the limit extrinsic curvature K̄_{∞} is pure trace, with |K̄_{∞}|^{2} = 3 on the limit Σ̄_{∞}.

Observe that consequently, not only does (3.4) hold, but in fact

H(Σ_{i}) · t(y_{i}) → 3,  (3.20)

for all y_{i} ∈ Σ_{i} such that distḡτ̄_{i} (x_{i}, y_{i}) ≤ D, for any fixed D < ∞.

Thus, we see that the only limit geometry on the CMC leaves arising in the non-collapse situation is the hyperbolic geometry.

Case B. (Collapse)

Suppose the sequence (Σ̄_{i}, ḡ_{τ_{i}}, x_{i}) is collapsing, so that (1.25) holds, again with ḡτ̄_{i} in place of g_{i}. By (3.12), it is then collapsing with bounded curvature, and thus, for any R < ∞, the geodesic R-balls about x_{i} in (Σ̄_{i}, ḡτ̄_{i}) are Seifert fibered spaces or torus bundles over an interval I, with collapsing fibers. In particular, this part of Σ̄_{i} is an (elementary) graph manifold, and corresponds to a piece in the decomposition of the graph manifold G, c.f. the discussion in §0.

As discussed following Proposition 1.5, we may choose a sequence R = R_{j} → ∞ and pass to a suitable diagonal subsequence, which, after unwrapping the collapse, converges in C^{1,β} ∩ L^{2,p} to a complete limit Riemannian manifold (Σ̄_{∞}, ḡ_{∞}, x_{∞}). Note that we use here the assumption σ(Σ) ≤ 0, so that Σ ≠ S^{3}/Γ.

The limit is either a Seifert fibered space or torus bundle over an interval, (i.e. a Sol manifold), and has either a locally free isometric S^{1}, S^{1} × S^{1}, T^{3} or Nil action. For the same reasons as above in Case A, preceding (3.18), the unwrapping of the collapse also gives rise to a limit space-time (M̄_{∞}, ḡ_{∞}, x_{∞}) and a corresponding maximal CMC foliation M̄F̄_{∞} with Σ̄_{∞} as a leaf. Thus the limit metric ḡ_{∞} has the form (3.18), although here of course ḡτ̄_{∞} is not (necessarily) hyperbolic.

Thus, combining the discussion in Cases A and B, we see that all based limits of (Σ̄_{i}, gτ̄_{i}, x_{i}) are either complete hyperbolic manifolds, complete Seifert fibered spaces or complete Sol manifolds, with a corresponding non-trivial group of isometries.

To obtain the decomposition (0.8), any Riemannian 3-manifold (Σ, g) has a thick-thin decomposition, i.e. for any fixed ϵ > 0, we may write

Σ = Σ^{ϵ} ∪ Σ_{ϵ},  (3.21)

where Σ^{ϵ} = {x ∈ Σ : vol_{g}B_{x}(1) ≥ ϵ}, Σ_{ϵ} = {x ∈ Σ : vol_{g}B_{x}(1) ≤ ϵ}. Of course these domains are not necessarily connected. Under a fixed curvature bound as in (3.13), this corresponds to a decomposition according to the size of the injectivity radius, as in Remark 1.6(ii). Now apply the decomposition (3.21) to each (Σ̄_{i}, ḡτ̄_{i}). By the discussion in Case A, for any fixed ϵ > 0, the domains (Σ̄_{i}^{ϵ}, ḡτ̄_{i}) ⊂ (Σ̄_{i}, ḡτ̄_{i}), when based at base point sequence x_{i} ∈ Σ̄_{i}^{ϵ}, (sub)-converge to domains (Σ̄_{∞}^{ϵ}, ḡ_{∞}). Next choose a sequence ϵ = ϵ_{j} → 0, so that one has a double sequence in (i, j). By choosing a suitable diagonal subsequence j = j(i), the domains Σ̄_{i}^{ϵ_{j}} converge to the complete limit (Σ̄_{∞}, ḡ_{∞}) ⊂ H, and this defines the complete hyperbolic part H ⊂ Σ. On the other hand, for ϵ = ϵ_{j} sufficiently small, Σ_{ϵ_{j}} is a graph manifold, and this collapses everywhere as ϵ_{j} → 0. This gives the decomposition (0.8).

Of course when G is empty, then Case A gives convergence to the hyperbolic metric on H = Σ, while if H is empty, then G = Σ and Case B implies that the metrics ḡ_{i} collapse everywhere, c.f. also Remark 3.3 below.

We note that the transition from the ϵ_{j}-thick part Σ̄_{i}^{ϵ_{j}} to the ϵ_{j}-thin part (Σ̄_{i})_{ϵ_{j}} takes larger and larger diameter as i and j = j(i) → ∞, so that the distance between these regions diverges to ∞. Thus choosing different base points gives different limits only when the distance between base points goes to ∞. Hence for instance if the diameter of (Σ̄_{i}, ḡ_{τ_{i}}) happens to remain uniformly bounded, then any limit has a unique geometry, i.e. independent of the base point.

In fact, we see that the decomposition (3.21) is naturally refined into a further decomposition of Σ_{ϵ} according to the rank of the collapse, i.e. according to the type of group action of the limits described in Proposition 1.5; this is part of the general theory of collapse along F-structures, c.f. [15],[16] and also [2] for further details. Thus, based limits of (Σ̄_{i})_{ϵ_{j}} which have a free isometric S^{1} action become infinitely distant (in space) from based limits which have a free isometric S^{1} × S^{1} action. Hence from the near limiting geometry of (Σ̄_{i}, ḡ_{i}), one also obtains a decomposition of the graph manifold G ⊂ Σ into Seifert fibered components, as discussed in §0.

The decomposition (3.21) and hence (0.8) could change with different choices of sequences τ_{i} → 0, as noted in §0; see also §3.3 for further discussion.

This completes the proof of Theorem 3.1.

Remark 3.2. An alternate argument to the use of the volume comparison result (Corollary 1.3) in Case A can be given based on the monotonicity of the reduced Hamiltonian of Fischer-Moncrief [24]. Thus, it is proved in [24] that the function

∫_{Σ_{τ}} H^{3}dV = τ^{3}vol_{g_{τ}} Σ_{τ}

is monotone non-increasing in τ, and is constant if and only if Σ is hyperbolic. Of course the monotonicity of this Hamiltonian and that of the volume ratio (1.21) are closely related via (1.15).

Remark 3.3. Suppose (3.20) is strengthened to the statement that |H(τ_{i})| · t_{min}(τ_{i}) → 3. Then together with (1.15) and fact that H is constant, it follows that t_{max}/t_{min} → 1 on Σ_{i} = Σ_{τ_{i}}. This means that the corresponding rescaled proper time t̄_{i} following (3.10) converges to 1 everywhere on Σ̄_{i} which in turn implies that the lapse ᾱ_{i} on Σ̄_{i} also approaches a constant function everywhere on Σ̄_{i}.

It follows that the limit Σ̄_{∞} of Σ̄_{i} is hyperbolic everywhere so that (Σ̄_{∞}, ḡ_{∞}) is a compact hyperbolic manifold, diffeomorphic to Σ, i.e. G = ∅.

We claim that in this situation, the rescalings (Σ̄τ̄, ḡτ̄) converge to the hyperbolic limit (Σ, ḡ_{∞}), for all sequences τ → 0, i.e. by Mostow rigidity, the limit is unique. For the monotonicity of (3.17) implies that any limit of a sequence gives rise to a complete hyperbolic manifold H′ embedded in Σ, as in (0.8). Now Thurston’s cusp closing theorem, c.f. [39], implies that the volume of any hyperbolic cusp H′ embedded in Σ has volume strictly larger than that of the compact hyperbolic metric (Σ, ḡ_{∞}). Since the volume of the graph manifold part G in (0.8) has non-negative volume in the limit, the claim follows from the volume monotonicity.

Similarly, if the ratio in (3.17) converges to 0 on some sequence τ_{i} → 0, then the monotonicity implies it converges to 0 on all such sequences. Hence, by Case B above, H = ∅, Σ = G always, and all sequences (Σ̄_{i}, ḡ_{i}) collapse as τ_{i} → 0.

§3.2. In this section, we prove that the assumption (3.4) always holds, at least when the curvature assumption (3.3) is strengthened to (0.11).

Theorem 3.4. Let (M, g) be a cosmological CMC space-time satisfying the curvature assumption (0.11) and M= M_{F}. Then there exists a constant δ = δ(M, g) > 0 such that

|H| ≥ δ/t_{min}.  (3.22)

Proof: The proof will proceed in several steps, but overall the proof proceeds by contradiction, and so we suppose there exists a sequence τ_{i} → 0, and so t_{min}(τ_{i}) → ∞, such that

t_{min}(τ_{i}) · |H_{Σ_{τ_{i}}} | → 0, as i → ∞.  (3.23)

Let x_{i} ∈ Σ_{i} be base points realizing t_{min}, so t(x_{i}) = t_{min}(τ_{i}). In contrast to (3.8), throughout this section we consider the rescaled space-time metrics

ḡ_{i} = t_{min}(τ_{i})^{−2} · g,  (3.24)

with renormalized proper time t̄_{i} = t/t_{min}(τ_{i}) and CMC time τ̄_{i} = τ · t_{min}(τ_{i}). As in the proof of Theorem 3.1, the argument in (3.12) shows that the curvature R̄_{i} of ḡ_{i} is uniformly bounded in the region where t̄_{i} ≥ t_{o}, for any given t_{o} > 0. It then follows from the arguments in §3.1, c.f. the discussion preceding (3.18), that the space-times (M, ḡ_{i}, x_{i}) have a subsequence converging in the weak L^{2,p} topology to a limit L^{2,p} ∩ C^{1,β} maximal vacuum space-time (M̄_{∞}, ḡ_{∞}, x_{∞}), where one must pass to covers in the case of collapse. In this latter case, (M̄_{∞}, ḡ_{∞}) has at least a free isometric space-like S^{1} action. The parameters t̄_{i} and τ̄_{i} converge to limit parameters t̄_{∞} and τ̄_{∞}.

The CMC surfaces Σ_{τ_{i}}, now labeled as Σ̄_{i}, are such that a subsequence of (Σ̄_{i}, ḡ_{τ_{i}}, x_{i}) also converges in the weak L^{2,p} topology, and uniformly on compact subsets, to a limit L^{2,p} ∩ C^{1,β} CMC hypersurface (Σ̄_{∞}, ḡ_{∞}, x_{∞}). Of course, by construction, t̄_{∞} ≥ 1 on Σ̄_{∞}. Similarly, the CMC foliation MF̄_{i} in the scale (3.24) converges to a limit L^{2,p} ∩ C^{1,β} CMC foliation MF̄_{∞} of (M_{∞}, ḡ_{∞}) with leaves Σ̄τ̄_{∞}. Again, in the case of collapse, the leaves are unwrapped and so have at least an isometric S^{1} action.

The assumption on |∇R| in (0.11) implies in the same way that |∇Rḡ_{i}| is uniformly bounded where t̄_{i} ≥ t_{o}. By elliptic regularity applied to the equation (2.2), it follows that the intrinsic curvature Ricḡ_{i} and extrinsic curvature Kḡ_{i} of the leaves are uniformly bounded in L^{1,p} and L^{2,p} respectively, (compare with the proof of Theorem 0.1). Thus, the convergence above is everywhere in the weak L^{3,p} topology and the limits are in L^{3,p} ∩ C^{2,β}.

Now the scale-invariance of (3.23) and the smoothness of the convergence imply that the limit Σ̄_{∞} is a complete maximal hypersurface, i.e. H = 0. In fact, since τ = H is monotone decreasing on (M, g), it follows that all the leaves Σ̄τ̄_{∞} of MF̄_{∞} are complete maximal hypersurfaces. In particular, the limit mean curvature parameter τ̄_{∞} is identically 0.

This means that the lapse function α on (M, g) satisfies

α(x_{i}) >> t_{min}(τ_{i})^{2},  (3.25)

so that the corresponding renormalized lapse functions ᾱ_{i} = α/α(x_{i}) on (M, ḡ_{i}, x_{i}) satisfy ᾱ_{i}(x_{i}) >> 1. To remedy this situation, we redefine the normalized lapse by setting

ᾱ_{i} = \frac{α}{α(x_{i})}.  (3.26)

The estimates (2.21) imply that ᾱ_{i} is uniformly bounded within ḡ_{i} bounded distance to x_{i} on Σ̄_{i}. Similarly, the arguments preceding (3.18) imply that ᾱ_{i} is uniformly bounded on bounded domains on leaves within bounded proper time distance to x_{i}. Hence ᾱ_{i} converges to the limit lapse function ᾱ_{∞} on (M̄_{∞}, ḡ_{∞}). Similarly, we redefine the CMC time parameter by

τ̄_{i} = (α(x_{i}))^{1/2} · τ.  (3.27)

Then τ̄_{i} converges smoothly to a parametrization, again denoted τ̄_{∞}, of the leaves of the limit foliation. It follows that the limit lapse function ᾱ_{∞} satisfies the following lapse equation

− Δα + |K|^{2}α = 0,  (3.28)

on each leaf Σ̄τ̄_{∞}, where α = ᾱ_{∞}. The equation (3.28) means that the variation vector field αT, where T is the unit normal, is a Jacobi field for the volume functional, i.e. the flow of αT preserves the volume of (domains on) the leaves.

With respect to these redefinitions of ᾱ_{∞} and τ̄_{∞}, the limit space-time (M̄_{∞}, ḡ_{∞}) has the form (3.18).

Our task is now to rule out this limiting or near limiting behavior of the space-time (M, g). The key to this is the following result.

Lemma 3.5. The limit maximal hypersurface Σ̄_{∞} cannot be flat, i.e.

|Ric|ḡ_{∞}(y) > 0  (3.29)

for some y ∈ Σ̄_{∞}.

Proof: Assuming (Σ_{∞}, g_{∞}) is flat, we will obtain a contradiction by the Cauchy stability theorem. Thus, suppose (Σ_{∞}, g_{∞}) is flat. By the constraint equation (1.2), since s̄_{∞} = 0 and H = 0, we have

K̄_{∞} = 0,

so that Σ̄_{∞} is totally geodesic, (i.e. time-symmetric).

It follows that for i sufficiently large, the CMC surfaces (Σ̄_{i}, ḡ_{i}, x_{i}) are almost flat and totally geodesic, i.e.

|Ric|ḡ_{i}(y) < ϵ, and |K|ḡ_{i}(y) < ϵ,  (3.30)

for all y ∈ B_{x_{i}}(R), where R may be made arbitrarily large and ϵ > 0 arbitrarily small by choosing i sufficiently large; here B_{x_{i}}(R) is the geodesic R-ball in (Σ̄_{i}, ḡ_{i}) about x_{i}. In fact, as discussed above following (3.24), by Remark 1.6(i), the curvature bounds (0.11) imply that (B_{x_{i}}(R), ḡ_{i}) is ϵ-close in the weak L^{3,p} topology to the flat metric, and K is L^{2,p} close to the 0-form.

It follows that the Cauchy data (ḡ_{i}, K_{i}) on the CMC surfaces Σ̄_{i} in the region (B_{x_{i}}(R), ḡ_{i}) are ϵ-close in weak (L^{3,p}, L^{2,p}) to trivial Cauchy data. By Sobolev embedding, L^{3,p} is compactly contained in H^{s}, for any s < 3, where H^{s} is the Sobolev space with s derivatives in L^{2}. Similarly, L^{2,p} is compactly contained in H^{s−1}. This means that the Cauchy data (ḡ_{i}, K_{i}) are strongly close to trivial data in (H^{s}, H^{s−1}), s < 3.

Choosing s > 2.5, the Cauchy stability theorem, c.f. [23], then implies that the maximal Cauchy development of B_{x_{i}}(R/2) ⊂ Σ̄_{i} to the past exists for a proper time T = T (ϵ), where T may be made arbitrarily large if ϵ is chosen sufficiently small. However, t̄_{i}(x_{i}) = 1 and by the remarks in §0, the space-time (M, g) has a singularity, i.e. fails to be globally hyperbolic, within g-bounded proper time to the past of Σ_{τ_{o}}. Hence (M, ḡ_{i}) has a singularity to the past of Σ̄_{i} within proper time at most 2, for i large. This contradiction proves the result.

Remark 3.6. The following generalization of Lemma 3.5 will be needed in the work to follow. Namely, the proof above leads to the same contradiction if there is an ϵ > 0 sufficiently small, a number D = D(ϵ) sufficiently large, and a point y ∈ Σ̄_{∞} such that the metric

g′ = t̄_{∞}(y)^{−2} · ḡ_{∞}  (3.31)

is ϵ-close to the flat metric in the H^{s} topology, s > 2.5, in the ball (B_{y}(D), g′).

We remark that this use of the Cauchy stability theorem is the only place in the proof that the assumption on the derivative of the curvature in (0.11) is needed.

Thus, to prove Theorem 3.4, it suffices to prove the limit maximal surface (Σ̄_{∞}, ḡ_{∞}) must either be flat or have a point y satisfying the weaker assumption in Remark 3.6. The proof of this needs to be divided into non-collapse and collapse cases, as in §3.1.

Case A. (Non-Collapse).

Since the space-time (M̄_{∞}, ḡ_{∞}) is a rescaled limit at infinity of (M, g), there exists a future directed time-like geodesic ray γ in (M, g) whose rescalings in the metrics (3.24) converge to a geodesic ray γ̄_{∞} in (M̄_{∞}, ḡ_{∞}). As in the proof of Proposition 1.2, let S(s) = t^{−1}(s) be the time s geodesic sphere about Σ_{τ_{o}} in (M, g) and let S̄_{i}(s) = t̄_{i}^{−1}(s) be its rescaling in (M̄_{i}, ḡ_{i}) from (3.24). Let z_{i} = γ(t_{i}) ∈ S(t_{i}), where t_{i} = t(x_{i}). Thus z_{i} ∈ S̄_{i}(1) and x_{i} ∈ S̄_{i}(1).

The geodesic spheres (S̄_{i}(1), z_{i}) converge, (in subsequences), to a Lipschitz, achronal surface S̄_{∞}(1) = t̄_{∞}^{−1}(1) ⊂ (M̄_{∞}, ḡ_{∞}) containing the base point x_{∞}. The limit proper time function t̄_{∞} on (M̄_{∞}, ḡ_{∞}) induces a Lipschitz 3+1 splitting of the limit space-time metric ḡ_{∞}, (which of course is not well-defined on the cut locus of t̄_{∞}).

For any fixed R < ∞, let D_{z_{i}}(R · t_{i}) be the geodesic ball of radius R · t_{i} about z_{i} in S(t_{i}), and D̄_{z_{i}}(R) be its rescaling in S̄_{i}(1).

In this case, we assume that there exists ν_{o} > 0 such that

volD̄_{z_{i}}(1) ≥ ν_{o} > 0, as i → ∞,  (3.32)

so that the domains D̄_{z_{i}}(1) are non-collapsing and hence converge, (in the Hausdorff topology), to the limit domain D̄_{z_{∞}}(1) ⊂ S̄_{∞}(1).

Now as in (1.16), the expansion θ of the congruence formed by time-like geodesics normal to the spheres S(s) ⊂ (M, g) satisfies (1.15), (see the proof of Proposition 1.2). Hence, if dV_{σ}(s) denotes the infinitesimal volume of the family S(s) along any geodesic σ in this congruence, then dV_{σ}(s)/s^{3} is monotone non-increasing as s → ∞.

It then follows from (3.32) as with (3.19) that the domain in (M̄_{∞}, ḡ_{∞}) formed by the geodesics normal to D̄_{z_{∞}}(1) ⊂ S̄_{∞}(1) is a volume cone, and hence this domain is contained in a flat Lorentz cone as in (0.12). This flat structure extends past the cone on D̄_{z_{i}}(1) and implies that all of (M̄_{∞}, ḡ_{∞}) is a flat Lorentz cone. In particular, the universal cover of (M̄_{∞}, ḡ_{∞}) is flat Minkowski space (ℝ^{4}, η).

Now the limit surface (Σ_{∞}, g_{∞}) is a complete maximal hypersurface in (M̄_{∞}, ḡ_{∞}), which lifts to a complete maximal hypersurface in (ℝ^{4}, η). It is an easy consequence of the maximum principle applied to (2.4), as in the proof of Proposition 2.2, that the only complete maximal hypersurfaces in (ℝ^{4}, η) are flat and totally geodesic; this is also proved in [17]. Hence Σ̄_{∞} ⊂ (M̄_{∞}, ḡ_{∞}) is flat. Lemma 3.5 thus rules out the possibility of this case.

Case B. (Collapse).

In this case, we suppose (3.32) does not hold, so that

volD̄_{z_{i}}(1) → 0, as i → ∞.  (3.33)

It follows from the monotonicity used above that for any fixed s_{o} > 0, and s ∈ [s_{o}, s_{o}^{−1}], the domains D̄_{z_{i}}(s) ⊂ S̄_{i}(s) also satisfy (3.33).

This implies that the CMC leaf Σ̄′_{i} containing z_{i} is also collapsing when based at z_{i} as i → ∞. By the uniform control on the geometry following (3.24), it follows that all CMC leaves within ḡ_{i} bounded distance, in space and proper time, to (Σ̄′_{i}, z_{i}), are collapsing as i → ∞. In particular, the leaves Σ̄_{i} containing the base points x_{i} are collapsing everywhere within bounded distance to x_{i}.

Thus, as described in the beginning of the proof, we unwrap the collapse of Σ̄_{i} and the spacetimes to obtain the limit Σ̄_{∞} and the limit space-time (M̄_{∞}, ḡ_{∞}, x_{∞}). These limits have a non-trivial group of isometries, and M̄_{∞} has a foliation by maximal hypersurfaces.

Now unfortunately, we need to divide this situation into two further subcases, according to the size of the isometry group.

Case B(I). (S^{1} × S^{1}).

Suppose the limit (Σ̄_{∞}, ḡ_{∞}) has a free isometric S^{1} × S^{1} action, so that Σ̄_{∞} is a T^{2}-bundle over ℝ. By the constraint equation (1.2), the metric ḡ_{∞} is a complete metric of non-negative scalar curvature on Σ̄_{∞}. Further, any orbit of the S^{1} × S^{1} action on Σ̄_{∞} is an incompressible torus in Σ̄_{∞}. However, by a result of [27], any complete 3-manifold of non-negative scalar curvature which has an incompressible torus must be flat. Hence we see that (Σ̄_{∞}, ḡ_{∞}) is in fact flat. Lemma 3.5 again rules out this possibility. Of course the same arguments apply, (and are even easier), if the limit Σ̄_{∞} has a locally free T^{3} or Nil action.

Case B(II). (S^{1}).

Suppose the limit (Σ̄_{∞}, ḡ_{∞}) has (at most) a free isometric S^{1} action. Thus, we may assume that Σ̄_{∞} is an S^{1} bundle over a surface V, with induced complete Riemannian metric g_{V} . By the result of [27] above, it follows that V must be simply connected, and hence topologically Σ̄_{∞} is a solid torus D^{2} × S^{1}.

In this case, we are not able to prove directly that (Σ̄_{∞}, ḡ_{∞}) is flat, since a solid torus carries many complete metrics of positive scalar curvature. Instead, we will prove that the hypothesis concerning (3.31) in Remark 3.6 holds, which again gives a contradiction.

To do this, we need to understand the asymptotic geometry of the maximal hypersurface (Σ̄_{∞}, ḡ_{∞}). To simplify, we drop the bar and subscript everywhere from the notation for the rest of Case B(II); thus let (Σ, g) denote (Σ̄_{∞}, ḡ_{∞}), (M, g) denote (M̄_{∞}, ḡ_{∞}), t(x) denote t̄_{∞}(x), etc. The scale-invariant bound (0.11) gives

|R| + t|∇R| ≤ \frac{C}{t^{2}} ,  (3.34)

on (Σ, g). Since by construction t ≥ 1 on Σ, the curvature R and its derivative ∇R are thus uniformly bounded on Σ. By Proposition 2.2 it follows that

|Ric| ≤ \frac{C}{t^{2}} , |K| ≤ \frac{C}{t} ,  (3.35)

on (Σ, g). One derives (3.35) most easily from (3.34) by using scale-invariance and working in the scale where t = 1. In the same way, applying elliptic regularity to the equation (2.2) gives the scale-invariant bounds

t^{−3/p}|t^{3}∇Ric|_{L^{p}(B_{t})} ≤ C, t^{−3/p}|t^{3}∇^{2}K|_{L^{p}(B_{t})} ≤ C,  (3.36)

where the L^{p} norms are taken over geodesic balls B_{t} = B_{q}(\frac{1}{2}t(q)) about any q ∈ (Σ, g). In the case of collapse, these norms are understood to be taken in the corresponding covering spaces.

Let r : Σ → ℝ be the (Riemannian) distance function on (Σ, g) from the given base point x_{∞} ∈ Σ. We need the following Lemma.

Lemma 3.7. The functions t and r on (Σ, g) satisfy

t << r,  (3.37)

on any sequence y_{i} with r(y_{i}) → ∞.

Proof: If t(y_{i}) is bounded, then (3.37) is obvious, so suppose t(y_{i}) → ∞ as r(y_{i}) → ∞. Then by (3.34)-(3.35) the metric g on Σ becomes flat everywhere as t → ∞ and (M, g) thus approaches empty Minkowski space (ℝ^{4}, η), (or a discrete quotient of it). The maximal foliation of (M, g) approaches the constant foliation of (ℝ^{4}, η) where all leaves are parallel, i.e. time-equidistant, since this is the only maximal foliation of (ℝ^{4}, η), c.f. [17].

Hence, for i sufficiently large, the lapse function α = ᾱ_{∞}, (c.f. (3.28)), approaches a constant function on domains of bounded diameter about the base points y_{i} while the light cones of (M, g) have axis becoming perpendicular to the leaves. This means that the time function t also approaches a constant on such domains. Since r obviously continues to increase linearly, (3.37) follows.

We are now ready to examine the asymptotic structure of (Σ, g). Thus for any sequence y_{i} in Σ with r(y_{i}) → ∞, consider the rescaled metrics

g_{i} = t(y_{i})^{−2} · g.  (3.38)

Hence, t_{i}(y) = t(y)/t(y_{i}) satisfies t_{i}(y_{i}) = 1 and by Lemma 3.7, r_{i}(y) = r(y)/t(y_{i}) → ∞. Thus the Riemannian distance of y_{i} to the base point x_{∞} of Σ in the g_{i} metric diverges to ∞, while the time function t_{i} remains uniformly bounded at y_{i}. We may alter the choice of y_{i} if necessary so that it satisfies all of the estimates above and in addition

t_{i}(z_{i}) ≥ \frac{1}{2},  (3.39)

for all z_{i} ∈ Σ such that dist_{g_{i}}(y_{i}, z_{i}) ≤ C, where C may be made arbitrarily large if i is sufficiently large. To see this, for any sequence q_{i} with r(q_{i}) → ∞, let ρ_{i} = r(q_{i}) and let A_{i} = A(\frac{1}{2}ρ_{i}, 2ρ_{i}) be the g-geodesic annulus centered at x_{∞} of inner and outer radii \frac{1}{2}ρ_{i}, 2ρ_{i}. Choose the points y_{i} ∈ A_{i} to realize the minimal value of the scale-invariant ratio t(y)/dist_{g}(y, ∂A_{i}). By (3.37), this minimal value converges to 0 as i → ∞. Since t_{i}(y_{i}) = 1, we must have dist_{g_{i}}(y_{i}, ∂A_{i}) → ∞ and the estimate (3.39) then follows easily from the minimality of the choice of y_{i}.

Thus, by (3.36), the metrics g_{i} based at y_{i} have a subsequence converging in the weak L^{3,p} topology, uniformly on compact subsets to a complete L^{3,p} limit (Σ^{∞}, g^{∞}), unwrapping in the case of collapse. We claim that any such limit must be flat.

The proof of this is essentially the same as in [5, Prop. 4.1-4.3]; for brevity, we refer to these results for some further details. Suppose first the metrics (g_{i})_{V} on the base space V of (Σ, g_{i}, y_{i}) are collapsing at y_{i}. By unwrapping this collapse, (and any possible collapse in the invariant S^{1} fiber direction), the limit (Σ^{∞}, g^{∞}) is thus a complete metric of non-negative scalar curvature on the limit manifold Σ^{∞} = T^{2} × ℝ, i.e. one obtains a second S^{1} factor from the collapse of the base metric at infinity. As above in Case B(I), by [27], the metric g^{∞} must be flat; this is the same argument as in [5, Prop.4.3, Case II].

If the base metrics (g_{i})_{V} are not collapsing, then the limit (Σ^{∞}, g^{∞}) is complete and of nonnegative scalar curvature on ℝ^{2} × S^{1}. In this case, the proof in [5, Prop.4.3, Case I], together with the control given by (3.36), proves again that (Σ^{∞}, g^{∞}) is flat. Here we note that although [5, Prop.4.3] applies to complete S^{1} invariant metrics on ℝ^{2} × S^{1} of non-negative scalar curvature which satisfy certain elliptic equations, (the Z_{c}^{2} equations), the proof only requires these equations to obtain regularity estimates; in the case at hand, these are provided by (3.36). This proves the claim that the limits (Σ^{∞}, g^{∞}) are flat.

By Sobolev embedding, the convergence to any such flat limit (Σ^{∞}, g^{∞}) is in the strong H^{s} topology, for any s < 3. Hence there are arbitrarily large domains in (Σ, g_{i}, y_{i}) on which g_{i} is ϵ-close to a flat metric in the H^{s} topology.

Thus, the hypothesis of Lemma 3.6 is satisfied, and one obtains a contradiction from the Cauchy stability theorem as in Lemma 3.5.

This completes the proof of Case B(II), which thus also completes the proof of Theorem 3.4. Together with Theorem 3.1, this also completes the proof of Theorem 0.3.

§3.3. Discussion on collapse behavior.

(I). The collapse behavior arising in Theorems 0.3 and 3.1, c.f. §3.1 Case B, does actually occur in numerous examples. In fact, let (M, g) be a Bianchi space-time, i.e. (M, g) locally has a 3-dimensional local isometry group whose action is free and space-like. All such space-times admit a global CMC foliation F whose leaves Σ_{τ} are invariant under the action, i.e. each leaf Σ_{τ} is locally homogeneous. Further, the foliation F exists for all allowable CMC time, for any value of σ(Σ), and M = M_{F}, c.f. [20].

The Bianchi space-times correspond very closely to the eight Thurston geometries arising in the geometrization program. In particular, all eight of the Thurston geometries occur as space-like Cauchy surfaces in a corresponding Bianchi space-time, with the interesting exception of S^{2} × ℝ, which corresponds to a Kantowski-Sachs space-time. We refer to [6] for a discussion of this Bianchi-Thurston correspondence and for further references.

The five geometries ℝ^{3}, Nil, H^{2} × ℝ, SL(2, ℝ) and Sol all give 3-manifolds Σ with σ(Σ) = 0 and the corresponding Bianchi space-times are future geodesically complete. The curvature assumption (0.11) is satisfied and the space-times collapse in the sense of §3.1, Case B, as τ → 0 or t_{τ} → ∞.

The simplest example is the Kasner metric, given by

g = −dt^{2} + ∑_{i=1}^{3} t^{2p_{i}}dθ_{i} ⊗ dθ_{i},  (3.40)

where {p_{i}} are any constants satisfying ∑ p_{i} = 1, ∑ p_{i}^{2} = 1 and θ_{i} are parameters for T^{3}. This has ℝ^{3} geometry, with Σ a flat 3-torus, H = τ = −t^{−1}. In this case, for any sequence τ_{i} → 0, the rescalings Σ̄_{i} of Σ_{i} as in (3.9) collapse with bounded curvature and diameter converging to 0. Hence, the unwrapping in covers give rise to a T^{3} action on the limit surface, i.e. one obtains the Kasner metric again, (with the same exponents).

For the remaining 3 geometries, the S^{3} and S^{2}×ℝ geometries have σ(Σ) > 0 and give Bianchi and Kantowski-Sachs space-times which recollapse, as in the recollapse conjecture, while the hyperbolic geometry H^{3}(−1) gives the flat Lorentz cone (0.12) and so evolves trivially, i.e. just by rescalings.

(II). As discussed in §3.1 Case B, if the rescalings (3.9) collapse at x_{i}, then one may unwrap the collapse of both the space and space-time, and pass to a limit space-time (M̄_{∞}, ḡ_{∞}, x_{∞}), with geodesically complete CMC Cauchy surface (Σ̄_{∞}, ḡ_{∞}, x_{∞}) and a corresponding limit CMC foliation M̄F̄_{∞}. The leaves Σ̄τ̄_{∞} of M̄F̄_{∞} have at least a free isometric S^{1} action, and possibly a locally free isometric action of a 2 or 3-dimensional group. The leaves of M̄F̄_{∞} may be either compact, or complete and non-compact. The limit space-time (M̄_{∞}, ḡ_{∞}) is time-like geodesically complete to the future of Σ̄_{∞}.

In the case of an S^{1}×S^{1} action, these limit space-times (M̄_{∞}, ḡ_{∞}) include the Gowdy space-times and have been quite well studied, at least when Σ̄_{∞} is compact, c.f. [6] and references therein.

One can now study (again) the further long-time evolution of these limit space-times. It is an interesting open question whether these limits (M̄_{∞}, ḡ_{∞}) evolve as t̄_{∞} → ∞ to a Bianchi spacetime as in (I), and if so whether in fact the space-times (M̄_{∞}, ḡ_{∞}) themselves are already Bianchi. To prove this, it seems likely one would need some kind of monotonicity structure to replace the volume monotonicity of Corollary 1.3, (or the Hamiltonian monotonicity [24]), used in the proof of Theorem 0.3.

(III). Finally, we discuss the relation between weak and strong geometrizations, and the recollapse conjecture.

Let Σ = H ∪ G be a weak geometric decomposition as given by Theorem 0.3 or Theorem 3.1. The manifolds G and H have toral boundary T = ∪T_{i} and the decomposition (0.8) is strong, and consequently unique, if each torus T_{i} is incompressible in Σ. Now of course since the hyperbolic part H has no incompressible tori, T_{i} is incompressible in Σ if and only if T_{i} is incompressible in G.

It is essentially a standard result in 3-manifold topology, c.f. [31, Ch.II.2], [36], that each boundary component T_{i} is incompressible in G if and only if G has no solid tori factors, i.e. in the decomposition of G into Seifert fibered spaces S_{j} discussed in §0, no S_{j} is a solid torus D^{2} × S^{1}.

Thus, the question is whether solid torus factors D^{2} × S^{1} can arise asymptotically from the collapse behavior, (compare with Case B(II) in the proof of Theorem 3.4). At least in a special case, this question is closely related to the recollapse conjecture.

Thus suppose one has a D^{2} × S^{1} factor in G arising from the asymptotic collapsing geometry of (M, g) as discussed in the proof of Theorem 3.1. The limiting metric ḡ_{∞} on D^{2} × S^{1} is complete and invariant under a free isometric S^{1} action.

Suppose, for some t_{τ_{i}} sufficiently large, i.e. τ_{i} sufficiently small, there is D^{2} ×S^{1} region contained in (Σ̄_{i}, ḡ_{τ_{i}}) with a totally geodesic boundary T^{2} in the ḡ_{i} metric. One can then double or reflect across the boundary to obtain a closed 3-manifold S^{2} × S^{1} with an isometric ℤ_{2} action, fixing the boundary. Of course in this process we have created a different space-time. Suppose further the extrinsic curvature K̄ of Σ̄_{i} at the boundary T^{2} is invariant under this ℤ_{2} action so that the Cauchy data on Σ̄_{i} are ℤ_{2} invariant. Now σ(S^{2} × S^{1}) > 0 and so the recollapse conjecture, if true, implies that this S^{2} × S^{1} must evolve into a singularity in finite future time. Since the evolution of Cauchy data preserves isometries, this implies that the D^{2} × S^{1} factor in (Σ̄_{i}, ḡ_{τ_{i}}) also evolves to a singularity in finite future time. This is of course a contradiction.

It would obviously be interesting if this argument could be strengthened to prove in general that the recollapse conjecture implies that a weak decomposition of Σ must necessarily be a strong decomposition.

4. The Future Boundary of M_{F} in M.

In this section, we discuss the situation where M_{F} is strictly contained in M, c.f. §0. In this case, Theorem 0.1 implies that M has global existence in CMC time both to the past τ → −∞ and to the future τ → 0, (or τ → +∞ when σ(Σ) > 0). In particular, by the discussion at the end of §2, the past singularity is a crushing singularity. Thus, in this section, we only consider the boundary

∂_{o}M_{F} ⊂ M  (4.1)

of M_{F} formed when τ → 0, i.e. the boundary to the future of the initial surface Σ_{τ_{o}} from (0.2). The condition (4.1) means that the space-time (M, g), i.e. the maximal Cauchy development of initial data on Σ, extends a definite amount locally, i.e. in a neighborhood of any point in ∂_{o}M_{F}, to the future of the CMC foliation F, so that Σ_{τ} does not approach anywhere ∂M as τ → 0.

We begin with the following general result.

Theorem 4.1. Suppose σ(Σ) ≤ 0 and (4.1) holds. Then each component Σ_{o} of ∂_{o}M_{F} is a smooth, complete non-compact maximal hypersurface Σ_{o} ⊂ M, weakly embedded in Σ, c.f. (1.27).

Proof: Referring to the estimates in §2, by Proposition 2.2, |K|^{2} is uniformly bounded on any compact subset Ω of M and hence by (1.9),

inf_{K}α ≥ α_{o} = α_{o}(Λ_{o}, H_{o}) > 0,  (4.2)

i.e. there is no collapse of the lapse function on compact subsets. Further, we recall that α satisfies the Harnack inequality (2.21), i.e.

\frac{sup_{B(r_{o})}α}{inf_{B(r_{o})}α} ≤ C,  (4.3)

where B(r_{o}) is any geodesic r_{o}-ball in (Σ_{τ} , g_{τ}) and C is independent of τ, for τ ∈ [−τ_{o}, 0].

The estimate (1.8) also gives an upper bound on α when H is bounded away from 0, (as in (2.28)). However, when H = τ → 0, the lapse function α may well blow-up, i.e. diverge to +∞. Applying the Harnack inequality iteratively to a collection of balls covering a domain of bounded diameter, it follows that α either remains uniformly bounded on uniformly bounded domains within Σ_{τ} , or α diverges uniformly to +∞ on such domains, as τ → 0.

Given these preliminary remarks, let τ_{i} be any sequence with τ_{i} → 0 monotonically and consider the behavior of {Σ_{τ_{i}}}. We divide the discussion into three cases.

Suppose first that there is a constant T_{o} < ∞ such that

sup_{Σ_{τ_{i}}} t = t_{τ_{i}} ≤ T_{o}, as i → ∞.  (4.4)

As in the proof of Theorem 0.1, c.f. also Remark 2.6, it then follows that Σ_{τ_{i}} converges smoothly to a compact maximal hypersurface Σ_{o}, and Σ_{o} is diffeomorphic to Σ. Of course, this situation implies σ(Σ) > 0, contradicting the assumption. (This case will be discussed further below).

Next suppose

inf_{Σ_{τ_{i}}} t = t_{min}(τ_{i}) → ∞, as i → ∞,  (4.5)

so that t tends uniformly to +∞ on Σ_{τ_{i}} as i → ∞. Thus, the surfaces Σ_{τ_{i}} diverge uniformly to ∞ in infinite proper time. In this case, we then clearly have

M = M_{F}

to the future of Σ_{τ_{o}}, and so ∂_{o}M_{F} = ∅ again.

It follows that there are base points y_{i} ∈ Σ_{τ_{i}} such that limsup_{i→∞}t(y_{i}) < ∞, and other points z_{i} ∈ Σ_{τ_{i}} where liminf_{i→∞}t(z_{i}) = ∞. Analogous to the decomposition (3.21), for any T_{o} < ∞ and any i, write

Σ = Σ_{τ_{i}} = Σ^{T_{o}} ∪ Σ_{T_{o}},  (4.6)

where Σ^{T_{o}} = {x ∈ Σ : t(x) ≥ T_{o}}, Σ_{T_{o}} = {x ∈ Σ : t(x) ≤ T_{o}}. Hence for all i and T_{o} sufficiently large, both Σ^{T_{o}} and Σ_{T_{o}} are non-empty. Note that these domains need not be connected. Lemma 2.3 implies that the domains Σ_{T_{o}} = (Σ_{i})_{T_{o}} ⊂ (Σ_{τ_{i}}, g_{τ_{i}}) do not collapse anywhere, i.e. at any sequence of base points. Further, as in the proof of Theorem 0.1, elliptic regularity applied to (2.2) gives uniform local control over the derivatives of Ric and K on (Σ_{i})_{T_{o}}. It then follows exactly as in the proof of Theorem 0.1, that for any fixed T_{o}, and at any sequence of base points, the domains (Σ_{i})_{T_{o}} have a subsequence converging smoothly and uniformly on compact subsets to a limit surface (Σ_{o})_{T_{o}}; the limit obtained of course depends on the choice of base points. As following (3.20), choose then a sequence T_{j} → ∞ and consider the double sequence (i, j). A suitable diagonal subsequence of (Σ_{i})_{T_{j}} then converges smoothly and uniformly on compact subsets to a limit (∂_{o}M_{F}, g_{o}). The limit ∂_{o}M_{F} is a collection of maximal hypersurfaces in (M, g) and forms the boundary of M_{F} (to the future of Σ_{τ_{o}}). Since each Σ_{τ_{i+1}} lies to the future of Σ_{τ_{i}}, the limit ∂_{o}M_{F} is unique. By construction, the components Σ_{o} of ∂_{o}M_{F} are weakly embedded in Σ. Of course the complementary domains Σ_{i}^{T_{j}} diverge to infinity and hence have no limit in (M, g) itself.

Remark 4.2. Theorem 4.1 extends to the case σ(Σ) > 0 with only a minor modification, but for clarity we have separated the cases. Thus, suppose σ(Σ) > 0.

The first case above, where (4.4) holds, gives a limit Σ_{o} which is compact, and so diffeomorphic to Σ. By Theorem 0.1, the foliation M_{F} then extends a definite amount to the future in CMC time, i.e. the range of τ extends a definite amount past 0 into ℝ^{+}. Thus, as in Remark 2.7, either the leaves of M_{F} approach ∂M somewhere in the finite proper and CMC future of Σ_{o} or the CMC foliation Σ_{τ} extends to all values of τ ∈ [0, ∞). For the second (4.5) and third cases (4.6), the proof of Theorem 4.1 proceeds in the same way. Of course the recollapse conjecture would imply that neither of these last two cases actually occurs.

Remark 4.3. We have not asserted that the lapse functions α_{i} on Σ_{i} converge to the lapse function α on Σ_{o} as τ_{i} → 0 in Theorem 4.1 or Remark 4.2. If α_{i}(x_{i}) remains uniformly bounded as i → ∞, then the lapse functions converge smoothly to the limit lapse function α on Σ_{o}, and α satisfies the lapse equation (1.6) on Σ_{o}. We will call this the ’non-degenerate’ situation.

However, it is possible, (although probably only in very special situations), that α_{i} → ∞ uniformly as τ_{i} → 0 even under the bound (4.4). Thus suppose α_{i}(x_{i}) → ∞ with t(x_{i}) ≤ T_{o}, x_{i} ∈ Σ_{τ_{i}}. Then exactly as in the proof of Theorem 3.4, c.f. (3.26), renormalize α_{i} by setting ᾱ_{i} = α_{i}/α_{i}(x_{i}). As discussed there, using (4.3), the functions ᾱ_{i} converge smoothly to a limit lapse ᾱ_{∞} satisfying, for α = ᾱ_{∞},

− Δα + |K|^{2}α = 0.  (4.7)

This ’degenerate’ situation occurs exactly when the limit maximal surface Σ_{o} is only weak maximum for the volume functional and not a strict maximum. Thus, there is a deformation of Σ_{o}, namely the flow of the vector field ᾱ_{∞} · T, where T is the unit normal, which preserves the volume of Σ_{o} to first order.

If Σ_{o} is compact, (as in the case (4.4) when σ(Σ) > 0) then the maximum principle applied to (4.7) shows that K = 0 and ᾱ_{∞} = const on Σ_{o}. This situation does actually occur, for example in the Taub-NUT metric with m = 0, c.f. [30, §5.8]. However, if Σ_{o} is non-compact, (corresponding to the situation (4.6)), it is no longer clear that necessarily K = 0.

We make some further observations on the geometry of ∂_{o}M_{F} in (M, g). By Remark 4.2, we may assume that each component Σ_{o} is non-compact. First, the constraint equation (1.2) shows that s ≥ 0 everywhere on Σ_{o}, so that g is a complete metric of non-negative scalar curvature on Σ_{o}. Each component Σ_{o} is a partial Cauchy surface for the region M^{+} in M to the future of Σ_{o} and ∂_{o}M_{F} is a Cauchy surface for M^{+}.

The proper time t(x) is a proper, unbounded exhaustion function on Σ_{o}, as is the lapse function α, (or its renormalization ᾱ_{∞} in the degenerate case).

Now suppose the curvature bound (2.1), i.e.

|R| ≤ Λ_{o},  (4.8)

holds globally on M_{F} to the future of Σ_{τ_{o}} and hence also on ∂_{o}M_{F}. (This assumption, which is automatic on compact subsets Ω of M, has not been used in the discussion above in §4). The estimates (2.8) and (2.23) thus give a uniform bound on |K| and |∇logα| on M_{F}. Hence, by integration of the 0^{th} order Bel-Robinson estimate (2.34) from τ_{o} to 0, we obtain the bound

∫_{∂_{o}M_{F}} |Ric|^{2} + |dK|^{2} < ∞.  (4.9)

The integrand |Ric|^{2} + |dK|^{2} is of course pointwise bounded on ∂_{o}M_{F} by (2.8) and (1.7), (4.8). Further, one certainly expects that each component Σ_{o} is not volume collapsing anywhere, i.e.

volB_{x}(1) ≥ v_{o} > 0,  (4.10)

for all geodesic balls B_{x}(1) ⊂ Σ_{o} and some v_{o} > 0. If (4.10) does hold, then (4.9) implies that

|Ric| → 0, |K| → 0,  (4.11)

uniformly at infinity in the components Σ_{o}. Hence the surfaces Σ_{o} become flat and totally geodesic at infinity in this case. Thus, in a certain sense, Σ_{o} is weakly asymptotically flat and time-symmetric.

Remark 4.4. There are numerous interesting open questions concerning the structure of the components Σ_{o}. For example, does Σ_{o} admit a complete metric of uniformly positive scalar curvature, c.f. [27] for results on the structure of such manifolds. Do the components Σ_{o} have at least two ends, and is there an essential 2-sphere S^{2} ⊂ Σ_{o}? Are the ends of Σ_{o} asymptotically flat?

It is also interesting to speculate if there is any relation of the structure of the manifolds Σ_{o} weakly embedded in Σ with the sphere or prime decomposition of Σ. To explain this, as noted in §0, not every closed oriented 3-manifold admits a geometric structure in the sense of Thurston, nor does every 3-manifold admit a strong geometrization in the sense that (0.8) holds with incompressible T . The essential 2-spheres in Σ are obstructions to the strong decomposition (0.8) and the geometrization conjecture [39] implies that these are the only obstructions, i.e. if Σ has no essential 2-sphere, then Σ admits a (unique) strong decomposition (0.8). Thus, one might speculate that a reducible 3-manifold Σ, i.e. a 3-manifold with an essential 2-sphere, cannot satisfy the assumptions of Theorem 0.3, and in particular that Σ_{τ}, τ ≤ 0, does not evolve to the future with proper time everywhere diverging to ∞. Hence Σ_{o} may be non-empty and it is then natural to consider the question if essential 2-spheres of Σ are captured in the topology of Σ_{o}.

We also make the following conjecture on the structure of the region M^{+} to the future of ∂_{o}M_{F} in M.

Non-Compact CMC Conjecture. If ∂_{o}M_{F} ⊂ M is non-empty, then either there exists a CMC foliation F^{+} of M^{+} by non-compact leaves Σ_{τ} , defined for all τ ∈ (0, ∞), and satisfying

M_{F^{+}} = ∪Σ_{τ} = M^{+},  (4.12)

or there are leaves Σ_{τ_{i}} ∈ M_{F^{+}} wich approach ∂M in finite proper and CMC time.

This conjecture is a non-compact analogue of Theorem 0.1. Using the estimates in §2, together with the results and methods of [25], the conjecture can be proved provided the existence of suitable barrier surfaces S_{i} can be established, i.e. surfaces in M^{+} whose mean curvature H(S_{i}) diverges everywhere to +∞ as i → ∞.

Of course, one could consider a strengthening of this conjecture, asserting that ∂M is not reachable in finite CMC time, so that F^{+} covers all of M^{+}; this is the analogue of the CMC global existence problem in M^{+}.

Note also that this conjecture implies the singularity formation conjecture of §0, since if the CMC foliation F^{+} is defined for all τ ∈ [0, ∞) then the leaves Σ_{τ} limit on a crushing singularity as τ → ∞. This behavior must also occur at a finite proper future time from ∂_{o}M_{F}, by the same argument given at the end of §2. Hence (M, g) cannot be future geodesically complete.

5. Remarks on the Curvature Assumption.

In this section, we discuss the curvature assumption (0.11), which is the strongest assumption in Theorem 0.3. Of course one would like to replace this by the assumption that (M, g) is merely future geodesically complete and derive (0.11) as a consequence. We will ignore the assumption on |∇R| in (0.11), since it is of lesser importance.

Let q be any point in ∂M, i.e. at the boundary of the maximal Cauchy development, (assumed not to be at infinity). Thus, near q, either the space-time (M, g) is inextendible, for example because the curvature blows up in a parallely propagated frame, or (M, g) is extendible as a space-time, (possibly not vacuum), near q and so (M, g) has a non-empty Cauchy horizon passing through q.

For any point x ∈ M, let t_{∂M}(x) be the supremum of the times t such that

B_{x}(t(x)) ⊂ M.  (5.1)

The condition (5.1) means that any past or future directed time-like geodesic starting at x, which has proper length less than t_{∂M}(x), has its endpoint in M. Thus, t_{∂M}(x) is essentially the proper time distance, to the past or future, to the time-like nearest point of ∂M.

Curvature Blow-up Problem. Let (M, g) be a CMC cosmological vacuum space-time. Does there exist a constant C = C(M, g) < ∞, such that,

|R| ≤ C/t_{∂M}^{2}?  (5.2)

As in (0.11), the estimate (5.2) is scale-invariant. Since the vacuum Einstein equations are also scale-invariant, the estimate (5.2) can be viewed as an upper bound on the rate of curvature blowup on approach to a singularity, i.e. a point in ∂M, or as an upper bound on the decay of the curvature in (large) distances from ∂M. By scale invariance, these statements are equivalent. Of course the estimate (5.2) implies the curvature assumption (0.11) without the |∇R| bound, when (M, g) is geodesically complete to the future of Σ_{τ_{o}}.

There are numerous examples where the opposite inequality holds, i.e. |R| ≥ c/t_{∂M}^{2}, so that (5.2) cannot be sharpened significantly, except perhaps by removing the dependence of C on the space-time (M, g).

We note that the estimate (5.2) has been proved for time-independent, i.e. stationary spacetimes, in [3], where ∂M is interpreted as the region where the associated Killing field becomes null.

We first observe that (5.2) is in fact false for general globally hyperbolic space-times, i.e. without compact CMC Cauchy hypersurfaces. For suppose (M′, g′) is a geodesically complete globally hyperbolic vacuum space-time. For any such space-time, ∂M′ = ∅, so that t_{∂M′} = ∞. Hence, (5.2) would imply that R′ = 0, i.e. (M′, g′) is flat.

However, there exist such vacuum space-times which are not flat, namely the Christodoulou-Klainerman (CK) space-times arising as global perturbations of Minkowski space, [18]. Note that the CK space-times in fact have a global foliation by maximal hypersurfaces. There seem to be no other known examples and it is of basic importance to understand the general class of geodesically complete globally hyperbolic space-times.

Now for cosmological space-times, ∂M must of course be non-empty by the Hawking-Penrose singularity theorem. However, the failure of the estimate (5.2) is closely related to the existence of the CK and any related space-times. Namely, if (5.2) were false, then there must exist a sequence of globally hyperbolic cosmological space-times (M_{i}, g_{i}), (possibly identical), and points x_{i} ∈ M_{i} with t_{i}(x_{i}) → 0, t_{i}(·) = t_{∂M_{i}}(·), (this may be arranged by a rescaling if necessary), such that the curvature at x_{i} satisfies

|R_{i}|(x_{i}) · t_{i}(x_{i})^{2} → ∞.  (5.3)

We may assume without loss of generality that the points x_{i} realize the maximal value of |R_{i}| on the level set L_{i} = t_{i}^{−1}(x_{i}). By a standard argument, (the same as that establishing (3.39) but with time in place of space), we may choose the points x_{i} to realize an approximate maximal value of |R_{i}|t_{i}^{2} in small time-like annuli about L_{i}.

Next, rescale the metrics g_{i} by setting g′_{i} = |R_{i}|(x_{i}) · g_{i}, so that

|R′_{i}|(x_{i}) = 1,  (5.4)

and consequently |R′_{i}| ≤ C < ∞ within g′_{i} bounded time distance to x_{i}. This has the effect of making t′_{i}, the proper distance to the boundary w.r.t. g′_{i}, tend to ∞. As discussed in §3, a subsequence then converges weakly in L^{2,p} to a L^{2,p} limit (M_{∞}, g_{∞}, x_{∞}), (unwrapping in the case of collapse). The limit is necessarily a geodesically complete globally hyperbolic space-time. Further, if the condition (5.4), or some weakening of it, passes to the limit, then (M_{∞}, g_{∞}) is not flat. (This is not at all obvious however, and may be difficult to prove).

In addition, if (M_{i}, g_{i}) are CMC cosmological space-times and x_{i} ∈ (M_{F})_{i}, then the apriori bound (1.15) and the arguments from §3 show that the limit (M_{∞}, g_{∞}) has global foliation by maximal hypersurfaces, as in the CK space-times.

Thus, the problem is essentially whether such non-flat CK or related space-times can actually arise in this way, or more precisely whether the evolution from smooth Cauchy data can lead to singularities which have this blow-up character.

We conclude this paper with the following remark or caveat. The discussion above presents relations between the geometrization of 3-manifolds, (at least in case σ(Σ) ≤ 0), and the Einstein evolution equations. Thus, one may ask if a detailed study of the long-time behavior of the hyperbolic Einstein equations is a possible avenue of approach to the solution of the geometrization conjecture when σ(Σ) ≤ 0. And conversely, if the resolution of the geometrization conjecture has any direct bearing on the resolution of some of the fundamental problems on global existence and asymptotics for the Einstein evolution.

With regard to the first problem, this seems to be an extremely difficult approach. Approaches to the solution of the geometrization conjecture by study of a suitable sequence of elliptic PDEs are given in [1], [2] for example, or by study of a parabolic PDE, the Ricci flow, in [28] for example. Both the elliptic and the parabolic approaches appear to be much simpler than the hyperbolic approach via the Einstein flow and stand a much better chance of resolving geometrization. For instance, an exact analogue of Theorem 0.3 has been proved in both of these approaches, c.f. [1],

[2] and [29], with only the bound on R in (0.11), i.e. without any bound on |∇R|. Further, strong geometrization in the sense of Definition 0.2 has been proved in this case in these approaches.

Similarly, it is not at all clear that the (affirmative) resolution of the geometrization conjecture would have any concrete implications on the resolution of any of the global existence, asymptotics, or singularity formation problems in general relativity. Nevertheless, it is clear that there are many interesting relations between these two subjects.

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June 2000

Department of Mathematics S.U.N.Y. at Stony Brook Stony Brook, N.Y. 11794-3651 anderson@math.sunysb.edu