Lorentzian Gromov–Hausdorff theory and finiteness results

Cheeger–Gromov finiteness results, asserting that there are only finitely many diffeomorphism types of manifolds satisfying certain geometric bounds, feature among the most prominent results in Riemannian geometry. To transplant those into Lorentzian geometry, one could use a functor between a Lorentzian and a Riemannian category, which, however, can be shown not to exist if the former contains Minkowski space and its isometries. Here, we construct a functor from a restricted category of Lorentzian manifolds-with-boundary (regions between two Cauchy surfaces) to a category of Riemannian manifolds-with-boundary that preserves geometric bounds and obtain, as a corollary, the first known Lorentzian Cheeger–Gromov type finiteness result.


Introduction and statement of the main results
Considering the tremendous success of Gromov's theory of metric spaces in Riemannian geometry [4,Sect. 14.6], it appears worthwhile to look for a Lorentzian version B Olaf Müller mullerol@math.hu-berlin.de 1 Institut für Mathematik, Humboldt-Universität zu Berlin, Unter den Linden 6, 10099 Berlin, Germany of this concept. This is also motivated by the idea to approach non-perturbative Quantum Gravity in the spirit of geometric quantization (see [26] for an overview on the latter), where the space of global solutions and its topology are central objects, on which various constructions depend. The classical phase spaces of relativistic field theories contain the space of all Ricci-flat globally hyperbolic spacetimes, thus it matters how to functorially topologize sets of spacetimes and their isometry classes. A central object in Gromov's theory is the set M(r , s, D) of all isometry classes of (r , s, D)-Alexandrov spaces (for (r , s, D) ∈ R 3 ), which are locally compact complete (thus geodesic) length spaces with Hausdorff dimension r , curvature ≥ s and diameter ≤ D (and cardinality less than a fixed bound). Gromov [12] showed that M(≤ r , s, D) := u≤r M (u, s, D), if equipped with the Gromov-Hausdorff metric (defined first by Edwards [8], for historical details see [24]), is compact. Then Perelman's stability theorem (cf [14]) (stating that each converging sequence in M(r , s, D) is eventually in one homeomorphism class) implies the finiteness result of [11]: M(r , s, D) contains at most finitely many pairwise non-homeomorphic Riemannian manifolds.
On Minkowski space R 1,n , there is no metric compatible with the topology and covariant w.r.t. the Poincaré group [19,p. 6,Item 3], essentially because the orbit of every point in ∂ J + ( p) under a group of boosts accumulates at p. Thus there is no injective functor N from globally hyperbolic (g.h.) spacetimes to metric spaces preserving the topology, i.e., with F = F • N for the forgetful functor F to topological spaces. This makes the transplantation of Riemannian finiteness results to the Lorentzian world a nontrivial issue. Luckily, there is a Riemannian finiteness result for manifolds-with-boundary due to Wong [25, see Theorem 8], indicating a way out by restricting our category to one without boosts, which will open the door for the construction of the desired functor.
For n ∈ N, let C − n be the category of n-dimensional Cauchy slabs: Obj(C − n ) consists of the g.h. n-dimensional manifolds-with-boundary X whose boundary are two disjoint smooth spacelike Cauchy surfaces, the morphisms being isometric, oriented, time-oriented diffeomorphisms, and the category C + n is the one of complete connected n-dimensional C 2 oriented Riemannian manifolds-with-boundary and isometries (those preserve the boundaries). To set up geometric bounds, we now define four real functions on the set of objects (X , G) of C + n ∪ C − n (so G is of unspecified signature): < 0, exp p (w) exists}. For G Riemannian, cdiam(X , G) = diam(X , G), for G Lorentzian, cdiam(X , G) ≥ sup{ (c)|c : R → X causal} by compactness of J + ( p) for each p ∈ X and the first variational formula.
For objects (X , g) of C − n we moreover define, with J g ( p) of C ± n . To be able to reconstruct the Lorentzian metric from the Riemannian data (see Theorem 1(ii)), we need a slightly richer category K n consisting of objects (X , G) of C + n carrying in addition a Lipschitz function X × X → R 3 , the morphisms acting by pull-back on them. We will find an injective functor F : C n → K n mapping some subsets If X is a g.h. manifold-with-boundary, then σ g is finite and continuous [3]. In analogy to Kuratowski's classical aproach to identify points with their distance functions, we apply the same to σ g x := σ g (x, ·). For f ∈ M := { f : R → R| f measurable and locally essentially bounded} and p ∈ [1, ∞], we define pseudo- We apply the functor λ from metric spaces to length spaces (revised in Sect. 2) to d g, f , p . One example of this scheme, for p = 1, leads to a metric defined by Beem (see [2,16]), who suggested the corresponding topology on the future causal boundary, another one, for p = ∞, to a metric defined by Noldus [20]. Both are briefly reviewed in Sect. 2.
From now on we choose p = 2, and let h : . For each f , the metric λ(d g, f ,2 ) is the metric of the pullback of the scalar product of L 2 (X ) via g, f . In Sect. 3 we show: is an injective functor C − n → K n whose push-down to isometry classes is injective, too.
The last item shows that the range of applicability of the theorem is quite large. To prove the third item, we use certain links between second fundamental form, intrinsic and extrinsic diameter for submanifolds of Hilbert spaces that might be of independent interest (Theorems 6 and 7). The above bounds do not, by restricting the metric to ∂ X , imply Riemannian bounds on ∂ X sufficient for finiteness results. Still, via Theorem 1(iii) and Wong's Riemannian result [25] we obtain: Theorem 2 For every n ∈ N and for all b ∈ R 7 with b 1 ≥ 0, there are only finitely many homeomorphism classes of compact Cauchy slabs in C − n (b).
Theorem 2 is, to the author's knowledge, the first known Cheeger-type finiteness result in Lorentzian geometry. A comparison with other approaches to this topic can be found in Sect. 3.6.

Preliminaries on , metrics with p = 2
We first revise well-known facts: Proof First assume w.l.o.g. that not both c(0) and c(1) are contained in the upper boundary {1} × R (otherwise reverse the time orientation). Then let, for a > 0,

So everything boils down to calculating the
Then the argument is completed by calculating Thus X with the Noldus metric d N is not locally connected by rectifiable paths, so λ(d N ) does not generate the topology of X . The same holds for X = [0, 1] × S 1 , with essentially the same proof.

Theorem 4 Each g-causal curve in X is geodesic for the neutral Beem metric
Proof As the characteristic function χ R\{0} of R \{0} encodes timelike future and past, we get

Proof of Theorem 1 (i)
The maps g, f : . Keep in mind that the causal cut locus is of vanishing measure [3], and that y ∈ Cut( .

well-defined and smooth on the complement of the causal cut locus Cut(x) of x). The Hessian Hess
: where K v,y is the Jacobi vector field along the unique maximal geodesic c y from Proof As g and f are fixed, we write = g, f . Let x ∈ X . We will frequently need an inverse of exp x , which is a priori only defined on a subset exp x (S x ) ⊂ X of full Lebesgue measure, which we complement by exp −1 x is continuous a.e. and is therefore measurable. As it is furthermore bounded and exp −1 x (X ) ∩ I x is compact, all involved functions are square-integrable. By the assumptions on f , the same is true of the expressions given for d x g, f and Hess x ( g, f ): To see this, first assume that y = exp p (w) x. Then the first variational formula implies the formula for d , correspondingly for x y. For the Hessian Hess (V , W ) := V (W ( )) − (∇ V W )( ) of g, f taking values in a Hilbert space with its trivial connection, which is a tensorial symmetric bilinear map, by polarization we only have to compute its values on V = W = e i where e i are p-synchronous vector fields such that e i ( p) is a g-orthonormal basis of T p X .
The second variational formula [21,Prop. 10.8] determines the second derivative of the length l(c) for a geodesic variation around a non-null geodesic c of signature ε and speed κ in the direction V (which is a Jacobi vector field along c) as where ∇ * denotes the pull-back connection and A is the transverse acceleration ∇ * ∂s ∂ s F(0, ·) for the variation F : (−ε, ε) s × [a, b] t → X . For a synchronous vector field we have A = 0.
The contribution of the shifting of the boundary of J (x), which is of Lebesgue measure 0, vanishes, because f (s) s 2 , f (s) s 3 → s→0 0 and due to the formulas d (x)·v = ( f • σ x ) · (σ x · v) (here the first variational formula implies that the second factor is proportional to (σ x ) −1 ) and (σ x (v ⊗v)) (here the second variational formula gives a pole of first order in σ x for the second factor in the second term, whereas by the first variational formula the second factor in the first term has a pole of second order in σ x ). This fact is opposed to the situation in [7], Lemma 3.1, e.g., where the boundary term is central. Now linearity of the solution K of the Jacobi equation in its values at the endpoints gives bilinearity in v and we conclude σ g ∈ C 2 (X , L 2 (X )).
Injectivity of d x follows: Finally, is closed (and thus an embedding) for (X ) > 0, csec finite: First we see along the lines of Lemma 3 that there is a global (on X ) positive lower bound on vol(J(p)); defining recursively C n ∈ ∂ − X with J − (J + (C n )) C n+1 and A n := J + (C n ) we get for any p n ∈ A n , q n ∈ A n+2 \ A n that J ( p n ) ∩ J (q n ) = ∅, so that there is U > 0 s.t. the d( p n , q n ) ≥ U ∀n ∈ N.

Proof of Theorem 1 (ii)
We now suppress the dependence of g for a moment in our notation. The fact that λ(d g 0 ) is a Riemannian metric follows from the previous item. For x ∈ X , let σ + x resp. σ − x denote the positive resp. negative part of σ x . With the one-parameter family is a metric on X \ ∂ ± X , it vanishes identically on ∂ ± X ×∂ ± X . For fixed points p and q of X , let V be the two-dimensional linear span span(u + , u − ) of u + := σ + p − σ + q and u − := σ − p − σ − q . Then the L 2 scalar product on V is uniquely given by the corresponding quadratic form on three vectors any two of which are non-collinear, thus given by d −1/2 ( p, q), d 0 ( p, q) and d 1/2 ( p, q). In other words, the datum of those three recovers the whole family d r . Then we can identify future and past boundary as ∂ ± X = {x ∈ X |∃y ∈ X \ {x} : d g ±1 (x, y) = 0}. In the following we want to recover the causal structure. For p, q ∈ X we define λ ± p := (σ ± p ) 4 and consider the following expressions only depending on the metrics d r : where the last two terms vanish due to causality of X . This implies as p q ⇔ σ + p , σ − q L 2 = 0. The distinction between the two relevant cases can be made by taking into account p q ⇔ σ + p > σ + q , so we only have to pick a point r ∈ ∂ + X (for which d 1 ( p, r ) = |σ + p | L 2 and d 1 (q, r ) = |σ + q | L 2 ) and then The causal structure can be recovered from the chronological structure as usual by thus we can identify the future and past subsets. These in turn form a subbasis for the manifold topology and allow to identify the conformal structure [3, p. 6, Thm. 2.3, Cor. 2.4, Prop. 3.11], so everything is reduced to reconstructing the volume form. Now let g and h be two Lorentzian metrics g, h on X with F(X , g) = F(X , h). By the above we can conclude that g and h are conformally related to each other, by, say, g = e 2u · h for a smooth function u. Assume that u does not vanish everywhere, then w.l.o.g. let u(x) > 0 for some x ∈ X . Due to continuity, there is an open neighborhood U of x and a real number ε > 0 such that u(y) > ε > 0 ∀y ∈ U . As (X , g) is g.h., But we can reconstruct J + from the given data as above and exclude u = 0. Thus g = h. Of course, F is also well-defined and injective on isomorphism classes, as each morphism on the right-hand side induces an isomorphism on the left-hand side.

Proof of Theorem 1 (iii)
We first define := pr 1 • F : (X , g) → (X , λ(d g )). First of all, the first item of the theorem establishes that the maps are closed embeddings, thus the pull-back metric is complete.
The next four lemmas show how transfers uniform Lorentzian bounds to Riemannian bounds. Recall that cospacelike sectional curvature is sectional curvature on planes with spacelike orthogonal complement, which in the Lorentzian case are the Lorentzian planes.
Thus we have to find a bound of the Hessian given by Theorem 5, which is the integral over g(v = K v (0), K v (0)) for Jacobi fields with K v (1) = 0. Let J be a Jacobi field along timelike geodesics c : I → X , w.l.o.g. orthogonal to c . The crucial term in the definition of the Hessian is u(v, w) := |K vw | (0) where K vw is the Jacobi field along c : t → exp x (tw) with K vw (1) = 0, K vw (0) = v. Now if csec(X , g) ≤ −μ ≤ 0 then with κ := μ · |w| we get g (R(V , c ) Thus, for u := g(J , J ) and μ = 0 we get u ≥ 0. Then Eq. 2 with where the equality (**) is due to positivity of the basis elements.
Proof The difficulty here is that we do not have a two-sided bound on the sectional curvature (and thus the Hessian of ). Otherwise, to bound the second fundamental form S X ∂ X , we could use S X ∂ X = S , the fact that the second fundamental form is the normal part of the Hessian, the triangle inequality and the fact that ∂ ± X are Cauchy surfaces whose second fundamental form w.r.t. g is uniformly bounded, allowing us to bound the transverse acceleration in Eq. 2. So we have to find another way: We need an upper bound on the integral · K e j ,y (0), K e j ,y (0) =:h(y)

dy.
We find such a bound not via pointwise bounds but using Stokes' Theorem (observe that the last factor in the integral changes sign) in Federer's version [9, Thm. 4.5.6. (5)] for Lipschitz vector fields. In Federer's book the theorem is formulated for open subsets of Euclidean half-spaces, but we can directly transfer it to pseudo-Riemannian manifolds-with-boundary via standard techniques like partition of unity and transformation formulas for tensors, and in our application we easily see via the choice of a Cauchy temporal function that the involved vector fields are Lipschitz). Defining the vector field V (y) := P c y (w y ) = c y (|w(y)|) = d w(y) exp ·w(y) (recall that w is smooth a.e.), we see that the second factor h in the above integral is the divergence of V : and K e j ,y (0) = e j , thus for a parallelly extension of the orthonormal basis (e 0 = w(y), . . . , e n ) denoted by the same symbol we get g(K e j ,y (0), K e j ,y (0)) = g(∇ e j V (y), e j (y)), and summing up gives the divergence of V : In consequence, we get The first term in Eq. 3 is tangential to c, non-geometric and bounded by the the bounds on volumes of the causal cones and cdiam, the second one by the bounds on cdiam and the volume of causal cones intersected with ∂ X .

Proof
The Ricci curvature bound implied by the sectional curvature bound ensures that we find E > 0 s.t. for all (X , g) ∈ A and all y ∈ X we have exp * y vol ≥ E·volˇ(for volˇ= exp * Q x vol Mˇi n the Lorentzian model space M κ of constant sectional curvature κ) within the domain of injectivity, via the Ricatti equation (cf [23]). For x ∈ X and v ∈ T x X , we denote by J + x resp. I + x the causal resp. timelike future cone in T x X and for v ∈ for all (X , g) ∈ A and for every point is w ∈ T z M timelike future with g(w, w) < −E 2 and exp |V w is a diffeomorphism onto its image for V w := {v ∈ T z X |v future, 0 ≤ v ≤ w}. Now let us consider Eq. 1. Let vol(U) > 0/2 and injrad + g (y) > /2 for all y ∈ U , let x ∈ U . We consider a g-pseudo-ONB e 0 , . . . , e n at x where ae 0 = w for some a > 0. For k ∈ N, k ≤ n, we consider the open subsets W k := {w ∈ T x X |g(w, w) < 0, w 0 < 1, w k > 1/2}. They have A-uniformly large volˇin T x X . Thus (λ(d g o )) kk ≥ C 4 (g + V ) kk , where V = e 0 and g + V is the metric obtained by a Wick rotation around the normalized vector field V , which has the same volume form as g. Taking together the estimates, we get a bound λ(d g 0 ) > C 3 ·g + V at every point x ∈ X , so we find constants C 1 , C 2 > 0 s.t. all g with timelike sectional curvature ≥ s, ric ≤ R and ≥ e u satisfy vol(X,˘(d g 0 )) ≥ C 1 vol(U, g + V ) > C 2 .
For the proof of Lemma 4 below, we will need some variant of the following result about the connection between intrinsic and extrinsic diameter, which is of independent interest:  (4) and consequently, by the Cauchy-Schwartz inequality, we get We are interested in the question when h(t) becomes greater than r and therefore calculate the maximum locus of h. Then we apply the first part of the theorem. The bound on the second fundamental form of the boundary is needed as the connecting curve could have a part along the boundary where it is in general not a geodesic and has two normal parts, one in M and one in H , whose sum is c . Now, the direct application of the preceding lemma in our context would impose too severe restrictions, as two-sided bounds on cospacelike sectional curvature imply that sectional curvature is constant [13, Prop. A.1]. Instead, we pursue a different course: We first observe that by the uniform upper estimates e a 4 resp. e a 3 on volume resp. timelike diameter on a subset A ⊂ C, we have that g, f (X ) ∈ P ∩ (v − P), for v : X → R, x → e a 3 . Furthermore, L := P ∩(v − P) is compact, as explained below. Then the following theorem ensures that the set {diam( g, f (X ), λ(d g, f ))|g ∈ A} is bounded: U := max{| w, c − (t) | : w ∈ L} ∈ [0, ∞[, then D := max{D 0 , 8U } implies, by convexity of ·, c − (t) along c| [0,t] , that for all l > D we have ||c(0)−c(t)|| ≥ 1 2 · l 2 > 2U , in contrast to the definition of U . If the other subinterval was of length ≥ l/2, then in the last computation we replace c − (t) with c + (t). 2 , a 3 , a 4 , a 6 ∈ R∃b ∈ R : k 4 ( (C − n (0, a 2 , a 3 , a 4 , , ∞, a 6 , ∞))) ⊂ (0, b].

Lemma 4 ∀a
Proof The bounds on the volume and the timelike diameter of (X , g) imply a bound on the extrinsic diameters. Then the condition H L X ∈ P on the Hessians H L X of g, f : X → L := L 2 (X ) found in Lemma 1 together with the condition v − H L X ( p) ∈ P ∀ p ∈ ∂ − X bound the intrinsic diameter in terms of the extrinsic one by means of Theorem 7: Let x, y ∈ g, f (X ), then by completeness of X there is a geodesic c (modulo ∂ X ) from x to y, whose second covariant derivative is the sum of a part normal to g, f (X ), which is in the image of the Hessian taking values in P and on the boundary of another part collinear to the exterior normal, (which is also a nonnegative function, as varying a point to the exterior increases σ + ), thus c (t), w ≤ 0. To show the last requirement c (s) ∈ v − K , we first notice that arc-length parametrization of c implies that for y c(s) and for any orthonormal basis w(y) = e 0 , e 1 , . . . , e n at c(s) we get as in the Lemma before that we find a uniform bound on g(e j , c (s)), and then we have to find a pointwise bound on y → 4|w(y)| 3 · v, w uniform in C n (0, a 2 , a 3 , a 4 , a 5 , ∞). Indeed, for w := w 0 e 0 + W and v := v 0 e 0 + W (where V , W ⊥ e 0 ) we calculate |w| 3 which yields the desired uniform upper bound.

Proof of Theorem 2
The finiteness result of Theorem 2 now follows directly from Theorem 1 and a result by Wong (where C a n is denoted by M(n, K − , λ ± , vol ≥ v > 0, d), and for ease of comparison we note K − := a 1 , λ ± := ±e a 2 , v := e a 3 , d := e a 4 ): This concludes the Proof of Theorem 2 seems quite unlikely that one could establish something comparable to the Gromov Hausdorff distance with the Gromov compactness result, let alone an analogon to Perelman stability-at least there is no obvious way towards those results. Consequently, this does not seem to be a viable approach to finiteness results.