Lorentzian Gromov–Hausdorff theory and finiteness results

Abstract

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.

1 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 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 $(r ,s,D) \in \mathbb {R}^3$), which are locally compact complete (thus geodesic) length spaces with Hausdorff dimension r, curvature $\ge s$ and diameter $\le D$ (and cardinality less than a fixed bound). Gromov [12] showed that $M(\le r, s,D ):= \bigcup _{u \le r} M(u, s, D)$, 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 $\mathbb {R}^{1,n}$, 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 $\partial J^+(p)$ 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 $\mathcal {F} = \mathcal {F } \circ N$ for the forgetful functor $\mathcal {F}$ 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 $n \in \mathbb {N}$, let $C^-_n$ be the category of n-dimensional Cauchy slabs: $\mathrm{Obj} (C_n^-)$ 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 $C_n^+ \cup C_n^-$ (so G is of unspecified signature):

• $k_1: (X,G) \mapsto \mathrm{csec}(X,G) := \inf \{ \mathrm{sec}(A) \vert x \in X , A \subset T_xX$ linear two-dimensional subspace, $A^\perp \mathrm{\ spacelike}\}$, inf of cospacelike sectional curvature (i.e. timelike sectional curvature on $C^-$);

• $k_2: (X, G) \mapsto \mathrm{sup}_{x \in \partial X} \{ \vert \nabla ^G \nu (x) \vert _G : x \in \partial X \}$ norm of second fundamental form (here $\nu (x)$ is the outer normal vector at x, past on $\partial ^- X$ and future on $\partial ^+ X$ if $(X,G) \in \mathrm{Obj} (C_n^-)$);

• $k_3: (X,G) \rightarrow \mathrm{Vol}^G (X) = \int _X \mathrm vol^G$;

• $k_4: (X,G) \mapsto \mathrm{cdiam}((X,G)) := \mathrm{sup}\{ G(\nu (p) , w (p)) \vert p \in \partial X, G(w(p), w(p)) <0, \exp _p (w) \mathrm{\ exists } \}$. For G Riemannian, $\mathrm{cdiam}(X,G) = \mathrm{diam}(X,G)$, for G Lorentzian, $\mathrm{cdiam}(X,G) \ge \mathrm{sup}\{ \ell (c) \vert c : \mathbb {R}\rightarrow X \mathrm{\ causal } \}$ by compactness of $J^+(p)$ for each $p \in X$ and the first variational formula.

For objects (X, g) of $C_n^-$ we moreover define, with $J_g(p) := J_g^-(p) \cup J_g^+(p)$:

• $\mathrm{Jvol} (X) := \mathrm{sup}\{ \mathrm vol(J (p) ) \vert p \in X \}, {\mathrm{Jvol }}$ $({\partial } X) = \mathrm{sup}\{ \mathrm vol(J (p) \cap$ ${\partial } X ) \vert p \in X \} ,$

• $\mathrm{injrad}^\pm _g (x) := \mathrm{sup}\{ \sqrt{- g(w,w)} : w \in T_xX, w \ll 0, \exp \vert _ {I^\pm (0) \cap I^\mp (w)}$ is a diffeomorphism},

• $\Gamma (X) := \inf _{x \in X} \max \{ \mathrm{injrad}^+_g (x) , \mathrm{injrad}^-_g (x) \} .$

Then, for each $a \in (\mathbb {R}\cup \{ \infty \})^4$ and each $b \in (\mathbb {R}\cup \{ \infty \})^7$, we define subcategories

$$\begin{aligned} C_n^\pm (a):= & {} \left. \left\{ (X,g) \in C_n \vert - \mathrm{csec}(X) \le a_1 \wedge \vert \nabla \nu \vert _{\partial X} \right. \right. \\\le & {} \left. e^{a_2} \wedge (\mathrm{vol} (X))^{-1} \le e^{a_3} \wedge \mathrm{cdiam}(X) \le e^{a_4} \right\} ,\\ C_n^- (b):= & {} \left\{ (X,g) \in C_n^- (b_1, b_2, b_3, b_4) ) \vert \mathrm{Jvol}^g (X) \le e^{b_5} \wedge (\Gamma (X))^{-1} \right. \\\le & {} \left. e^{b_6} \wedge \mathrm{Jvol}^g ({\partial } X ) \le e^{b_7} \right\} , \end{aligned}$$

of $C_n ^\pm$. 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 $X \times X \rightarrow \mathbb {R}^{3}$, the morphisms acting by pull-back on them. We will find an injective functor $F: C_n \rightarrow K_n$ mapping some subsets $C^-_n(b)$ into some $C_n^+ (a)$. Let $\sigma ^g: X \times X \rightarrow \mathbb {R}$ the signed Lorentzian distance $\sigma ^g : X \times X \rightarrow \mathbb {R}\cup \{ \infty \} \cup \{ - \infty \}$ defined by $\sigma ^g(x,y) := \mp \mathrm{sup}\{ \ell (c) \vert c: x \leadsto y \mathrm{\ causal \ curve} \}$ for $x \in J^\pm ( y)$ and $\sigma ^g (x,y) = 0$ otherwise. Here, the length $\ell (c)$ of a $C^1$ causal curve $c: [a,b] \rightarrow X$ is $\ell (c):= \int _{a}^b \sqrt{- g(c'(t) , c'(t))} dt$. If X is a g.h. manifold-with-boundary, then $\sigma ^g$ is finite and continuous [3]. In analogy to Kuratowski’s classical aproach to identify points with their distance functions, we apply the same to $\sigma ^g_x:= \sigma ^g(x, \cdot )$. For $f \in M:= \{ f: \mathbb {R}\rightarrow \mathbb {R}\vert f \mathrm{\ measurable \ and \ locally \ essentially \ bounded} \}$ and $p \in [1, \infty ]$, we define pseudo-metrics $d_{g,f,p} (x,y) := \vert f \circ \sigma ^g_x - f \circ \sigma ^g_y \vert _{L^p (X)}$. Compactness of each J(x) implies $d_{g,f,p} < \infty$. We apply the functor $\lambda$ 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 $p = \infty$, to a metric defined by Noldus [20]. Both are briefly reviewed in Sect. 2.

From now on we choose $p=2$, and let $h: \mathbb {R}\rightarrow \mathbb {R}$ with $h(x) := x^4 \ \forall x \in \mathbb {R}$, its crucial properties (see Sect. 3) being $h \vert _{\mathbb {R}\setminus \{ 0 \}}, h'' \vert _{\mathbb {R}\setminus \{ 0\}} >0$ and $h(0) = h'(0) = h''(0) = h^{(3)} (0) = 0$. Furthermore, for $r \in [-1,1]$ we define $f_r \in M$ by $f_r(x) := (\frac{1}{2} + \frac{r}{2} \cdot \mathrm sgn(x) ) \vert x \vert \cdot x^3 \ \ \forall x \in \mathbb {R}$. We define $\Phi ^{g,f}: x \mapsto f \circ \sigma _x^g$, $\Phi ^g:= \Phi ^{g,h}$ and $\Phi ^g_r:= \Phi ^{g, f_r}$, $d^g:= d_{g,h,2}$ and $d^g_r:= d_{g,f_r,2}$. For each f, the metric $\lambda ( d_{g,f,2} )$ is the metric of the pullback of the scalar product of $L^2(X)$ via $\Phi ^{g,f}$. In Sect. 3 we show:

Theorem 1

Let $n \in \mathbb {N}$ and $X \in \mathrm{Obj} (C_n^-)$.

• (i) $\Phi ^{g}: x \mapsto h \circ \sigma _x^g$ is a $C^2$ injective immersion of X into $L^2(X)$, closed if $k_1(X)> - \infty \wedge \Gamma (X) >0$.

• (ii) $F: (X,g) \mapsto (X, \lambda (d^{g}_0) = (\Phi ^{g} )^* (\langle \cdot , \cdot \rangle _{L^2(X)} ), (d^g_{-1/2}, d^g_0, d^g_{1/2}) )$ is an injective functor $C_n^- \rightarrow K_n$ whose push-down to isometry classes is injective, too.

• (iii) For all $b \in \mathbb {R}^7$ with $b_1 \ge 0$, there is $a \in \mathbb {R}^4$ s.t. $(X, (\Phi ^g)^* \langle \cdot , \cdot \rangle ) \in C_n^+ (a)$ for all $(X,g) \in C_n^-(b)$.

• (iv) $\forall \ b \in \mathbb {R}^7 : b_6 \le b_4 \wedge b_3 \le b_5 \Rightarrow C_n^-(b)$ contains an open set in $C_n^-$.

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 $\partial X$, imply Riemannian bounds on $\partial X$ sufficient for finiteness results. Still, via Theorem 1(iii) and Wong’s Riemannian result [25] we obtain:

Theorem 2

For every $n \in \mathbb {N}$ and for all $b \in \mathbb {R}^7$ with $b_1 \ge 0$, 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.

2 Preliminaries on $\lambda$, metrics with $p \ne 2$

We first revise well-known facts: Let (Y, d) be a metric space, let $c \in C^0 ([a,b], Y)$. A partition of [a, b] is a finite subset $Z= \{ y_i \vert i \in \mathbb {N}_N \}$ of [a, b] with $a,b \in Z$, which we always number monotonically. We denote the set of partitions of [a, b] by P(a, b) and put $\ell _Z (c) := \sum _{i=1}^N d(c_{i-1}, c_i)$ and $(L (d))(c) := \mathrm{sup}\{ \ell _Z(c) \vert Z \in P(a,b) \}$, whose finiteness defines the set of rectifiable curves, containing all Lipschitz curves. Conversely, there is a map K from length structures to $\infty$-metric spaces given by $K(Y,\ell ) := (Y,d)$ with $d(x,y) := \inf \{ \ell (c) \vert c: x \leadsto y\}$, then $\lambda := K \circ L$ is a non-injective functor from metric spaces to length spaces (with extended metrics, i.e. $\infty$-metric spaces), $\lambda \ge \mathrm Id$ (i.e., $\lambda (d) \ge d \ \forall d$), $L \circ K = \mathrm Id$ (see e.g. [5, Prop. 2.3.12]), and d-geodesics are $\lambda (d)$-geodesics.

Theorem 3

In $(X:= [-1,1] \times \mathbb {R}, g= -dt^2 + ds^2)$ equipped with the Noldus metric $d_{\mathrm{N}} := d_{g, \vert \cdot \vert , \infty }$, no continuous curve $c: [0,1] \rightarrow X$ for which $x_2 (c(0)) \ne x_2(c(1))$ is $d_{\mathrm{N}}$-rectifiable.

Proof

First assume w.l.o.g. that not both c(0) and c(1) are contained in the upper boundary $\{ 1 \} \times \mathbb {R}$ (otherwise reverse the time orientation). Then let, for $a >0$,

$$\begin{aligned} A_a:= \{ t \in [0,1] \vert x_1(c(t)) \le 1-a \} \end{aligned}$$

and note that due to the continuity of c there is a neighborhood [0, t] of 0 contained in some $A_a$. With $u:= x_2 (c(0)), v:= x_2(c(t))$ we obtain $\ell (c\vert _{[0, t]}) \ge \ell (k)$, where $k: [0, t] \rightarrow X$ is defined by

$$\begin{aligned} k(s) := (1-a, (1-t^{-1}s) u + t^{-1} s v) . \end{aligned}$$

So everything boils down to calculating the $L^\infty$-distance D(t) between $\sigma _{(1-a, 0)}$ and $\sigma _{(1-a, t)}$ which is $\sqrt{t(2a-t)}$. Then the argument is completed by calculating

$$\begin{aligned} n \cdot D(t/n)= \sqrt{t} \sqrt{2an-t } \rightarrow _{n \rightarrow \infty } \infty . \end{aligned}$$

$\square$

Thus X with the Noldus metric $d_N$ is not locally connected by rectifiable paths, so $\lambda (d_N)$ does not generate the topology of X. The same holds for $X= [0,1 ] \times {\mathbb {S}}^1$, with essentially the same proof.

Theorem 4

Each g-causal curve in X is geodesic for the neutral Beem metric $d^{(0)}_{\mathrm{B}} := d_{g,\chi _{\mathbb {R}\setminus \{ 0 \}},1}$.

Proof

As the characteristic function $\chi _{\mathbb {R}\setminus \{ 0 \}}$ of $\mathbb {R}\setminus \{0\}$ encodes timelike future and past, we get

$$\begin{aligned} d_{\mathrm{B}} (x,y)= \mathrm vol(J^- (x) \triangle J^-(y) ) + \mathrm vol(J^+(x) \triangle J^+(y) ). \end{aligned}$$

For $p \le q \le r$ we get

$$\begin{aligned} J^\pm (p) \triangle J^\pm (r)= & {} (J^\pm (p) \triangle J^\pm (q) ) {\dot{\cup }} (J^\pm (q) \triangle J^\pm (r)),\\&\hbox {so}\quad d_{\mathrm{B}} (p,r) = d_{\mathrm{B}} (p,q) + d_{\mathrm{B}} (q,r). \end{aligned}$$

$\square$

Thus all g-causal curves are geodesics for $d_{\mathrm{B}}$ and $\lambda (d_{{B}})$, so they split, unlike in Alexandrov spaces. The same is true for the positive/negative Beem metric $d_B^\pm := d_{g, \chi _{\pm ]0; \infty [},1}$ and linear combinations of $d_B^+$, $d_B^-$ and $d_B^{(0)}$.

3 Proofs of Theorems 1 and 2, discussion

3.1 Proof of Theorem 1 (i)

The maps $\Phi ^{g,f} : X \rightarrow L^2 (X)$ are obviously injective for every $f \in M$ with $f^{-1} (0) = \{ 0 \}$: A left inverse of $\Phi ^{g,f}$ is given by $k \mapsto p$ if p is the unique point in $\mathrm{cl}(k^{-1} (] - \infty , 0[)) \cap \mathrm{cl}(k^{-1} (]0, \infty [))$. Keep in mind that the causal cut locus is of vanishing measure [3], and that $y \in \mathrm{Cut}(x) \cap J^+(x) \Leftrightarrow x \in \mathrm{Cut} (y) \cap J^-(x)$.

Theorem 5

Let $f \in C^4(\mathbb {R}, \mathbb {R})$ with $f^{-1} (0) = \{ 0 \}$ and $f^{(j)} (0) = 0 \forall j \in \mathbb {N}_3$, let (X, g) be a g.h. manifold-with-boundary, then $\Phi ^{g,f} : X \rightarrow L^2(X), x \mapsto f \circ \sigma ^{g}_{x}$ is a $C^2$ embedding. For $w \in T_xX$, $\vert w \vert := (- g(w,w) )^{1/2}$ and $w(y):= \exp _x^{-1} (y)$ we get

$$\begin{aligned} d_x \Phi ^{g,f} \cdot v =\left( y \mapsto {\left\{ \begin{array}{ll} f'( \vert w(y) \vert ) \cdot \vert w(y) \vert ^{-1} \cdot g( v, w (y) ) &{} \mathrm{for}\quad w(y) \ \mathrm{future \ timelike }\\ - f'( - \vert w(y) \vert ) \cdot \vert w(y) \vert ^{-1} \cdot g( v, w (y) ) &{} \mathrm{for}\quad w(y) \ \mathrm{past \ timelike }\\ 0 &{}\mathrm{else } \end{array}\right. }\right) \end{aligned}$$

(well-defined and smooth on the complement of the causal cut locus $\mathrm{Cut} (x)$ of x). The Hessian $\mathrm{Hess}_x (\Phi ^{g,f})$ of $\Phi ^{g,f}$ at x is defined on $X \setminus \mathrm{Cut}(x)$ by (for $v \in T_xX$, $I_x:= \{v \in T_xX: g(v,v) <0 \}$):

$$\begin{aligned}&\mathrm{Hess}_x (\Phi ^{g,f}) \cdot (v \otimes v)\\&\quad =\left( y \mapsto {\left\{ \begin{array}{ll} f'(\vert w \vert )\cdot \vert w (y) \vert ^{-1} g( v, K_{v,y}'(0) )&{}\\ \quad + f''(\vert w (y) \vert ) \cdot (\vert w(y) \vert )^{-2} \cdot g( v, w )^2, &{} w (y) \in I_x \\ 0 &{}\mathrm{else } \end{array}\right. }\right) \end{aligned}$$

where $K_{v,y}$ is the Jacobi vector field along the unique maximal geodesic $c_y$ from $x= c_y(0)$ to $y=\exp _x(w) = c_y(1)$ with $K_{v,y}(1) = 0$ and $K_{v,y}(0)=v$. The pull-back Riemannian metric is

$$\begin{aligned} (\Phi ^{g,f})^* (\langle \cdot , \cdot \rangle _{L^2(X)})_x (a,b) = \int _{J(x) } \vert w (y) \vert ^{-1} (f'(\vert w (y) \vert ))^2 \cdot g( a , w(y) ) \cdot g( b , w(y) ) dy \nonumber \\ \end{aligned}$$

(1)

Remark

Despite of being $C^2$, in general $\Phi ^{g,f}$ does not take values in $W^{1,p} (X)$.

Proof

As g and f are fixed, we write $\Phi = \Phi {g,f}$. Let $x \in X$. We will frequently need an inverse of $\exp _x$, which is a priori only defined on a subset $\exp _x (S_x) \subset X$ of full Lebesgue measure, which we complement by $\exp ^{-1}_x (q) := 0 \in T_xX$ for all $q \in X \setminus \exp _x (S_x)$. As $\exp _x^{-1}$ is smooth on $\exp _x (S_x)$, $\exp _x^{-1}$ is continuous a.e. and is therefore measurable. As it is furthermore bounded and $\exp _x^{-1} (X) \cap I_x$ is compact, all involved functions are square-integrable. By the assumptions on f, the same is true of the expressions given for $d_x \Phi ^{g,f}$ and $\mathrm{Hess}_x (\Phi ^{g,f})$: To see this, first assume that $y = \exp _p(w) \gg x$. Then the first variational formula implies the formula for $d \Phi$, correspondingly for $x \gg y$.

For the Hessian $\mathrm{Hess} \Phi (V,W) := V(W ( \Phi )) - (\nabla _V W) (\Phi )$ of $\Phi ^{g,f}$ 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 $T_pX$.

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 $\varepsilon$ and speed $\kappa$ in the direction V (which is a Jacobi vector field along c) as

$$\begin{aligned} V(V(l(0))) = \frac{\varepsilon }{\kappa } g(\nabla ^*_{\partial t} V^\perp (t), V^\perp (t) ) \vert _a^b + g(c'(t) , A(t)) \vert _a^b \end{aligned}$$

where $\nabla ^*$ denotes the pull-back connection and A is the transverse acceleration $\nabla ^*_{\partial s} \partial _s F(0, \cdot )$ for the variation $F: (- \varepsilon , \varepsilon )_s \times [a,b]_t \rightarrow X$. 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 $\frac{f''(s)}{s^2} , \frac{f'(s)}{s^3} \rightarrow _{s \rightarrow 0} 0$ and due to the formulas $d \Phi (x) \cdot v = (f' \circ \sigma _x ) \cdot (\sigma _x ' \cdot v)$ (here the first variational formula implies that the second factor is proportional to $(\sigma _x)^{-1}$) and $d^2 \Phi (x) \cdot (v \otimes v) = (f'' \circ \sigma _x) \cdot (\sigma _x' \cdot v)^{\otimes 2} + (f' \circ \sigma _x) \cdot (\sigma _x''(v \otimes v))$ (here the second variational formula gives a pole of first order in $\sigma _x$ 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 $\sigma _x$). 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 $\sigma ^g \in C^2(X, L^2(X))$.

Injectivity of $d_x \Phi$ follows: Let $v \ne 0$, then we find $w \in J^+_x \subset T_xX$ with $g_x(v,w) \ne 0$ and $t \in ]0, \infty [$ with $t w \in U_x$, which is the open subset of $T_xX$ from which $\exp _x$ is a diffeomorphism, then $d_x \Phi (v) : \exp _x (tw) \mapsto (\frac{1}{2} + \frac{r}{2})^2 \langle v, w \rangle \ne 0$, that is, $(d_x \Phi ) (v) \ne 0$.

Finally, $\Phi$ is closed (and thus an embedding) for $\Gamma (X) >0, \mathrm{csec}$ finite: First we see along the lines of Lemma 3 that there is a global (on X) positive lower bound on $\mathrm vol(J(p))$; defining recursively $C_n \in \partial ^- X$ with $J^-(J^+(C_n)) \subsetneq C_{n+1}$ and $A_n := J^+(C_n)$ we get for any $p_n \in A_n, q_n \in A_{n+2} \setminus A_n$ that $J(p_n) \cap J(q_n) = \emptyset$, so that there is $U >0$ s.t. the $d(p_n,q_n) \ge U \ \forall n \in \mathbb {N}$. $\square$

3.2 Proof of Theorem 1 (ii)

We recall $d_r^g (p,q) := \vert f_r \circ \sigma ^g_p - f_r \circ \sigma ^g_q \vert _{L^2(X)}$ for $p,q \in X$. We now suppress the dependence of g for a moment in our notation. The fact that $\lambda (d^g_0)$ is a Riemannian metric follows from the previous item. For $x \in X$, let $\sigma _x^+$ resp. $\sigma _x^-$ denote the positive resp. negative part of $\sigma _x$. With the one-parameter family

$$\begin{aligned} f_r: x \mapsto \left( \frac{1}{2} + \frac{r}{2} \cdot \mathrm sgn(x) \right) \cdot x^3 \vert x \vert , \end{aligned}$$

$d_r$ interpolates between the past metric (taking into account only the past cones) $d_{-1}$ with $d_{-1} (x,y) := \vert \vert (\sigma _x^-)^4 - (\sigma _y^-)^4 \vert _{L^2}$ for $f_{-1} = (1- \theta _0) \cdot \mathrm Id ^4$ and the future metric $d_1$ (taking into account only the future cones) for $f_1 = \theta _0 \mathrm Id ^4$, passing through $d_0 (x,y) = \frac{1}{2} \big \vert \sigma _x^3 \cdot \vert \sigma _x \vert - \sigma _y^3 \cdot \vert \sigma _y \vert \big \vert _{L^2}$. Whereas $d_{\pm 1}^g$ is a metric on $X \setminus \partial ^\pm X$, it vanishes identically on $\partial ^\pm X \times \partial ^\pm X$. For fixed points p and q of X, let V be the two-dimensional linear span $\mathrm{span} (u^+, u^-)$ of $u^+ := \sigma _p^+ - \sigma _q^+$ and $u^- := \sigma _p^- - \sigma _q^-$. 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 $\partial ^\pm X = \{ x \in X \vert \exists y \in X \setminus \{ x \} : d_{\pm 1}^g(x,y) = 0 \}$. In the following we want to recover the causal structure. For $p,q \in X$ we define $\lambda _p^\pm := (\sigma _p^\pm )^4$ and consider the following expressions only depending on the metrics $d_r$:

$$\begin{aligned} (d_{\pm 1 } (p,q) )^2= & {} \langle \lambda _p^\pm - \lambda _q^\pm , \lambda _p^\pm - \lambda _q^\pm \rangle _{L^2}\\ (d_0 (p,q))^2= & {} \langle \lambda _p^+ - \lambda _p^- - \lambda _q^+ + \lambda _q^- , \lambda _p^+ - \lambda _p^- - \lambda _q^+ + \lambda _q^- \rangle _{L^2} , \end{aligned}$$

we calculate

$$\begin{aligned}&(d_0 (p,q))^2 - (d_{-1 } (p,q) )^2 - (d_1 (p,q))^2 = 2 \bigg ( \langle \lambda _q^+ , \lambda _p^- \rangle _{L^2} + \langle \lambda _p^- , \lambda _q^+ \rangle _{L^2} \\&\left. \quad - \underbrace{\langle \lambda _p^+, \lambda _p^- \rangle _{L^2}}_{=0} - \underbrace{\langle \lambda _q^+, \lambda _q^- \rangle _{L^2}}_{=0}\right. \bigg ), \end{aligned}$$

where the last two terms vanish due to causality of X. This implies

$$\begin{aligned} (d_0 (p,q))^2 - (d_{-1 } (p,q) )^2 - (d_1 (p,q))^2 \ne 0 \Leftrightarrow p \ll q \vee q \ll p . \end{aligned}$$

as $p \ll q \Leftrightarrow \langle \sigma _p^+ , \sigma _q^- \rangle _{L^2} \ne 0$. The distinction between the two relevant cases can be made by taking into account $p \ll q \Leftrightarrow \sigma _p^+ > \sigma _q^+$, so we only have to pick a point $r \in \partial ^+ X$ (for which $d_1(p,r ) = \vert \sigma _p^+ \vert _{L^2}$ and $d_1(q,r ) = \vert \sigma _q^+ \vert _{L^2}$) and then

$$\begin{aligned} p \ll q \Leftrightarrow (d_0 (p,q))^2 - (d_{-1 } (p,q) )^2 - (d_1 (p,q))^2 \ne 0 \wedge d_1(p,r) > d_1(q,r) . \end{aligned}$$

The causal structure can be recovered from the chronological structure as usual by

$$\begin{aligned} p \le q \Leftrightarrow \forall r \in X: (q \ll r \Rightarrow p \ll r) , \end{aligned}$$

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, $g= e^{2u} \cdot h$ for a smooth function u. Assume that u does not vanish everywhere, then w.l.o.g. let $u (x)>0$ for some $x \in X$. Due to continuity, there is an open neighborhood U of x and a real number $\varepsilon >0$ such that $u (y)> \varepsilon >0 \ \forall y \in U$. As (X, g) is g.h., there are $p,q \in X$ with $J_g^+(p) \cap J_g^-(q) \subset U$. Then

$$\begin{aligned}&\int _{J_g^+ (p) \cap J_g^-(q)} (\sigma ^g_p)^4 (x) \cdot (\sigma ^g_q)^4 (x) d \mathrm vol^g (x) \ge (e^{\varepsilon })^{n/2} \cdot (e^\varepsilon )^8 \\&\int _{J_h^+ (p) \cap J_h^-(q)} (\sigma _p^{h})^4 (x) \cdot (\sigma _q^{h})^4 (x) d \mathrm vol^h (x) , \end{aligned}$$

But we can reconstruct $\int _{J_g^+ (p) \cap J_g^-(q)} (\sigma _p^{g})^4 (x) \cdot (\sigma _q^{g})^4 (x) d \mathrm vol^g (x)$ from the given data as above and exclude $u \ne 0$. 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. $\square$

3.3 Proof of Theorem 1 (iii)

We first define $\Phi := \mathrm{pr}_1 \circ F: (X, g) \mapsto (X, \lambda (d^g))$. First of all, the first item of the theorem establishes that the maps $\Phi$ are closed embeddings, thus the pull-back metric is complete.

The next four lemmas show how $\Phi$ 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.

Lemma 1

$k_1 (\Phi (k_1^{-1} ( [0, \infty [ ))) \subset [0, \infty [$ (in other words: when $k_1(X,g) \ge 0$ then $k_1 ((X, \lambda (d_0^g))) \ge 0$).

If $k_1(X) \ge 0$ then the Hessian $H_X^L$ of $\Phi ^{g,f} : X \rightarrow L^2(X)$ takes values in the cone P(X) of positive functions on X.

Proof

The Gauss equation for the immersion $\Phi _r^g: X \rightarrow L:=L^2(X)$ with Hessian $H^L_X$ reads:

$$\begin{aligned} 0= & {} \langle R^L (V,Y)Z, W \rangle = \langle R^X (V,Y) Z , W \rangle + \langle H_X^L(V,Z) , H_X^L(Y,W) \rangle \\&- \langle H_X^L(Y,Z), H_X^L(V,W) \rangle \end{aligned}$$

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 $c: I \rightarrow X$, w.l.o.g. orthogonal to $c'$. The crucial term in the definition of the Hessian is $u(v,w):= \vert K_{vw} \vert '(0)$ where $K_{vw}$ is the Jacobi field along $c: t \mapsto \exp _x (tw)$ with $K_{vw} (1) =0$, $K_{vw} (0) = v$. Now if $\mathrm{csec}(X,g) \le -\mu \le 0$ then with $\kappa := \mu \cdot \vert w \vert$ we get $g(R(V, c') c' , V) \ge \kappa \langle V, V )$, and

$$\begin{aligned} g( J, J )'' + \kappa g( J, J )= & {} g( J'', J ) + \kappa g( J,J ) + 2 g( J', J' ) \nonumber \\\ge & {} g (R(J, c') c', J ) + \kappa g( J,J ) \ge 0 . \end{aligned}$$

(2)

Thus, for $u:= g( J, J )$ and $\mu = 0$ we get $u'' \ge 0$. Then Eq. 2 with $u(0) = g( v, v) >0$ and $u(\vert w \vert ) = 0$ means $0 >u'(0) = g( J(0), J'(0) )$. So $H_X^L:= \mathrm{Hess}(\Phi ^{g,f})$ takes values in P. The Gauss equation in $L^2(X)$ implies

$$\begin{aligned} 0 = \langle R^X (V,Y) Z,W \rangle _{L^2} + \langle H_X^L (V,Z), H_X^L (Y,W) \rangle _{L^2} - \langle H_X^L (Y,Z), H_X^L (V,W) \rangle _{L^2} , \end{aligned}$$

As $\mathrm{sec}(A,B) = \frac{\langle R(A,B)B, A \rangle }{\langle A,A \rangle \langle B,B \rangle - \langle A,B \rangle ^2}$, so for an orthonormal pair (A, B) we have

$$\begin{aligned} \mathrm{sec}(A,B) = \langle R(A,B)B, A \rangle = \underbrace{\langle H_X^L (B,B), H_X^L (A,A) \rangle }_{=: W \ge 0} - \underbrace{\langle H_X^L (A,B) , H_X^L (A,B) \rangle }_{\vert \cdot \vert \le _{(*)} W} \ge 0 \end{aligned}$$

where the starred inequality is due to the fact that for an orthonormal basis $(e_i)_{i \in \mathbb {N}}$ of positive functions we have (defining $W_i (U,V) := \langle H_X^L (U,V), e_i \rangle _{L^2}$):

$$\begin{aligned} \left| \langle H_X^L (A,B) , H_X^L (A,B) \rangle _{L^2} \right|= & {} \left| \sum _i \langle H_X^L (A,B) , e_i \rangle _{L^2} \langle e_i , H_X^L (A,B) \rangle _{L^2} \right| \\\le & {} \sum _i \vert W_i (A,B) \cdot W_i (A,B) \vert \\\le & {} \sum _i \vert W_i (A,A) \vert \cdot \vert W_i (B,B) \vert \\&=_{(**)}&\sum _i W_i (A,A) W_i(B,B) \\= & {} \langle H_X^L (A,A), H_X^L (B,B) \rangle _{L^2} \end{aligned}$$

where the equality (**) is due to positivity of the basis elements. $\square$

Lemma 2

For every $a_2, a_4, a_5, a_7 \in \mathbb {R}$, $k_2$ is bounded above on $\Phi (C_n^- (\infty , a_2, \infty , a_4, a_5, \infty , a_7))$.

Proof

The difficulty here is that we do not have a two-sided bound on the sectional curvature (and thus the Hessian of $\Phi$). Otherwise, to bound the second fundamental form $S_{\partial X}^X$, we could use $S_{\partial X}^X = S_{\partial X}^{L^2(X)} - S_X^{L^2(X)}$, the fact that the second fundamental form is the normal part of the Hessian, the triangle inequality and the fact that $\partial ^\pm X$ are Cauchy surfaces whose second fundamental form w.r.t. g is uniformly bounded, allowing us to bound the transverse acceleration in Eq. . So we have to find another way: We need an upper bound on the integral

$$\begin{aligned} A := \int _X \underbrace{\vert w(y) \vert ^k \langle w(y), \nu \rangle }_{=: f(y)} \cdot \underbrace{\langle K_{e_j, y} ' (0), K_{e_j, y} (0) \rangle }_{=: h(y)} dy. \end{aligned}$$

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 $V(y):= P_{c_y} (w_y) = c_y'(\vert w(y) \vert ) = d_{w(y)} \exp \cdot w(y)$ (recall that w is smooth a.e.), we see that the second factor h in the above integral is the divergence of V:

$$\begin{aligned} K_{e_j, y}' (0) = \frac{\nabla }{dt} K_{e_j, y} (0) = \frac{\nabla }{dt} \frac{d}{ds} V(0,0) = \frac{\nabla }{ds} \frac{d}{dt} V(0,0) = \nabla _{e_j} (w(y)) , \end{aligned}$$

and $K_{e_j,y} (0) = e_j$, thus for a parallelly extension of the orthonormal basis $(e_0 = w(y),\ldots , e_n)$ denoted by the same symbol we get

$$\begin{aligned} g( K_{e_j, y}' (0) , K_{e_j,y} (0) ) = g( \nabla _{e_j} V(y) , e_j (y) ) , \end{aligned}$$

and summing up gives the divergence of V: In consequence, we get

$$\begin{aligned} A= \int _X V( \vert w \vert ^k \cdot g( w , \nu ) ) + \int _{\partial ^+ X } f(y) \cdot g( V(y) , \nu (y) ) . \end{aligned}$$

(3)

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 $\mathrm{cdiam}$, the second one by the bounds on $\mathrm{cdiam}$ and the volume of causal cones intersected with $\partial X$. $\square$

Lemma 3

$\forall a_1, a_3, a_6 \in \mathbb {R}\ \exists b \in \mathbb {R}: k_4 (\Phi (C_n^-(a_1, \infty , a_3, \infty , \infty , a_6, \infty ))) \subset [e^b , \infty [$.

Proof

The Ricci curvature bound implied by the sectional curvature bound ensures that we find $E>0$ s.t. for all $(X,g) \in A$ and all $y \in X$ we have $\exp _y^* \mathrm vol\ge E \cdot \mathrm vol_{\kappa }$ (for $\mathrm vol_{\kappa }= \exp _{\tilde{x}}^* \mathrm vol_{M_\kappa }$ in the Lorentzian model space $M_\kappa$ of constant sectional curvature $\kappa$) within the domain of injectivity, via the Ricatti equation (cf [23]). For $x \in X$ and $v \in T_xX$, we denote by $J^+_x$ resp. $I^+_x$ the causal resp. timelike future cone in $T_xX$ and for $v \in I_x^+$ define $J^{+,v}_x := \{ u \in J_x^+ \vert g( u,v)/g(v,v) \le 1 \wedge g( u, u) \ge \frac{1}{3} g ( v, v ) \}$. By definition, there is $E_2>0$ s.t. for all $(X,g) \in A$ and for every point z in $(\mathrm{injrad}^+_g)^{-1}( ]b, \infty [)$ there is $w \in T_zM$ timelike future with $g(w,w) < -E_2$ and $\exp \vert {V_w}$ is a diffeomorphism onto its image for $V_w := \{ v \in T_zX \vert v {\ \mathrm future , \ } 0 \le v \le w \}$. Now let us consider Eq. 1. Let $\mathrm vol(U) > \Gamma /2$ and $\mathrm{injrad}^+_g (y) > \Gamma /2$ for all $y \in U$, let $x \in U$. We consider a g-pseudo-ONB $e_0,\ldots , e_n$ at x where $ae_0 = w$ for some $a >0$. For $k \in \mathbb {N}$, $k \le n$, we consider the open subsets $W_k := \{ w \in T_xX \vert g(w,w)<0, w_0<1, w_k>1/2 \}$. They have A-uniformly large $\mathrm vol_\kappa$ in $T_xX$. Thus $(\lambda (d_o^g))_{kk} \ge C_4 (g^+_V)_{kk}$, 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 $\lambda (d_0^g)> C_3 \cdot g^+_V$ at every point $x \in X$, so we find constants $C_1, C_2>0$ s.t. all g with timelike sectional curvature $\ge s$, $\mathrm{ric}\le R$ and $\Gamma \ge e^u$ satisfy $\mathrm vol(X, \lambda (d^g_0)) \ge C_1 \mathrm vol(U, g_V^+) >C_2$. $\square$

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:

Theorem 6

Let H be a Hilbert space. Let $U(r,s):= \{ c \in C^2([0,b], H ) : b \in \mathbb {R}, \vert c' \vert =1, d(c(0) , c(t))<r , \vert c''(t) \vert < s \}$.

• 1. If $s < \frac{2 \sqrt{2}}{3r} =: \rho$ then $\forall c \in U(r,s) : l(c) < \frac{3}{2} r$.

• 2. Let $p \in H$ and let M be a closed submanifold of B(p, r/2) . If $\sqrt{\langle S_M^H , S_M^H \rangle + \langle S_{\partial M}^M, S_{\partial M}^M \rangle \cdot \chi _{\partial M}} < \frac{2 \sqrt{2}}{3r}$, then the intrinsic diameter $\mathrm{diam}(M)$ satisfies $\mathrm{diam}(M) < \frac{3}{2} r$.

Proof of Theorem 6

To prove the first item, assume w.l.o.g. that c is parametrized by arc length. The mean value theorem asserts that for $\Delta (t):= c'(t) - c'(0)$ we get $\vert \Delta (s) \vert < t \cdot \rho$. For $\alpha := \arccos (\langle c'(t) , c'(0) \rangle )$ and $\beta := \alpha /2$, we get by elementary trigonometry $\sin \beta = \Delta (t) /2$, thus

$$\begin{aligned} \langle c'(t), c'(0) \rangle= & {} \cos (2 \beta ) = \cos ^2 \beta - \sin ^2 \beta \nonumber \\= & {} 1 - 2 \sin ^2 \beta = 1 - 2 \frac{\Delta ^2 (t)}{4} = 1 - \frac{\Delta ^2 (t)}{2} \ge 1 - \frac{1}{2} \rho ^2 t^2 \end{aligned}$$

(4)

and consequently, by the Cauchy–Schwartz inequality, we get

$$\begin{aligned} \vert c(t) - c(0) \vert\ge & {} \langle c(t) - c(0) , c'(0) \rangle \ge \int _0^t \left( 1 - \frac{1}{2} \rho ^2 \tau ^2\right) d \tau \\= & {} t - \frac{1}{6} \rho ^2 t^3 =: h(t) \end{aligned}$$

We are interested in the question when h(t) becomes greater than r and therefore calculate the maximum locus of h. The equation $0= h'(b_0) = 1 - \frac{1}{2} \rho ^2 b_0^2$ is solved by $b_0 = \frac{\sqrt{2}}{\rho }$. Inserted into h this gives

$$\begin{aligned} h(b_0 ) = \frac{\sqrt{2}}{\rho } - \frac{2 \sqrt{2}}{6\rho } = \frac{2\sqrt{2}}{3\rho } =r , \end{aligned}$$

and thus, if c is defined on $[0, \sqrt{2}/\rho ] = [0, \frac{3}{2}r ]$, then $\vert c(\sqrt{2}/\rho ) - c(0) \vert > r$.

For the second assertion, let $x, y \in M$. Closedness of M and completeness of H imply that there is a geodesic c from x to y. Then the extrinsic diameter of c(I) is $\le r$ by the triangle inequality. As c is geodesic im M, we have $\nabla _t c' (t) \in (TM)^\perp$ and $\langle \nabla _t c'(t) , Y \rangle = \langle S_M^H (c'(t) , c'(t) ) , Y\rangle \ \forall t \in I$, so $\vert c'' (t) \vert = \vert S_M^H (c'(t), c'(t)) \vert \le \vert S_M^H \vert$. 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 $c''$. $\square$

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 $A \subset C$, we have that $\Phi ^{g,f}(X) \in P \cap (v - P)$, for $v: X \rightarrow \mathbb {R}, x \mapsto e^{a_3}$. Furthermore, $L:= P \cap (v-P)$ is compact, as explained below. Then the following theorem ensures that the set $\{ \mathrm{diam}( \Phi ^{g,f} (X), \lambda (d^{g,f})) \vert g \in A \}$ is bounded:

Theorem 7

Let H be a separable Hilbert space and let $L \subset H$ be compact. Let K be a convex self-dual cone in H and $v \in K$. Then there is $D>0$ such that for all curves $c \in C^2( [a,b] ,L)$ parametrized by arc length with $c' ([a,b]) \subset v - K$ and $c''([a,b]) \subset K$ we have $l (c) \le D$.

Remark

The hypothesis of self-duality is indispensable here, as otherwise for $v \in H \setminus \{ 0 \}$ and $K:= \{ u \in H \vert \langle u,v \rangle >0\}$ any periodic curve (or one contained in a finite-dimensional subspace) in the affine hyperplane $\{ u \in H \vert \langle u,v \rangle = \frac{1}{2} \langle v,v \rangle \}$ satisfies the hypotheses. Also D has to depend on both K and L, see the example given $H:= \mathbb {R}^2 \ni v$, $K:= (0, \infty [ ^2$, $L:= K \cap (v-K)$ and the curves c being restrictions of $k: ]0, \infty [ \rightarrow L, k(t) := (t, 1/t)$.

Proof

Let $n \in \mathbb {N}\cup \{ \infty \}$, $\mathbb {N}_{n} := \{ m \in \mathbb {N}\vert m \le n \}$ and $\ell ^2 (n) := \{ a: \mathbb {N}_{n} \rightarrow \mathbb {R}\vert \sum _{k=1}^n a_k^2 < \infty \}$. Then for any self-dual cone in the n-dimensional separable Hilbert space there is a unitary map $U: H \rightarrow \ell ^2 (n)$ with $U(K) := \{ a \in \ell ^2 (n) \vert a(k) \ge 0 \ \forall n \in \mathbb {N}_n \} \subset [0,\infty [^n$. Thus U(L) is compact, in particular contained in B(0, r) for some $r >0$, and there is an orthogonal projection onto U(K) by just taking the positive part $w_+ \in U(K)$ of any vector w, and with $w_- := w_+ - w \in U(K)$ we get $w= w_+ - w_-$ and $\langle w_+, w_- \rangle = 0$ (in the example of $H= L^2 (X)$ this coincides with taking pointwise the positive and negative part of the functions). For all $u \in v-K$ there is some $E \in ]0, \infty [$ with $E \cdot \vert \vert u \vert \vert \le \langle u, v \rangle \le E \cdot \vert \vert u \vert \vert$. Thus, by arc-length parametrization of c and orthogonality of the projection on K, we see

$$\begin{aligned} 1 = \langle c', c' \rangle = \langle c_+' , c_+' \rangle + \langle c_-', c_-'\rangle \le E^{-2} (\langle c_+', v \rangle ^2 + \langle c_-', v \rangle ^2) , \end{aligned}$$

(5)

which implies $\forall t \in [a,b]: (\langle c_+'(t) , v \rangle \ge E/\sqrt{2} ) \vee (\langle c_+'(t) , v \rangle \ge E/\sqrt{2} ).$

With $L:= l(c) = b-a$, Cauchy’s mean value theorem implies existence of some $t \in ]a,b[$ with

$$\begin{aligned} l:= & {} l(c) = \langle c_+'(t) , v \rangle - \langle c_-'(t) , v \rangle = \langle c'(t) , v \rangle = \frac{1}{l} ( \langle c(b) , v \rangle - \langle c(a), v \rangle )\nonumber \\&\le 2 r\vert \vert v \vert \vert /l. \end{aligned}$$

(6)

Now choose $D_0 \in \mathbb {R}$ such that

$$\begin{aligned} \forall l \ge D_0: \frac{E}{\sqrt{2}} - \frac{2}{l} r \vert \vert v \vert \vert > E/2 . \end{aligned}$$

(7)

Then Eq. 6 implies that both parts of the derivative satisfy a lower estimate: $(\langle c_+' (t) , v \rangle \ge E/2) \wedge (\langle c_-' (t) , v \rangle \ge E/2)$.

Assume that c has length $\ge l$, then one of the two subintervals [a, t] and [t, b] has length $\ge l/2$. Assume first that this is true for [a, t]. Then

$$\begin{aligned} \langle c'(t), c_-'(t) \rangle = - \langle c_- '(t) , c_-'(t) \rangle \le - E^{-1} \langle c_-'(t) , v \rangle \le - E^{-1} \cdot E/2 = 1/2. \end{aligned}$$

and therefore

$$\begin{aligned} \vert \vert c(0) - c(t) \vert \vert \ge \langle c(t) , c_- (t) \rangle - \langle c(0) , c_- (t) \rangle = \int _0^t \langle c'(s) , c_- (t) \rangle ds \ge \frac{l}{2} \cdot \frac{1}{2} \end{aligned}$$

(here the last inquality holds as for $u(s):= \langle c(s), c_-' (t) \rangle$ we get $u'(s) \ge 1/2$ and $c'' (s) \in K$ implies $u''(s) <0$ and thus $\langle c'(s) , c_-' (t) \rangle \le \langle c'(t) , c_-' (t) \rangle \le 1/2$). Let $U:= \max \{ \vert \langle w, c_-' (t) \rangle \vert : w \in L \} \in [0, \infty [$, then $D:= \max \{ D_0, 8U \}$ implies, by convexity of $\langle \cdot , c_-'(t) \rangle$ along $c \vert _{[0,t]}$, that for all $l >D$ we have $\vert \vert c(0) - c(t) \vert \vert \ge \frac{1}{2} \cdot \frac{l}{2} >2U$, in contrast to the definition of U. If the other subinterval was of length $\ge l/2$, then in the last computation we replace $c'_- (t)$ with $c_+' (t)$. $\square$

Lemma 4

$\forall a_2, a_3, a_4 , a_6 \in \mathbb {R}\exists b \in \mathbb {R}: k_4(\Phi (C_n^- (0, a_2, a_3, a_4, , \infty , a_6, \infty ))) \subset (0, b].$

Proof

The bounds on the volume and the timelike diameter of (X, g) imply a bound on the extrinsic diameters. Then the condition $H_X^L \in P$ on the Hessians $H_X^L$ of $\Phi ^{g,f}: X \rightarrow L:= L^2(X)$ found in Lemma 1 together with the condition $v - H_X^L (p) \in P \ \forall p \in \partial ^- X$ bound the intrinsic diameter in terms of the extrinsic one by means of Theorem 7: Let $x,y \in \Phi ^{g,f} (X)$, then by completeness of X there is a geodesic c (modulo $\partial X$) from x to y, whose second covariant derivative is the sum of a part normal to $\Phi ^{g,f} (X)$, 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 $\sigma ^+$), thus $\langle c'' (t) , w \rangle \le 0$. To show the last requirement $c'(s) \in v - K$, we first notice that arc-length parametrization of c implies that for $y \gg c(s)$ and for any orthonormal basis $w(y) = e_0, e_1,\ldots , e_n$ 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 $y \mapsto 4 \vert w(y) \vert ^3 \cdot \langle v, w \rangle$ uniform in $C_n (0, a_2, a_3, a_4, a_5 , \infty )$. Indeed, for $w:= w_0 e_0 + W$ and $v:= v_0 e_0 + W$ (where $V, W \perp e_0$) we calculate $\vert w \vert ^3 \cdot g( v,w ) = (\underbrace{w_0^2 - g(W,W)}_{>0} )^{3/2} \cdot (v_0 w_0 + \underbrace{g(V,W)}_{\le \sqrt{g(V,V) \cdot g(W,W)}})\le \vert w \vert ^4 \cdot \underbrace{\vert v_0 + \sqrt{g(V,V)} \vert }_{< \sqrt{2} \vert v \vert _+}$, which yields the desired uniform upper bound. $\square$

3.4 Proof of Theorem 1 (iv)

Let $S:= S_r:= \mathbb {S} (r)$ for $r > \pi ^{-1}$, then $l:= \mathrm{diam}(S) >2$ and $\mathrm{diam}(S^{n-1}) = \sqrt{n-1} l$. Let $X_r := ([-1, 1] \times S_r^{n-1} , g := -dt^2 + g^{n-1}_S)$, then $\mathrm{cdiam}(X_r)=2$, $\mathrm vol(X_r) = 2 \textit{l}^{n-1}$, $\mathrm vol$ $(\partial X_r) = \textit{l}^{n-1}$, $\mathrm{injrad}^\pm (x) = \pm 1 \mp x_0$, $V(b) := \mathrm vol((\mathrm{injrad}^+)^{-1} ((b, \infty )) ) = \mathrm vol$ $(\partial X_r) - b \cdot \mathrm vol(X_r)/2$, and the claim is proven by perturbing around $X_r$ noticing that all involved geometric data are continuous real functions in the metric w.r.t. the $C^2$ topology.

3.5 Proof of Theorem 2

The finiteness result of Theorem 2 now follows directly from Theorem 1 and a result by Wong (where $C_n^a$ is denoted by $\mathcal {M} (n,K^-,\lambda ^\pm , \mathrm vol\ge v>0,d)$, and for ease of comparison we note $K^- := a_1, \lambda ^\pm := \pm e^{a_2}, v:= e^{a_3}, d:= e^{a_4}$):

Theorem 8

[25, Thm. 1.4] For every $n \in \mathbb {N}$ and every $a \in \mathbb {R}^4$, the number of homeomorphism classes in $C^+_n(a)$ is finite. $\square$

This concludes the Proof of Theorem 2$\square$

3.6 Indispensibility of the bounds in Theorem 1, comparison with other approaches

Recall that in Cheeger-Gromov type finitenes results, the volume bound is needed to prevent dimension loss, as Perelman’s theorem refers to Alexandrov spaces of Hausdorff dimension exactly n, whereas in Gromov’s theorem all dimensions $\le n$ are included. Due to dimensional homogeneity of Alexandrov spaces [5, Thm. 10.6.1] the dimension loss in a GH limit has to occur globally, if it occurs. Now, this dimension loss can actually happen in our context: For a sequence of thinner and thinner Lorentzian cylinders $([0,1] \times \mathbb {S}^1, -dt^2 \oplus \varepsilon \cdot ds^2)$, the corresponding Riemannian cylinders become also infinitely thin in the limit $\varepsilon \rightarrow 0$, so there is dimensional degeneracy in this example. Therefore the lower bound on $\Gamma$ is indispensable in Theorems 1 and 2 . It could not be replaced with a mere bound on the total volume of X either, which can be seen by considering a sequence of cylinders concentrating the volume at the boundary: There are functions $f_n: [-1, 1] \rightarrow [0, \infty [$ with $f_n (-x) = f_n(x)$ and $\lim _n f_n (\pm 1 ) = \infty$ such that $\mathrm vol(X_n) \rightarrow _{n \rightarrow \infty } 1$ for $X_n := ([-1, 1] \times \mathbb {S}^1, -dt^2 + f_n ds^2)$ but still the limit of $\mathrm{pr}_1 (F (X_n))$ (F being the metrization functor of Theorem 1) is one-dimensional. The same holds for flat cylinders, $f_n = n$ and [0, 1/n] instead of [0, 1]. One way to obstruct this phenomenon is precisely the above combination $\Gamma$ of volume and injectivity radius.

The bound on the volume is seen to be indispensable considering the examples of the semi-Riemannian products of long surfaces of genus $\rightarrow \infty$ and the Lorentzian line.

It seems worthwhile to search for similar finiteness results using the notions of Lorentzian injectivity radius developed in [1, 6], keeping in mind that, in order to obtain a genuinely Lorentzian result, it is fundamental to renounce the use of ad hoc Riemannian metrics or, equivalently, ad hoc temporal vector fields, the only known natural such being gradients of CMC Cauchy temporal functions t, i.e. such that for all $a \in t(X)$ its a-level set is a Cauchy surface of constant mean curvature a. To construct those, e.g. as in [10], we need $\partial X$ to be CMC itself—quite a nongeneric condition.

It would, of course, also be desirable to replace the condition on timelike sectional curvature with a Ricci bound $c g (v,v) \le \mathrm{ric}(v,v) \le C g(v,v)$ on causal vectors v, more natural in the context of mathematical relativity. This looks feasible because of the definition of the metric as an integral over Jacobi fields, but here well-posedness of the initial-value problem for the vacuum Einstein equations puts strong limitations. In any case, replacing bounds on sec by bounds on Ric would need to be complemented including bounds on the injectivity radius (cf. [1, Thm. 1]).

If we try to parallel the Riemannian constructions by building up a Gromov–Hausdorff space directly on the Lorentzian length function (or its synthesized counterpart in the fascinating framework of Lorentzian length spaces as in [15]), possibly defining curvature bounds via the theory of optimal transport (for the Lorentzian version of which see [22]), we encounter the difficulty that the respective main structures satisfy the inverse triangle inequality instead of the triangle inequality,¹ thus it 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.