Volume comparison for $C^{1,1}$ metrics

We establish volume comparison results for balls in Riemannian manifolds with $C^{1,1}$-metrics with a lower bound on the Ricci tensor and for the evolution of spacelike, acausal, causally complete hypersurfaces with an upper bound on the mean curvature in spacetimes with $C^{1,1}$-metrics with a lower bound on the timelike Ricci curvature. These results are then used to give proofs of Myers' theorem and of Hawking's singularity theorem in this regularity.


Introduction
There are many similarities between the ideas used in the proof of Riemannian comparison theorems (in particular Myers' theorem) and the singularity theorems in Lorentzian geometry. Both use curvature conditions to obtain that in some sense the maximal length of a geodesic without conjugate points is bounded: in the case of Myers' theorem one assumes completeness and obtains a bound on the diameter of the manifold (as the distance between two points is given by the length of a minimizing geodesic, which can not have conjugate points) and in the case of, e.g., the Hawking singularity theorem the assumptions together with geodesic completeness would imply compactness of a certain Cauchy horizon which then gives a contradiction. While there has been some interest in developing Lorentzian analogues to many results from Riemannian comparison geometry in general (see e.g. [1], [2] and [10]) this close connection to the singularity theorems was explored further by Treude and Grant in their recent paper [25], where they use Riccati comparison techniques to prove area and volume monotonicity theorems in Lorentzian geometry (with respect to fixed Lorentzian warped product manifolds). These are then applied to give a new proof of the classical Hawking singularity theorem.
We will show that many of these results carry over to C 1,1 (locally Lipschitz continuous first derivatives) regularity by showing volume monotonicity results for both Riemannian and Lorentzian C 1,1metrics with appropriate curvature bounds and applying them to prove a version of Myers' theorem and Hawking's singularity theorem, respectively.
In general, for a (semi-)Riemannian metric the class C 1,1 is the lowest differentiability class of the metric where one still has local existence and uniqueness of solutions of the geodesic equation. Also by Rademacher's theorem all curvature terms still exist almost everywhere and are locally bounded, which allows the definition of curvature bounds in the following way. We say that the Ricci curvature tensor Ric is bounded from below (by κ) if for every smooth, local vector field X ∈ X(U ) for some open and relatively compact U ⊂ M one has that the function (1.1) p → Ric(p)(X p , X p ) − (n − 1)κg(p)(X p , X p ) is non-negative as an element of L ∞ (U ) (i.e., is non-negative almost everywhere). If M is Lorentzian we say that the timelike Ricci curvature is bounded from below (by −κ) if the above holds for any smooth, local timelike vector field. Clearly this coincides with the usual notion for smooth metrics. As further motivation for studying metrics of this regularity we give a brief overview about the specific situations in the Riemannian and the Lorentzian setting.
In Riemannian geometry there are ways to generalize curvature bounds to even lower regularity, however this requires -at first glance -very different definitions (see e.g. [24,16], where metric measure spaces with lower bounds on the Ricci curvature are studied). While these definitions are equivalent for smooth metrics this has not yet been shown for C 1,1 -metrics, so at least for now those two approaches are independent.
In Lorentzian geometry there has recently been an increased interest and many advances in the understanding of low regularity spacetimes (i.e. C 1,1 -instead of C 2 -metrics, see [7,18,12,13]), which allowed the proof of both the Hawking and the Penrose singularity theorem in this regularity (see [14,15]), a problem that had been open for a long time (cf. [22]). From the viewpoint of general relativity, the importance of this regularity is that it allows for a finite jump in the matter variables via the Einstein equations. It is also worth noting that many of the standard results fail dramatically when lowering the regularity further, for example it is shown in [7] that for any α ∈ (0, 1) there exist 'bubbling metrics' (of regularity C 0,α ), whose lightcones have nonempty interior.
The plan of the paper is as follows. In section 2 we study Riemannian manifolds with C 1,1 -metrics with a lower bound on the Ricci curvature and show a C 1,1 version of the Bishop-Gromov volume comparison theorem for Riemannian manifolds with a lower bound on the Ricci curvature. This also serves as a preparation for the Lorentzian case as it requires significantly less technical details but the ideas remain largely the same. In section 3 we first give the definition of the cosmological comparison condition (as introduced in [25]) and a brief overview of relevant results from causality theory for C 1,1metrics, in particular concerning global hyperbolicity and maximizing geodesics to a subset. Then we show the existence of suitable approximating metrics (using results from [7,13,14]) and in section 3.3 we show that for C 1,1 -metrics the cut locus still has measure zero. As a last preparation we define our comparison spacetimes (again introduced in [25]) as Robertson-Walker spacetimes with constant Ricci curvature and study their dependence on the curvature quantities κ and β. This then allows us to show (as a generalization of [25,Thm. 9] to C 1,1 -metrics) Theorem 1.1 (Volume comparison). Let κ, β ∈ R, g ∈ C 1,1 and assume (M, g, Σ) is globally hyperbolic and satisfies CCC(κ, β) (see Def. 3.9). Let A ⊂ Σ be compact with µ Σ (∂A) = 0, B ⊂ Σ κ,β (with finite, non-zero area) and T > 0 such that all timelike, future directed, unit speed geodesics starting orthogonally to A exist until at least T . Then the function Finally, in section 4, as applications we give a proof of a C 1,1 -Myers' theorem in the Riemannian and two C 1,1 -singularity theorems (one of them being an alternative proof of the C 1,1 -version of Hawking's theorem proved in [14, Thm. 1.1]) in the Lorentzian case.
Notation. Throughout M will always be a connected, Hausdorff and second countable smooth manifold of dimension n ≥ 2. For a semi-Riemannian metric g on M the curvature tensor of the metric is defined with the convention R(X, Y )Z = [∇ X , ∇ Y ] − ∇ [X,Y ] Z and we denote the Ricci tensor of g by Ric.
2. Volume comparison for Riemannian C 1,1 -metrics The goal of this first section is to show a C 1,1 version of the Bishop-Gromov volume comparison theorem.
where B κ (r) denotes any ball of radius r in the n-dimensional simply-connected Riemannian manifold with constant sectional curvature equal to κ, is a nonincreasing function on (0, ∞) and volB p (r) ≤ vol κ B κ (r).
A proof of the classical result (for smooth metrics) can be found, e.g., in [26,Cor. 3.3]. The idea of the proof for C 1,1 -metrics is to apply the classical result to some smooth approximating metrics, so we first have to show that we can find approximations such that (M, g ε ) is a complete Riemannian manifold and that for any compact K ⊂ M and δ > 0 we have Ric ε | K ≥ (n − 1) (κ − δ)g ε | K (where Ric ε denotes the Ricci tensor of g ε ) for ε small enough.

Lemma 2.2.
Let g ∈ C 1,1 be a (geodesically) complete Riemannian metric on M . Then there exist smooth complete Riemannian metrics g ε on M such that g ε → g in C 1 , the approximations have locally uniformly bounded second derivatives and Proof. It is well known that one can construct smooth, symmetric (0, 2)-tensor fieldsg ε ∈ T 0 2 (M ) with g ε → g in C 1 and locally uniformly bounded second derivatives by gluing together componentwise convolutions via a partition of unity: Let (U α , ψ α ) be a (countable) atlas and {χ α } a partition of unity subordinate to the U α and choose functions ζ α ∈ C ∞ (U α ) with compact support in U α such that 0 ≤ ζ α ≤ 1 and ζ α ≡ 1 on an open neighborhood of supp(χ α ) in U α . Given a locally integrable (p, q)-tensor field T we set where T α ∈ L 1 loc ψ α (U α ), R n p+q denotes the chart representation of T ,χ α := χ α • ψ −1 α and the convolution is to be understood componentwise. Note that this construction also ensures that the map (ε, p) →g ε (p) is smooth. Now let δ > 0. By locally uniform convergence we get that for any K ⊂ M compact, w.l.o.g. K ⊂ U α for some chart domain U α (otherwise we may cover K by finitely many of those), there exists ε K such that for all ε ≤ ε K (here . e denotes the euclidean norm on R n and we used Cauchy's inequality, AX e ≤ n max i,j≤n |A ij | X e and that X e |X|g < C, where C = sup {X∈T M|K :|X|g=1} X e < ∞, for any X ∈ T M | K ). But then the globalization lemma [13,Lem. 2.4] allows us to construct (new) approximations g ε : p →g u(ε,p) (p) such that for each compact set K ⊂ M there exists ε K such that g ε (p) =g ε (p) for all ε ≤ ε K and p ∈ K (in particular the g ε still satisfy g ε → g in C 1 and have locally uniformly bounded second derivatives) and such that for each δ > 0 there exists ε 0 (δ) such that d(g, g ε ) < δ for all ε ≤ ε 0 , i.e., d(g, g ε ) → 0.
From this it immediately follows that for ε small enough positive definiteness of g implies positive definiteness of g ε , hence the approximations are Riemannian, and it also immediately gives for any (locally Lipschitz) curve γ. But this implies that for ε ≤ ε 0 we have is relatively compact for all p ∈ M and r > 0, so (M, g ε ) is a complete Riemannian manifold by the Hopf-Rinov theorem.
The next Lemma deals with the Ricci curvature estimate and its proof is largely analogous to the Lorentzian version shown in [14, Lem. 3.2] for κ = 0, but a bit less involved. Lemma 2.3. Let g ∈ C 1,1 be a complete Riemannian metric on M that satisfies Ric ≥ (n − 1) κg. Then there exist smooth approximations g ε with all properties of the previous Lemma and such that for any compact K ⊂ M and δ > 0 there exists ε 0 such that Proof. We first note that (2.6) Ric ε −Ric ε → 0 uniformly on compact sets, whereRic ε is defined as in (2.2). This is established by the same arguments as in the proof of [14, Lem. 3.2]: Clearly the only problematic terms are the ones involving second derivatives of the metric (all other terms converge to the respective ones of Ric in C 0 ). Now on every compact set g ε =g ε for ε small enough by construction, so the terms involving second derivatives of g are dealt with using a variant of the Friedrichs lemma, showing that for any f ∈ C 0 (R n ) and g ∈ L ∞ loc the difference Now let δ > 0 and K ⊂ M compact (and w.l.o.g. contained in some chart domain). If we define A ε := Ric ε − (n − 1) κg ε and A := Ric − (n − 1) κg, then clearly also A ε −Ã ε → 0 uniformly on K.
for ε small (this follows by similar estimates as in (2.3)). So if we can show thatÃ ε (X, X) ≥ 0 for all X ∈ T M | K the claim follows. By constructionÃ ε | K is a finite sum of terms of the form ζ α ψ * α ((χ α A ij ) * ρ ε ) (see (2.2)) so it suffices to show that ((χ α A ij ) * ρ ε ) (p) is a positive semi-definite matrix for any p ∈ ψ α (suppζ α ) (note that (χ α A ij ) * ρ ε is well defined on an open neighborhood U of ψ α (suppζ α ) contained in ψ α (U α ) for ε small enough). Now let p ∈ ψ α (suppζ α ) and X p ∈ R n and let X be the constant vector field x → X p on ψ α (U α ). Then These preparations now enable us to show: Theorem 2.4 (Volume comparison). Let (M, g) be a complete Riemannian manifold with g ∈ C 1,1 and Ric ≥ (n − 1) κ g. Then is a nonincreasing function on (0, ∞) and volB p (r) ≤ vol κ B κ (r).
Proof. Let p ∈ M and 0 < r 1 < r 2 < R. Using the approximating metrics g ε constructed in Lem. 2.2 and 2.3 we see that for any δ > 0 there exists some ε 0 such that (B p (R), g ε ) (as a submanifold of M ) satisfies the conditions of the classical Bishop-Gromov volume comparison (Thm. 2.1) with Ric ε ≥ (n − 1) (κ − δ) g ε for all ε ≤ ε 0 . This gives us . Now by (2.5) from the proof of Lem. 2.2 it follows that d gε (p, q) → d g (p, q) and hence for any r > 0 one has that χ Bε,p(r) → χ Bp(r) almost everywhere (because the sphere S p (r) ⊂ exp p (r · {v ∈ T p M : |v| g = 1}), which has measure zero since exp p is still locally Lipschitz and r · {v ∈ T p M : |v| g = 1} ⊂ T p M has measure zero). So by dominated convergence (note that B ε,p (r) ⊂ B p ( r √ 1−δ ) by (2.5) for ε small, hence the support of all characteristic functions is contained in a common compact set) vol ε B p (r) → volB p (r) for all r > 0. Calculating the volumes of balls in the comparison spaces shows for δ → 0. Altogether this proves the theorem.

The Lorentzian case
In this section the goal is to use volume comparison results (as developed in [25]) for smooth, globally hyperbolic spacetimes M with timelike Ricci curvature bounded from below and containing a spacelike hypersurface Σ (satisfying some additional causality and completeness conditions) that has mean curvature bounded from above to establish analogous results for C 1,1 -metrics. It should be noted that these conditions are very similar to those of the Hawking singularity theorem and [25] includes proofs of this theorem using the new comparison techniques therein. So one of the motivations of this paper was to also give an alternative proof of Hawking's singularity theorem in C 1,1 -regularity (which was first shown in [14]). This will be done in section 4.2.
However, there are some additional difficulties (compared to the Riemannian result from the previous section) arising due to the metric being Lorentzian: First, one has to be more careful when choosing approximating metrics and simple convolution is no longer sufficient since it need not preserve the causal structure. Here the pioneering work was done by Chruściel and Grant in [7], and from there on causality theory for C 1,1 metrics has been developed (see, e.g., [18,13,14]). Additionally, the concept of global hyperbolicity for continuous metrics has recently been explored in [21]. This will be helpful in establishing certain results from causality theory for globally hyperbolic spacetimes with a C 1,1 -metric in section 3.1.
Second, while there is no assumption of (geodesic) completeness needed for the smooth result, an assumption on the minimal time of existence of geodesics starting orthogonally to the hypersurface with unit speed has to be made to ensure that everything plays out in relatively compact sets.
Third, showing that the volumes of the balls in the approximating metrics actually converge to the volumes in the C 1,1 -metric is a bit more involved and will need a result regarding the cut locus of Σ with respect to the C 1,1 -metric, namely that it has measure zero. This will be shown in section 3.3.
3.1. Basic definitions and results. Throughout this section M will always be a Lorentzian manifold with a time orientation. While we will generally assume C 1,1 regularity of the metric, we will often include this assumption explicitly to highlight its importance (many of our results will be both wellknown in higher and not true, or at least unproven, in lower regularity). We also fix once and for all a (complete) Riemannian background metric h on M .
For p, q ∈ M we write p ≪ q if there exists a future directed (f.d.) timelike curve from p to q and p ≤ q if either p = q or there exists a f.d. causal curve from p to q. We also define I − and J − are defined analogously. Note that for a C 1,1 -metric it does not matter whether one allows Lipschitz causal curves or one requires causal curves to be piecewise C 1 (or even broken geodesics) in the definition of I + and J + (see [18, Thm. 1.27] or [13,Cor. 3.10]). Note also that most results from smooth causality theory carry over to C 1,1 -metrics, we refer to [18,13] and [14, Appendix A] for an overview.
We will mainly work with globally hyperbolic manifolds and as for smooth metrics one may use any of the following equivalent properties as definition. (1) (M, g) is causal and for all p, q ∈ M the set J(p, q)

a set S ⊂ M that is met exactly once by every inextendible timelike curve) and (3) (M, g) is causal and C(p, q) (the space of equivalence classes of future directed causal curves from p to q with the compact-open topology) is compact If any of these conditions holds, we say that (M, g) is globally hyperbolic.
Proof. In [21] it was shown that these are equivalent even for continuous metrics, if one replaces causality with the slightly stronger assumption of (M, g) being non-totally imprisoning. So it only remains to show that for a C 1,1 -metric both (1) and (3) already imply M being non-totally imprisoning. This follows as for smooth metrics so we will only present a brief outline: From compactness of J(p, q) (respectively C(p, q)) one obtains that J ± (p) is closed for all p, see [19,Prop. 3.71], respectively [21,Prop. 3.3] (note that the proof only actually uses compactness of C(p, q)). Since g ∈ C 1,1 one can still use the exponential map to show that then already J ± (p) = I ± (p) ( [19] show the existence of a time function and Prop. 3.57 gives strong causality). That strong causality is stronger than non-totally imprisoning follows again as in the smooth case (see e.g. [20, Lem. 14.13]) as was already remarked in [15].
Remark 3.2. The previous proof also shows that for C 1,1 -metrics this definition of global hyperbolicity is equivalent to the one in [21].
where L(γ) denotes the Lorentzian arc-length of γ, i.e., for a curve γ : |g(γ(t),γ(t))|dt. Similarly one defines the future time separation to a subset Σ by If M is globally hyperbolic with a continuous metric then any two causally related points can be connected by a maximizing curve (  Remark 3.5. In a globally hyperbolic manifold the sets J ± (p) are closed ( [21,Prop. 3.3]) and hence for any FCC subset Σ and p ∈ J + (Σ) we have that J − (p) ∩ Σ is compact and Σ itself is closed. Furthermore, from [21,Cor. 3.4], it then follows that As a preparation for Prop. 3.7 we prove the following limit-curve lemma (that will also be needed again later on), which is a slight modification of Thm. 1.5 in [21] (which is in turn based on [17]): Lemma 3.6. Let M be globally hyperbolic and γ n : [0, 1] → M be a sequence of causal curves and K ⊂ M compact such that γ n ⊂ K for all n ∈ N. Then there exists a subsequence γ n k that converges (h-)uniformly to a causal curve γ : In particular, if the γ n are maximizing, then γ is as well.
Proof. By [21, Lem. 2.7] we get an upper bound on the Lipschitz constants of the γ n . And so, since the sequence must have an accumulation point, the convergence result follows from Thm. 1.5 of [21]. It remains to show (3.4) and that γ is maximizing if the γ n are. By [21, Thm. 6.3] the length func- h-uniform convergence as defined above (note that while the statement there only deals with a special subset of causal curves defined on [0, 1], the proof works for any set of such curves with an upper bound on the Lipschitz constants), so L(γ) ≥ lim sup L(γ n k ). Using this and lower semi- For an acausal, spacelike FCC hypersurface in a globally hyperbolic manifold the following holds (which is shown largely analogous to the smooth case ([25, Thm. 2]), only using Lem. 3.6 instead of other limit curve results, we nevertheless include a complete proof): Proposition 3.7. Let (M, g) with g ∈ C 1,1 be globally hyperbolic and let Σ ⊂ M be an acausal, FCC subset. Then the future time-separation τ Σ to Σ is finite-valued and continuous on M and for any p ∈ J + (Σ) \ Σ there exists q ∈ Σ and a causal curve γ from q to p with τ Σ (p) = τ (q, p) = L(γ). Any such maximizing curve γ has to be a (reparametrization of) a geodesic, which is timelike for p ∈ I + (Σ) and null otherwise. If Σ ⊂ M is, additionally, a spacelike hypersurface, then for p ∈ I + (Σ) any maximizing geodesic has to start orthogonally to Σ.
Then there exists a causal curve γ from p to q ∈ Σ and if p / . So assume p ∈ I + (Σ). By definition of τ Σ there exist q n ∈ Σ such that τ (q n , p) → τ Σ (p). Since p ∈ I + (Σ) we have τ Σ (p) > 0 and hence τ (q n , p) > 0 for n large, so q n and p are causally related and can be connected by a maximizing curve γ n (see [21,Prop. 6.4] which is compact by Rem. 3.5. Therefore (after maybe reparametrizing and passing to a subsequence), Lem. 3.6 gives a uniform limit curve γ that is causal, satisfies q = γ(0) ∈ Σ (note that Σ is closed by Rem. 3.5) and p = γ(1) and is maximizing, so by upper semi-continuity of the length functional we get τ (p, q) = L(γ) ≥ lim sup L(γ n ) = lim sup τ (q n , p) = τ Σ (p). Consequently, γ maximizes the distance from Σ to p and τ Σ (p) is finite.
Regarding continuity we show lower and upper semi-continuity separately, starting with lower semicontinuity. Let p ∈ M . We have to show that for every ε there exists a neighborhood U ε of p such that for all If τ Σ (p) = 0, there is nothing to prove due to non-negativity of τ Σ . Let γ : Next we show upper semi-continuity, i.e., for every ε there exists a neighborhood U ε of p such that for all q ∈ U ε τ Σ (q) ≤ τ Σ (p) + ε. Assume to the contrary that there exists ε > 0 and p n → p such that and let γ pn : [0, 1] → M be causal curves from Σ to p n with τ Σ (p n ) = L(γ pn ) (such curves exist, since τ Σ (p n ) > τ Σ (p) + ε > 0 and so p n ∈ I + (Σ)). Let p + ∈ I + (p), then p n ∈ J − (p + ) eventually and thus , which is compact by Rem. 3.5. So we can apply Lem. 3.6 to obtain (after passing to a subsequence) a curve γ from Σ to p = lim p n with Since causal geodesics are locally maximizing (by [18, Thm. 6]), any maximizing curve must be (a reparametrization of) a geodesic and if p ∈ I + (Σ) then τ Σ (p) > 0, so it has to be timelike. Now let Σ be an acausal, FCC, spacelike hypersurface. We show that all timelike geodesics that start in Σ and maximize the distance to Σ must start orthogonally: First note that if γ : [0, b] → M maximizes the distance then also γ| [0,ε] must maximize the distance to Σ, so this is a local question and we may assume that M = R n , Σ ⊂ R n is a hypersurface and γ : [0, 1] → R n is a timelike unit-speed geodesic with γ(0) = 0 ∈ Σ that maximizes the distance to Σ. Now for any v ∈ T 0 Σ we can find a smooth curve α : [0, ε] → Σ such thatα(0) = v and α(0) = 0. We use this to define a C 2,1 (note that γ is a geodesic, hence C 2,1 by the geodesic equation) variation Since γ is timelike this is a timelike variation for small enough ε and we may use the first variation of arc-length (see [20,Prop. 10.2] and note that s → L(σ(., s)) is still C 1 ) to obtain This shows thatγ(0) ⊥ v for all v ∈ T 0 Σ, so γ starts orthogonally.
Note that the part of the proof that shows that γ has to start orthogonally to Σ really only works for p ∈ I + (Σ) and not for p ∈ J + (Σ) since in that case one could not guarantee that the constructed variation consists only of timelike curves. However, the next remark shows that J + (Σ)\ (Σ ∪ I + (Σ)) = ∅ anyways.
Note that even though basically all of the upcoming results (except for the C 1,1 version of Hawking's theorem at the very end, see Thm. 4.5) will additionally require global hyperbolicity, we choose not to include this in the definition of the comparison condition.
Now we show that we may additionally demand the following: Proof. We first show that we can construct approximations g ε that retain the properties of theǧ ε from above but additionally satisfy that Σ is g ε spacelike for ε small, i.e. g ε | T Σ is positive definite.
To do this, we show that for every compact set K ⊂ Σ there exists ε K such that this holds for thě g ε for all ε ≤ ε K and then apply the globalization lemma ( [13,Lem. 2.4]). This gives us metrics g ε (p) :=ǧε (ε,p) (p) that satisfy g ε | K =ǧ ε | K for all ε ≤ ε K and g ε | T Σ is positive definite. Since g| T Σ is a Riemannian metric on Σ we have that g(X, X) = 1 implies X h ≤ C for all X ∈ T Σ| K and hence sup {X∈T Σ|K :g(X,X)=1}ǧε (X, X)−g(X, X) → 0 by the previous proposition. Soǧ ε (X, X) > c g(X, X) > 0 for any nonzero X ∈ T Σ| K for all ε small (depending on K), showing positive definiteness. The other properties follow because by the above construction g ε ≺ g (since g ε (p) =ǧε (ε,p) (p) anď g ε ≺ g): By Prop. 3.1, global hyperbolicity is equivalent to the existence of a Cauchy hypersurface and by definition any Cauchy hypersurface for g also has to be a Cauchy hypersurface for any g ′ ≺ g. This shows that (M, g ε ) is globally hyperbolic. Similarly Σ being g-FCC implies g ε -FCC and g-acausality of Σ implies g ε -acausality.
From now on g ε will always denote smooth approximating metrics as constructed above, in particular satisfying Prop. 3.11, Lem. 3.12 and g ε ≺ g. The next Lemma shows properties of the Ricci curvature Ric ε of this approximations (which is basically [14, Lem. 3.2], except also explicitly covering the case κ = 0, and the proof proceeds similarly). Lemma 3.13. Let g ∈ C 1,1 and h be a background Riemannian metric. Suppose that Ric g (X, X) ≥ −n κ g(X, X) for any local smooth g-timelike vector field X ∈ X(U ). Then for any compact set K ⊂ M , C > 0 and δ > 0 there exists ε 0 = ε 0 (K, C, δ) such that Proof. Fix K ⊂ M (w.l.o.g. contained in a chart domain), C > 0 and δ > 0. As in the proof of Lem. 2.3 we proceed similarly to [14, Lem. 3.2]. By the argument given there g ε −g ε → 0 in C 2 (note that by construction g ε =ǧ ε on K for ε small). As in (2.6) we have Ricg ε −Ric ε → 0 uniformly on compact sets and so (3.6) Ric ε −Ric ε → 0 uniformly on compact sets. Now we define A ε := Ric ε − (n − 1) κg ε and A := Ric − (n − 1) κg. Clearly A ε −Ã ε → 0 uniformly on compact sets and thus (for ε small enough) As in Lem. 2.3 it now suffices to show this for every term ofÃ ε of the form ζ α ψ * α ((χ α A ij ) * ρ ε ). Again we may assume M = R n andÃ ε = A * ρ ε . Now choose ε 0 such that |g ε (X, X) − g (X, X)| < 1 2 for all X ∈ T M | K with X h ≤ C and all ε < ε 0 . Since g is uniformly continuous on K there exists some ε 0 > r > 0 such that for any p, x ∈ K with x − p h < r and any 2 . This implies that for any p ∈ K and X p ∈ R n with X p h ≤ C and g ε (p)(X p , X p ) = −1 the constant vector fieldX : x → X p is g timelike on on the open ball B p (r) and thus by our assumption A(X,X) = Ric(X,X) − (n − 1) κg(X,X) ≥ 0 almost everywhere on B p (r). So for ε < r we get since ρ ε ≥ 0 and suppρ ε ⊂ B 0 (ε).
Lemma 3.15. Let g ∈ C 1,1 and assume that the mean curvature of Σ ⊂ M is bounded from above by β. Then there exist approximations g ε such that for any compact set A ⊂ Σ and η > 0 there exists ε 0 such that H ε | A < β + η for all ε < ε 0 .
Proof. Since H = tr g| T Σ S n (see Rem. 3.10) and the Christoffel symbols of g ε converge to those of g uniformly on compact sets it suffices to show that the the g ε unit normal vector field n ε to Σ converges to n in C 1 . Because Σ is a smooth hypersurface it is locally given as the zero set of a submersion f : U → R n−1 and hence in C 1 , proving the claim.
We need two further properties of this approximations.
Proof. This follows from a local argument using a standard result on the comparison of solutions to ODE ([8, 10.5.6 and 10.5.6.1]): Note that the Γ k g,ij are locally Lipschitz continuous, the Γ k ε,ij (p) depend smoothly on ε and p for ε > 0 and Γ k ε,ij → Γ k g,ij locally uniformly for ε → 0. Given any γ v (t 1 ) andγ ε v (t 1 ), which will be close if v andṽ were), U 2 and so forth gives the claim.
For compact A ⊂ Σ each S + ε N ε A is compact for any ε ≥ 0 (since the respective future pointing unit normal vector fields n ε are continuous and S + ε N ε A = n ε (A) by definition). The following lemma shows that this remains true for their union over 0 ≤ ε ≤ ε 0 .

Lemma 3.18. Let A ⊂ Σ be compact. Then for any neighborhood
and is compact.

3.3.
The cut locus of Σ has measure zero. As a further preparation we will now show that for an acausal, spacelike, FCC hypersurface Σ in a globally hyperbolic spacetime with C 1,1 -metric the (future) cut locus Cut + (Σ) ⊂ M has measure zero. This will be vital in the proof of Lem. 3.31.

Definition 3.19 (Cut function). Let (M, g)
with g ∈ C 1,1 be globally hyperbolic and Σ ⊂ M be an acausal, spacelike, FCC hypersurface. The function We first show measurability of the cut function. Proof. To begin with we rewrite the cut function in a form that makes it possible to use Prop. A.6. Define the set-valued map F : S + N Σ → P(R) by where D denotes the maximal domain of definition of the flow of the (normal) exponential map. Note that D is open. Then (using Prop. 3.7) is measurable. This in turn follows immediately if we can show that both the map (v, t) → τ Σ (γ v (t)) and (v, t) → L(γ v | [0,t] ) are continuous on D. The first continuity follows from continuity of τ Σ on M (see Lem. 3.7) and continuity of (v, t) → γ v (t) on D (by continuous dependence of ODE solutions on the initial data). For the second one, note that (v, t) →´t 0 g(γ v (τ ))(γ v (τ ),γ v (τ ))dτ is continuous because the integrand is.
Remark 3.21. Note that B µg is actually independent of g: Ifg is any other semi-Riemannian metric on M then B µg = B µg because the Borel sets of measure zero are the same for µ g and µg as locally any such measure is given by the Lebesque measure multiplied by a positive function (cf. [9, 16.22.2]).
Also, for smooth metrics measurability is a direct consequence of lower semi-continuity of the cut function, but the proof of lower semi-continuity uses the characterization of the cut points as either conjugate points or meeting points of two maximizing geodesics (see, e.g., [3,Prop. 9.7]), which one does not have in the C 1,1 case and it is unclear whether lower semi-continuity even remains true for C 1,1 -metrics.
Using this identification we have So from measurability of the cut function (Lem. 3.20) and Prop. A.7 and Prop. A.8 from the appendix, we obtain that the tangential cut locus Cut + T (Σ) ⊂ N Σ has measure zero. Now the normal exponential map exp N : N Σ → M is a locally Lipschitz continuous map from the n− 1 + 1 = n-dimensional manifold N Σ to the n-dimensional manifold M , hence its chart representations (with relatively compact domains) can be extended to Lipschitz continuous maps from R n → R n . Using a compact exhaustion K n of N Σ and covering each K n by finitely many charts (with relatively compact domains) we see from Prop. A.9 that exp N (K n ∩Cut + T (Σ)) has measure zero. Thus Cut + (Σ) = n exp N (K n ∩ Cut + T (Σ)) has measure zero.
3.4. The comparison manifolds. For any given κ, β we use the comparison manifolds M κ,β defined in [25,Sec. 4.2]. To make this work more self contained we will briefly review their definition and properties. These comparison manifolds were constructed to satisfy CCC(κ, β) with equality in both the Ricci as well as the mean curvature estimates and are given by certain warped products M κ,β = (a κ,β , b κ,β )× N κ,β for 0 ∈ (a κ,β , b κ,β ) ⊂ R, where N κ,β is the (n − 1)-dimensional simply connected Riemannian manifold with constant sectional curvature of 0, 1, or −1, depending on κ and β (so either R n−1 , the unit sphere S n−1 , or hyperbolic space H n−1 ), with metric where h denotes the standard Riemannian metric on N κ,β and f κ,β : (a κ,β , b κ,β ) → R is a positive smooth function.
The warping functions for each pair κ, β are summarized in the table below, which is based on [25, Table 1], but we use that the mean curvature H 0 of the hypersurface Σ κ,β := {0} × N κ,β ⊂ M κ,β is equal to β to express their constant b in terms of β and we also include the respective constants b κ,β that specify the upper bound of the interval where f 2 κ,β > 0. Also note that our base manifold is assumed to be n-dimensional (whereas it is (n + 1)-dimensional in [25]) and for notational simplicity some of the f κ,β listed in Table 1 for all t > 0 and any measurable B n ⊂ Σ κ−δn,β+ηn and measurable B ⊂ Σ κ,β .
Proof. The first statement is an immediate consequence of the previous Lemma and f κ,β (0) ) n−1 for any κ, β (see [25, eq. (15)] and note that S + B (t) = ∅ for t ≥ b κ,β ). For (3.10) note that and vol κ,β B + B (t) =´t 0 area κ,β S + B (τ )dτ and that we may apply dominated convergence since for κ ≤ 0 and β ∈ R one has f κ−δn,β+ηn (τ ) for all τ ≤ t and for κ > 0, β ∈ R one has Remark 3.27. The reason we only show this for specific sequences δ n and η n lies in our somewhat incomplete treatment of the dependence offκ,β/f κ,β (0) on κ, β in Lem. 3.24: While it does seem reasonable that the result remains true for all such sequences, that would require many additional cases of possible convergence to be checked in Lem. 3.24, which is rather tedious and completely unnecessary for the rest of this work.
3.5. Volume Comparison. We first need to show area and volume comparison statements for the approximating metrics and to do so we need to define future spheres that avoid the cut locus.
Similarly, but using the approximations g ε (from section 3.2), the g ε -time separation τ ε,Σ and the ε-cut locus, we define S + ε,A (t). Using results from [25] we are now able to prove area and volume comparison statements for the approximating metrics. Proposition 3.29 (Area comparison for approximations). Let κ, β ∈ R, g ∈ C 1,1 and assume (M, g, Σ) is globally hyperbolic and satisfies CCC(κ, β). Let A ⊂ Σ be compact, η, δ > 0, B ⊂ Σ κ−δ,β+η (with finite, non-zero area) and T > 0 such that all timelike, f.d. unit speed g-geodesics starting in A orthogonally to Σ exist until at least T . Then there exists ε 0 > 0 (depending on η, δ, A, T ) such that for all ε < ε 0 the function Proof. We would like to use [25, Thm. 8], however we have to argue that the bounds on Ricci and mean curvature from Lemma 3.13 and 3.15 are sufficient to show this for smooth metrics ( [25] requires global bounds while we only have them on compact subsets of T M respectively Σ). First we note that by compactness of S + N A there exists a neighborhood U of S + N A in T M such that all g-geodesics starting in U exist until at least T . Then by Lem. 3.18, for ε 0 small the set K ε0 := 0≤ε≤ε0 S + ε N ε A ⊂ T M is compact and contained in U , hence any g-geodesic starting in it exists until T . So by Prop. 3.16 there exists From here the proof proceeds analogously to [25,Thm. 8]. Let 0 < t 1 < t 2 < min(T, b κ−δ,β+η ) and choose a sequence of compact sets K ε,j ⊂ S + ε,A (t 2 ) with area K ε,j ր area S + ε,A (t 2 ). Now, as in [25], we get sets H ε,t (q)dµ ε,t (q).
Note that this is all that is needed to apply the Riccati comparison argument used in the proof of Thm. 7 and it is the only place where the curvature estimates enter the proof. So we get (3.11). The remainder of the proof is completely analogous to [25,Thm. 8].

t) We now show that in cases (1) − (3) the characteristic functions converge in p:
In case (3) we have L g (γ p ) > t. But then for ε small this γ p is also g ε timelike and Lem. 4.2 from [14] gives that for any small δ > 0 there exists ε 0 such that for all ε ≤ ε 0 Thus p / ∈B + ε,A (t) for ε small. Now for case (1) Let γ ε be a g ε -geodesic between q ε ∈ Σ and p with L ε (γ ε ) = τ ε,Σ (p). From q ε ∈ J − (p) ∩ Σ, it follows that γ ε ⊂ J − (p) ∩ J + (J − (p) ∩ Σ) for all ε, which is compact by Rem. 3.5. This allows us to use Lem. 4.2 from [14] to obtain that for any small δ > 0 there exists ε 0 such that for all ε ≤ ε 0 . This shows that if τ Σ (p) < t, then τ ε,Σ (p) < t for small ε. Now let U ⊂ A • be a neighborhood of γ p (0) in Σ. It remains to show that q ε ∈ U ⊂ A • for small ε. Assume the contrary and let γ εj be a subsequence with q εj / ∈ U . By our limit curve Lemma 3.6, we may assume (after reparametrizing and passing to a further subsequence) that γ εj converges to a causal curveγ from q :=γ(0) = lim q εj / ∈ U to p with L g (γ) ≥ lim sup j→∞ L g (γ εj ). Using (3.13) and (3.12) gives for any δ > 0 and letting δ → 0 shows thatγ is also maximizing the distance between p and Σ, giving a contradiction, sinceγ(0) = γ p (0) but γ p is the unique causal curve realizing the distance from Σ to p by definition. Altogether, p ∈ B + ε,A • (t) ⊂ B + ε,A (t) for small enough ε. Next we look at case (2), i.e., γ p (0) / ∈ A (and thus p / ∈ B + A (t)). Let U be a neighborhood of γ p (0) in Σ with U ∩ A = ∅ (this exists since A is closed). By the argument presented when dealing with case (1), we have that for ε small enough γ ε (0) ∈ U , hence not in A and so p / ∈ B + ε,A (t). It remains deal with cases (4) and (5). Here we show that both S + A (t) andB + ∂A (t) are contained in sets of measure zero.
Regarding S + A (t), letñ be a C 1,1 -extension of n to some small neighborhood U of A (in M ) and consider the map h : p → exp p (tñ(p)). For U small enough this is well defined on U (by a standard ODE argument) and because the exponential map is locally Lipschitz continuous, this map is as well. Now since S + A (t) ⊂ h(A), µ(A) = 0 (because A ⊂ Σ), A is compact and any Lipschitz map from R n → R n maps sets of (Lebesgue-)measure zero to sets of measure zero (see Prop. A.9 in the appendix), we have that h(A) has measure zero.
Finally, forB + ∂A (t), note that ∂A ⊂ A and hence all f.d., unit-speed, normal geodesics starting in ∂A exist until at least . Now since [0, t] · ∂A ⊂ N Σ has measure zero (because by assumption µ Σ (∂A) = 0) and is compact (by compactness of A) and exp N is locally Lipschitz, the desired result follows again from Prop. A.9.
Altogether this shows that indeed We are now ready to prove Thm. 1.1.
. By Lem. 3.31 we have for all t ∈ (0, T ]. So using Prop. 3.30 and letting ε → 0 shows that for all η, δ > 0 and B δ, Now by (3.10) there exist sequences δ n , η n → 0 such that .  for all x ∈ U , so U ⊂ B p (d(p, q) + 1) and U ∩ B p ( π √ κ ) = ∅. But this shows that This result is not very surprising since it is known that there are generalizations of Myers' theorem even for metric measure spaces (see Cor. 2.6 in [24]). However, these do not immediately imply Thm. 4.1 above, because for metric measure spaces the needed curvature bound is (by necessity) defined in a different manner from Ric ≥ (n − 1) κ g in L ∞ loc .

4.2.
Hawking's singularity theorem for C 1,1 -metrics. We first show a general result concerning geodesic incompleteness of globally hyperbolic manifolds.
Proof. First note that for these values of κ and β we have b κ,β < ∞ (see Table 1), so τ Σ (p) ≤ b κ,β for all p ∈ I + (Σ) implies L (γ) ≤ b κ,β for all timelike, f.d. geodesics γ starting in Σ, which implies incompleteness of M . Now assume to the contrary that there exists p ∈ I + (Σ) with τ Σ (p) > b κ,β . We first argue that we may w.l.o.g. assume p / ∈ Cut + (Σ): By continuity of τ Σ (see Lem. 3.7) there is a neighborhood U of p such that τ Σ (q) > b κ,β for all q ∈ U and since Cut + (Σ) has measure zero (see 3.23) but U does not there existsp / ∈ Cut + (Σ) with τ Σ (p) > b κ,β . Now, if we have p / ∈ Cut + (Σ) then, by the same argument as in the proof of Lem. 3.31, there exists a unique unit-speed geodesic γ p from γ p (0) ∈ Σ to p with L (γ p ) = τ Σ (p) > b κ,β (and this geodesic has to start orthogonally to Σ by Lem. 3.7). In particular, γ p exists until at least some T > τ Σ (p) > b κ,β . Let A be a neighborhood of γ p (0) in Σ such that all unit-speed geodesics starting in A orthogonally to Σ also exist until at least T . We may choose A to be compact with µ Σ (∂A) = 0 (e.g. as the pre-image of a small, closed ball in R n−1 under a chart of Σ).
We now show that there exists a neighborhood U of p such that for any q ∈Ũ := U \ Cut + (Σ) we have b κ,β < τ Σ (q) < T (which follows immediately from continuity of τ Σ ) and that the unique unit-speed geodesic γ q from γ q (0) ∈ Σ to q with L (γ q ) = τ Σ (q) satisfies γ q (0) ∈ A. This is done via contradiction in a similar way to case (1) in the proof of Lem. 3.31: Let p + ∈ I + (p), then there exists a small neighborhood U of p such that γ q (0) ∈ J − (p + ) ∩ Σ for all q ∈Ũ . Assume there exist p j ∈Ũ with p j → p but γ pj (0) / ∈ A. Then γ pj ⊂ J − (p + ) ∩ J + (J − (p + ) ∩ Σ) and since this set is compact by Rem. 3.5 our limit curve Lemma 3.6 shows that there existsγ with p =γ(1) andγ(0) = γ p (0) and by continuity of τ Σ (see Lem. 3.7). Soγ is also maximizing the distance between p and Σ, but this a contradiction sinceγ = γ p and γ p was unique since p / ∈ Cut + (Σ). We now apply Thm. 1.1 to obtain that which is a contradiction to t → vol B + A (t) vol κ,β B + B (t) being nonincreasing on (0, T ]. Remark 4.3. If Ric ≥ κ(n − 1)g with κ > 0 the mean curvature of Σ is irrelevant, hence any globally hyperbolic spacetime satisfying such a curvature bound is necessarily geodesically incomplete: By [21,Thm. 4.5] there exists a smooth metric g ′ ≻ g such that (M, g ′ ) is globally hyperbolic as well and by [4, Thm. 1.1] there exists a smooth, spacelike Cauchy hypersurface Σ for g ′ . This Σ is then necessarily acausal ([20, Lem. 14.29 and 14.42]) and FCC (see [25,Rem. 1]) and thus also a smooth, spacelike, acausal, FCC Cauchy hypersurface Σ for g (by arguments similar to the ones in Lem. 3.12) and τ Σ ≤ π √ κ : On every compact subset A ⊂ Σ the mean curvature is bounded from above by some β ∈ R (and this is all that is actually needed to show Thm. 1.1 for this fixed A) and since b κ,β ր π √ κ for β → ∞ one arrives at a contradiction by the same construction as in Thm. 4.2. This even shows that L(γ) ≤ 2 π √ κ for any timelike curve γ since any inextendible timelike curve must meet Σ. Of course, the smooth version of this result is well-known and can be proven without this detour ([3, Thm. 11.9]).
If (M, g) is not globally hyperbolic, we cannot apply Thm. 1.1 directly, but if (M, g, Σ) satisfies CCC(κ, β) with κ, β as in Thm. 4.2 and Σ is additionally compact we can still use it to prove compactness of the Cauchy development D + (Σ). The case κ = 0, β < 0 of the previous Lemma provides an alternative proof of Hawking's singularity theorem for C 1,1 -metrics: Already in the smooth case the proof of Hawking's singularity theorem splits into two distinct parts, namely an analytic bit, which shows relative compactness of D + (Σ), and a part using causality theory. This second part proceeds in the same way whether one deals with smooth or merely C 1,1 metrics, so we will not repeat it here (see e.g. [20,Thm. 14.55A and 14.55B] for the smooth case or [14, Thm. 1.1] for the C 1,1 proof 1 ). Thus we obtain: There seem to be several advantages of this new approach. First, it illustrates the interdependence of the two curvature bounds κ and β very nicely (see conditions (1) to (3) in Thm. 4.2): The parameter β describes the initial focusing (β < 0) or defocusing (β > 0) of geodesics emanating orthogonally to Σ (looking at the comparison manifolds in Table 1 we see that |f κ,β | is initially decreasing if β < 0 and increasing if β > 0 and by the formula for the areas in the proof of (3.9) the same remains true for area κ,β S + A (t)), while κ describes a global focusing (κ > 0) or defocusing (κ < 0) effect for timelike geodesics. Depending on their relative strength there exists a time t = b κ,β where f κ,β becomes zero (and the comparison manifold becomes singular) or not. By the volume comparison Theorem 1.1 (and its application in Thm. 4.5) this time gives a universal bound on the maximal time of existence of geodesics starting orthogonally to Σ in globally hyperbolic manifolds satisfying the respective curvature bounds. While of course this behavior is also present in the Rauchaudhuri argument used in [14] (and for the smooth case in e.g. [22]) and an analogous argument would also suffice to show cases (1) and (3) from Thm. 4.2, it seems that it is somewhat more explicit in the comparison treatment given here.
Second, while the proof of Thm. 1.1 again relies on approximation arguments, the volume comparison result itself now provides a tool which works directly in C 1,1 and allows us to prove other important results (e.g., Thm. 4.2 and Thm. 4.5) without returning to the smooth case.
And, perhaps most importantly, the volume comparison theorem Thm. 1.1 itself is of considerable interest: As already pointed out by the authors of [25], their results are remarkably close to the corresponding Riemannian ones and thus might lend themselves to generalizations of curvature bounds to even lower regularity, a hope that may be strengthened by the C 1,1 version of their volume comparison result ([25, Thm. 9]) proven here.
This statement uses the following definitions (see [ Universal σ-algebra). Let (Ω, A) be a measurable space. Given any finite measure µ we denote the completion of A with respect to µ by A µ . Then the universal σ-algebraÂ is defined aŝ Remark A.5. If µ is a σ-finite measure on (Ω, A) then there exists an equivalent (i.e. having the same zero-measure sets) measureμ that is finite. So one haŝ This shows that any universally measurable set is measurable with respect to every complete σ-finite measure µ on (Ω, A).
This allows us to show the following: Another statement we need concerns itself with the measurability of the graph of a measurable function (and can, e.g., be found in [23,Prop. 3

.1.21])
Proposition A.7. Let (Ω, A) be a measurable space and X a second countable topological space satisfying the T 1 separation axiom (for every pair of distinct points there exists a neighborhood for each that does not contain the other) with Borel-σ-algebra B(X). If f : Ω → X is measurable, then graph(f ) ∈ A ⊗ B(X). Furthermore, if the graph of a function between two σ-finite measure spaces is measurable (and points have measure zero), it has measure zero:
Finally, we will state a result concerning images of sets of measure zero under Lipschitz continuous functions on R n (which can be found, e.g., in [11,Prop. 3.2] for differentiable maps, but the proof only uses the Lipschitz property) that is needed in various proofs of this work. Proposition A.9. Let f : R n → R n be Lipschitz continuous. If A ⊂ R n has (Lebesgue-)measure zero, then f (A) ⊂ R n has (Lebesgue-)measure zero as well.