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On the regularity of Cauchy hypersurfaces and temporal functions in closed cone structures

We complement our work on the causality of upper semi-continuous distributions of cones with some results on Cauchy hypersurfaces. We prove that every locally stably acausal Cauchy hypersurface is stable. Then we prove that the signed distance dS from a spacelike hypersurface S is, in a neighborhood of it, as regular as the hypersurface, and by using this fact we give a proof that every Cauchy hypersurface is the level set of a Cauchy temporal (and steep) function of the same regularity of the hypersurface. We also show that in a globally hyperbolic closed cone structure compact spacelike hypersurfaces with boundary can be extended to Cauchy spacelike hypersurfaces of the same regularity. We end the work with a separation result and a density result.


Introduction
In a recent work [20] we developed the causality theory for "closed cone structures" i.e. upper semi-continuous distributions p → C p of closed sharp nonempty convex cones. This theory developed as a generalization of causality theory for Lorentzian geometry [9,21], the previous works on conic distributions being mostly concerned with the existence of various type of increasing functions [3,8].
Since causality theory is quite an extensive subject the work [20] could only cover some portions of it. Still it was unfortunate that some important topics were left aside. Recently, James Vickers and Günther Hörmann asked me some questions on whether smooth spacelike Cauchy hypersurfaces are stable and whether they are level sets of smooth Cauchy temporal functions. These questions originated in their study of quantum field theory in globally hyperbolic spacetime with non-C 2 metric [10]. A positive answer would generalize a well known theorem by Bernal and Sánchez in Lorentzian geometry [2,Thm. 1.2].
The closely related question on whether globally hyperbolic cone structures admit smooth Cauchy temporal functions was answered affirmatively in [3] and [20] (the same question under continuity was answered in [8]), by using different smoothing techniques. This generalized another result by Bernal and Sánchez [1].
Bernard and Suhr used some smoothing techniques they developed in [3] that differ in philosophy from the approach introduced by Chruściel, Grant and the author [6], later refined in [20]. Our approach consisted in first showing, through topological arguments, that the causal cones can be widened while preserving some key causality properties (e.g. global hyperbolicity or the Cauchy property of the hypersurface). This first step makes the 'room' needed for the subsequent step which consists in the smoothing of a suitable continuous increasing function. On the contrary Bernard and Suhr work out the smoothing first, a step which is more technically demanding, and then infer the stability properties more easily, but a posteriori, thanks to the obtained smoothness.
In a recent preprint Bernard and Suhr [5] show that the generalization of [2, Thm. 1.2] can indeed be obtained via their smoothing technique, namely smooth spacelike Cauchy hypersurfaces are indeed level sets of smooth Cauchy temporal functions. It is natural to ask whether our approach [20] can also be used to answer this question. In this work we prove that it can and, as it turns out, the proof is conceptually simple as it consists in the single application of our smoothing theorem [20,Thm. 3.9] to a signed distance function from the Cauchy hypersurface, cf. Thm. 2.14. Our analysis clarifies that in the nonsmooth case the Cauchy temporal function can be found of the same regularity of the hypersurface.
We start with the next section in which we prove the equivalence between the locally stably acausal Cauchy hypersurfaces and the stable Cauchy hypersurfaces (Thm. 2.5). This result, being propaedeutic for the proof of the other results, is also quite interesting because it applies to general Lipschitz hypersurfaces, not just to smooth ones. Then we introduce the notion of signed distance and obtain a result on its regularity (Thm. 2.11) that is analogous to a not so well known result in Riemannian geometry. We show how to obtain the Cauchy temporal function adapted to the Cauchy hypersurface (Thm. 2.14). Next, we prove a generalization of [2, Thm. 1.1] namely that in a globally hyperbolic closed cone structure compact spacelike hypersurfaces with boundary can be extended to Cauchy spacelike hypersurfaces of the same regularity (Thm. 2.17). Finally, we prove a smooth separation result (Thm. 2.20) and a density result (Thm. 2.21) for stably causal closed cone structures.
Our conventions, terminology and notations are the same of [20]. The reader is assumed to be familiar with that work, particularly with the definition of closed (and proper) cone structure, and with the definition of closed (and proper) Lorentz-Finsler space.

Stability, signed distance and smoothing
We start with the topological proof that cones can be widened preserving the Cauchy property. In this way we shall make 'the room' needed for the subsequent smoothing by convolution.

Stability and Cauchy hypersurfaces
We need a notion that replaces that of spacelikeness in a rough setting.
Definition 2.1. Let (M, C) be a closed cone structure. A topological hypersurface S is locally stably acausal if every point p ∈ S admits an open neighborhood U and a locally Lipschitz proper cone structure C > C on U , such that S ∩ U is acausal in (U, C ).
On a closed cone structure (M, C) a C 1 hypersurface S is spacelike if for every p ∈ S, T p S ∩ C p = ∅ (remember that C p does not include the zero vector). The C 1 spacelike hypersurfaces are the simplest examples of locally stably acausal hypersurfaces.
We recall some definitions from [20]. Notice that if C is a closed cone structure and C is a locally Lipschitz proper cone structure, then we can find an intermediate locally Lipschitz proper cone structure C , C < C < C , cf. [20,Thm. 2.23]. As a consequence, if S is a stable Cauchy hypersurface for (M, C) then we can find a locally Lipschitz proper cone structure C > C, such that S is a stable Cauchy hypersurface for (M, C ).
Our first objective is to show that the locally stably acausal Cauchy hypersurfaces coincide with the stable Cauchy hypersurfaces.
By definition a closed cone structure is globally hyperbolic if it is acausal and the causally convex hull of compact sets is compact [11] (this definition improves the previous ones discussed in [3,20] there is a past inextendible continuous causal curve η ending at p and not intersecting S. There must exist a compact neighborhood K p such that every future inextendible continuous causal curve issued from K reaches S. Indeed, suppose not, then we can find a sequence of future inextendible continuous causal curves σ n with staring point p n → p, not intersecting S. By the limit curve theorem there is a limit future inextendible continuous causal curve σ to which a subsequence (here denoted in the same way) σ n converges (uniformly on compact subsets for a suitable parametrization, see details in [20,Thm. 2.14]). If σ intersects (and hence crosses) S then so do σ n for sufficiently large n (due to [20,Thm. 2.33]), which is a contradiction. If σ does not intersect S, then σ • η provides an inextendible continuous causal curve passing through p not intersecting S, a contradiction. Thus K exists, but then since p ∈ J + (S) we can find q ∈ K ∩ J + (S). Chosen any future inextendible continuous causal curve starting from q (it exists [20, Thm. 2.1]), it intersects S, which proves that S is not acausal. The contradiction proves that J + (S) is closed.
Since through every point of S passes an inextendible continuous causal curve the sets J ± (S)\S are indeed non-empty thus M \S is indeed disconnected as stated.
The next result was almost completely proved in [20,Thm. 2.41], where it was shown that the cones can be widened preserving the property that every inextendible continuous causal curve intersects S. Unfortunately, the proof that S was acausal was missing. Example 2.6. Not all Cauchy hypersurfaces are locally stably acausal consider Minkowski 1+1 spacetime of metric g = −dt 2 +dx 2 and the Cauchy hypersurface S given by the graph of the function t = tanh x. The hypersurface S is not locally stably acausal in a neighborhood of (0, 0). Notice that S is C 1 but the tangent space at the origin is null. Proposition 2.22 in [20] now simplifies as follows Proposition 2.7. Let (M, C) be a closed cone structure. Any two locally stably acausal C k , 0 ≤ k ≤ ∞, Cauchy hypersurfaces are C k diffeomorphic.

The signed distance function and its regularity
The proof of the next result is similar to the Lorentzian one. Proof. Otherwise there is a sequence of continuous casual curves σ n connecting S to K and points p n ∈ σ n , p n → ∞ (i.e. escaping every compact set). A standard limit curve argument [20, Thm. 2.14] would produce a past inextendible continuous causal curve σ ending at K. Thus by the stable local acausality of S it would cross S entering, [20,Thm. 2.33], the open set J − (S)\S. But then a subsequence σ k → σ starting from S, would have to enter it, contradicting the acausality of S. The function d S is called the signed (Lorentz-Finsler) distance function.
Proof. By the upper semi-continuity of the length functional [20, Thm. 2.54] through a standard limit curve argument one gets that the distance is realized, namely there is a continuous causal curve connecting S to p whose Lorentz-Finsler length coincides the distance d(S, p). In particular d + S is finite. The same argument shows that d + S is upper semi-continuous. For the lower semicontinuity one has to use the locally Lipschitz property of the cone distribution. The proof is analogous to that of [20,Thm. 2.53].
The next result proves that the signed distance is a rushing time in the sense of [21]. This anti-Lipschitz property will be of fundamental importance for the application of our smoothing theorem.
Proof. If p and q belong to J + (S), then d S = d + S = d(S, ·), and we have for gives what we wanted to prove. For p ∈ J − (S)\S and q ∈ J + (S)\S consider the maximizing continuous causal curve σ connecting p to q and let r be its intersection point with S. Then Let C be a smooth distribution of proper sharp strongly convex closed cones. Let us consider a positive homogeneous function F : C → [0, +∞) such that F −1 (0) = ∂C, L := −F 2 /2 is such that the vertical Hessian on C, d 2 y L , is smooth and Lorentzian, and dL = ∅ on ∂C. This function exists by [19,Prop. 13], see also [12,Cor. 5.8]. With these properties (M, F ) is certainly a locally Lipschitz proper Lorentz-Finsler space [20, Thm. 2.52] so the previous results hold true.
We want to know how the regularity C k of a spacelike Cauchy hypersurface S reflects itself on the regularity of the signed distance d S . For this reason we are considering a quite regular Lorentz-Finsler structure, in this way all the non-differentiability aspects of d S will be due to the hypersurface. Notice that under this regularity the timelike geodesics make sense and they coincide with the locally maximizing continuous causal curve with non-vanishing length [17,Thm. 6] (the formalism of [17] can be applied because the Finsler Lagrangian can be extended to all of T M \0, see [18]). Thus the maximal curves that realize the distance d + S or −d − S are timelike geodesics normal to S (all our continuous causal curves are future-directed).
Here normality has to be understood as follows. A vector v is normal to S if it can be written in the form v = rn, where r ∈ R and n is a (future-directed) normalized timelike vector n ∈ IntC (necessarily unique) such that F (n) = 1, T S = kerg n (n, ·).
The normal bundle N ⊂ T M can be identified with the image of a map , is smooth because the spray derived from the Finsler Lagrangian is smooth. Thus the map S × R → T M , (p, t) → γ np (t) is C k−1 as it is the composition of a C k−1 map and of a smooth map.
Notice that γ n (r) = π • γ n (r), where π : T M → M . It is well known that the map (n, r) → γ n (r) is C k−1 and locally invertible (with the identification of S × R with N this map is basically exp S ), in the sense that there is an open neighborhood V of the zero section of S × R and an open neighborhood U of S that are C k−1 diffeomorphic according to the map. The regularity of the map that assigns to q ∈ U , q = π(γ n (r)) the vector w q = γ n (r) ∈ T q M is C k−1 because one has first to apply the C k−1 inverse of the map (n, r) → γ n (r) to find (n, r) and then to apply the C k−1 map (n, r) → γ n (r) to find w q .
The next theorem generalizes a well known result on the regularity of the signed distance function in Riemannian spaces [13,15].
Theorem 2.11. Let (M, F ) be a sufficiently regular proper Lorentz-Finsler space as described above. If S is a C 1,1 or a C k , 2 ≤ k ≤ ∞, spacelike Cauchy hypersuface, then d S has the same regularity in a neighborhood U of S.
Weaker regularity options for the hypersurface could be considered, but one would need to introduce the notion of spacelike hypersurface of positive reach.
Proof. This fact follows from the previous discussion and from the next formula (analogous to Eq. (13) in [17], see also Remark 7 in the same reference). At Since q → w q is C k−1 and non-zero, and since g (or dL ) is smooth over the slit tangent bundle we have that d S is C k (in the C 1,1 case the argument is similar but exp S really establishes a lipeomorphism not a diffeomorphism [17, Thm. 13] [14]).

Smoothing signed distance functions
We recall our smoothing theorem [20, Thm. 3.9] [6] which we slightly improve by adding the penultimate statement.
Theorem 2.12. Let (M, C) be a closed cone structure and let τ : M → R be a continuous function. Suppose that there is a C 0 proper cone structureĈ > C and continuous functions homogeneous of degree one in the fiber F , F :Ĉ → R such that for everyĈ-timelike curve x : Let h be an arbitrary Riemannian metric, then for every continuous function α : M → (0, +∞) there exists a smooth functionτ such that |τ − τ | < α and for Let A be a closed set and G ⊃ A an open neighborhood. If τ is already C 1 and satisfying (2) on G thenτ can be found such that it is C 1 and (2) holds everywhere, |τ − τ | < α on M ,τ coincides with τ on A,τ has the regularity of τ on G, and is smooth outside G. Similar versions, in which some of the functions F , F do not exist hold true. One has just to drop the corresponding inequalities in (2).
Proof. We added just the penultimate sentence with respect to the previous version. In order to prove it we modify the original proof by taking, for p / ∈ A, (p) so small that B p (3 (p)) ∩ A = ∅, while for p ∈ G, (p) so small that B p (3 (p)) ⊂ G. Recall that in the original proof by using σ-compactness we pass to a locally finite covering of M given by the open sets O i = B pi ( i ), i = (p i ), we introduce a partition of unity ϕ i subordinate to the covering {O i }, in O i we smooth τ getting a function τ i , and defineτ = i ϕ i τ i . Instead of always constructing τ i through convolution with a smooth kernel χ, it is sufficient to set τ i = τ whenever p i ∈ G, while define τ i as in the original proof for p i / ∈ G. Notice that each τ i is at least as regular as τ , so the same can be said ofτ . Given the support of ϕ i ,τ is smooth outside G and coincides with τ on A. The calculation on p. 110 of the original proof easily pass through with this modification, so Eq. (2) does indeed hold everywhere.
The next results hold also for closed cone structures (M, C) just set F = 0. They are obtained smoothing the signed distance function d S for a suitable F > F .
The first result is well known [20, Thm. 2.45,3.12] [3,8]. Compared to the proof in [20] here we use the signed distance to produce the anti-Lipschitz function to be smoothed, while there we made use of a product trick (which gave also information on stable spacetimes). We include this proof because it shows the effectiveness of the smoothing theorem coupled with the signed distance idea. Let F be chosen such that F (∂C ) = 0, and sufficiently large so that F > 2F on C, and such that F −1 (1) ∩ C, is contained in the unit ball of 4h. As a consequence, for every v ∈ C, 2 v h ≤ F (v).
Given a Cauchy hypersurface S for (M, C ), the function t : M → R, t = d S , satisfies over every C -timelike curve x : By Theorem 2.12 there is a smooth function τ , |τ − t| < 1, that satisfies dτ The Cauchy property follows from h-steepness and the fact that h is complete. Proof. The hypersurface S is locally stably acausal and hence a stable Cauchy hypersurface. Let C be a smooth strongly convex proper cone structure such that C > C is globally hyperbolic and S is a stable Cauchy hypersurface for C . As mentioned in the previous section we can find a smooth positive homogeneous functionF : C → [0, +∞) such thatF (∂C ) = 0,L := −F 2 /2 has Lorentzian vertical Hessian on IntC , and dL = ∅ on ∂C , so that (M,F ) is a locally Lipschitz proper Lorentz-Finsler space. Let F = ϕF and choose the factor ϕ : M → (0, +∞) so that F is so large that F > 2F on C, and such that F −1 (1) ∩ C is contained in the unit ball of 4h. As a consequence, for every v ∈ C, 2 v h ≤ F (v). Notice that F retains all the mentioned properties of F . The function t : M → R, t = d S , satisfies Eq. (3) over every C -timelike curve x : [0, 1] → M . By Thm. 2.11 there is a neighborhood U of S such that t| U has the same regularity of S. In particular, t| U is C 1 thus using the inequality (3) we have on U , (just consider a F -geodesic with tangent v) dt(v) ≥ F (v) for every C -timelike vector v, and hence the inequality holds for every v ∈ C. We conclude that dt(v) ≥ F (v) − v h for every v ∈ C| U . Let A, V , be closed and open neighborhoods of S, respectively, such that A ⊂ V ⊂ U . Let α : M → R be a positive continuous function such that α| M \A < |t| M \A |. By Theorem 2.12 there is a continuous function τ , |τ − t| < α, that coincides with t on A, has the regularity of t and hence of S on V , is smooth outside V , and satisfies The Cauchy property follows from h-steepness and the fact that h is complete. Finally, since τ coincides with t on A, and has the same sign of t outside A, we have τ −1 (0) = S.
The next result is analogous to the previous one for rough S. Proof. Let (M, F ) be any locally Lipschitz proper Lorentz-Finsler space such that C > C is globally hyperbolic and S is a stable Cauchy hypersurface for C . Moreover let F be chosen such that F (∂C ) = 0, and sufficiently large so that F > 2F on C, and such that F −1 (1) ∩ C, is contained in the unit ball of 4h. As a consequence, for every Similarly, we obtain τ − and then the function τ equal to zero on S, to τ + on J + (S)\S and to τ − on J − (S)\S has the desired properties (the Cauchy property follows from h-steepness and the fact that h is complete).
A stable Cauchy time function for a closed cone structure (M, C) is a function which is a Cauchy time function for some locally Lipschitz proper cone structure C > C, and hence for C. Let t be a stable Cauchy time function, then there is a locally Lipschitz proper cone structure C > C such that t is a Cauchy time function for C . In particular, t −1 (0) is a Cauchy hypersurface for C . Thus the zero level set of a stable Cauchy time function is a stable Cauchy hypersurface.
In the next result by C k closed acausal spacelike hypersurface with boundary we understand an acausal spacelike hypersurface S which is a closed set for the topology of M (and hence includes its edge) and can be slightly extended to an acausal spacelike hypersurfaceS ⊃ S, which is a C k submanifold, does not comprise its edge, and is such that edgeS ∩ edgeS = ∅. In other words S can be slightly enlarged so as to comprise the previous edge in the new relative interior points but maintaining the causality and differentiability properties. This result holds also for Lipschitz regularity of S, provided S is locally stably acausal (rather than spacelike) and provided it can be slightly extended over the edge preserving its causality properties. The proof is obtained through a minimal modification of the next one.
Proof. LetŜ, S ⊂Ŝ ⊂ S be a compact acausal spacelike hypersurface with boundary, such that edge S ∩edgeŜ = ∅. By the stability of global hyperbolicity we can find a locally Lipschitz proper cone structureC > C such that (M,C) is globally hyperbolic. Moreover, we can find a locally Lipschitz proper cone structure C , C < C <C so narrow thatŜ is spacelike for C . Actually, C can be chosen so narrow thatŜ is acausal for C . Indeed, let us consider a sequence of locally Lipschitz proper cone structures C < C n <C, such thatŜ is spacelike for every C n and C n → C pointwise. Let H be a stable Cauchy hypersurface forC such thatŜ ∩J − (H) = ∅. If the statement were false we could find a sequence of continuous C n -causal curves σ n starting and ending at S. The curves cannot contract to a point (due to [20,Thm. 2.33] applied tõ S), thus by the limit curve theorem and the acausality ofŜ in (M, C), there is a past inextendibile limit continuous C-causal curve σ q , ending at some point q ∈Ŝ. But then σ q intersects H and hence enters J − (H) ⊂J − (H) and so do all σ n for sufficiently large n, a contradiction withJ + (Ŝ) ∩ H = ∅.
So let C > C be as established, and let us choose it smooth. Let h be a complete Riemannian metric. Moreover, let F , F −1 (0) = ∂C , be a Lorentz-Finsler fundamental function, as regular as it is described in Sec. 2.2 and with F so large that F −1 (1) ∩ C is contained in the unit ball of 4h. As a consequence, for every v ∈ C, 2 v h ≤ F (v). Let H be a stable Cauchy hypersurface for C such thatŜ ∩ J − (H) = ∅. SinceŜ is compact, the set J + (H) ∩ J −1 (Ŝ) is compact and hence cannot future imprison any continuous C -causal curve (remember that (M, C ) is globally hyperbolic hence non-total imprisoning). It follows that L = ∂[J − (H) ∪ J − (Ŝ)] is a Cauchy hypersurface for (M, C ) such thatŜ ⊂ L, due to the C -acausality ofŜ.
The function t : M → R, t = d L , satisfies Eq. (3) over every C -timelike curve x : [0, 1] → M . Let U ⊂ D(Ŝ\edgeŜ) be a neighborhood of S so small that t| U = d Ŝ \edgeŜ | U has the same regularity ofŜ\edgeŜ and hence of S, cf.
Thm. 2.11. In particular, t| U is C 1 thus using the inequality (3) we have on U , (just consider a F -geodesic with tangent v) dt(v) ≥ F (v) for every Ctimelike vector v, and hence the inequality holds for every v ∈ C. We conclude be closed and open neighborhoods of S, respectively. Let α : M → R be a positive continuous function. By Theorem 2.12 there is a continuous function τ , |τ − t| < α, that coincides with t on A, has the regularity of t and hence of S on V , is smooth outside V , and satisfies thus the function τ is Cauchy due to the h-steepness and the fact that h is complete. Notice that since τ coincides with t on A, Σ := τ −1 (0) ⊃ S, and Σ being the level set of a C k (smooth outside V ) Cauchy temporal function is a C k Cauchy hypersurface (smooth outside V ).

Smooth separation and density theorems
The sets introduced with the next definition are called trapping domains in the terminology of [3]. Let ϕ : M → [0, 1] be a smooth temporal function for (M,Ĉ). Let > 0. The function ϕ is also smooth temporal, thus there is a Riemannian metric g such that ϕ is g-steep on (M,Ĉ), just take the unit balls of g so large on T M that they contain (d ϕ) −1 (1) ∩Ĉ. The functionτ = τ + ϕ is g-anti Lipschitz on (M,Ĉ), since the term ϕ has this property (this trick, due to Fathi, was used in [6] and [4]). Thusτ can be smoothed using Thm. 2.12 with F (v) = v g , h = g/4, α = . We can find a smooth temporal (for C) function τ , |τ −τ | < , and hence |τ − τ | < 2 . Let us consider a smooth function φ : R → [0, 1] which is 0 on (−∞, 2 ], and 1 on [1 − 2 , +∞) and such that φ > 0 elsewhere, ([1 − 4 , 1]). Notice that φ •τ is temporal wherever it differs from 0 and 1. Let f k = φ 1/k •τ , and observe that for every p ∈ M \(A ∪ B) we can find a sufficiently large k such that 4/k < τ (p) < 1 − 4/k, so that 0 < f k (p) < 1 and hence f k is temporal at p. There are constants c k > 0 such that, k c k = 1 and t = k c k f k is smooth [7,Lemma 3.2], which concludes the proof.
Let > 0 and let ϕ : M → (0, 1) be a time function for (M, C), thenτ := τ + ϕ is also a time function. Let h be a complete Riemannian metric such that By [20,Lemma 3.5] we can find a locally Lipschitz proper cone structureC > C such that the openC-future set V =Ĩ + (τ −1 (2 , +∞)) is disjoint from Notice thatÃ := M \V is a closedC-past set such that In other words,Ã is a stable past set for (M, C). Dually,C > C can be further narrowed so that the openC-past set U =Ĩ − (τ −1 (−∞, 1 − 2 )) is disjoint from τ −1 ([1 − , +∞)) ⊃ B. Notice thatB = M \U is a closedC-future set such that In other words,B is a stable future set for (M, C). By Thm. 2.19 there is a smooth isotone function t : M → [0, 1], temporal on M \(Ã ∪B), such that t −1 (0) =Ã and t −1 (1) =B. There are constant c k > 0 such that k c k = 1 and t = k c k t 1/k is smooth, which concludes the proof.
We recall [20,Thm. 3.14] that if t is a time function for a closed cone structure we have: (p, q) ∈ J S \∆ ⇒ t(p) < t(q). The next density result improves the bound in [4,Cor. 9]. Interestingly, our proof is completely different.
Theorem 2.21. Let t be a continuous isotone function on the stably causal closed cone structure (M, C) and let : R → (0, +∞) be a smooth positive function. Then there is a smooth temporal function τ such that for every p ∈ M , |τ (p) − t(p)| < (t(p)).
Clearly, if t is Cauchy and is chosen bounded then τ is Cauchy.
Under stable causality we can find temporal functions adapted to stably acausal spacelike hypersurfaces.
Theorem 2.22. Let (M, C) be a stably causal closed cone structure. Let S be a C 1,1 or a C k , 2 ≤ k ≤ ∞ stably acausal spacelike hypersurface without edge (hence a partial Cauchy hypersurface), then there is a smooth temporal function t of the same regularity such that S ⊂ t −1 (0).
The stable acausality condition cannot be deduced from the other assumptions and is necessary (because the cones can be widened preserving the temporality of t).
Proof. LetC > C be a stably causal locally Lipschitz proper cone structure such that S is acausal forC. LetĈ, C <Ĉ <C be an intermediate locally Lipschitz proper cone structure. Since S isC-achronal there is a maximalCachronal set Σ, M = Σ ∪Ĩ + (Σ) ∪Ĩ − (Σ). The removal of Σ disconnects the spacetime. TheC-achronality of Σ implies that it isĈ-acausal [20,Thm. 2.24]. The maximal achronality implies that it does not have edge and that it is a topological hypersuface (see also [21,Thm. 2.87]). On the globally hyperbolic spacetimeD(Σ) we consider a signed distance d Σ for some locally Lipschitz proper Lorentz-Finsler structure withĈ < C <C. Then we proceed as in the proof of Thm. 2.14 but with A, V , U , neighborhoods of S (rather than Σ) all contained in a neighborhood of S where d S is C k . Thus we obtain a C k Cauchy temporal function τ such that τ −1 (0) ⊃ S. Let σ(x), be a smooth non-decreasing function equal to x in a neighborhood of 0, equal to 1 for x ≥ 1, and equal to −1 for x ≤ −1. The function σ • τ can be extended all over M to a C k isotone function by setting σ • τ = ±1 inĨ ± (Σ)\D(Σ). The set P := {q : τ (q) ≤ 0} is a stably past set thus by Thm. 2.19 there is a smooth function τ + , temporal on M \P , which vanishes on P . Similarly there is a smooth function τ − that vanishes on the stable future set F := {q : τ (q) ≥ 0} and is temporal in M \F . The function t = τ + τ + + τ − is the desired smooth temporal function.
By using the same set of ideas it is possible to prove the next result.
Theorem 2.23. Let (M, C) be a stably causal closed cone structure. Let S 0 and S 1 be two C 1,1 or two C k , 2 ≤ k ≤ ∞, stably acausal spacelike hypersurfaces without edge (hence partial Cauchy hypersurfaces), such that S 1 ⊂ J + S (S 0 )\S 0 . Then there is a smooth temporal function t of the same regularity such that S 0 ⊂ t −1 (0) and S 1 ⊂ t −1 (1).
In the proof one has to sum five smooth isotone functions which are temporal on suitable open sets.

Conclusions
By using an approach complementary to that proposed by Bernard and Suhr [5], we generalized the results by Bernal and Sánchez [2] to the framework of closed cone structures. Preliminarily we showed that the spacelike Cauchy hypersurfaces are stable and that the signed distance d S from a spacelike hypersurface S is, in a neighborhood of it, as regular as the hypersurface. Then we applied our smoothing theorem to the signed distance function of a Lorentz-Finsler structure having wider cones. The statements derived from our approach are rather detailed, in fact they are useful already in the Lorentzian metric theory, and the proofs are conceptually rather straightforward. Furthermore, they nicely pass through concepts, such as the signed distance, that deserved to be investigated.