aa r X i v : . [ m a t h - ph ] J a n A NOTE ON THE GANNON-LEE THEOREM
BENEDICT SCHINNERLROLAND STEINBAUER
University of Vienna, Faculty of MathematicsOskar-Morgenstern-Platz 1, A-1090 Wien, Austria
Abstract.
We prove a Gannon-Lee theorem for strongly causal and null pseu-doconvex Lorentzian metrics of regularity C . Thereby we generalize earlierresults to the non-globally hyperbolic setting in a natural way, while at thesame time lowering the regularity to the most general class currently availablein the context of the classical singularity theorems. Along the way we also provethat any maximizing causal curve in a C -spacetime is a geodesic and hence of C -regularity. Introduction
In the mid 1970-ies, several years after the appearance of the singularity theo-rems of Penrose and Hawking (see e.g. [HE73, Ch. 8]), D. Gannon [Gan75, Gan76]and C.W. Lee [Lee76] independently derived a body of results that relate the sin-gularities of a Lorentzian manifold to its topology. More precisely, in these results,often dubbed Gannon-Lee theorem(s), they established that under an appropri-ate asymptotic flatness assumption, a non-trivial fundamental group of a (partial)Cauchy surface Σ necessarily leads to the existence of incomplete causal geodesics.Most of these early results assumed global hyperbolicity, with the notable excep-tion of [Gan75, Thms. 2.1-2]. However, the proofs relied on the false deduction thatmaximizing geodesics in a covering spacetime project to maximizing geodesics, aspointed out in [Gal83]. In the same paper Galloway proved a Gannon-Lee theoremwithout assuming global hyperbolicity by invoking the Hawking-Penrose theoremand heavily using a result from geometric measure theory. The latter fact wasalso reflected in the formulation of the main theorem, which assumed an extrinsiccondition on the three surface Σ and the topological condition was that Σ is not a
E-mail addresses : [email protected], [email protected] . Date : January 12, 2021. handlebody . A related recent result for the globally hyperbolic case in a settingcompatible with a positive cosmological constant was given in [GL18a], providinga precise connection between the topology of a future expanding compact Cauchysurface and the existence of past singularities.Another issue with the earlier results was that the nontrivial topology was con-fined to a compact region of a (partial) Cauchy surface bounded by a topological2-sphere S . In the context of topological censorship [FSW95], S is naturally in-terpreted as a section of an event horizon and in four dimensions the S topologyis only natural in the light of Hawking’s black hole topology theorem [HE73, Sec.9.2]. However, since the latter fails to hold in higher dimensions, with more com-plicated horizon topologies occurring (see [ER08], but [GS06] for correspondingrestrictions), the demand for higher dimensional Gannon-Lee type results withmore natural assumptions on the topology of S arises. Such a result was indeedgiven by Costa e Silva in [CeS10] which also avoided the assumption of globalhyperbolicity. More precisely the results was given for causally simple spacetimes,i.e. causal spacetimes where the causality relation is closed. However, we haverecently discovered that this proof relies on an analogous false deduction: It usesthat causal simplicity lifts to coverings, which is not true in general, as was explic-itly shown in [MS20]. In the latter paper Minguzzi and Costa e Silva also presenta corrected result, which, however makes an assumption on the causality of thecovering spacetime.In this paper we extend the validity of the Gannon-Lee theorems in two respects.We provide a result that, on the one hand, avoids the assumption of globally hyper-bolic without making assumptions on the causality of the covering manifold. It isbased on the notion of null pseudoconvexity, which can be placed on the causal lad-der between causal simplicity and global hyperbolicity when, additionally, strongcausality is assumed. On the other hand, our result applies to Lorentzian metricsof low regularity, in particlular to (certain) C -spacetimes. Having low regularitysingularity theorems at hand is especially favorable in the context of extendingspacetimes, as already noted in [HE73, Ch. 8]. In particular, they rule out thepossibility to extend the spacetime to a complete one, even under mild regularityassumptions on the metric. Related results on C - and Lipschitz non-exendabilityhave recently been given in [Sbi18, GLS18, CK18, Sbi20].Indeed during the last couple of years the classical singularity theorems ofPenrose, Hawking and Hawking-Penrose have been extended to C , -regularityin [KSSV15], [KSV15], and [GGKS18]. These results built upon extensions ofLorentzian causality theory to low regularity [CG12, KSSV14, Min15, S¨a16]. Mostrecently, Graf in [Gra20] was able to further lower the regularity assumptions forthe Penrose and the Hawking theorem to C . We will extend her recent techniques,in particular to the non-globally hyperbolic setting, to prove our result. Under these conditions [MY82] guarantees the existence of a trapped surface within Σ.
NOTE ON THE GANNON-LEE THEOREM 3
This note is organized in the following way. In the next section we discusspreliminaries for our work, especially the intricacies arising in C -regularity andwe state our results. In section 3 we lay the analytical foundations of the proofof the main theorem. In particular, we will show that any causal maximizer in a C -spacetime is a geodesic and hence a C -curve. In section 4 we will be concernedwith causality theory and in particular, with pseudoconvex spacetimes. We relatepseudoconvexity to limiting properties of geodesics in C -spacetimes and also provethat under the assumption of strong causality, maximal null pesudoconvexity isequivalent to causal simplicity. Finally in section 5 we provide new focussingresults for null geodesics and collect all our results together to prove the maintheorem. 2. Preliminaries and results
We begin by introducing our main notations and conventions, as well as somebasic notions that are necessary to give a precise formulation of our results. Ourmain reference for all matters of Lorentzian geometry is [O’N83].We will assume that all manifolds M are smooth, Hausdorff, as well as secondcountable and of dimension n with n ě
3. We will consider Lorentzian metrics g on M that are of regularity at least C with signature p´ , ` , . . . , `q . Anotherimportant regularity class is g P C , which means that g is C and its firstderivatives are locally Lipschitz continuous. A spacetime is a Lorentzian manifoldwith a time orientation, which we assume to be induced by a smooth vector field.Throughout we will fix a complete Riemannian background metric h and use itsinduced norm } } h and distance d h . All local estimates will be independent of thechoice of h .We will denote the fundamental group of a manifold M by π p M q . Also if i : N Ñ M is a continuous map, the induced homomorphism of the fundamentalgroups is denoted by i : π p N q Ñ π p M q .2.1. Causality theory.
A curve γ : I Ñ M defined on some interval I is calledtimelike, causal, null, future or past directed, or spacelike, if γ is locally Lipschitzcontinuous and its velocity vector γ which exists Lebesgue almost everywhere hasthe respective property almost everywhere. We denote the timelike and causalrelation by p ! q and p ď q , respectively and write I ` p A q and J ` p A q for thechronological and causal future of a set A Ď M . Finally we denote the futurehorismos of A by E ` p A q : “ J ` p A qz I ` p A q . The respective past versions of thesesets will be denoted by I ´ , J ´ and E ´ , respectively. When we refer to these setswith regard to a particular metric g , it will appear in subscript, e.g. E ` g p A q denotesthe future horismos of A w.r.t. g .For Lorentzian metrics g , g one says that g has narrower lightcones than g (respectively, g has wider lightcones than g ), noted as g ă g , if g p X, X q ď g p X, X q ă “ X P T M . A NOTE ON THE GANNON-LEE THEOREM
One notion, central for our arguments is null pseudoconvexity which was in-troduced in [BP87]: Null curves connecting points in a compact set are entirelycontained in a larger compcact set. This property was used as a replacement forglobal hyperbolicity in the context of wave equations. Later in [BK92], some re-lations to causality conditions were proved. An essential aspect will be that in astrongly causal, null pseudoconvex spacetime a sequence of null geodesics with con-verging endpoints possesses a subsequence converging to a null geodesic connectingthe limits of the endpoints of the sequence. Further, in strongly causal spacetimesnull pseudoconvexity can be placed on the causal ladder in between causal sim-plicity and global hyperbolicity, as we will show even for certain C -spacetimes insection 4.2.2. Low regularity.
During the last couple of years the bulk of Lorentziancausality theory has been transferred to C , -spacetimes, where the exponentialmap and convex neighbourhoods are still available. While convexity fails belowthat regularity [HW51, SS18], nevertheless most aspects of causality theory canbe maintained even under Lipschitz regularity of the metric. Further below somesignificant changes occur [CG12, GKSS20], while some robust features continue tohold even in more general settings [Min19, KS18, BS18, GKS19].Here we will discuss some properties of C -spactimes. As compared to the C , -setting one loses two essential features: uniqueness of solutions of the geodesicequation and the local boundedness of the curvature tensor. Given the first fact,one has to make a choice concerning the definition of geodesic completeness. Wewill follow the natural approach of [Gra20] and define a spacetime to be time-like (respectively null or causal) geodesically complete if all inextendible timelike(respectively null or causal) geodesics are defined on R .Concerning the second issue, first note that C is well within the maximal classof spacetimes allowing for a (stable definition of) distributional curvature, whichis g locally in H X L [GT87, LM07, SV09]. So we may define the Riemann andRicci tensor by the usual coordinate formulaeRiem mijk : “ B j Γ mik ´ B k Γ mij ` Γ mjs Γ sik ´ Γ mks Γ sij , (1)Ric ij : “ B m Γ mij ´ B j Γ mim ` Γ mij Γ kkm ´ Γ mik Γ kjm . (2)The Riemann and the Ricci tensor are then tensor distributions in D T p M q and D T p M q , respectively, where we recall that D T rs p M q : “ Γ c p M, T sr M b Vol p M qq “ D p M q b C T rs p M q . (3) The alternative would be to only demand the existence of one complete geodesic for everytimelike (or null or causal) initial data to the geodesic equation.
NOTE ON THE GANNON-LEE THEOREM 5
Here Vol p M q is the volume bundle over M , Γ c denotes spaces of sections with com-pact support and D p M q is the space of scalar distributions on M , i.e. the topologi-cal dual space of the space of compactly supported volume densities Γ c p M, Vol p M qq .For all details on tensor distributions see [GKOS01, Ch. 3.1].Naturally, we define curvature bounds resp. energy conditions using the notionof positivity for distributions, i.e. D p M q Q u ě u ą
0) if x u, µ y ě ą
0) for allnon-negative (positive) volume densities µ P Γ c p M, Vol p M qq . Then p M, g q is saidto satisfiy the strong energy condition (resp. to have non-negative Ricci curvature)if the scalar distribution Ric p X , X q is non-negative for all smooth timelike vectorfields X . In the case of g being smooth this condition coincides with the classicalone, Ric p X, X q ě X P T p M and all p P M by the fact that allsuch X can be extended to smooth timelike vector fields on M . For the samereason the condition for g P C , is equivalent to the condition Ric p X , X q ě X , used in the context ofthe C , -singularity theorems. However, to generalize the null energy condition ismore tricky due to the obstacles one encounters when extending null vectors. Fora detailed discussion see [Gra20, Sec. 5], and we define following her: Definition 2.1 (Distributional null energy condition) . A C -metric g satisfies the distributional null energy condition , if for any compact set K Ď M and any δ ą ε p δ, K q such that Ric p X , X q ą ´ δ (in the sense of distributions) forany local smooth vector field X P X p U q , U Ď K with } X } h “ ε p δ, K q close to a C g -null vector field N on U , i.e. || X ´ N || h ă ε p δ, K q on U .Again this condition is equivalent to the classical null energy condition if themetric is smooth. Moreover, in case g P C , it is equivalent to the condition usedin the C , -setting i.e. Ric p X , X q ą X .One key technique in low regularity Lorentzian geometry is regularization. Morespecifically Chrusciel and Grant in [CG12] have put forward a technique to regu-larize a continuous metric g by smooth metrics ˇ g ε with narrower lightcones resp.by a net ˆ g ε with wider lightcones than g . The basic operation (denoted by ˚ ) ischartwise convolution with a standard molifier ρ ε , which is globalized using cut-offsand a partition of unity, cf. [GKOS01, Thm. 3.2.10]. To manipulate the lightconesin the desired way one has to add a “spacelike correction term”. The most recentversion of this construction which also quantifies the rate of convergence in termsof ε is [Gra20, Lem. 4.2], which we recall here. Lemma 2.2.
Let p M, g q be a spacetime with a C -Lorentzian metric. Then forany ε ą , there exist smooth Lorentzian metrics ˇ g ε and ˆ g ε with ˇ g ε ă g ă ˆ g ε , bothconverging to g in C loc . Additionally, on any compact set K , for all small ε } ˇ g ε ´ g ˚ ρ ε } ,K ď c K ε and } ˆ g ε ´ g ˚ ρ ε } ,K ď c K ε . (4) A NOTE ON THE GANNON-LEE THEOREM
A main step in the proof of singularity theorems in low regularity is to showthat the energy condition (Definition 2.1, in our case) implies that the regularizedmetrics ˆ g ε and/or ˇ g ε violate the classical energy conditions (the NEC, in our case)only by a small amount — small enough, such that null geodesics still tend tofocus. Technically this is done by a Friedrich-type lemma, which in the presentcase is [Gra20, Lem. 4.5], and draws essentially from (4). The corresponding resultis then [Gra20, Lem. 5.5]: Lemma 2.3 (Surrogate energy condition) . Let M be a C -spacetime where thedistributional null energy condition holds. Given any compact set K Ď M and c , c ą , then for all δ ą there is ε ą such that @ ε ă ε Ric r ˇ g ε sp X, X q ą ´ δ @ X P T M | K with ˇ g ε p X, X q “ and ă c ď || X || h ď c . (5)Here ˇ g ε is as in Lemma 2.2 and Ric r ˇ g ε s is its Ricci tensor.We will use this result in an essential way, when showing compactness of thehorismos of a certain set, which is needed for the causal/analytic part of the proofof our Gannon-Lee theorem. The difference to the arguments in [Gra20] is that wedo not assume global hyperbolicity, but are able to compensate for it by assumingnon-branching of null maximizers. Formally we define: Definition 2.4.
Let M be a spacetime and let γ : r , s Ñ M be a maximizingnull curve. We say that γ branches if there exists another maximizing null curve σ : r , s Ñ M such that γ p t q “ σ p t q for all 0 ď t ď a for some 0 ă a ă γ p t q ‰ σ p t q for all 1 ě t ą a . The point γ p a q is called branching point. If nomaximizing null curve branches, we say that there is no null branching in M .In light of the fact that (even) for C -metrics causal maximizers are geodesics(to be proven in Theorem 3.3, below), we see that if null branching were to occur in γ p a q , there would be (at least) two different, maximizing solutions to the geodesicequations with initial values γ p a q and γ p a q .Generally speaking, in the low regularity Riemannian setting, branching of max-imizers is associated with unbounded sectional curvature from below. More pre-cisely, in length spaces with a lower curvature bound branching does not occur[Shi93, Lem. 2.4]. In smooth Lorentzian manifolds sectional curvature bounds,although more delicate to handle, are still characterized by triangle comparison[AB08] and in the setting of Lorentzian length spaces [KS18] synthetic curvaturebounds extending sectional curvature bounds for smooth spacetimes have been in-troduced. In [KS18, Section 4] the authors show that a synthetic curvature boundfrom below prevents the branching of timelike maximizers. Unfortunately it is notclear at the moment how one could extend such a result to triangles with nullsides.However, in a merely C -spacetime the curvature is generically not locallybounded and so it seems natural that an additional condition as in Definition 2.4 NOTE ON THE GANNON-LEE THEOREM 7 is needed. Indeed, this condition enters in an essential way into our arguments.It will be a topic of future investigations to relate null branching to curvaturebounds.2.3.
Results.
Next we introduce the particular notions needed for the formulationof our results. In spirit they reflect Gannon’s [Gan75] assumptions on the space-time, inspired by asymptotic flatness, however, we stay close to the formulationsof [CeS10, Sec. 2].
Definition 2.5. An asymptotically regular hypersurface is a spacelike, smooth,connected partial Cauchy surface Σ which possesses an enclosing surface, i.e. acompact, connected submanfiold S of codim. 2 in M with the properties(1) S separates Σ into two (open, sub-) manifolds Σ ` , Σ ´ such that ¯Σ ` isnon-compact and Σ ´ connected,(2) the map h : π p S q Ñ π p ¯Σ ` q is surjective,(3) k ´ ą S , i.e. S is inner trapped.We further say that a surface Σ admits a piercing , if there exists a timelikevector field X on M such that every integral curve of X meets Σ exactly once.To elaborate on the last item above, we first fix some notation. Throughout letus denote the future directed timelike unit vector field perpendicular to Σ near S by U . Further as S is a hypersurface in Σ we denote by N ˘ the unit normals to S in Σ such that N ´ points into Σ ´ and N ` into Σ ` . We obtain future directednull normals to S via K ˘ : “ U | S ` N ˘ . We will refer to K ´ p p q as the ingoing nullvector. The convergence of a point in S is defined via k ˘ : “ g p H p , K ˘ p p qq , where H p is the mean-curvature vector field of Σ at p . Observe that since g P C , H p isstill continuous and all corresponding formulae hold “classically”.Also note that any piercing of Σ induces a continuous, open map ρ X : M Ñ Σ,which maps any point p in M to the unique intersection of the integral curve of X through p with Σ.We are now ready to state our main result, a Gannon-Lee theorem for stronglycausal, null pseudoconvex C -spacetimes without null branching. We will discussseveral of its special cases below. Theorem 2.6 ( C -Gannon-Lee theorem) . Let p M, g q be a strongly causal, nullpseudoconvex, null geodesically complete C -spacetime without null branching andsuch that the distributional null energy condition holds. Let Σ be an asymptoticallyregular hypersurface, which admits a piercing and contains an enclosing surface S Ď Σ such that M is homeomorphic to R ˆ Σ , then the map i : π p S q Ñ π p Σ q ,induced by the inclusion i : S Ñ Σ , is surjective. A NOTE ON THE GANNON-LEE THEOREM
Note that for C , -metrics the geodesic equation is uniquely solvable and hencethere can be no null branching. Moreover using that the distributional null en-ergy condition reduces to the “almost everwhere condition” for C , -metrics weimmediately obtain the following C , -Gannon-Lee theorem. Corollary 2.7 ( C , -Gannon-Lee theorem) . Let p M, g q be a strongly causal, nullpseudoconvex, null geodesically complete C , -spacetime such that the null energycondition Ric p X , X q ě holds for all local Lipschitz-continuous null vector fields X . Let Σ be an asymptotically regular hypersurface, which admits a piercing andcontains an enclosing surface S Ď Σ such that M is homeomorphic to R ˆ Σ , thenthe map i : π p S q Ñ π p Σ q is surjective. Going back to C and assuming global hyperbolicity we clearly can skip nullpseudoconvexity and, due to results in [Gra20], the assumption of no null branch-ing. Further in this case also some of the assumptions in [CeS10] can be dropped,as they are implied by the existence of a Cauchy surface. Corollary 2.8 (Globally hyperbolic C -Gannon-Lee theorem) . Let p M, g q be anull geodesically complete C -spacetime such that the distributional null energycondition holds. Further let Σ be an asymptotically regular Cauchy surface, con-taining an enclosing surface S Ď Σ , then the map i : π p S q Ñ π p Σ q is surjective. A simpler formulation is obtained when assuming that S is simply connected:the theorems then state that the entire spacetime is simply connected provided itis null complete. Originally the theorem of Gannon was given in contrapositiveform, saying that if S is (topologically) a sphere and Σ is not simply connected,then M has to be null incomplete.In proving our results we will follow the general layout of [CeS10]. The proofconsists of a causal and analytic part as well as a purely topological part. Atthe heart of the causal part lies Proposition 4.1 of [CeS10], which we will provefor C -spacetimes without null branching in Proposition 5.3. It essentially statesthat the inside region Σ ´ of an asymptotically regular hypersurface Σ is relativelycompact. Its proof is based on the notion of causal simplicity. But this resultneeds to be applied to a Lorentzian covering rather than to M itself. However,since causal simplicity does not lift to coverings, we bulid our arguments on nullpseudoconvexity. We will show in section 4 that this property indeed lifts tocoverings and that, together with strong causality, it implies causal simplicity.So, we will first establish the needed causality properties for C -spacetimes insections 3 and 4. 3. Maximizers and Causality in C In this section we establish properties of geodesics and results on causality in C -spacetimes. Building on recent results of [GL18b] and [Gra20] we will establish NOTE ON THE GANNON-LEE THEOREM 9 that causal maximizers in C -spacetimes are geodesics. Note that this result wasalso independently discovered very recently in [LLS20].First note that by [GL18b] maximizers have a causal character, even in Lipschitzspacetimes.We start by showing that also in a C -spacetime broken causal geodesics are notmazimizing. As a prerequisite we use a variational argument similar to the one in[O’N83], 10.45-46, which still holds true in our setting. Lemma 3.1.
Let c : r , s Ñ M be a causal pieceweise C -curve in a C -spacetime M and let X be a piecewise C -vector field along c . Then there is a piecewise C -variation c s of c with variation vector field X . Moreover, if g p X , c q ă along c then for any variation c s of c with variation vector field X and small enough s ,the longitudinal curve c s is timelike and longer than c .Proof. First we expand X to a vector field ˜ X in a neighbourhood of c and set c s p t q : “ Fl ˜ Xs p c p t qq , which is a variation of c with variation vector field X . For thesecond part note that g p c p t q , c p t qq ď t (except possible break points) as c is causal and further BB s | g p c s p t q , c s p t qq “ g p BB s | BB t c s p t q , c p t qq “ g p BB t X p t q , c p t qq ă s we have g p c s p t q , c s p t qq ă g p c p t q , c p t qq ď t ) and hence L p c s q “ ş p´ g p c s p t q , c s p t qqq dt ą ş p´ g p c p t q , c p t qqq dt “ L p c q . (cid:3) Lemma 3.2.
In a C -spacetime no broken causal geodesic is maximizing.Proof. Let γ : r , s Ñ M be a broken causal geodesic with a break point at γ p q .Hence v : “ lim t Ò γ p t q and w : “ lim t Ó γ p t q are linearly independent. Also sinceboth vectors are either future or past pointing, we have x v, w y ă x v, v y ´ x v, w y ą x v, w y ´ x w, w y ă . (6)If both v and w are null, this is clear by x v, w y ă
0. If both vectors are timelike wecan w.l.o.g. assume them to be unit vectors. By the Lorentz-Schwarz inequality wethen have |x v, w y| ą v and w are not colinear) and hence x v, w y ă ´ x v, v y ´ x v, w y ą
0, and in the same way it follows that x v, w y ´ x w, w y ă γ p q to γ p q longer than γ .First note that by continuity of the the Christoffel symbols we can solve the linearequations for parallel transport along γ (uniquely) and the solution is a C -vectorfield.We set y “ v ´ w and let Y and Y be the vector fields along γ | r , s and γ | r , s ,obtained by parallel transport of y along γ | r , s and γ | r , s , respectively. Next wedefine a piecewise C -vector field Y along γ by setting Y | r , s “ Y and Y | r , s “ Y . Since γ is a geodesic and Y is parallel we have on r , s using (6) x γ p t q , Y p t qy “ x v, v ´ w y “ x v, v y ´ x v, w y ą r , s we have x γ p t q , Y y ă f : r , s Ñ r , be a continuous, piecewise linear function such that f p q “ “ f p q , f | r , q “
1, and f | p , s “ ´ X p t q : “ f p t q Y p t q . Then X is a piecewise C -vector field along γ . Now consider the variation γ s of γ withvariation vector field X . By (7) we have x γ , X y ă s , such that γ s is longer than γ . By the choice of f theendpoints agree and we have shown the statement. (cid:3) Theorem 3.3.
Let p M, g q be a C -spacetime, then any maximizing causal curveis a causal geodesic and hence a C -curve.Proof. Observe that being a geodesic is a local property and that any part of amaximizing curve is maximizing. Moreover, since any point in a C -spacetime hasa globally hyperbolic neighbourhood , we can assume M to be globally hyperbolic.Let γ : r , s Ñ M be a maximizer and set p “ γ p q , q “ γ p q . By [Gra20],Proposition 2.13, there exists a maximizing causal geodesic from p to q of thesame causal character as γ . Also there exist maximizing, causal geodesics σ from p to γ p q and σ from γ p q to q . Note that since σ i are maximizing, we have L p σ q “ L p γ | r , s q and L p σ q “ L p γ | r , s q . This means L p σ ˝ σ q “ L p γ q . Sothe curve γ : “ σ ˝ σ is maximizing and hence by Lemma 3.2 it is an unbrokengeodesic.This procedure can be iterated to obtain a sequence of maximizing causalgeodesics γ n from p to q , which meet γ at all parameter values k n , for N Q k ď n .Observe that γ n converge to γ uniformly: First, for any ε ą γ by finitelymany open, causally convex, sets V εp i around p i P γ of h -diameter at most ε . Theunion V ε “ sup i V εp i is a neighbourhood of γ , and there exist dyadic numbers t m , m “ , . . . k , t “ t k “
1, such that γ p t m q and γ p t m ` q lie in a single V εp i forsome i . Moreover, there exists some N p ε q , such that for all n ě N all curves γ n meet every γ p t m q and hence the segments of γ n from t m to t m ` are contained in V εp i . So we conclude that d h p γ p t q , γ n p t qq ď ε , so γ n Ñ γ uniformly.We can now parameterize γ n , such that } γ n p q} h “ γ n ) such that γ n p q Ñ v . By [Har02], Sec. II Theorem3.2, there exists a subsequence of γ n which converges uniformly on compact setsto a geodesic σ with initial values σ p q “ p and σ p q “ v . But as γ n converges to γ , so must any subsequence and hence γ “ σ on the entire domain of γ . Finally σ reaches q since otherwise, σ would agree with γ on its entire maximal domain ofdefinition, would be future inextendible and contained in the compact set γ pr , sq , Actually there exists a smooth metric with wider lightcones, which has a neighbourhood baseof globally hyperbolic sets, but these are also globally hyperbolic for g . NOTE ON THE GANNON-LEE THEOREM 11 a contradiction to non-imprisonment which holds in any globally hyperbolic C -spacetime. (cid:3) Assuming non-branching we are able to prove results on limits of maximizersneeded in the following sections.
Proposition 3.4.
Let M be a C -spacetime without null branching. If two causalgeodesic segments contained in an achronal set intersect, they are segments of thesame geodesic or they intersect at the endpoints.Proof. Suppose such a segment intersetcs a second one in the interior of its do-main. Then either their tangents at the meeting point are not proportional and soby Lemma 3.2 their concatination, which is a broken causal geodesic, stops max-imising, contradicting the fact that both segments are contained in an achronalset. Or otherwise their tangents at the meeting point are proportional and hencenull branching would occur, again a contradiction. (cid:3)
Using Theorem 3.3 we may also give a slightly different formulation: If twodifferent initially maximizing null curves starting at the same point meet again,they stop maximizing.The final result of this section will be essential in the proof of the main theoremand it is the essential point at which we use the assumption that null branchingdoesn’t occur. Again, ˇ g ε is as in Lemma 2.2. Corollary 3.5.
Let p M, g q be a C -spacetime without null branching. Let S Ď M ,then any g -null curve γ : r , s Ñ E ` g p S q is a limit of ˇ g ε n -null curves contained in E ` ˇ g εn p S q for an appropriate subsequence of ˇ g ε n .Proof. Let γ be a future directed g -null S -maximizer starting at p “ γ p q P S ,so γ Ď E ` p p q . For any small δ ą q δn P B I ` ε n p S q : “ B I ` ˇ g εn p S q converging to γ p ´ δ q “ : q δ : To see this let U k be a sequence of connectednested neighbourhoods of q δ with Ş k U k “ t q δ u . Choose some q ek P U k z J ` p S q and q ik P U k X I ` p S q . For large n we can achieve q ik P I ` ε n p S q and since also q ek P U k z J ` ε n p S q there exists a curve from q ik to q ek which starts in I ` ε n p S q and leavesit and hence must meet B I ` ε n p S q in a point which we call q δn .Hence there are future directed ˇ g ε n -null maximizing geodesics γ δn ending at q δn and contained in B I ` g εn p S q . Note that the γ δn either meet S or are past inextendible.Now by [Har02, Sec. II, Thm. 3.2] there exists a subsequence of γ δn , denoted againby γ δn , converging to a maximizing g -null geodesic σ δ ending at q δ which is entirelycontained in B I ` p S q . By 3.3 γ is a geodesic and continues to be maximizing after q δ . Also, by construction σ δ is non-trivial, so γ and σ δ coincide on some γ | r a, ´ δ s (with a ă ´ δ ) by Proposition 3.4. As this works for any small δ , letting δ Ñ g ε n -geodesics denoted by γ n converging to a past-directed g -null geodesic σ starting at γ p q , and γ and σ agree where both are defined. There are now two possibilities: Either there exists a subsequence of ˇ g ε n suchthat any γ n meets S , in which case, by passing to this subsequence, also σ meets S and hence σ Ě γ and we obtain the desired property.The other possibility is that there is no subsequence, such that all γ n meet S .We show that this is impossible. If it were the case we may choose a subsequence γ n of past-inextendible curves. Then also σ is past-inextendible and hence mustleave γ pr , sq . Thus again σ Ě γ and it remains to show that γ n Ď E ` n p S q forlarge n . To this end let U be a globally hyperbolic neighbourhood of p “ γ p q P S ,which is w.l.o.g. also globally hyperbolic for ˇ g ε n for all n . Then there are points p n P γ n with p n Ñ p and so for large n , p n P B I ` n p S q X U “ E ` n p S q X U and hence γ n must meet S , a contradiction. (cid:3) Observe that Corollary 3.5 clearly holds true for C , -spacetimes but also forglobally hyperbolic C -spacetimes (where, in principle, null branching can occur)by [Gra20, 2.16], however, with stronger assumptions on S .4. On pseudoconvexity
In this section we discuss null pseudoconvexity, the central causality notion inour approach. As we work in C -spacetimes, we have to take more care in dealingwith convergence properties of geodesics, and we will bulid on the results of theprevious section. However, to our knowledge, the if-part of the main result of thissection, Theorem 4.4, was also unproven in the smooth case. Definition 4.1.
A spacetime p M, g q is called causal (null) pseudoconvex, if forany compact set K , there exists another compact set K , such that each causal(null) geodesic with endpoints in K must be entirely contained in K .It is said to be maximally null pseudoconvex, if each maximal null geodesic withendpoints in K is contained in K .Our aim is to relate this notion to convergence of geodesics, so let us first collectsome related facts in C -spacetimes. Lemma 4.2.
Let p n , q n be points in a C -spacetime and γ n : r , b n s Ñ M be futuredirected causal geodesics from γ n p q “ p n to γ n p b n q “ q n . Further let p n Ñ p and q n Ñ q “ p . Then there exists a causal geodesic γ : r , c q Ñ M starting at p , suchthat c is maximal and such that there is a subsequence of γ n , which converges in C on compact subsets of r , c q to γ . Either γ reaches q or c ď lim inf b n : “ b .In either case, if all γ n are maximizing, so is γ until it reaches q or up to c ,respectively.Proof. We can assume w.l.o.g. that } γ n p q} h “ γ n p q Ñ v P T p M .Re-writing the geodesic equations as a first order system in T M , we can apply In our case this follows since we approximate from the inside. However, global hyperbolicity isstable in the interval topology even for continuous metrics [S¨a16] and C -convergence is stronger. NOTE ON THE GANNON-LEE THEOREM 13 [Har02, Sec. II Theorem 3.2] to infer that there exists a solution γ : r , c q Ñ M of the geodesic equation with initial conditions γ p q “ p and γ p q “ v and asubsequence γ n k of γ n such that for any a ă c we have γ n k | r ,a s Ñ γ | r ,a s uniformlyin T M . Here r , c q is the maximal domain of γ to the future. If b “ lim inf b n ă c ,there is a subsequence such that b n l Ñ b ă c and hence q n l “ γ n l p b n l q Ñ γ p b q “ q .For the final point, note that by [S¨a16, Thm. 6.3] the length functional is upper-semi continuous and by [KS18, Prop. 5.7] the time-separation function τ is lower-semi continuous. We can thus apply the ususal estimates for any point γ p a q “ lim k γ n k p a k q on γτ p p, γ p a qq ď lim inf k τ p p n k , γ n k p a k qq ď lim sup k τ p p n k , γ n k p a k qqď lim sup k L p γ n k | r ,a k s q ď L p γ p , a qq ď τ p p, γ p a qq , and so γ | r ,a s is maximizing. (cid:3) One very useful characterisation of pseudoconvexity in strongly causal space-times, is via limits of geodesics. The proof is straightforward for C , -metrics,see e.g. [BP89, Lem. 3] [VPE19, Prop. 4] and will follow from Lemma 4.2 in the C -case. Lemma 4.3.
Let M be a strongly causal C -spacetime, then M is (maximally)null or causal pseudoconvex if and only if for any sequence of (maximal) null orcausal geodesics γ n with endpoints p n Ñ p and q n Ñ q “ p , there is a subsequenceof γ n which converges locally uniformly to a (maximal) null or causal geodesic γ from p to q , respectively.Proof. We only deal with the case of maximal null pseudoconvexity, the other casesfollow analogously.So let M be maximal null pseudoconvex and let p n Ñ p , q n Ñ q . Further let γ n be maximal null geodesics from p n to q n . By Lemma 4.2 there exists a maximalnull geodesic γ : r , c q and a subsequence of the γ n which converges to γ on any r , a s for a ă c . If γ reaches q we are done.Otherwise γ is maximal up to c and inextendible. Now consider the closed h -ball B r p p q for r ą d h p p, q q , which is a compact neighbourhood of p and q . By maximalnull pseudoconvexity there exists a compact set ˜ K such that for n large enoughall γ n are contained in ˜ K . But then γ is also contained in ˜ K contradicting strongcausality .Let us show the reverse implication. We assume that the limiting property formaximizing null geodesics holds, but M is not maximal null pseudoconvex. Take acompact exhaustion K i of M . By assumption there exists a compact set K and asequence of maximizing null geodesics γ n from p n P K to q n P K such that γ n leaves which implies non-imprisonment also for C metrics. K n . By passing to a subsequence we can assume p n Ñ p P K and q n Ñ q P K ,where p “ q due to strong causality. By the limiting property a subsequence, againdenoted by γ n , converges to a maximizing null geodesic γ from p to q . Hence thereexists some N such that γ Ď int p K N q and hence for n large, also γ n Ď int p K N q , acontradiction. (cid:3) The main result of this section is the following characterization of causally simplespacetimes.
Theorem 4.4.
Let p M, g q be a strongly causal C -spacetime without null branch-ing, then M is causally simple if and only if it is maximally null pseudoconvex.Proof. The fact that every causally simple smooth spacetime is maximally nullpseudoconvex is proved in [BK92, Thm. 1]. An inspection of the proof revealsthat it also holds in the C -case assuming no null branching and using Proposition3.4. For the converse direction we enhance the proof of [VPE19, Thm. 2] . Weassume indirectly that M is not causally simple. So there is p P M with J ` p p q notclosed and hence some q P J ` p p qz J ` p p q .Let U be a globally hyperbolic neighbourhood of q and choose r P I ` p q q X U then there exists some future directed timelike curve c : r , s Ñ U with c p q “ q , c p q “ r . Note that push-up also works in the case where q P I ` p p q , i.e. I ` p I ` p p qq Ď I ` p p q and so there exists a timelike curve γ from p to r .Let x P c pp , sq and note that r R I ´ p x q by chronology, but p P I ´ p x q by push-up and so there exists a unique intersection point q x of γ with B I ´ p x q (since thelatter is an achronal boundary). We will look at the subset of all points x P c pp , sq for which there exists a maximizing geodesic from x to q x and define A : “ t x P c pp , sq | γ XB I ´ p x q “ t q x u Ď E ´ p x qu and B : “ t t P p , s | c p t q P A u . First we will show that A is non-empty and closed and hence so is B by continuityof c . Now, A ‰ H , as r P A , and further, as long as the intersection q x “B I ´ p x q X γ is in U , we have x P A .Further A is closed by maximal null pseudoconvexity: Let x n P A , x n Ñ x , for all n there exists a maximal null geodesic segment from x n to q x n “ γ XB I ´ p x n q , whichis unique. By compactness of the image of γ we have that up to a subsequence q x n Ñ y P γ . By maximal null pseudoconvexity the geodesics from x n to q x n have a subsequence, which converges to a maximal null geodesic from x to y andhence y “ q x , and x P A . Besides the non-branching property the proof in [BK92, Thm. 1] requires that causal sim-plicity implies causal continuity, which also holds for C -metrics. In our opinion that proof is not conclusive since in step (2) it cannot be ruled out that s “ q x ,which forcloses the application of maximal null pseudoconvexity. Note that we can not a priori guarantee that y = q x , this would need the spacetime to bereflecting. NOTE ON THE GANNON-LEE THEOREM 15
To conclude the proof set t : “ inf B . If t “ B Q t n Ñ c p t n q “ x n to points q x n on γ . By maximal null pseudoconvexity there is a maximizing null geodesicfrom q “ c p q to an accumuliation point of q x n in γ . So p ď q , a contradiction.So assume t ą
0. Then B Ď r t , s and c p t q P A by closedness of A Ř c pp , sq .But as t ą t n Ò t and x n : “ c p t n q R A .Setting x “ c p t q and using the above notation we show that lim q x n : “ y “ q x :If this is not the case then as y P γ we either have y " q x or y ! q x and in eithercase we can w.l.o.g. assume all q x n " q x or all q x n ! q x . In the first case, since q x n P I ´ p x n q by push-up we obtain x " x n " q x , a contradiction. In the secondcase as q x P J ´ p x q we obtain y P I ´ p x q and by openness of ! also q x n P I ´ p x n q forlarge n , again a contradiction. Thus we have established q x “ y .Let α : r , s Ñ M be a future directed maximal null geodesic from q x to x , α p q “ q x and α p q “ x . We can assume γ p q “ p and γ p q “ q x and γ tobe future directed. Take any timelike variation of γ s of γ through p such that γ s p q “ α p s q . For all n there again exists a unique q sx n P B I ´ p x n q X γ s and futuredirected null geodesics η sn Ď B I ´ p x n q starting at q sx n .We claim there exists a subsequence of x n denoted in the same way, such thatall η sn reach x n , i.e. such that q sx n P E ´ p x n q . If not, then there exists some N suchthat @ n ě N the geodesics η sn do not reach x n , i.e. they are future inextendible andcontained in B I ´ p x n q . In the same way as for q x n we obtain q sx n Ñ α p s q and hence η sn Ñ η s , where η s : r , c q Ñ M is a future inextendible, maximizing null geodesicstarting at α p s q . We will show that p η s q p q and α p s q are linearly dependent. Inthis case by the assumption of no null branching we obtain α r s, s Ď η s . If they arelinearly independent, then by Lemma 3.2 the broken geodesic α | r ,s s Y η s is notmaximizing beyond α p s q , i.e. η s p t q P I ` p q x q for any t ą
0. But then by openness of ! we have q x n ! η sn p t q P B I ´ p x n q for large n , which by push-up implies q x n ! x n ,a contradiction. So we conclude that η s Ě α r s, s . But as α p q “ x also η sn mustcome close to x and hence enter U . Moreover, η sn is inextendible so it must reach x n , a contradiction.Let x n be a subsequence as above and let s m Ñ m Ñ 8 , then q s m x n Ñ q x n byconstruction of γ s and the fact that γ X B I ´ p x n q “ t q x n u . So, for any n we obtaina sequence of maximizing null geodesics η s m n from q s m x n to x n , which, by anotherappeal to maximal null pseudoconvexity, converge to a maximal null geodesic from q x n to x n and hence x n P A , a contradiction. So t ą (cid:3) Lorentzian coverings clearly preserve local geometric properties. However, max-imality of causal curves is not local and given a maximal causal curve in the covering, its projection need not be maximal. This is why maximal null pseudo-convexity and causal simplicity do not lift. However, we will show that one can liftnull pseudoconvexity to any Lorentzian covering of a strongly causal C -spacetime.As null pseudoconvexity is stronger than maximal null pseudoconvexity, by The-orem 4.4 strong causality and null pseudoconvexity imply causal simplicity. Onecan thus place strongly causal, null pseudoconvex spacetimes on the causal ladderbetween causal simplicity and global hyperbolicity, where in the C -case one alsoneeds to assume no null branching. Theorem 4.5.
Let M be a strongly causal, null pseudoconvex C -spacetime andlet π : ˜ M Ñ M a Lorentzian-covering. Then ˜ M is null pseudoconvex.Proof. First note that any Lorentzian covering of a strongly causal C -spacetimeis again strongly causal. We show the lmiting property of Lemma 4.3.Assume this property fails, so there exist ˜ p n , ˜ q n P ˜ M and ˜ γ n : r , b n s Ñ ˜ M futuredirected null geodesics from ˜ p n to ˜ q n and such that ˜ p n Ñ ˜ p and ˜ q n Ñ ˜ q “ ˜ p , suchthat no subsequence of ˜ γ n converges to a null geodesic from ˜ p to ˜ q .By Lemma 4.2 there exists a subsequence of ˜ γ n (denoted again by ˜ γ n ) which con-verges to a null geodesic ˜ γ : r , c q Ñ ˜ M , starting at ˜ p . By our indirect assumptionit never meets ˜ q , and, moreover, it is inextendible, and c ď lim inf b n .Now set p “ π p ˜ p q , q “ π p ˜ q q , p n “ π p ˜ p n q , q n “ π p ˜ q n q and γ n “ π ˝ ˜ γ n . Then p ‰ q ,since otherwise p “ q possess an evenly covered, causally convex neighbourhood W and ˜ γ n would remain in a single sheet of π ´ p W q , implying ˜ p “ ˜ q . Since M isnull pseudoconvex γ n has a subsequence (which we again denote by γ n ) convergingto a null geodesic γ : r , b s Ñ M from p to q . Let σ : r , b s Ñ ˜ M be the unique liftof γ through ˜ p , then σ is a null geodesic and moreover we claim that it is equal to ˜ γ on its entire domain r , b s . To see this we cover γ by finitely many neighbourhoods U i and σ by neighbourhoods ˜ U i , such that π : ˜ U i Ñ U i is an isometry. As γ | U i isthe limit of γ n | U i , its lift σ | ˜ U i is the limit of ˜ γ n | ˜ U i . This means that, wherever both σ and ˜ γ are defined, their images agree. But as ˜ γ is inextendible it must leave σ pr , b sq and hence ˜ γ pr , c qq Ś σ pr , b sq , so ˜ γ exists longer than σ .Now since γ reaches q , σ reaches a point ˜ r P π ´ p q q and hence, by the aboveso does ˜ γ , and by assumption ˜ r ‰ ˜ q . Let V be a causally convex, evenly coveredneighbourhood of q and let ˜ V be the isometric copy in ˜ M containing ˜ r and ˜ V theone containing ˜ q . We have ˜ V X ˜ V “ H and for some large n we have ˜ γ n p t n q P ˜ V for a suitable t n and ˜ γ n p b n q P ˜ V . Hence the causal curve ˜ γ n | r t n ,b n s leaves ˜ V andenters ˜ V . Since π p ˜ V q “ π p ˜ V q “ V , γ n | r t n ,b n s is a causal curve which leaves andre-enters V , a contradiction to V being causally convex. (cid:3) Proof of the main result
We will split the proof of Theorem 2.6 into two parts. The first, analytic partwill be concerned with showing that the set Σ ´ is relatively compact. Here we will NOTE ON THE GANNON-LEE THEOREM 17 generalize (the proof of) [CeS10, Prop. 4.1] by proving new focusing statementsfor null geodesics using the results from section 3.The second, topological part uses the results from causality theory detailed insection 4 to generalize (the proof of) [CeS10, Thm. 2.1]. The key argument isthat null pseudoconvexity and strong causality lift to the covering to imply causalsimplicity.To be self-contained we will briefly sketch also those parts of the original (smooth)proofs which do not need major revision.5.1.
Analytic aspects.
The main analytical ingredient of our proof is a gen-eralization of [CeS10, Prop. 4.1] to C -spacetimes, which we will give below inProposition 5.3.In order to do so we need a focusing results for null geodesics in smooth space-times which violate the null energy condition by a small margin δ , as in Lemma2.3. To this end we apply a result of [FK20] , which itself is a generalization of[O’N83, Prop. 10.43]. Proposition 5.1. ( [FK20, Prop. 2.7] ) Let S be a spacelike submanifold of co-dimension in a smooth spacetime and let γ be a null geodesic joining p P S to q P J ` p S q . If there exists a smooth f on γ which is nonvanishing at p but vanishesat q and so that ż γ ` p n ´ qp f q ´ f Ric p γ , γ q ˘ ď p n ´ q x f γ , H qy | p , (8) then there is a focal point to S along γ . Lemma 5.2.
Let S be a C -spacelike submanifold of codimension 2 in a smoothspacetime. Let γ be a geodesic starting at some p P S such that ν : “ γ p q is afuture pointing null normal to S . Let the convergence c : “ k S p ν q : “ x H γ p q , ν y ą and choose some b ą c and ă δ p b, c q “ : δ ď b p n ´ qp bc ´ q . Now, ifRic p γ , γ q ě ´ δ along γ , then γ | r ,b s cannot be maximizing to S , provided it existsthat long. Proof.
We set f p t q : “ ´ tb and check condition (8). For our choice of δ we obtain ż b p n ´ q b dt ´ ż b ˆ ´ tb ` t b ˙ Ric p γ p t q , γ p t qq dt ď n ´ b ` δ b ď p n ´ q c “ p n ´ q k S p ν q “ p n ´ q x f γ , H qy | p , (10)and hence γ | r ,b s cannot be maximizing. (cid:3) Observe the opposite signature convention used there. One can specify the choice of b further to allow for bigger violations of δ , but we do notneed this here. Recall that for C -metrics the mean curvature and the convergence of S are stillcontinuous. The core of the following proof is in the spirit of [Gra20, Thm. 5.7],however, we have to make up for the lack of global hyperbolicitiy. Proposition 5.3.
Let p M, g q be an n -dimensional (with n ě ), causally simple,null geodesically complete C -spacetime without null branching which satisfies thedistributional null energy condition and admits an asymptotically regular hyper-surface Σ with a piercing. Then for any enclosing surface S Ď Σ the closure of itsinside, Σ ´ “ S Y Σ ´ is compact. Using the notation T “ B I ` p Σ ` qz Σ ` , the proof consists in successively estab-lishing the following three claims:(1) T Ď E ` p S q , (2) T is compact, (3) ρ X p T q “ Σ ´ .Note that it is steps (1) and (3) that rely on causal simplicity, while step (2)depends upon the study of conjugate points along null geodesics. As announcedwe will be sketchy in the proofs of (1) and (3). Proof. (1) By causal simplicity E ` p S q “ B J ` p S q “ B I ` p S q and we show that T Ď B I ` p S q . Let p P T , and note that if p P Σ ` z Σ ` “ S Ď E ` p S q we are done.So let p R Σ ` and take any neighbourhood U of p with U X ¯Σ ` “ H . Then wepick p ˘ P I ˘ p p q X U and we show that p ` P I ` p S q and p ´ P M z I ` p S q , which willprove (1).By definition of B I ` p Σ ` q we know, p ´ R I ` p Σ ` q (if it were, then p P I ` p Σ ` q )and so p ´ R I ` p S q . By achronality of B I ` p Σ ` q and the existence of a piercing wehave p P I ` p Σ q and we may assume U Ď I ` p Σ q . This yields p ´ P I ` p Σ qz I ` p Σ ` q hence p ´ P I ` p Σ ´ q , while clearly p ` P I ` p Σ ` q . Now for C : “ I ´ p p ` q X Σ we have C X Σ ` ‰ H and C X Σ ´ ‰ H . Also C is connected since any pair of points in C can first be connected via a path in I ´ p p ` q which projects via the piercing to apath in C . But this implies that C X S “ H and so p ` P I ` p S q .(2) First, T is closed: Assume it were not, then there would exist a sequence p n P T such that p n Ñ p R T . We know p P B I ` p Σ ` q as this set is closed, and hence p P Σ ` “ B I ` p Σ ` qz T . Further as E ` p S q is closed and T Ď E ` p S q , also p P E ` p S q and hence there must exist a causal curve from S to p , which is a contradiction toacausality of Σ.Any point p on T lies on a null geodesic emanating from S by Theorem 3.3 andits initial tangent vector is inward pointing, i.e. proportional to K ´ p q q for some q P S : That such a tangent vector is perpendicular to S is shown for C , -metricsin [GGKS18, Remark 6.6(ii)] by constructing a timelike variation starting in S if it were not perpendicular, however an inspection reveals that this method alsoworks for C -metrics. NOTE ON THE GANNON-LEE THEOREM 19
Further since the two normal future null congruences starting from S are dis-joint, see e.g. [Gan75, Lemma 1.1] , the only other possibility is that the tangentis outward pointing, i.e. proportional to K ` , but then the geodesic would enter I ` p Σ ` q , hence not be in T . This can again be shown in a similar fashion as aboveby constructing a variation as in [GGKS18, Remark 6.6(ii)].By continuity k ´ possesses a minimum c : “ min p P S k ´ p p q “ min p P S x H p , K ´ p p qy on S . Also, the set K : “ tp p, λK ´ p p qq P T S K | ď λ ď c u Ď T M is compact and,by [Gra20, Prop. 2.11] (or rather a simplified version without ε ) the set F : “ ď γ with γ p qP K im p γ | r , s q (11)is relatively compact.We will show that T Ď π p F q , where π : T M Ñ M is the projection. As-sume not and let p P T z π p F q . By Theorem 3.3 there exists a null geodesic γ : r , s Ñ M from S to p maximizing the distance to S . As γ Ď T we knowthat γ p q “ µK ´ p γ p qq for some µ ą c , since γ p q R K . This means that k S p γ p qq “ x H p γ p qq , µK ´ p γ p qqy ě µc ą g ε k be as in Lemma 2.2. By Corollary 3.5 γ is the C -limit of ˇ g ε k -nullgeodesics γ ε k : r , s Ñ M maximizing the ˇ g ε k -distance to S and contained in E ` ˇ g εk p S q . Further we can assume that all γ ε k are contained in a compact neigh-bourhood ˜ K of γ and that k ă } γ ε k } h ă k for some k i ą
0. Additionally for k large enough we have k ε k S p γ ε k p qq : “ c k ą r g ε k sp γ ε k , γ ε k q ě ´ p n ´ q . In order to apply Lemma 5.2, in our case for b “ ´ δ k “ ´ b p n ´ qp b c k ´ q “ p n ´ qp c k ´ q . Since c k ą k ,we have ´ δ k ă ´ p n ´ q and hence Ric r g ε k sp γ ε k , γ ε k q ě ´ p n ´ q ą ´ δ k . So byLemma 5.2 γ ε k cannot be maximizing up to c k ă ă “ b , a contradiction.(3) First ρ X p T q Ď Σ ´ (since otherwise T X I ` p Σ ` q “ H ) and Σ ´ z ρ X p T q Ď Σ ´ (since S “ ρ X p S q Ď ρ X p T q ). Now assuming indirectly that Σ ´ z ρ X p T q “ H ,there is p P B Σ ρ X p T q X Σ ´ and we will reach a contradiction by showing that p P int ρ X p T q .By (2) there is q P T with ρ X p q q “ p , and q R S (otherwise p “ q P S X Σ ´ “ H ).So q P T z S “ B I ` p Σ ` qz Σ ` which is a topological hypersurface ( S being the edgeof the achronal set T ). So there is V , an M -neighbourhood of q with V X T open in T z S , ρ X p V q Ď Σ ´ , and V X Σ “ H . Next denote by Ψ the local flowof X and choose ε ą U , an open M -neighbourhood of q , so small thatΨ p U ˆ p´ ε, ε qq Ď V . Further set Ψ : “ Ψ |p U X T qˆp´ ε,ε q , and W : “ ImΨ . Byachronality of T and invariance of domain W is open and Ψ is a homeomorphism.But then p P ρ X p U ˆ p´ ε, ε qq “ ρ X p W q and the latter set is open by openness of ρ X ([O’N83, 14.31]) and so p P int ρ X p T q . (cid:3) The proof there is given for smooth spacetimes and one assumes S to be simply connected,however the method of proof also works in our case. Topological aspects.
Finally we invoke Theorem 4.5 and Proposition 5.3to prove our main result. Here we will be brief on the topological aspects laid outalready in the proof of [CeS10, Thm. 2.1].
Proof of theorem 2.6.
Let Φ : ˜ M Ñ M be connected (smooth) covering withΦ p π p ˜ M qq “ j p π p S qq , where j is the inclusion of S in M . By assumptionthe inclusion m of Σ in M induces an isomorphism m : π p Σ q Ñ π p M q . Inparticular, Φ Σ : “ Φ | ˜Σ : Φ ´ p Σ q : “ ˜Σ Ñ Σ is a Riemannian covering with ˜Σconnected.In Lemma 5.4 below we will establish that Φ Σ is trivial. Accepting this for themoment, we will show the theorem, i.e. for every y P π p Σ q there is x P π p S q with i p x q “ y . From the diagrams MS Σ ˜ M ˜Σ ij m ΦΦ Σ ˜ m j p π p S qq π p S q π p Σ q π p M q π p ˜Σ q π p ˜ M q i j m Φ we see that m p y q “ p Φ ˝ ˜ m ˝ Φ ´ q p y q “ Φ p ˜ m ˝ Φ ´ q p y q P Φ p π p ˜ M qq “ j p π p S qq . (12)Since j “ m ˝ i we have m p y q “ j p x q “ m p i p x qq , and since m is anisomorphism and we are done. (cid:3) Lemma 5.4. Φ Σ : “ Φ | ˜Σ : ˜Σ : “ Φ ´ p Σ q Ñ Σ is a trivial covering.Proof. First, by definition there is a local deformation F : S ˆ p´ , q Ñ Σ of S .Further set U ´ F : “ F p S ˆ p , qq and V : “ U ´ F Y S Y Σ ` . Then ¯Σ ` is a deformationretract of V and hence π p ¯Σ ` q – π p V q .Next we establish that on every connected component ˜ V of Φ ´ p V q the mapΦ V : “ Φ | ˜ V : ˜ V Ñ V is a diffeomorphism.We only have to show injectivity since Φ V is a local diffeomorphism. Take˜ p, ˜ q P ˜ V such that Φ V p ˜ p q “ Φ V p ˜ q q “ : p P V . Let ˜ α : r , s Ñ ˜ V be a pathconnecting these two points, then α : “ Φ ˝ ˜ α is a loop in V , homotopic to a loopin S since π p V q – π p ¯Σ ` q – π p S q . Further since Φ p π p ˜ M qq “ j p π p S qq , thereis a loop ˜ β in ˜ M , fixed endpoint-homotopic to ˜ α and so we must have ˜ p “ ˜ q .In order to show that Φ Σ is trivial we assume the converse, and, in particularthat Φ ´ p S q has more than one component. Since S Ď V each of these componentsis diffemorphic to S , and they separate ˜Σ. NOTE ON THE GANNON-LEE THEOREM 21
Let ˜ S , ˜ S be two such different components and let ˜ V , ˜ V be the respectivecopies of V containing them. Since ¯Σ ` Ď V , each ˜ V i ( i “ ,
2) contains a diffeo-morphic copy of ¯Σ ` called ˜ C i , which are closed and non-compact. Further since ˜Σis connected and separated by ˜ S , the set ˜ C is contained in ˜ S Y ˜Σ p q´ : “ ˜ S Yp ˜Σ z ˜ C q ,since otherwise ˜ V X ˜ V ‰ H .So ˜ C Ď ˜ S Y ˜Σ p q´ . Being local properties, both null geodesic completeness andthe distributional null convergence condition lift to ˜ M , as does null pseudoconvex-ity by Theorem 4.5. So the covering ˜ M is causally simple by Theorem 4.4. Furtheras null branching is a local property as well, it also cannot occur in ˜ M . Thus theassumptions of Proposition 5.3 are fulfilled for ˜ M , ˜Σ, ˜ C and ˜Σ z ˜ C implying that˜ S Y p ˜Σ z ˜ C q is compact. However this set contains the non-compact, closed set˜ C , a contradiction and we are done. (cid:3) Finally we sketch the proof of Corollary 2.8 , i.e. the globally hyperbolic C -Gannon-Lee theorem. We can proceed analogously to the one of Theorem 2.6,where most points will even be significantly easier. The only aspect one hast topay attention to is proving that T is compact, i.e. step (2) above: We have to provethat any maximizing g -null geodesic is a limit of maximizing ˇ g ε -null geodesics.Since Corollary 3.5 does not apply we have to replace it by the “limiting-result”[Gra20, Prop. 2.13]. Acknowledgement
We are greatful to Michael Kunzinger for sharing his experience and to MelanieGraf and Clemens S¨amann for helpful discussions. This work was supported byFWF-grants P28770, P33594 and the Uni:Docs program of the University of Vi-enna.
References [AB08] Stephanie B. Alexander and Richard L. Bishop. Lorentz and semi-Riemannian spaceswith Alexandrov curvature bounds.
Comm. Anal. Geom. , 16(2):251–282, 2008. 2.2[BK92] John K. Beem and Andrzej Kr´olak. Cosmic censorship and pseudoconvexity.
J. Math.Phys. , 33(6):2249–2253, 1992. 2.1, 4, 6[BP87] John K. Beem and Phillip E. Parker. Pseudoconvexity and general relativity.
J. Geom.Phys. , 4(1):71–80, 1987. 2.1[BP89] John K. Beem and Phillip E. Parker. Pseudoconvexity and geodesic connectedness.
Ann. Mat. Pura Appl. (4) , 155:137–142, 1989. 4[BS18] Patrick Bernard and Stefan Suhr. Lyapounov functions of closed cone fields: fromConley theory to time functions.
Comm. Math. Phys. , 359(2):467–498, 2018. 2.2[CeS10] I. P. Costa e Silva. On the Gannon-Lee singularity theorem in higher dimensions.
Classical Quantum Gravity , 27(15):155016, 13, 2010. 1, 2.3, 2.3, 2.3, 5, 5.1, 5.2[CG12] Piotr T. Chru´sciel and James D. E. Grant. On Lorentzian causality with continuousmetrics.
Classical Quantum Gravity , 29(14):145001, 32, 2012. 1, 2.2, 2.2[CK18] Piotr T. Chru´sciel and Paul Klinger. The annoying null boundaries.
Journal ofPhysics: Conference Series , 968:012003, feb 2018. 1 [ER08] Roberto Emparan and Harvey S. Reall. Black Holes in Higher Dimensions.
LivingRev. Rel. , 11:6, 2008. 1[FK20] Christopher J. Fewster and Eleni-Alexandra Kontou. A new derivation of sin-gularity theorems with weakened energy hypotheses.
Classical Quantum Gravity ,37(6):065010, 31, 2020. 5.1, 5.1[FSW95] John L. Friedman, Kristin Schleich, and Donald M. Witt. Comment on: “Topologicalcensorship” [Phys. Rev. Lett. (1993), no. 10, 1486–1489; MR1234452 (94e:83071)]. Phys. Rev. Lett. , 75(9):1872, 1995. 1[Gal83] Gregory J. Galloway. Minimal surfaces, spatial topology and singularities in space-time.
J. Phys. A , 16(7):1435–1439, 1983. 1[Gan75] Dennis Gannon. Singularities in nonsimply connected space-times.
J. MathematicalPhys. , 16(12):2364–2367, 1975. 1, 2.3, 5.1[Gan76] Dennis Gannon. On the topology of spacelike hypersurfaces, singularities, and blackholes.
General Relativity and Gravitation , 7(2):219–232, 1976. 1[GGKS18] Melanie Graf, James D. E. Grant, Michael Kunzinger, and Roland Steinbauer. TheHawking-Penrose singularity theorem for C , -Lorentzian metrics. Comm. Math.Phys. , 360(3):1009–1042, 2018. 1, 5.1[GKOS01] Michael Grosser, Michael Kunzinger, Michael Oberguggenberger, and Roland Stein-bauer.
Geometric theory of generalized functions with applications to general relativ-ity , volume 537 of
Mathematics and its Applications . Kluwer Academic Publishers,Dordrecht, 2001. 2.2, 2.2[GKS19] James D. E. Grant, Michael Kunzinger, and Clemens S¨amann. Inextendibility ofspacetimes and Lorentzian length spaces.
Ann. Global Anal. Geom. , 55(1):133–147,2019. 2.2[GKSS20] James D. E. Grant, Michael Kunzinger, Clemens S¨amann, and Roland Steinbauer.The future is not always open.
Lett. Math. Phys. , 110(1):83–103, 2020. 2.2[GL18a] Gregory J. Galloway and Eric Ling. Topology and singularities in cosmological space-times obeying the null energy condition.
Comm. Math. Phys. , 360(2):611–617, 2018.1[GL18b] Melanie Graf and Eric Ling. Maximizers in Lipschitz spacetimes are either timelikeor null.
Classical Quantum Gravity , 35(8):087001, 6, 2018. 3[GLS18] Gregory J. Galloway, Eric Ling, and Jan Sbierski. Timelike completeness as an ob-struction to C -extensions. Comm. Math. Phys. , 359(3):937–949, 2018. 1[Gra20] Melanie Graf. Singularity theorems for C -Lorentzian metrics. Comm. Math. Phys. ,378(2):1417–1450, 2020. 1, 2.2, 2.2, 2.2, 2.2, 2.2, 2.3, 3, 3, 3, 5.1, 5.1, 5.2[GS06] Gregory J. Galloway and Richard Schoen. A generalization of Hawking’s black holetopology theorem to higher dimensions.
Comm. Math. Phys. , 266(2):571–576, 2006.1[GT87] Robert Geroch and Jennie Traschen. Strings and other distributional sources in gen-eral relativity.
Phys. Rev. D (3) , 36(4):1017–1031, 1987. 2.2[Har02] Philip Hartman.
Ordinary differential equations , volume 38 of
Classics in AppliedMathematics . Society for Industrial and Applied Mathematics (SIAM), Philadelphia,PA, 2002. Corrected reprint of the second (1982) edition [Birkh¨auser, Boston, MA;MR0658490 (83e:34002)], With a foreword by Peter Bates. 3, 3, 4[HE73] S. W. Hawking and G. F. R. Ellis.
The large scale structure of space-time . CambridgeUniversity Press, London-New York, 1973. Cambridge Monographs on MathematicalPhysics, No. 1. 1[HW51] Philip Hartman and Aurel Wintner. On the problems of geodesics in the small.
Amer.J. Math. , 73:132–148, 1951. 2.2
NOTE ON THE GANNON-LEE THEOREM 23 [KS18] Michael Kunzinger and Clemens S¨amann. Lorentzian length spaces.
Ann. Global Anal.Geom. , 54(3):399–447, 2018. 2.2, 2.2, 4[KSSV14] Michael Kunzinger, Roland Steinbauer, Milena Stojkovi´c, and James A. Vickers. Aregularisation approach to causality theory for C , -Lorentzian metrics. Gen. Rela-tivity Gravitation , 46(8):Art. 1738, 18, 2014. 1[KSSV15] Michael Kunzinger, Roland Steinbauer, Milena Stojkovi´c, and James A. Vick-ers. Hawking’s singularity theorem for C , -metrics. Classical Quantum Gravity ,32(7):075012, 19, 2015. 1[KSV15] Michael Kunzinger, Roland Steinbauer, and James A. Vickers. The Penrose singular-ity theorem in regularity C , . Classical Quantum Gravity , 32(15):155010, 12, 2015.1[Lee76] C. W. Lee. A restriction on the topology of Cauchy surfaces in general relativity.
Comm. Math. Phys. , 51(2):157–162, 1976. 1[LLS20] Christian Lange, Alexander Lytchak, and Clemens S¨amann. Lorentz meets Lipschitz.9 2020. 3[LM07] Philippe G. LeFloch and Cristinel Mardare. Definition and stability of Lorentzianmanifolds with distributional curvature.
Port. Math. (N.S.) , 64(4):535–573, 2007. 2.2[Min15] E. Minguzzi. Convex neighborhoods for Lipschitz connections and sprays.
Monatsh.Math. , 177(4):569–625, 2015. 1[Min19] Ettore Minguzzi. Causality theory for closed cone structures with applications.
Rev.Math. Phys. , 31(5):1930001, 139, 2019. 2.2[MS20] Ettore Minguzzi and Ivan P. Costa e Silva. A note on causality conditions on coveringspacetimes. 5 2020. 1[MY82] William W. Meeks, III and Shing Tung Yau. The existence of embedded minimalsurfaces and the problem of uniqueness.
Math. Z. , 179(2):151–168, 1982. 1[O’N83] Barrett O’Neill.
Semi-Riemannian geometry with applications to relativity , volume103 of
Pure and Applied Mathematics . Academic Press, Inc. [Harcourt Brace Jo-vanovich, Publishers], New York, 1983. 2, 3, 5.1, 5.1[S¨a16] S¨amann, Clemens. Global hyperbolicity for spacetimes with continuous metrics.
Ann.Henri Poincar´e , 17(6):1429–1455, 2016. 1, 4, 4[Sbi18] Jan Sbierski. The C -inextendibility of the Schwarzschild spacetime and the spacelikediameter in Lorentzian geometry. J. Differential Geom. , 108(2):319–378, 2018. 1[Sbi20] Jan Sbierski. On holonomy singularities in general relativity and the C , -inextendibility of spacetimes. 7 2020. 1[Shi93] Katsuhiro Shiohama. An introduction to the geometry of Alexandrov spaces , volume 8of
Lecture Notes Series . Seoul National University, Research Institute of Mathematics,Global Analysis Research Center, Seoul, 1993. 2.2[SS18] Clemens S¨amann and Roland Steinbauer. On geodesics in low regularity.
J. Phys.Conf. Ser. , 968:012010, 14, 2018. 2.2[SV09] R. Steinbauer and J. A. Vickers. On the Geroch-Traschen class of metrics.
ClassicalQuantum Gravity , 26(6):065001, 19, 2009. 2.2[VPE19] Mehdi Vatandoost, Rahimeh Pourkhandani, and Neda Ebrahimi. On null and causalpseudoconvex space-times.