Geometry and topology of complete Lorentz spacetimes of constant curvature
GGEOMETRY AND TOPOLOGY OF COMPLETE LORENTZSPACETIMES OF CONSTANT CURVATURE
JEFFREY DANCIGER, FRANC¸ OIS GU´ERITAUD, AND FANNY KASSEL
Abstract.
We study proper, isometric actions of nonsolvable discretegroups Γ on the 3-dimensional Minkowski space R , as limits of actionson the 3-dimensional anti-de Sitter space AdS . To each such action isassociated a deformation of a hyperbolic surface group Γ inside O(2 , is convex cocompact, we prove that Γ acts properly on R , ifand only if this group-level deformation is realized by a deformation ofthe quotient surface that everywhere contracts distances at a uniformrate. We give two applications in this case. (1) Tameness: A completeflat spacetime is homeomorphic to the interior of a compact manifold.(2) Geometric transition: A complete flat spacetime is the rescaled limitof collapsing AdS spacetimes. Introduction
A Lorentzian 3-manifold of constant negative curvature is locally mod-eled on the anti-de Sitter space AdS = PO(2 , / O(2 , RP as the set of negative points with respect to a quadraticform of signature (2 , Minkowski space R , , which is the affine space R endowed withthe Lorentzian structure induced by a quadratic form of signature (2 , identifies with R , ;this basic fact motivates the point of view of this paper that a large classof manifolds modeled on R , (convex cocompact Margulis spacetimes) areinfinitesimal versions of manifolds modeled on AdS . We consider only com-plete Lorentzian manifolds which are quotients of AdS or R , by discretegroups Γ of isometries acting properly discontinuously.The following facts, specific to dimension 3, will be used throughoutthe paper. The anti-de Sitter space AdS identifies with the manifold G = PSL ( R ) endowed with the Lorentzian metric induced by (a multiple of)the Killing form. The group of orientation and time-orientation preservingisometries is G × G acting by right and left multiplication: ( g , g ) · g = g gg − . The Minkowski space R , can be realized as the Lie algebra J.D. is partially supported by the National Science Foundation under the grant DMS1103939. F.G. and F.K. are partially supported by the Agence Nationale de la Rechercheunder the grants DiscGroup (ANR-11-BS01-013) and ETTT (ANR-09-BLAN-0116-01),and through the Labex CEMPI (ANR-11-LABX-0007-01). a r X i v : . [ m a t h . G T ] J un OMPLETE LORENTZ SPACETIMES OF CONSTANT CURVATURE 2 g = sl ( R ). The group of orientation and time-orientation preserving isome-tries is G (cid:110) g acting affinely: ( g, v ) · w = Ad( g ) w + v .Examples of groups of isometries acting properly discontinuously on AdS are easy to construct: one can take Γ = Γ × { } where Γ is any discretesubgroup of G ; in this case the quotient Γ \ AdS identifies with the unittangent bundle to the hyperbolic orbifold Γ \ H . Such quotients are calledstandard. Goldman [Go1] produced the first nonstandard examples by de-forming standard ones, a technique that was later generalized by Kobayashi[Ko2]. Salein [Sa2] then constructed the first examples that were not defor-mations of standard ones.On the other hand, although cyclic examples are readily constructed, itis not obvious that there exist nonsolvable groups acting properly discon-tinuously on R , . The Auslander conjecture in dimension 3, proved byFried–Goldman [FG], states that any discrete group acting properly discon-tinuously and cocompactly on R , is solvable up to finite index, generalizingBieberbach’s theory of crystallographic groups. Milnor [Mi] asked if the co-compactness assumption could be removed. This was answered negativelyby Margulis [Ma1, Ma2], who constructed the first examples of nonabelianfree groups acting properly discontinuously on R , ; the quotient manifoldscoming from such actions are often called Margulis spacetimes . Drumm[Dr1, Dr2] constructed more examples of Margulis spacetimes by introduc-ing crooked planes to produce fundamental domains.1.1.
Proper actions and contraction.
A discrete group Γ acting on AdS by isometries that preserve both orientation and time orientation is deter-mined by two representations j, ρ : Γ → G = PSL ( R ), called the first projec-tion and second projection respectively. We refer to the group of isometriesdetermined by ( j, ρ ) using the notation Γ j,ρ . By work of Kulkarni–Raymond[KR], if such a group Γ j,ρ acts properly on AdS and is torsion-free, then oneof the representations j, ρ must be injective and discrete; if Γ is finitely gen-erated (which we shall always assume), then we may pass to a finite-indexsubgroup that is torsion-free by the Selberg lemma [Se, Lem. 8]. We assumethen that j is injective and discrete. When j is convex cocompact, Kassel[Kas] gave a full characterization of properness of the action of Γ j,ρ in termsof a double contraction condition. Specifically Γ j,ρ acts properly on AdS if and only if either of the following two equivalent conditions holds (up toswitching j and ρ if both are convex cocompact): • ( Lipschitz contraction ) There exists a ( j, ρ )-equivariant Lipschitz map f : H → H with Lipschitz constant < • ( Length contraction )(1.1) sup γ ∈ Γ with λ ( j ( γ )) > λ ( ρ ( γ )) λ ( j ( γ )) < , where λ ( g ) is the hyperbolic translation length of g ∈ G (defined to be 0 if g is not hyperbolic, see (2.1)). This was extended by Gu´eritaud–Kassel [GK] OMPLETE LORENTZ SPACETIMES OF CONSTANT CURVATURE 3 to the case that the finitely generated group j (Γ) is allowed to have para-bolic elements. The two (equivalent) types of contraction appearing aboveare easy to illustrate in the case when ρ is also discrete and injective: the Lip-schitz contraction criterion says that there exists a map j (Γ) \ H → ρ (Γ) \ H (in the correct homotopy class) that uniformly contracts all distances on thesurface, while the length contraction criterion says that any closed geodesicon ρ (Γ) \ H is uniformly shorter than the corresponding geodesic on j (Γ) \ H .Lipschitz contraction easily implies length contraction, but the converse isnot obvious. One important consequence that can be deduced from eithercriterion is that for a fixed convex cocompact j , the representations ρ thatyield a proper action form an open set. In Section 6 (which can be readindependently), we derive topological and geometric information about thequotient manifold directly from the Lipschitz contraction property.We remark that Γ j,ρ does not act properly on AdS in the case thatΓ is a closed surface group and j, ρ are both Fuchsian ( i.e. injective anddiscrete). For Thurston showed, as part of his theory of the asymmetricmetric on Teichm¨uller space [T2], that the best Lipschitz constant of maps j (Γ) \ H → ρ (Γ) \ H (in the correct homotopy class) is ≥
1, with equalityonly if ρ is conjugate to j . However, Γ j,ρ does act properly on a convexsubdomain of AdS ; the resulting AdS manifolds are the globally hyperbolicspacetimes studied by Mess [Me].We now turn to the flat case. A discrete group Γ acting on R , byisometries that preserve both orientation and time orientation is determinedby a representation j : Γ → PSL ( R ) and a j -cocycle u : Γ → sl ( R ), i.e. amap satisfying u ( γ γ ) = u ( γ ) + Ad( j ( γ )) u ( γ )for all γ , γ ∈ Γ. We refer to the group of isometries determined by ( j, u )using the notation Γ j,u , where j (Γ) is the linear part and u the transla-tional part of Γ j,u . The cocycle u may be thought of as an infinitesimaldeformation of j (see Section 2.3). Fried–Goldman [FG] showed that if Γacts properly on R , and is not virtually solvable, then j must be injectiveand discrete on a finite-index subgroup of Γ; in particular j (Γ) is a sur-face group (up to finite index). Unlike in the AdS case, here j (Γ) cannotbe cocompact (see Mess [Me]). In the case that it is convex cocompact,Goldman–Labourie–Margulis [GLM] gave a properness criterion in terms ofthe so-called Margulis invariant . Given the interpretation of this invariantas a derivative of translation lengths [GM], the group Γ j,u (with j convexcocompact) acts properly on R , if and only if, up to replacing u by − u ,the deformation u contracts the lengths of group elements at a uniform rate:sup γ ∈ Γ with λ ( j ( γ )) > dd t (cid:12)(cid:12)(cid:12) t =0 λ ( e tu ( γ ) j ( γ )) λ ( j ( γ )) < . (1.2)As a consequence, for a fixed j , the set of j -cocycles u giving a proper actionis open. The proof involves an extension of the Margulis invariant to thespace of geodesic currents on j (Γ) \ H and the dynamics of the geodesic flow OMPLETE LORENTZ SPACETIMES OF CONSTANT CURVATURE 4 on the affine bundle Γ j,u \ ( G (cid:110) g ) → j (Γ) \ G , where j (Γ) \ G identifies withthe unit tangent bundle of j (Γ) \ H .It is natural to view the properness criterion (1.2) for R , as an infin-itesimal version of the length contraction criterion (1.1) for AdS , with ρ approaching j along the cocycle u . In the first part of this paper, we pursuethis analogy further by developing an R , version of the Lipschitz theoryof [Kas, GK], replacing equivariant Lipschitz maps with their infinitesimalanalogues, namely deformation vector fields that change distances in a uni-formly controlled way. This yields an infinitesimal version of the Lipschitzcontraction criterion as well as a new proof of the infinitesimal length con-traction criterion (1.2) of [GLM].1.2. A new properness criterion for R , . As before, let j : Γ → G =PSL ( R ) be a convex cocompact representation. An infinitesimal deforma-tion of the hyperbolic surface S := j (Γ) \ H is given by a vector field X on the universal cover (cid:101) S = H and a j -cocycle u : Γ → g such that X is( j, u ) -equivariant : for any p ∈ H and γ ∈ Γ, X ( γ · p ) = γ ∗ X ( p ) + u ( γ )( γ · p ) , where Γ acts on H via j , and elements of g such as u ( γ ) are interpretedas Killing vector fields on H in the usual way. A ( j, u )-equivariant vectorfield X should be thought of as the derivative of a family of developingmaps f t : (cid:101) S = H → H describing a varying family of hyperbolic surfaces j t (Γ) \ H , with t = 0 corresponding to the original hyperbolic structure S (hence the map f is the identity of H ). The failure of the vector field X = dd t (cid:12)(cid:12) t =0 f t to descend to the surface is measured by the derivative of theholonomy representation, which is precisely the g -valued cocycle u : u ( γ ) = dd t (cid:12)(cid:12)(cid:12) t =0 j t ( γ ) j ( γ ) − ∈ g . We call u the holonomy derivative of the deformation X .We say an infinitesimal deformation X is k -lipschitz (with a lowercase ‘l’)if for any p (cid:54) = q in H ,dd t (cid:12)(cid:12)(cid:12) t =0 d (cid:0) exp p ( tX ( p )) , exp q ( tX ( q )) (cid:1) ≤ k d ( p, q ) . The lipschitz constant lip( X ) will refer to the infimum of k ∈ R such that X is k -lipschitz. We shall see (Proposition 7.3) that under appropriateconditions, lip( X ) is the derivative of the Lipschitz constants of a family ofdeveloping maps f t tangent to X as above. The lowercase ‘l’ is not intendedto diminish the work of Rudolf Otto Sigismund Lipschitz (K¨onigsberg 1832– Bonn 1903), but rather to distinguish this notion from the traditional onewhile emphasizing its infinitesimal nature (as in the notational conventionfor Lie groups and their Lie algebras). While a Lipschitz section of thetangent bundle is always lipschitz, the converse is false: for example, if χ is the characteristic function of the negative reals, then the 0-lipschitz OMPLETE LORENTZ SPACETIMES OF CONSTANT CURVATURE 5 vector field x (cid:55)→ χ ( x ) ∂∂x on R is not even continuous. The number lip( X )can be negative: this means that the vector field X is in an intuitive sense“contracting”.With this terminology, here is what we prove. Theorem 1.1.
Let Γ be a discrete group, j ∈ Hom(Γ , G ) a convex cocompactrepresentation, and u : Γ → g a j -cocycle. The action of Γ j,u on R , isproperly discontinuous if and only if, up to replacing u by − u , one of theequivalent conditions holds:(1) (Infinitesimal lipschitz contraction) For some k < , there exists a k -lipschitz infinitesimal deformation of j (Γ) \ H with holonomy de-rivative u .(2) (Infinitesimal length contraction) As in [GLM] : sup γ ∈ Γ with λ ( j ( γ )) > dd t (cid:12)(cid:12)(cid:12) t =0 λ ( e tu ( γ ) j ( γ )) λ ( j ( γ )) < . We note that our proof of Theorem 1.1 and the resulting applications isindependent of [GLM]. As in the AdS case, the geometric and topologicaldescriptions of flat Lorentzian manifolds that we give in this paper (The-orem 1.2 and Corollary 1.3 below) are derived directly from the lipschitzcontraction criterion; it is not clear to us that they could be derived fromlength contraction only. The proof of Theorem 1.1 requires j (Γ) to be con-vex cocompact. However, we believe that in the future similar techniquescould be applied, with some adjustment, to the case when j (Γ) containsparabolic elements (as has already been done by [GK] in the AdS case).1.3. The topology of quotients of
AdS and R , . Theorem 1.1 and itsAdS predecessor from [Kas, GK] allow for a complete characterization of thetopology of the quotient manifold when j is convex cocompact. We prove: Theorem 1.2.
Let Γ be a torsion-free discrete group and j ∈ Hom(Γ , G ) aconvex cocompact with quotient surface S = j (Γ) \ H .(1) Let ρ ∈ Hom(Γ , G ) be any representation such that Γ j,ρ acts properlyon AdS . Then the quotient manifold Γ j,ρ \ AdS is a principal S -bundle over S .(2) Let u : Γ → g be any j -cocycle such that Γ j,u acts properly on R , .Then the quotient manifold Γ j,u \ R , is a principal R -bundle over S .In both cases the fibers are timelike geodesics. Based on a question of Margulis, Drumm–Goldman [DG1] conjecturedin the early 1990s that all Margulis spacetimes should be tame , meaninghomeomorphic to the interior of a compact manifold. Since then, Charette–Drumm–Goldman have obtained partial results toward this conjecture, in-cluding a proof in the special case that the linear holonomy is a three-holedsphere group [CDG2]. In the context of Theorem 1.2, we obtain tamenessin both the flat and negatively-curved case as a corollary:
OMPLETE LORENTZ SPACETIMES OF CONSTANT CURVATURE 6
Corollary 1.3. (1) Any manifold which is the quotient of
AdS by agroup of isometries with convex cocompact first projection is Seifertfibered over a hyperbolic orbifold.(2) A complete flat Lorentzian manifold with convex cocompact linearholonomy is homeomorphic to the interior of a handlebody. In the compact case, Corollary 1.3.(1) follows from Kulkarni–Raymond’sdescription of the fundamental groups of quotients of AdS and from clas-sical results of Waldhausen [W] and Scott [Sc] (see [Sa1, § . The noncompact quotients appearing here arefinitely covered by the tame manifolds of Theorem 1.2.(1) and are there-fore tame (e.g. using Tucker’s criterion [Tu]); the Seifert-fibered statementthen follows from classical results of Waldhausen. We note also that The-orem 1.2.(1) and Corollary 1.3.(1) actually hold in the more general casethat j is any finitely generated surface group representation; indeed, theproperness criterion of [Kas] on which we rely holds in this more generalsetting, by [GK].Choi–Goldman have recently announced a different proof of the tame-ness of complete flat Lorentzian manifolds with convex cocompact linearholonomy [Ch, CG]. Their proof, which builds a bordification of the R , spacetime by adding a real projective surface at infinity, is very differentfrom the proof given here. In particular, we do not use any compactificationand our proof is independent of [GLM].1.4. Margulis spacetimes are limits of AdS manifolds.
We also de-velop a geometric transition from AdS geometry to flat Lorentzian geometry.The goal is to find collapsing AdS manifolds which, upon zooming in on thecollapse, limit to a given Margulis spacetime. We obtain two statements thatmake this idea precise, the first in terms of convergence of real projectivestructures, the second in terms of convergence of Lorentzian metrics.The projective geometry approach follows work of Danciger [Da] in de-scribing the transition from hyperbolic to AdS geometry. Both AdS and R , are real projective geometries: each space can be represented as a domainin RP , with isometries acting as projective linear transformations. As such,all manifolds modeled on either AdS or R , naturally inherit a real projec-tive structure. We show that every quotient of R , by a group of isometrieswith convex cocompact linear holonomy is (contained in) the limit of a col-lapsing family of complete AdS manifolds, in the sense that the underlyingreal projective structures converge. Note that collapsing AdS manifolds neednot (and in this case do not) collapse as projective manifolds, because thereis a larger group of coordinate changes that may be used to prevent collapse. Theorem 1.4.
Let M = Γ j,u \ R , be a Margulis spacetime such that S = j (Γ) \ H is a convex cocompact hyperbolic surface. Let t (cid:55)→ j t and t (cid:55)→ ρ t besmooth paths with j = ρ = j and dd t (cid:12)(cid:12) t =0 ρ t j − t = u . OMPLETE LORENTZ SPACETIMES OF CONSTANT CURVATURE 7 (1) There exists δ > such that for all t ∈ (0 , δ ) the group Γ j t ,ρ t actsproperly discontinuously on AdS .(2) There is a smooth family of ( j t , ρ t ) -equivariant diffeomorphisms (de-veloping maps) H × S → AdS , defined for t ∈ (0 , δ ) , determiningcomplete AdS structures A t on the fixed manifold S × S .(3) The real projective structures P t underlying A t converge to a pro-jective structure P on S × S . The Margulis spacetime M is therestriction of P to S × ( − π, π ) , where S = R / π Z . In order to construct this geometric transition very explicitly, we arrangefor the geodesic fibrations of the A t , given by Theorem 1.2, to change con-tinuously in a controlled manner. In particular, the geodesic fibrations ofthe collapsing AdS manifolds converge to a geodesic fibration of the limitingMargulis spacetime. The surface S × { π } in P compactifies each timelikegeodesic fiber, making each fiber into a circle.As a corollary, we derive a second geometric transition statement in termsof convergence of Lorentzian metrics. Corollary 1.5.
Let M be a complete flat Lorentzian -manifold with convexcocompact linear holonomy j (Γ) . Let S = j (Γ) \ H be the associated surface.Then there exist complete anti-de Sitter metrics (cid:37) t on S × S , defined for allsufficiently small t > , such that when restricted to S × ( − π, π ) , the met-rics t − (cid:37) t converge uniformly on compact sets to a complete flat Lorentzianmetric (cid:37) that makes S × ( − π, π ) isometric to M . This second statement is proved using the projective coordinates givenin Theorem 1.4. Note that the convergence of metrics is more delicate inthe Lorentzian setting than in the Riemannian setting. In particular, evenusing the topological characterization of Theorem 1.2, it would be difficultto prove Corollary 1.5 directly.1.5.
Maximally stretched laminations.
The key step in the proof ofTheorem 1.1 is to establish the existence of a maximally stretched lamina-tion . We recall briefly the corresponding statement in the AdS setting, asestablished in [Kas, GK]. Let j, ρ : Γ → G be representations with j convexcocompact (or more generally geometrically finite) and let K be the infi-mum of all possible Lipschitz constants of ( j, ρ )-equivariant maps H → H .If K ≥
1, then there is a nonempty geodesic lamination L in the convexcore of j (Γ) \ H that is “maximally stretched” by any K -Lipschitz ( j, ρ )-equivariant map f : H → H , in the sense that f multiplies arc length byexactly K on the leaves of the lift to H of L . (In fact, a similar resultholds when replacing H by H n and PSL ( R ) (cid:39) SO(2 , by SO( n, .)Now let u : Γ → g be a j -cocycle and consider ( j, u )-equivariant vectorfields X on H . Assume that the infimum k of lipschitz constants of allsuch X satisfies k ≥
0. By analogy, one hopes for the existence of a geodesiclamination that would be stretched at rate exactly k by any k -lipschitz X .This turns out to be true, but there is a crucial problem: it is not clear OMPLETE LORENTZ SPACETIMES OF CONSTANT CURVATURE 8 that a vector field X with the best possible lipschitz constant exists. In-deed, bounded k -lipschitz vector fields on a compact set are not necessarilyequicontinuous, and so the Arzel`a–Ascoli theorem does not apply. In fact, alimit of lipschitz vector fields is something more general that we call a convexfield . A convex field is a closed subset of T H such that the fiber above eachpoint of H is a convex set (Definition 3.1); in other words, a convex fieldis a closed convex set-valued section of the tangent bundle which is uppersemicontinuous in the Hausdorff topology. Theorem 1.6.
Let Γ be a discrete group, j ∈ Hom(Γ , G ) a convex cocompactrepresentation, and u : Γ → g a j -cocycle. Let k ∈ R be the infimum oflipschitz constants of ( j, u ) -equivariant vector fields on H . If k ≥ , thenthere exists a geodesic lamination L in the convex core of S := j (Γ) \ H thatis maximally stretched by any ( j, u ) -equivariant, k -lipschitz convex field X ,meaning that dd t (cid:12)(cid:12)(cid:12) t =0 d (cid:0) exp p ( tx p ) , exp q ( tx q ) (cid:1) = k d ( p, q ) for any distinct points p, q ∈ H on a common leaf of the lift to H of L and any vectors x p ∈ X ( p ) and x q ∈ X ( q ) ; such convex fields X exist. Let us describe briefly how Theorem 1.6 implies Theorem 1.1. When theinfimum k of lipschitz constants is <
0, the action of Γ j,u on R , is proper:this relatively easy fact is the content of Proposition 6.3, whose proof canbe read independently. Theorem 1.6 implies that the converse also holds: if k ≥ L gives a sequence( γ n ) n ∈ N of pairwise distinct elements of Γ that fail to carry a compact subsetof R , off itself under the ( j, u )-action.We mention also that Theorem 1.6 recovers the result, due to Goldman–Labourie–Margulis–Minsky [GLMM], that the length contraction criterion(1.2) still holds when the supremum is taken over simple closed curves ratherthan the entire fundamental group Γ.1.6. Organization of the paper.
In Section 2 we introduce some notationand recall elementary facts about affine actions, the Margulis invariant, andgeodesic laminations. In Section 3, we define and give some basic propertiesabout convex fields. In Section 4 we develop the main tool of the paper,namely the extension theory of lipschitz convex fields on H , in the spiritof the more classical theory of the extension of Lipschitz maps on H . InSection 5 we give a proof of Theorem 1.6, which is needed for the moredifficult direction of Theorem 1.1. In Section 6, we give the connectionbetween geodesic fibrations and contracting lipschitz fields and prove bothdirections of Theorem 1.1, as well as Theorem 1.2. Finally, Section 7 isdedicated to Theorem 1.4, showing how to build AdS manifolds that limitto a given Margulis spacetime. OMPLETE LORENTZ SPACETIMES OF CONSTANT CURVATURE 9
Acknowledgements.
We are very grateful to Thierry Barbot for the keyidea of Proposition 6.3, which gives the connection between Lipschitz mapsof H and fibrations of AdS . This paper was also nurtured by discussionswith Bill Goldman and Virginie Charette during the 2012 special programon Geometry and analysis of surface group representations at the InstitutHenri Poincar´e in Paris. We would like to thank the Institut Henri Poincar´e,the Institut CNRS-Pauli (UMI 2842) in Vienna, and the University of Illi-nois in Urbana-Champaign for giving us the opportunity to work togetherin very stimulating environments. We are also grateful for support from theGEAR Network, funded by the National Science Foundation under grantnumbers DMS 1107452, 1107263, and 1107367 (“RNMS: GEometric struc-tures And Representation varieties”). Finally, we thank Olivier Biquard forencouraging us to derive the metric transition statement, Corollary 1.5, fromour projective geometry formulation of the transition from AdS to R , .2. Notation and preliminaries
Anti-de Sitter and Minkowski spaces.
Throughout the paper, wedenote by G the group SO(2 , (cid:39) PSL ( R ) and by g its Lie algebra. Let (cid:104)·|·(cid:105) be half the Killing form of g : for all v, w ∈ g , (cid:104) v | w (cid:105) = 12 tr (cid:0) ad( v )ad( w ) (cid:1) . As mentioned in the introduction, we identify AdS with the 3-dimensionalreal manifold G , endowed with the bi-invariant Lorentzian structure inducedby (cid:104)·|·(cid:105) . Then the identity component of the group of isometries of AdS is G × G , acting by right and left multiplication:( g , g ) · g := g gg − . (Letting g act on the right and g on the left ensures later compatibilitywith the usual definition of a cocycle.)We identify R , with the Lie algebra g , endowed with the Lorentzianstructure induced by (cid:104)·|·(cid:105) . Then the identity component of the group ofisometries of R , is G (cid:110) g , acting by affine transformations:( g, v ) · w := Ad( g ) w + v. In the rest of the paper, we will write g · w for Ad( g ) w . We shall use the usual terminology for rank-one groups: a nontrivialelement of G is hyperbolic if it has exactly two fixed points in the boundaryat infinity ∂ ∞ H of H , parabolic if it has exactly one fixed point in ∂ ∞ H ,and elliptic if it has a fixed point in H . If g ∈ G is hyperbolic, we willdenote its (oriented) translation axis in H by A g . For any g ∈ G , we set(2.1) λ ( g ) := inf p ∈ H d ( p, g · p ) ≥ . This is the translation length of g if g is hyperbolic, and 0 if g is parabolic,elliptic, or trivial. OMPLETE LORENTZ SPACETIMES OF CONSTANT CURVATURE 10
Affine actions.
Recall that a
Killing field on H is a vector field whoseflow preserves the hyperbolic metric. Each X ∈ g defines a Killing field p (cid:55)−→ dd t (cid:12)(cid:12)(cid:12) t =0 ( e tX · p ) ∈ T p H on H , and any Killing field on H is of this form for a unique X ∈ g .We henceforth identify g with the space of Killing fields on H , writing X ( p ) ∈ T p H for the vector at p ∈ H of the Killing field X ∈ g . Under thisidentification, the adjoint action of G on g coincides with the pushforwardaction of G on vector fields of H :(2.2) (Ad( g ) X )( g · p ) = ( g ∗ X )( g · p ) = g ∗ ( X ( p )) = d p ( L g )( X ( p )) , where L g : H → H is the left translation by g . We can express Killingfields directly in Minkowski space: if g (cid:39) R , is seen as R with the qua-dratic form x + y − z and H as the upper hyperboloid { ( x, y, z ) ∈ R , | x + y − z = − , z > } , then(2.3) X ( p ) = X ∧ p ∈ T p H ⊂ g for all p ∈ H , where ∧ is the natural Minkowski crossproduct on R , :( x , x , x ) ∧ ( y , y , y ) := ( x y − x y , x y − x y , − x y + x y ) . (In Lie-theoretic terms, if we see H as a subset of g as above, then X ( p ) =ad( X ) p for all X ∈ g and p ∈ H ⊂ g .) The cross-product is Ad( G )-equivariant: g · ( v ∧ w ) = ( g · v ) ∧ ( g · w ) for all g ∈ G and v, w ∈ g . Note that ingeneral (cid:104) p | q ∧ r (cid:105) = det( p, q, r ) is invariant under cyclic permutations of p, q, r .Let Γ be a discrete group and j ∈ Hom(Γ , G ) a convex cocompact repre-sentation. By convex cocompact we mean that j is injective and that j (Γ) isa discrete subgroup of G acting cocompactly on the convex hull C j (Γ) ⊂ H ofthe limit set Λ j (Γ) ⊂ ∂ ∞ H (the image of C j (Γ) in j (Γ) \ H is called the convexcore of j (Γ) \ H ); equivalently, the hyperbolic orbifold j (Γ) \ H has finitelymany funnels and no cusp. By definition, a j -cocycle is a map u : Γ → g such that(2.4) u ( γ γ ) = u ( γ ) + j ( γ ) · u ( γ )for all γ , γ ∈ Γ. A j -coboundary is a j -cocycle of the form u X ( γ ) = X − j ( γ ) · X where X ∈ g . The condition (2.4) means exactly that Γ acts on g by affineisometries:(2.5) γ • X = j ( γ ) · X + u ( γ ) . This action fixes a point X ∈ g if and only if u is the coboundary u X . Definition 2.1.
We say that u is a proper deformation of j (Γ) if the Γ-action (2.5) on g is properly discontinuous. OMPLETE LORENTZ SPACETIMES OF CONSTANT CURVATURE 11
Small deformations.
The above terminology of proper deformation comes from the fact that j -cocycles u : Γ → g are the same as infinitesimaldeformations of the homomorphism j , in the following sense. Suppose t (cid:55)→ j t ∈ Hom(Γ , G ) is a smooth path of representations with j = j . For γ ∈ Γ,the derivative dd t (cid:12)(cid:12) t =0 j t ( γ ) takes any p ∈ H to a vector of T j ( γ ) · p H , thesevectors forming a Killing field u ( γ ) as p ranges over H . As above, we see u ( γ ) as an element of g ; if we also see H and its tangent spaces as subsetsof g , then (2.3) yields the formula(2.6) dd t (cid:12)(cid:12)(cid:12) t =0 j t ( γ ) · p = u ( γ ) ∧ ( j ( γ ) · p )for all p ∈ H ⊂ g . Equivalently, Ad ∗ ( dd t j t ( γ )) = ad( u ( γ )) ◦ Ad( j ( γ )). Themultiplicativity relation in Γ is preserved to first order in t if and only if forall γ , γ ∈ Γ,dd t (cid:12)(cid:12)(cid:12) t =0 j t ( γ γ ) · p = dd t (cid:12)(cid:12)(cid:12) t =0 (cid:0) j t ( γ ) · ( j t ( γ ) · p ) (cid:1) = (cid:16) dd t (cid:12)(cid:12)(cid:12) t =0 j t ( γ ) (cid:17) · ( j ( γ ) · p ) + j ( γ ) · (cid:16) dd t (cid:12)(cid:12)(cid:12) t =0 j t ( γ ) · p (cid:17) = u ( γ ) ∧ (cid:0) j ( γ γ ) · p (cid:1) + j ( γ ) · (cid:0) u ( γ ) ∧ ( j ( γ ) · p ) (cid:1) = (cid:0) u ( γ ) + j ( γ ) · u ( γ ) (cid:1) ∧ (cid:0) j ( γ γ ) · p (cid:1) . Since the left-hand side is also equal to u ( γ γ ) ∧ ( j ( γ γ ) · p ), this is equivalentto the fact that u is a j -cocycle. Given X ∈ g , it is easy to check that if j t isthe conjugate of j by g t where dd t (cid:12)(cid:12) t =0 g t = X ∈ g , then the cocycle dd t (cid:12)(cid:12) t =0 j t is the coboundary u X .2.4. The Margulis invariant.
Let u : Γ → g be a j -cocycle. We nowrecall the definition of the Margulis invariant α u ( γ ) for γ ∈ Γ with j ( γ )hyperbolic (see [Ma1, Ma2]). The adjoint action of j ( γ ) ∈ G on g ∼ = R , has three distinct eigenvalues µ > > µ − . Let c + , c − be eigenvectors inthe positive light cone of g , for the respective eigenvalues µ, µ − , and let c ∈ g be the unique positive real multiple of c − ∧ c + with (cid:104) c | c (cid:105) = 1. Forinstance, if j ( γ ) = (cid:18) a a − (cid:19) ∈ PSL ( R ) = G with a >
1, then µ = a andwe can take c + = (cid:18) (cid:19) , c − = (cid:18) − (cid:19) , c = 12 (cid:18) − (cid:19) . By definition, the Margulis invariant of γ is(2.7) α u ( γ ) := (cid:104) u ( γ ) | c (cid:105) . It is an easy exercise to check that α u is invariant under conjugation andthat α u ( γ n ) = | n | α u ( γ ) for all n ∈ Z . The affine action of γ on g by γ • X = j ( γ ) · X + u ( γ ) preserves a unique affine line directed by c , and α u ( γ ) is the (signed) translation length along this line. Since the image ofId g − Ad( j ( γ )) is orthogonal to c , we have α u ( γ ) = (cid:104) γ • X | c (cid:105) for all X ∈ g . OMPLETE LORENTZ SPACETIMES OF CONSTANT CURVATURE 12
In particular, if u is a coboundary, then α u ( γ ) = 0 for all γ . If u and u are two j -cocycles and t , t ∈ R , then α t u + t u = t α u + t α u . Thus α u depends only on the cohomology class of u .Note that the projection of the Killing field u ( γ ) ∈ g to the orientedtranslation axis A j ( γ ) ⊂ H is the same at all points p of A j ( γ ) , equal to α u ( γ ). Indeed, the unit tangent vector to A j ( γ ) at p is c ∧ p , and so the A j ( γ ) -component π A j ( γ ) of u ( γ )( p ) = u ( γ ) ∧ p ∈ T p H is π A j ( γ ) (cid:0) u ( γ )( p ) (cid:1) = (cid:104) c ∧ p | u ( γ ) ∧ p (cid:105) = (cid:104) u ( γ ) | p ∧ ( c ∧ p ) (cid:105) = (cid:104) u ( γ ) | c (cid:105) = α u ( γ )(2.8)since c and p are mutually orthogonal (respectively spacelike and timelike)with unit norms. More generally, any Killing field always has a constantcomponent along any given line.2.5. Length derivative.
Finally, we recall that if the j -cocycle u is thederivative dd t (cid:12)(cid:12) t =0 j t of a smooth path t (cid:55)→ j t ∈ Hom(Γ , G ) with j = j , thenthe Margulis invariant α u ( γ ) associated with u is also the t -derivative of thelength λ ( j t ( γ )) of the geodesic curve in the class of γ : α u ( γ ) = dd t (cid:12)(cid:12)(cid:12) t =0 λ ( j t ( γ )) = dd t (cid:12)(cid:12)(cid:12) t =0 λ (cid:0) e tu ( γ ) j ( γ ) (cid:1) . This was first observed by Goldman–Margulis [GM]. Here is a short expla-nation. By conjugating j t ( γ ) by a smooth path based at 1 ∈ G , we mayassume that the translation axis of j t ( γ ) is constant; indeed, λ is invari-ant under conjugation, and conjugation changes u by a coboundary. Thekey point is that j ( γ ) = e λ ( j ( γ )) c , where c is the unit spacelike vector ofSection 2.4. Since j t ( γ ) has the same translation axis as j ( γ ), we can write j t ( γ ) = e λ ( j t ( γ )) c = e [ λ ( j t ( γ )) − λ ( j ( γ ))] c j ( γ ) . Thus u ( γ ) = dd t (cid:12)(cid:12) t =0 λ ( j t ( γ )) c . The formula follows: since (cid:104) c | c (cid:105) = 1, α u ( γ ) = (cid:104) u ( γ ) | c (cid:105) = dd t (cid:12)(cid:12)(cid:12) t =0 λ ( j t ( γ )) . By rigidity of the marked length spectrum for surfaces (see [DG3]) wethus have α u = 0 if and only if u is a coboundary.2.6. Geodesic laminations.
Let Ω be an open subset of H . In this paper,we call geodesic lamination in Ω any closed subset ˜ L of Ω endowed witha partition into straight lines, called leaves . We allow leaves to end at theboundary of Ω in H . Note that the disjointness of the leaves implies thatthe collection of leaves is closed in the C sense: any limiting segment σ ofa sequence of leaf segments σ i is a leaf segment (otherwise the leaf of ˜ L through any point of σ would intersect the σ i ). If we worked in H n with n >
2, then C -closedness would have to become part of the definition. OMPLETE LORENTZ SPACETIMES OF CONSTANT CURVATURE 13
When ˜ L and Ω are globally invariant under some discrete group j (Γ), wealso call lamination the projection of ˜ L to the quotient j (Γ) \ Ω (it is a closeddisjoint union of injectively immersed geodesic copies of the circle and/orthe line). In such a quotient lamination, if some half-leaf does not escape toinfinity, then it accumulates on a sublamination which can be approachedby a sequence of simple closed geodesics. Geodesic laminations are thus insome intuitive sense a generalization of simple closed (multi)curves.3.
Vector fields and convex fields
Some fundamental objects in the paper are equivariant, lipschitz vectorfields, as well as what we call convex fields . Definition 3.1. A convex field on H is a closed subset X of the tangentbundle T H whose intersection X ( p ) with T p H is convex for any p ∈ H .Equivalently, a convex field is a subset X of H whose intersection X ( p )with T p H is convex and closed for any p and such that X ( p ) depends in anupper semicontinuous way on p for the Hausdorff topology. For instance,any continuous vector field X on H is a convex field; we shall assume allvector fields in the paper to be continuous . In general, we do allow certainfibers X ( p ) to be empty, but say that the convex field X is defined over aset A ⊂ H if all fibers above A are nonempty. We shall use the notation X ( A ) := (cid:91) p ∈ A X ( p ) ⊂ T H and(3.1) (cid:107) X ( A ) (cid:107) := sup x ∈ X ( A ) (cid:107) x (cid:107) . Definitions and basic properties of convex fields.
For any convexfields X and X and any real-valued functions ψ and ψ , we define the sum ψ X + ψ X fiberwise:( ψ X + ψ X )( p ) = (cid:8) ψ ( p ) v + ψ ( p ) v | v i ∈ X i ( p ) (cid:9) . It is still a convex field.
Definition 3.2.
Let Γ be a discrete group, j ∈ Hom(Γ , G ) a representation,and u : Γ → g a j -cocycle. We say that a convex field X on H is ( j, u ) -equivariant if for all γ ∈ Γ and p ∈ H , X ( j ( γ ) · p ) = j ( γ ) ∗ ( X ( p )) + u ( γ )( j ( γ ) · p ) . A ( j, j -invariant .If t (cid:55)→ j t is a deformation of j tangent to u , then the ( j, u )-equivarianceof a vector field X expresses the fact that whenever j ( γ ) · p = q , the relationpersists to first order under the flow of X , that is, d ( j t ( γ ) · p t , q t ) = o ( t )where p t = exp p ( tX ( p )) and q t = exp q ( tX ( q )). OMPLETE LORENTZ SPACETIMES OF CONSTANT CURVATURE 14
We can rephrase Definition 3.2 in terms of group actions. The group Γ actson convex fields via the pushforward action ( γ ∗ X )( j ( γ ) · p ) = j ( γ ) ∗ ( X ( p )),and also via the affine u -action(3.2) γ • X = γ ∗ X + u ( γ )(which is a group action, due to (2.2) and the cocycle condition (2.4)). Aconvex field X is ( j, u )-equivariant (resp. j -invariant) if and only if γ • X = X (resp. γ ∗ X = X ) for all γ ∈ Γ. Definition 3.3.
A convex field X is k -lipschitz (lowercase ‘l’) if for anydistinct points p, q ∈ H and any vectors x p ∈ X ( p ) and x q ∈ X ( q ), the rateof change of the distance between p and q satisfies(3.3) d (cid:48) ( x p , x q ) := dd t (cid:12)(cid:12)(cid:12) t =0 d (cid:0) exp p ( tx p ) , exp q ( tx q ) (cid:1) ≤ k d ( p, q ) . The lipschitz constant of X , denoted by lip( X ), is the infimum of k ∈ R such that X is k -lipschitz. For A ⊂ H , we setlip A ( X ) := lip( X ( A )) . Finally, for p ∈ H , we define the local lipschitz constant lip p ( X ) to be theinfimum of lip U ( X ) over all neighborhoods U of p in H . We shall often usethe notation d (cid:48) X ( p, q ) := sup (cid:8) d (cid:48) ( x p , x q ) | x p ∈ X ( p ) , x q ∈ X ( q ) (cid:9) . The inverse of the map exp p : T p H → H will be written log p .A diagonal argument shows that the “local lipschitz constant” function p (cid:55)→ lip p ( X ) is upper semicontinuous: for any converging sequence p n → p ,lip p ( X ) ≥ lim sup n → + ∞ lip p n ( X ) . In order to compute (or estimate) lipschitz constants, we will often makeuse of the following observation (see Figure 1).
Remark 3.4.
The quantity d (cid:48) ( x p , x q ) is the difference of the (signed) pro-jections of x p and x q to the geodesic line ( p, q ) ⊂ H , oriented from p to q . p q ( p, q ) x p x q Figure 1.
The quantity d (cid:48) ( x p , x q ) may be calculated as thedifference of signed projections of x p and x q to the line ( p, q ).Here the contribution from x p is negative ( x p pushes p to-wards q ), while the contribution from x q is positive ( x q pushes q away from p ). OMPLETE LORENTZ SPACETIMES OF CONSTANT CURVATURE 15
In the case that X is a smooth vector field, the local lipschitz constant isgiven by the formula(3.4) lip p ( X ) = sup y ∈ T p H (cid:104)∇ y X, y (cid:105) , where ∇ is the Levi-Civita connection and T p H denotes the unit tangentvectors based at p . This is the vector field analogue of the formula Lip p ( f ) = (cid:107) d p f (cid:107) for smooth maps f .The following remarks are straightforward. Observation 3.5.
Let X be a convex field.(i) If X is ( j, u ) -equivariant and if X is a j -invariant convex field, thenthe convex field X + X is ( j, u ) -equivariant.(ii) If ( X i ) i ∈ I is a family of ( j, u ) -equivariant convex fields with (cid:83) i ∈ I X i ( p ) bounded for all p ∈ H , and ( µ i ) i ∈ I a family of nonnegative reals sum-ming up to , then the convex field (cid:80) i ∈ I µ i X i is well defined and ( j, u ) -equivariant.(iii) If in addition X i is k i -lipschitz, with ( k i ) i ∈ I bounded, then the convexfield (cid:80) i ∈ I µ i X i is (cid:0)(cid:80) i ∈ I µ i k i (cid:1) -lipschitz.(iv) Subdivision: if a segment [ p, q ] is covered by open sets U i such that lip U i ( X ) ≤ k for all i , then d (cid:48) X ( p, q ) ≤ k d ( p, q ) .(v) In particular, if A ⊂ H is convex, then lip A ( X ) = sup p ∈ A lip p ( X ) .(vi) If d (cid:48) X ( p, q ) = k := lip( X ) , then d (cid:48) X ( p, r ) = d (cid:48) X ( r, q ) = k for any point r in the interior of the geodesic segment [ p, q ] ; in this case we say thatthe segment [ p, q ] is k -stretched by X .(vii) Invariance: if X is ( j, u ) -equivariant, then lip j ( γ ) · A ( X ) = lip A ( X ) and lip j ( γ ) · p ( X ) = lip p ( X ) for all γ ∈ Γ , all A ⊂ H , and all p ∈ H .(viii) The map d (cid:48) is subscript-additive: d (cid:48) X ( p, q ) + d (cid:48) Y ( p, q ) = d (cid:48) X + Y ( p, q ) .(ix) The map d (cid:48) X ( · , · ) is uniformly if and only if X is a Killing field. If k <
0, then any k -lipschitz vector field X on H tends to bring pointscloser together; in particular, X has a positive inward component on theboundary of any large enough round ball of fixed center. By Brouwer’stheorem, X therefore has a zero in H , necessarily unique since k <
0. Infact, this extends to convex fields:
Proposition 3.6.
Any k -lipschitz convex field X with k < , defined on allof H , has a unique zero (that is, there is a unique fiber X ( p ) containing ∈ T p H ).Proof. We prove this by contradiction: suppose X is a counterexample; letus construct a vector field Y on a large ball B of H with no zero, but withpositive inward component everywhere on ∂B . Fix p ∈ H and x ∈ X ( p ). If B is a large enough ball centered at p , of radius R , then every vector y ∈ X ( q )for q ∈ ∂B is inward-pointing because d (cid:48) ( x, y ) ≤ k d ( p, q ) = kR (cid:28)
0. Since X has no zero and is closed in T H , with convex fibers, we can find for any q ∈ B a neighborhood V q of q in H and a vector field Y q defined on V q , OMPLETE LORENTZ SPACETIMES OF CONSTANT CURVATURE 16 such that Y ( q (cid:48) ) has positive scalar product with any vector of X ( q (cid:48) ) when q (cid:48) ∈ V q . Moreover, we can assume that the fields Y q are all inward-pointingin a neighborhood of ∂B . Extract a finite covering V q ∪ · · · ∪ V q n of B and pick a partition of unity ( ψ i ) ≤ i ≤ n adapted to the V q i . Then the vectorfield Y := (cid:80) ni =1 ψ i Y q i is continuous, defined on all of B , inward-pointing on ∂B , and with no zero (it everywhere pairs to positive values with X ). Thisvector field Y cannot exist by Brouwer’s theorem, hence X must have a zero— necessarily unique since X is k -lipschitz with k < (cid:3) Proposition 3.6 and its proof may be compared to Kakutani’s fixed pointtheorem for set-valued maps [Kak]. Here are two related results, which willbe important throughout the paper:
Proposition 3.7.
Any Hausdorff limit of a sequence of convex fields thatare uniformly bounded and k -lipschitz over a ball B , is a k -lipschitz convexfield defined over B .Proof. Let ( X n ) n ∈ N be such a sequence of convex fields, and X ∞ their Haus-dorff limit (a closed subset of T H ). For any p ∈ B , the closed set X ∞ ( p ) isnonempty because ( X n ( p )) n ∈ N is uniformly bounded. To see that X ∞ is k -lipschitz, we fix distinct points p, q ∈ B and consider sequences x n ∈ X n ( p n )converging to x ∈ X ∞ ( p ) and y n ∈ X n ( q n ) converging to y ∈ X ∞ ( q ). Then d (cid:48) ( x n , y n ) ≤ k d ( p n , q n ) for all n , and taking the limit as n → + ∞ gives d (cid:48) ( x, y ) ≤ k d ( p, q ).We now check that X ∞ ( p ) is convex for all p ∈ B . By adding a Killingfield, it is enough to show that if the zero vector lies in the convex hullConv( X ∞ ( p )) of X ∞ ( p ) in T p H , then 0 ∈ X ∞ ( p ). Consider the vector field W : q (cid:55)→ log q ( p ) that points toward p with strength equal to the distancefrom p . By convexity of the distance function in H , the vector field W is − Y n = X n + cW , where c (cid:29) n , the convex field Y n is − p ∈ ∂B , all vectorsof Y n ( p ) point strictly into B . By the proof of Proposition 3.6, the convexfield Y n has a zero at a point q n ∈ B , and after taking a subsequence wemay assume that ( q n ) n ∈ N converges to some q ∈ B . Then Y ∞ = X ∞ + cW is − ∈ Y ∞ ( q ). The fiberwise convex hull of Y ∞ is still − ∈ T q H n . Note that X ∞ ( p ) = Y ∞ ( p ). Therefore,if 0 ∈ Conv( X ∞ ( p )), then p = q and we have 0 ∈ X ∞ ( p ) = Y ∞ ( p ). (cid:3) Proposition 3.8.
Let X be a -lipschitz convex field defined over H . Forany Killing field Y on H , the set C := { p ∈ H | Y ( p ) ∈ X } is convex.Proof. We may assume that Y is the zero vector field 0, up to replacing X with X − Y (which is still 0-lipschitz). Consider two distinct points p, q ∈ C and a point r on the segment [ p, q ]. Thinking of [ p, q ] as thehorizontal direction, consider a point r (cid:48) ∈ H very close to r above [ p, q ].Let x r (cid:48) ∈ X ( r (cid:48) ). Since d (cid:48) (0( p ) , x r (cid:48) ) ≤ d (cid:48) (0( q ) , x r (cid:48) ) ≤
0, the vector x r (cid:48) must belong to a narrow angular sector around the vertical, downward OMPLETE LORENTZ SPACETIMES OF CONSTANT CURVATURE 17 direction (see Figure 2). In the limit as r (cid:48) approaches r from above [ p, q ],we find that X ( r ) must contain a vector x orthogonal to [ p, q ] pointing(weakly) down. By letting r (cid:48) approach r from below, we find that X ( r ) alsocontains a vector x orthogonal to [ p, q ] pointing (weakly) up. Since 0( r ) isin the convex hull of { x , x } , we have 0( r ) ∈ X ( r ), hence r ∈ C . (cid:3) r (cid:48) H p qrx r (cid:48) Figure 2. If d (cid:48) X ( p, q ) = 0 = lip( X ), then X contains therestriction of a Killing field to [ p, q ].3.2. Computing the Margulis invariant from an equivariant vectorfield.
The Margulis invariant α u and the map d (cid:48) : T H × T H → R of Defini-tion 3.3 both record rates of variation of hyperbolic lengths. We now explainhow one can be expressed in terms of the other via equivariant convex fields.Fix a ( j, u )-equivariant convex field X on H . For γ ∈ Γ with j ( γ ) hyper-bolic, choose a point p on the oriented axis A j ( γ ) ⊂ H of j ( γ ) and a vector x p in the convex set X ( p ). Define also x j ( γ ) · p := j ( γ ) ∗ ( X p ) + u ( γ )( j ( γ ) · p ),which belongs to X ( j ( γ ) · p ) by equivariance of X . If π A j ( γ ) ( x ) denotes the A j ( γ ) -component of any vector x ∈ T H based at a point of A j ( γ ) , then (2.8)implies d (cid:48) ( x p , x j ( γ ) · p ) = π A j ( γ ) ( x j ( γ ) · p ) − π A j ( γ ) ( x p )(3.5) = π A j ( γ ) (cid:0) u ( γ )( j ( γ ) · p ) (cid:1) = α u ( γ ) . In particular,(3.6) α u ( γ ) λ ( j ( γ )) ≤ lip( X ) . We now assume that X is a smooth equivariant vector field . The function ν X : A j ( γ ) → R defined by ν X ( p ) = π A j ( γ ) ( X ( p )) satisfies ν X ( j ( γ ) · p ) = ν X ( p ) + α u ( γ ), hence the derivative ν (cid:48) X is periodic and descends to a scalar function on the geodesic loop c γ representing the isotopy class of γ on thehyperbolic orbifold S = j (Γ) \ H . By construction, d (cid:48) X ( p, j ( γ ) · p ) is just theintegral of ν (cid:48) X along c γ for the Lebesgue measure d µ γ . Therefore,(3.7) α u ( γ ) = (cid:90) c γ ν (cid:48) X d µ γ . This formula holds independently of the choice of the smooth equivariantvector field X .Moreover, we can generalize this process and extend α u to the space ofgeodesic currents. This extension was described, in different terms, in [L]and [GLM]. First, the functions ν X : A j ( γ ) → R above, for γ ∈ Γ with
OMPLETE LORENTZ SPACETIMES OF CONSTANT CURVATURE 18 j ( γ ) hyperbolic, piece together and extend to a smooth function on theunit tangent bundle T H of H , which we again denote by ν X : it takes y ∈ T p H to (cid:104) X ( p ) , y (cid:105) ∈ R . By construction, the derivative ν (cid:48) X : T H → R of ν X along the geodesic flow satisfies(3.8) d (cid:48) X ( p, q ) = (cid:90) [ p,q ] ν (cid:48) X for any distinct p, q ∈ H , where the geodesic flow line [ p, q ] ⊂ T H from p to q is endowed with its natural Lebesgue measure. In terms of the Levi-Civita connection ∇ , ν (cid:48) X ( y ) = (cid:104)∇ y X, y (cid:105) T p H for any unit vector y in the Euclidean plane T p H . Remarkably, the function ν (cid:48) X is j (Γ)-invariant, because j ( γ ) ∗ X and X differ only by a Killing field u ( γ ), and Killing fields have constant component along any geodesic flowline. Therefore ν (cid:48) X descends to the unit tangent bundle T S of S = j (Γ) \ H ,and (3.7) can be rewritten in the form(3.9) α u ( γ ) = (cid:90) T S ν (cid:48) X d µ γ , which extends to all geodesic currents d µ on T S since ν (cid:48) X is continuous.Here is a useful consequence of this construction. Proposition 3.9.
Suppose there exists a ( j, u ) -equivariant convex field Y that k -stretches a geodesic lamination L in the convex core of S , in thesense that d (cid:48) Y ( p, q ) = k d ( p, q ) for any distinct p, q ∈ H on a common leafof preimage ˜ L ⊂ H of L . Then for any sequence ( γ n ) n ∈ N of elementsof Γ whose translation axes A j ( γ n ) converge to (a sublamination of ) ˜ L inthe Hausdorff topology, lim n → + ∞ α u ( γ n ) λ ( j ( γ n )) = k. Proof.
Let X be any smooth, ( j, u )-equivariant vector field. Then Y − X is j -invariant, hence bounded over the convex core by convex cocompactness.Thus for any distinct p, q ∈ H on a common leaf of ˜ L , the difference be-tween d (cid:48) Y ( p, q ) = k d ( p, q ) and d (cid:48) X ( p, q ) is bounded. Hence the average valueof ν (cid:48) X over a segment of length L of L is k + O ( L − ). On the other hand, ν (cid:48) X is a uniformly continuous function on T S . Since the loops representing γ n lift to long segments c n ⊂ H , of length λ ( j ( γ n )), uniformly close to leavesof ˜ L , uniform continuity of ν (cid:48) X implies α u ( γ n ) = (cid:82) c n ν (cid:48) X ∼ k λ ( j ( γ n )). (cid:3) A priori bounds inside the convex core.
Let j ∈ Hom(Γ , G ) beconvex cocompact and let U ⊂ H be the preimage of the interior of theconvex core of j (Γ) \ H . Let u : Γ → g be a j -cocycle. In this section weprove the following. OMPLETE LORENTZ SPACETIMES OF CONSTANT CURVATURE 19
Proposition 3.10.
For any compact subset C of U and any k ∈ R , thereexists R > such that for any k -lipschitz, ( j, u ) -equivariant convex field X ,any vector x ∈ X ( C ) satisfies (cid:107) x (cid:107) < R .Proof. Consider γ ∈ Γ such that j ( γ ) is hyperbolic, with translation axis A = A j ( γ ) . Let p ∈ H (cid:114) A and q := j ( γ ) · p be at distance r > A . Letˆ n be the unit vector field pointing away from A in the direction orthogonalto A , and θ ∈ (0 , π ) the angle at p (or q ) between − ˆ n and the segment [ p, q ](see Figure 3). A classical formula gives tan θ = coth λ ( j ( γ )) / r . We claim thatfor any x ∈ T p H ,(3.10) d (cid:48) ( x, j ( γ ) ∗ x ) = 2 (cid:104) x, ˆ n (cid:105) cos θ. Indeed, let ˆ a be the unit vector field along the segment [ p, q ], oriented to-wards q . Let ˆ e be the unit vector field orthogonal to ˆ n such that at p , thevectors ˆ a and ˆ e form an angle θ (cid:48) = π − θ . By symmetry, at q , the vec-tors ˆ a and ˆ e form an angle − θ (cid:48) . Note that the fields ˆ e and ˆ n are invariantunder j ( γ ). We have p q ˆ e − ˆ n ˆ a ˆ n ˆ a ˆ er r λ ( j ( γ )) A θ θ Figure 3.
Computing d (cid:48) ( x, j ( γ ) ∗ x ) in the proof of Proposi-tion 3.10. (cid:104) x, ˆ a (cid:105) = (cid:104) x, ˆ e (cid:105) cos θ (cid:48) − (cid:104) x, ˆ n (cid:105) sin θ (cid:48) , (cid:104) j ( γ ) ∗ x, ˆ a (cid:105) = (cid:104) x, ˆ e (cid:105) cos θ (cid:48) + (cid:104) x, ˆ n (cid:105) sin θ (cid:48) , hence d (cid:48) ( x, j ( γ ) ∗ x ) = 2 (cid:104) x, ˆ n (cid:105) sin θ (cid:48) = 2 (cid:104) x, ˆ n (cid:105) cos θ (by Remark 3.4), prov-ing (3.10).Now, given a point p ∈ U , we choose three elements γ , γ , γ ∈ Γ suchthat j ( γ i ) is hyperbolic and its translation axis A i = A j ( γ i ) does not con-tain p , for each 1 ≤ i ≤
3. Set q i = j ( γ i ) · p and λ i = λ ( j ( γ i )), and define r i , θ i , ˆ n i similarly to above. Since p ∈ U , we may choose the γ i so that thepositive span of the ˆ n i is all of T p H , i.e. the ˆ n i are not all contained ina closed half-plane. Let ϑ ii (cid:48) ∈ (0 , π ) be the angle between ˆ n i and ˆ n i (cid:48) (seeFigure 4). Set Q := max ≤ i ≤ (cid:107) u ( γ i )( q i ) (cid:107) .Now consider a ( j, u )-equivariant convex field X and a vector x ∈ X ( p ).For 1 ≤ i ≤
3, the vector γ i • x = j ( γ i ) ∗ x + u ( γ i )( q i ) ∈ T q i H also belongs OMPLETE LORENTZ SPACETIMES OF CONSTANT CURVATURE 20 A A A ˆ n ˆ n ˆ n ϑ ϑ ϑ p Figure 4.
Proof of Proposition 3.10: the angles ϑ ii (cid:48) arebounded away from π for p lying in a compact subset of theinterior of the convex core.to X . By (3.10), d (cid:48) ( x, γ i • x ) ≥ (cid:104) x, ˆ n i (cid:105) cos θ i − Q. However, d ( p, q i ) ≤ λ i + 2 r i . If X is k -lipschitz, it follows that2 (cid:104) x, ˆ n i (cid:105) ≤ k ( λ i + 2 r i ) + Q cos θ i . Now, for some i, i (cid:48) the vector x makes an angle ≤ ϑ ii (cid:48) / n i . Thus (cid:107) x (cid:107) ≤ max i,i (cid:48) k ( λ i + 2 r i ) + Q cos θ i cos (cid:0) ϑ ii (cid:48) (cid:1) . Since θ i = arctan coth λ i / r i is bounded away from π when r i is bounded awayfrom 0, and Q is bounded by a continuous function of p ∈ C , this givesa uniform bound in an open neighborhood of p where r i is bounded awayfrom 0 and ϑ ii (cid:48) is bounded away from π . (cid:3) Standard fields in the funnels.
We now focus on the exterior of theconvex core, namely on the so-called funnels of the hyperbolic surface (ororbifold) j (Γ) \ H . We define explicit vector fields in the funnels which willbe used in Section 5 to extend a k -lipschitz convex field on the interior ofthe convex core to the entire surface.We work explicitly with Fermi coordinates. Let A be an oriented geodesicline in H and p a point on A . For q ∈ H , let p (cid:48) be the point of A closest to q ;we define ξ ( q ) ∈ R and η ( q ) ∈ R to be the signed distance from p to p (cid:48) andfrom p (cid:48) to q , respectively. The numbers ξ ( q ) and η ( q ) are called the Fermicoordinates of q with respect to ( A , p ). Note that the Fermi coordinate map F : R ∼ → H sends R × { } isometrically to A , and { ξ } × R isometrically tothe geodesic line orthogonal to A at F ( ξ, F ( R × { η } ), whichlie at constant (signed) distance η from A , are called hypercycles . OMPLETE LORENTZ SPACETIMES OF CONSTANT CURVATURE 21
For k, r ∈ R , we define the ( k, r ) -standard vector field with respect to( p, A ) by(3.11) X kr : F ( ξ, η ) (cid:55)−→ kξ ∂F∂ξ + rη ∂F∂η . It is a smooth vector field on H . Proposition 3.11.
Suppose that r < min( k, . Then the vector field X kr is k -lipschitz. Further, for any η > there exists ε > such that at any p ∈ H with d ( p, A ) ≥ η , the local lipschitz constant satisfies lip p ( X kr ) ≤ k − ε . Inparticular, d (cid:48) X kr ( p, q ) < k d ( p, q ) for all distinct p, q ∈ F ( R × R ∗− ) .Proof. For any tangent vector y = a ∂F∂η + b ∂F∂ξ ∈ T F ( ξ,η ) , where a, b, ξ, η ∈ R ,direct computation yields(3.12) (cid:104)∇ y X kr , y (cid:105) = ra + kb cosh η + rb η sinh η cosh η. In particular, (cid:104)∇ y X kr , y (cid:105) ≤ k ( a + b cosh η ) = k (cid:107) y (cid:107) , hence by (3.4) wehave lip F ( ξ,η ) ( X kr ) ≤ k , which implies that lip( X kr ) ≤ k (Observation 3.5).If we assume that | η | ≥ η , then (3.12) gives the more precise estimate (cid:104)∇ y X kr , y (cid:105) < max { r, k + rη tanh η } (cid:107) y (cid:107) , hence lip F ( ξ,η ) ( X kr ) is uniformlybounded away from k by (3.4). To deduce that d (cid:48) X kr ( p, q ) < k d ( p, q ) for all p, q ∈ F ( R × R ∗− ), we use Observation 3.5.(v). (cid:3) Now let j : Γ → G be convex cocompact and let u : Γ → g be a j -cocycle.For γ ∈ Γ with j ( γ ) hyperbolic, let F : R → H be a Fermi coordinatemap with respect to the translation axis A j ( γ ) and let X kr be the standardvector field given by (3.11). If u ( γ ) is an infinitesimal translation along A j ( γ ) (which we can always assume after adjusting u by a coboundary), then X kr is ( j | (cid:104) γ (cid:105) , u | (cid:104) γ (cid:105) )-equivariant if and only if k = k γ := α u ( γ ) /λ ( j ( γ )). In thecase that γ is a peripheral element, we orient A j ( γ ) so that F ( R × R ∗− ) is acomponent of the complement of the convex core; this region covers a funnelof the quotient j (Γ) \ H . Definition 3.12. • For peripheral γ , we say that a ( j | (cid:104) γ (cid:105) , u | (cid:104) γ (cid:105) )-equi-variant convex field X on H is standard in the funnel F ( R × R ∗− ) ifthere exists η < X coincides on F ( R × ( −∞ , η )), up toaddition of a Killing field, with a vector field of the form X kr . • We say that a ( j, u )-equivariant convex field X on H is standard inthe funnels if it is standard in every funnel.The following proposition, in combination with the lipschitz extensiontheory of Section 4, will be used in Section 5 to extend k -lipschitz convexfields defined on the interior of the convex core to k -lipschitz convex fieldsdefined over all of H . Proposition 3.13.
For γ ∈ Γ with j ( γ ) hyperbolic, let X be a locallybounded, ( j | (cid:104) γ (cid:105) , u | (cid:104) γ (cid:105) ) -equivariant convex field defined over a j ( (cid:104) γ (cid:105) ) -invariantsubset Ω (cid:54) = ∅ of H . OMPLETE LORENTZ SPACETIMES OF CONSTANT CURVATURE 22 (1) We have lip( X ) ≥ k γ := α u ( γ ) /λ ( j ( γ )) .(2) Suppose Ω is a hypercycle F ( R × { η } ) with η > , and let N := F ( R × ( −∞ , η ]) for some η < . There is an extension Y of X to Ω ∪ N such that • Y is standard on N (in particular, lip N ( Y ) ≤ k γ ), • lip( Y ) = lip( X ) , • d (cid:48) Y ( p, q ) < k γ d ( p, q ) for all p ∈ N and q ∈ Ω . Note that unlike in Section 3.2, here we do not assume X to be definedover A j ( γ ) , hence Proposition 3.13.(1) does not follow directly from (3.5). Proof.
We first prove (1). Without loss of generality, we may assume thatΩ is compact modulo j ( (cid:104) γ (cid:105) ). Up to adjusting u by a coboundary, we mayassume that u ( γ ) is an infinitesimal translation along the axis A j ( γ ) . Set k := k γ and fix r < min(0 , k ). The convex field X − X kr is j | (cid:104) γ (cid:105) -invariantand locally bounded, hence globally bounded on Ω: there exists b > (cid:107) ( X − X kr )(Ω) (cid:107) < b (notation (3.1)). For p, q ∈ Ω, d (cid:48) X ( p, q ) d ( p, q ) ≥ d (cid:48) X kr ( p, q ) d ( p, q ) − bd ( p, q ) . However, for points p, q ∈ Ω further and further apart on fixed hypercycles,the ratio d (cid:48) X kr ( p, q ) /d ( p, q ) limits to k . Indeed, if p and q both belong toΩ ∩ A j ( γ ) , then d (cid:48) X kr ( p, q ) = k d ( p, q ). Otherwise, note that if p and q arevery far apart, the segment [ p, q ] spends most of its length close to theaxis A j ( γ ) ; we can then conclude using (3.8) and the uniform continuity of ν (cid:48) X kr near A j ( γ ) . As a consequence, lip( X ) ≥ k , proving (1).For (2), choose R < r − b | η tanh η | and define Y to be X on Ω and X kR on N . Then Y is k -lipschitz on N by Proposition 3.11. Thus we only need tocheck that d (cid:48) Y ( p, q ) < k d ( p, q ) for all p ∈ N and q ∈ Ω. Let θ ≤ π/ p between [ p, q ] and ∂F∂η ( p ). Then cos θ ≥ tanh | η | by a standardtrigonometric formula. In particular, d (cid:48) (cid:16) ∂F∂η ( p ) , q ) (cid:17) ≤ − tanh | η | . Thenfor x ∈ X ( q ), d (cid:48) ( Y ( p ) , x ) = d (cid:48) ( X kR ( p ) , x )= d (cid:48) (cid:18) X kr ( p ) + ( R − r ) η ( p ) ∂F∂η ( p ) , X kr ( q ) + x − X kr ( q ) (cid:19) = d (cid:48) (cid:0) X kr ( p ) , X kr ( q ) (cid:1) + d (cid:48) (cid:0) p ) , x − X kr ( q ) (cid:1) + d (cid:48) (cid:18) ( R − r ) η ( p ) ∂F∂η ( p ) , q ) (cid:19) ≤ k d ( p, q ) + b + ( R − r ) η ( − tanh | η | ) , which is < k d ( p, q ) by choice of R . (cid:3) OMPLETE LORENTZ SPACETIMES OF CONSTANT CURVATURE 23 Extension of lipschitz convex fields
To produce the lipschitz convex fields promised in Theorem 1.6, we willneed to extend lipschitz convex fields that are only defined on part of H .This falls into the subject of Lipschitz extension, a topic initiated by Kirsz-braun’s theorem [Ki] to the effect that a partially-defined, K -Lipschitz mapfrom a Euclidean space to another always extends (with the same Lipschitzconstant K ) to the whole space. An analogue of Kirszbraun’s theorem in H n (when K ≥
1) was proved by Valentine [V]. We need a generalization ofthis in several directions: • at the infinitesimal level ( k -lipschitz fields, not K -Lipschitz maps); • with local control, i.e. information on which pairs of points achievethe lipschitz constant (eventually, pairs belonging to a leaf of somelamination for k ≥ • in an equivariant context.Negative curvature is responsible for the sharp divide taking place at K = 1(resp. k = 0). The “macroscopic case” of maps from H n to H n , in anequivariant context and with a local control of the Lipschitz constant, wastreated in [GK], refining [Kas]. For context, we quote: Theorem 4.1 [GK, Th. 1.6 & 5.1] . Let Γ be a discrete group and j, ρ : Γ → Isom( H n ) two representations with j convex cocompact. Suppose ρ (Γ) does not have a unique fixed point in ∂ ∞ H n . Let C (cid:54) = ∅ be a j (Γ) -invariantcocompact subset of H n and ϕ : C → H n a ( j, ρ ) -equivariant Lipschitzmap with Lipschitz constant K . Then there exists an equivariant extension f : H n → H n of ϕ with (cid:26) Lip( f ) < if K < , Lip( f ) = K if K ≥ . Moreover, if
K > (resp. K = 1 ), then the relative stretch locus E C ,ϕ ( j, ρ ) isnonempty, contained in the convex hull of C , and is (resp. contains) theunion of the stretch locus of ϕ and of the closure of a geodesic lamination ˜ L of H n (cid:114) C that is maximally stretched by any K -Lipschitz ( j, ρ ) -equivariantextension f : H n → H n of ϕ . By maximally stretched we mean that distances are multiplied by K onevery leaf of ˜ L . The stretch locus of ϕ is by definition the set of points p ∈ C such that the Lipschitz constant of ϕ restricted to U ∩ C is K (and no smaller)for all neighborhoods U of p in H n . The relative stretch locus E C ,ϕ ( j, ρ ) isthe set of points p ∈ H n such that the Lipschitz constant of any K -Lipschitzequivariant extension of ϕ is K on any neighborhood of p in H n .Note that when K = 1, a K -Lipschitz equivariant extension of ϕ may beforced to be isometric on a larger set (for instance, if j = ρ and ϕ = Id C , then f must be the identity map on the convex hull of C ). Theorem 5.1 of [GK]describes precisely which pairs of points p, q ∈ H n achieve d ( f ( p ) , f ( q )) = OMPLETE LORENTZ SPACETIMES OF CONSTANT CURVATURE 24 Kd ( p, q ) for all K -Lipschitz equivariant maps f when K = 1, and also allowsfor geometrically finite j (Γ) (with parabolic elements) when K ≥ H with no para-bolic elements in j (Γ), and to special “ C ” and “ ϕ ”. However all macroscopicideas of [GK] should generalize.To work out a microscopic analogue of Theorem 4.1, the starting point isto consider a sequence of equivariant convex fields with lipschitz constantsconverging to the infimum. Note however that to an equivariant field, wecan always add an invariant field pointing strongly towards the convex core,without increasing the lipschitz constant: this is why minimizing sequenceswill usually not converge outside the convex core. We therefore resort toimposing a “standard” form (as in Section 3.4) to the convex field inside thefunnels, and minimize under that constraint.In Section 4.1, we prove a local lipschitz extension theorem with a localcontrol: this is where k -stretched lines appear for the first time. In Sec-tion 4.2, we turn this into a global equivariant extension result for vectorfields defined away from the convex core (typically the standard vector fieldsof Section 3.4). These tools will be used in Section 5 to prove Theorem 1.6.4.1. Local lipschitz extensions of convex fields with a local control.
We say that a convex field Y is an extension of a convex field X if Y ⊃ X as subsets of T H and if X ( U (cid:48) ) = Y ( U (cid:48) ) for any open set U (cid:48) ⊂ H on which X is defined. This means that Y is defined at least on the largest domainwhere X is and that X and Y coincide on the interior of this domain.The following key theorem lets us extend convex fields locally withoutloss in the lipschitz constant. Theorem 4.2.
Let C ⊂ H be a compact set and X a compact ( i.e. bounded)convex field defined over C . Suppose X is k -lipschitz with k ≥ . Then X admits a k -lipschitz compact extension to the convex hull Conv( C ) . The proof will be simplified by the following lemma, which is also usefulfor several other arguments in the paper. We use the notation (3.1).
Lemma 4.3.
Consider the vertices p , . . . , p m ∈ H of a convex polygon Π ,vectors x i ∈ T p i H , and a compact subset C (cid:48) contained in the interior of Π .For any k ∈ R , there exists R > such that (cid:107) Y ( C (cid:48) ) (cid:107) ≤ R for any k -lipschitzconvex field extension Y of { x , . . . , x m } to C (cid:48) ∪ { p , . . . , p m } .Proof. Consider the vectors log p ( p i ) ∈ T p H pointing towards p i . By com-pactness of C (cid:48) , the maximum angle 2 ϑ between log p ( p i ) and log p ( p i (cid:48) ) for1 ≤ i, i (cid:48) ≤ m and p ∈ C (cid:48) is < π . For any p ∈ C (cid:48) and x ∈ T p H , there exists1 ≤ i ≤ m such that the angle between x and log p ( p i ) lies between zeroand ϑ . By Remark 3.4, d (cid:48) ( x, x i ) ≥ (cid:107) x (cid:107) cos ϑ − (cid:107) x i (cid:107) . OMPLETE LORENTZ SPACETIMES OF CONSTANT CURVATURE 25
Therefore, if x = Y ( p ) for some k -lipschitz extension Y of { x , . . . , x m } ,then(4.1) (cid:107) x (cid:107) ≤ (cid:107) x i (cid:107) + k d ( p, p i )cos ϑ ≤ max i (cid:48) ( (cid:107) x i (cid:107) ) + | k | max i (cid:48) ,p ( d ( p, p i (cid:48) ))cos ϑ , where 1 ≤ i (cid:48) ≤ m and p ranges over C (cid:48) . This gives the desired uni-form bound. (cid:3) Proof of Theorem 4.2.
Let p , . . . , p m ∈ H be the vertices of a convex poly-gon Π containing Conv( C ) in its interior. Since X is bounded we mayextend X to each p i by choosing a large vector x i pointing into Π alongthe bisector of the angle at p i , so that the extension remains k -lipschitz on C ∪ { p , . . . , p m } . As we extend X , maintaining the k -lipschitz property, topoints of Conv( C ), these helper points, via Lemma 4.3, guarantee that ourextension will be bounded: (cid:107) X (Conv( C )) (cid:107) ≤ R .First we extend X to a single point p of Conv( C ) (cid:114) C . To choose X ( p ) optimally with respect to the lipschitz property, consider the map ϕ p : T p H → R defined by ϕ p ( x ) := sup ( q,y ) ∈ X ( C ) d (cid:48) ( x, y ) d ( p, q ) . By Lemma 4.3, the function ϕ p is proper. In particular, it has a minimum k (cid:48) ,achieved at a vector x ∈ T p H with (cid:107) x (cid:107) ≤ R . For any ( q, y ) ∈ X ( C ), theratio d (cid:48) ( x, y ) /d ( p, q ) is an affine function of x ∈ T p H , of gradient intensity1 /d ( p, q ). Since affine functions are convex, so is ϕ p . Let us show that k (cid:48) ≤ k . Since the convex field X is compact in T H , the supremum defining ϕ p is achieved. Let { ( q i , y i ) | i ∈ I } be the (compact) set of all vectors( q, y ) ∈ X ( C ) such that d (cid:48) ( x , y ) = k (cid:48) d ( p, q ). Suppose, for contradiction,that the convex hull of the q i does not contain p . Then there is an openhalf-plane H ⊂ H that is bounded by a line through p and contains allthe q i . By compactness, max ( q,y ) ∈ X ( C (cid:114) H ) d (cid:48) ( x , y ) d ( p, q ) < k (cid:48) . Since the gradient intensities 1 /d ( p, q ) for q ∈ C are bounded from above, itfollows that ϕ p ( x + ξ ) < k (cid:48) for any short enough vector ξ pointing orthog-onally into H : a contradiction with the minimality of k (cid:48) . Therefore, p liesin the convex hull of the q i . There are two cases to consider. • Case (i):
Suppose p lies on a segment [ q i , q i (cid:48) ] with i, i (cid:48) ∈ I . Then k d ( q i , q i (cid:48) ) ≥ d (cid:48) ( y i , y i (cid:48) ) = d (cid:48) ( y i , x ) + d (cid:48) ( x , y i (cid:48) )= k (cid:48) d ( q i , p ) + k (cid:48) d ( p, q i (cid:48) )= k (cid:48) d ( q i , q i (cid:48) ) , showing that k (cid:48) ≤ k . • Case (ii): If p does not lie on such a segment, then it lies in the interiorof a nondegenerate triangle q i q i (cid:48) q i (cid:48)(cid:48) with i, i (cid:48) , i (cid:48)(cid:48) ∈ I . We write { i, i (cid:48) , i (cid:48)(cid:48) } = OMPLETE LORENTZ SPACETIMES OF CONSTANT CURVATURE 26 { , , } to simplify notation. By adding a Killing field, we may assumethat x = 0( p ) and that y is parallel to the segment [ p, q ], so that y = − k (cid:48) log q ( p ). The geodesic rays from p passing through each of q , q , q divide H into three connected components (see Figure 5). There is a pair ofdistinct indices a, b ∈ { , , } such that y a and y b point (weakly) away fromthe component bordered by the rays from p through q a and through q b . Then d (cid:48) ( y a , y b ) ≥ d (cid:48) ( ˆ y a , ˆ y b ) where ˆ y i is the projection of y i to the [ p, q i ] direction.Note that ˆ y i = − k (cid:48) log q i ( p ) by Remark 3.4, because d (cid:48) ( x , y i ) = k (cid:48) d ( p, q i ).Now set q ta := exp q a ( t ˆ y a ) and q tb := exp q b ( t ˆ y b ). Since the angle (cid:91) q a pq b isdifferent from 0 and π , the distance function ψ : t (cid:55)→ d ( q ta , q tb ) is strictlyconvex (a feature of negative curvature) and vanishes at t = − /k (cid:48) as longas k (cid:48) > k (cid:48) ≤
0, then we already have k (cid:48) ≤ k ). Thus k (cid:48) < ψ (cid:48) (0) ψ (0) = d (cid:48) (ˆ y a , ˆ y b ) d ( q a , q b ) ≤ d (cid:48) ( y a , y b ) d ( q a , q b ) ≤ k. (4.2) pq y = ˆ y C H Conv( C ) q y q y ˆ y ˆ y Figure 5.
In this illustration of case (ii), the vectors y and y point weakly away from the sector q pq , hence d (cid:48) ( y , y ) ≥ d (cid:48) (ˆ y , ˆ y ). Next, d (cid:48) (ˆ y , ˆ y ) > k (cid:48) d ( q , q ) by convexity of thefunction t (cid:55)→ d (exp q ( t ˆ y ) , exp q ( t ˆ y )).We have shown that X admits a k -lipschitz extension to C ∪ { p } . Replac-ing C with C ∪ { p } , we can extend to a second point p (cid:48) of Conv( C ) (cid:114) C ,then to a third, and eventually to a dense subset S of Conv( C ). We takeour final extension Y to be the fiberwise convex hull of the closure of X ( S )in T H . That is, for any p ∈ Conv( C ), we define Y ( p ) to be the (closed)convex hull in T p H of all limits of sequences ( p n , y n ) ∈ X ( S ) with p n → p .Note that Y ( p ) (cid:54) = ∅ because X ( S ) was bounded uniformly at the beginningof the proof. By construction, Y is closed in T H , and k -lipschitz because d (cid:48) is continuous. It agrees with the original convex field X on the interior of C ,but may have larger fibers above points of the boundary of C in H . (cid:3) As mentioned at the beginning of Section 4, we will also need a versionof Theorem 4.2 with locally improved lipschitz constant: it is given by the
OMPLETE LORENTZ SPACETIMES OF CONSTANT CURVATURE 27 following proposition, which is a simple consequence of the proof of Theo-rem 4.2. Recall the definition 3.3 of the local lipschitz constant lip p ( X ). Proposition 4.4.
Under the hypotheses of Theorem 4.2, for any point p ∈ Conv( C ) (cid:114) C , the convex field X admits a k -lipschitz extension Y to Conv( C ) such that lip p ( Y ) < k , unless one of the following holds:(1) k > and p lies in the interior of a geodesic segment [ q , q ] with q , q ∈ C and d (cid:48) X ( q , q ) = k d ( q , q ) , the direction of this segmentat p being unique;(2) k = 0 and p lies in the convex hull of three (not necessarily dis-tinct) points q , q , q ∈ C such that X contains the restriction ofa Killing field to { q , q , q } (in particular, d (cid:48) X ( q i , q i (cid:48) ) = 0 for all ≤ i < i (cid:48) ≤ ).In case (2), any k -lipschitz extension of X to Conv( C ) restricts to a Killingfield on the interior of the triangle q q q . The segments [ q i , q i (cid:48) ] are k -stretched by X in the sense of Observation 3.5. Proof.
Fix p ∈ Conv( C ) (cid:114) C . As in the proof of Theorem 4.2, we first define Y ( p ) to be a vector at which ϕ p achieves its minimum k (cid:48) . If k (cid:48) < k , thenwe can extend Y as we wish in a continuous and locally k (cid:48) -lipschitz waynear p without destroying the global k -lipschitz property, and then continueextending Y to the rest of Conv( C ) as in the proof of Theorem 4.2. Weneed therefore only understand the case k (cid:48) = k .Suppose k (cid:48) = k >
0. Then the strict inequality (4.2) shows that we cannotbe in case (ii) in the proof of Theorem 4.2. Therefore, we are in case (i) ,and so p lies in the interior of a k -stretched segment [ q , q ] with q , q ∈ C .To see that (1) holds, suppose that p lies in the interior of a k -stretchedsegment [ q , q ] of a different direction, with q , q ∈ C . Then for 1 ≤ i ≤ y i ∈ X ( q i ) such that d (cid:48) ( Y ( p ) , y i ) = k d ( p, q i ) and we may argueas in case (ii) of the proof of Theorem 4.2 (with four directions instead ofthree) that d (cid:48) ( y a , y b ) > k d ( q a , q b ) for some 1 ≤ a < b ≤
4, a contradiction.Now suppose k (cid:48) = k = 0. If we are in case (i) in the proof of Theorem 4.2,then p belongs to the interior of a 0-stretched segment [ q , q ] with q , q ∈ C ;in particular, X contains the restriction to { q , q } of a Killing field. Supposewe are in case (ii) , i.e. p lies in the interior of a nondegenerate triangle q q q such that for any 1 ≤ i ≤ d (cid:48) ( x , y i ) = 0 for some x ∈ T p H and y i ∈ X ( q i ). Up to adding a Killing field, we may assume x = 0( p ) and y =0( q ). Then, for i ∈ { , } , the component ˆ y i of y i in the direction [ p, q i ] mustbe zero, since d (cid:48) ( x , y i ) = d (cid:48) (0( p ) , ˆ y i ) = 0. The component of y i orthogonalto [ p, q i ] must also be zero for each i or else d (cid:48) ( y a , y b ) > ≤ a < b ≤
3, contradicting that X is 0-lipschitz. This proves that (2) holds.In general, if X contains the restriction of a Killing field Z to { q , q , q } ,then Proposition 3.8 shows that any 0-lipschitz extension Y of X to the fulltriangle q q q contains the restriction of Z to q q q . Further , Y = Z onthe interior of q q q (apply Remark 3.4 with q = q i for 1 ≤ i ≤ (cid:3) OMPLETE LORENTZ SPACETIMES OF CONSTANT CURVATURE 28
Equivariant extensions of vector fields defined in the funnels.
We next derive a technical consequence of Theorem 4.2, namely that equi-variant lipschitz vector fields defined outside the convex core, with niceenough inward-pointing properties, can be extended equivariantly to allof H . This will be applied in the next section to standard vector fields(Definition 3.12). Proposition 4.5.
Let Γ be a discrete group, j : Γ → G a convex cocompactrepresentation, and u : Γ → g a j -cocycle. Let U ⊂ H be the preimage ofthe convex core of j (Γ) \ H , let U be the open -neighborhood of U , and let N := H (cid:114) U be its complement. Let X be a ( j, u ) -equivariant lipschitzvector field on N and set k := lip( X ) . Suppose that there exists ε > suchthat for all distinct p, q ∈ N , ( ∗ ) d (cid:48) X ( p, q ) < k d ( p, q ) (strict inequality); ( ∗∗ ) lip p ( X ) ≤ k − ε ; ( ∗∗∗ ) if p ∈ ∂N then p has a neighborhood V such that X ( V ∩ N ) admitsa vector field extension to V with lipschitz constant < k .Then there exists a ( j, u ) -equivariant convex field Y on H , extending X ,such that(1) if k < , then lip( Y ) < ;(2) if k ≥ , then lip( Y ) = k and there is a j (Γ) -invariant geodesiclamination ˜ L in U that is maximally stretched by Y , in the sensethat d (cid:48) X ( p, q ) = k d ( p, q ) for any p, q on a common leaf of ˜ L ; inparticular, k = k α := sup γ with λ ( j ( γ )) > α u ( γ ) λ ( j ( γ )) . Proof.
The idea of the proof is to construct an extension Y of X that is in acertain sense “optimal”. Then the lamination ˜ L will arise as the union of the k -stretched segments of Proposition 4.4.(1) at points p where lip p ( Y ) = k .Considering segments with endpoints in N that spend most of their lengthnear ˜ L , this will imply lip( Y ) = k .We first show that equivariant lipschitz extensions of X exist. The fol-lowing claim holds in general, independently of the regularity assumptions( ∗ ) , ( ∗∗ ) , ( ∗∗∗ ), and even if X is a convex field instead of a vector field. Claim 4.6.
There exist ( j, u ) -equivariant lipschitz convex field extensionsof X to H (possibly with very bad lipschitz constant).Proof. Let B , . . . , B m be open balls of H such that the sets j (Γ) · B i for1 ≤ i ≤ m cover U . We take them small enough so that j ( γ ) · B i is eitherequal to or disjoint from B i for all γ ∈ Γ and 1 ≤ i ≤ m . Let ( ψ i ) ≤ i ≤ m be a j (Γ)-invariant partition of unity on U , with each ψ i supported in j (Γ) · B i .We require the restriction of ψ i to B i to be Lipschitz. For any i , Theorem 4.2gives a compact extension Z i of X | B i ∩ N to B i with lip B i ( Z i ) ≤ max { k, } ;in case the stabilizer Γ i ⊂ Γ of B i is nontrivial, we can assume that Z i OMPLETE LORENTZ SPACETIMES OF CONSTANT CURVATURE 29 is ( j | Γ i , u | Γ i )-equivariant after replacing it with i (cid:80) γ ∈ Γ i γ • Z i (notation(3.2)), using Observation 3.5. We then extend it to a ( j, u )-equivariantconvex field Z i on j (Γ) · B i . The extension Z := X ∪ m (cid:88) i =1 ψ i Z i . of X is ( j, u )-equivariant (Observation 3.5). Let us check that Z is lipschitz.By subdivision and equivariance (Observation 3.5), we only need to checkthat Z is lipschitz on each of the balls B i (cid:48) for 1 ≤ i (cid:48) ≤ m . Consider twodistinct points p, q ∈ B i (cid:48) and vectors z p ∈ Z ( p ) and z q ∈ Z ( q ). We can write z p = m (cid:88) i =1 ψ i ( p ) x i and z q = m (cid:88) i =1 ψ i ( q ) y i where x i ∈ Z i ( p ) and y i ∈ Z i ( q ). By Observation 3.5, d (cid:48) ( z p , z q ) = d (cid:48) (cid:32) m (cid:88) i =1 ψ i ( p ) x i , m (cid:88) i =1 ψ i ( q ) y i (cid:33) = d (cid:48) (cid:32) m (cid:88) i =1 ψ i ( p ) x i , m (cid:88) i =1 ψ i ( p ) y i (cid:33) + d (cid:48) (cid:32) p ) , m (cid:88) i =1 (cid:0) ψ i ( q ) − ψ i ( p ) (cid:1) y i (cid:33) ≤ m (cid:88) i =1 ψ i ( p ) d (cid:48) ( x i , y i ) + (cid:13)(cid:13)(cid:13)(cid:13)(cid:13) m (cid:88) i =1 (cid:0) ψ i ( q ) − ψ i ( p ) (cid:1) y i (cid:13)(cid:13)(cid:13)(cid:13)(cid:13) ≤ m (cid:88) i =1 ψ i ( p ) lip B i (cid:48) ( Z i ) d ( p, q ) + m (cid:88) i =1 Lip( ψ i ) (cid:107) y i (cid:107) d ( p, q ) ≤ (cid:32) sup ≤ i ≤ m lip B i (cid:48) ( Z i ) + (cid:16) sup ≤ i ≤ m Lip( ψ i ) (cid:17) m (cid:88) i =1 sup z i ∈ Z i ( B i (cid:48) ) (cid:107) z i (cid:107) (cid:33) d ( p, q ) , where the last supremum is < + ∞ because B i (cid:48) meets only finitely many j (Γ)-translates of B i and Z i ( j ( γ ) · B i ) is compact for all γ ∈ Γ. Thus d (cid:48) Z ( p, q ) /d ( p, q ) is uniformly bounded for p, q ∈ B i (cid:48) and Z is lipschitz. (cid:3) Returning to the proof of Proposition 4.5, let k ∗ ∈ [ k, + ∞ ) be the infimumof lipschitz constants over all ( j, u )-equivariant convex field extensions of X to H . If k ∗ <
0, then k <
0; we may choose an extension Y with lip( Y )arbitrarily close to k ∗ , in particular with lip( Y ) <
0. This proves (1).From now on, we assume k ∗ ≥
0. Let ( Z n ) n ∈ N be a sequence of ( j, u )-equivariant extensions of X with lip( Z n ) → k ∗ . Note that U is coveredby the j (Γ)-translates of some polygon with vertices in N . Therefore, byLemma 4.3 (using Observation 3.5.(iv) and (vii)), the convex fields Z n areuniformly bounded over any compact set. By Proposition 3.7, we may ex-tract a subsequence which is Hausdorff convergent to a convex field Z ∞ on H with lip( Z ∞ ) = k ∗ . Thus the set Z of ( j, u )-equivariant convex field OMPLETE LORENTZ SPACETIMES OF CONSTANT CURVATURE 30 extensions of X to H with minimal lipschitz constant k ∗ is nonempty. For Z ∈ Z , we set E Z := { p ∈ H | lip p ( Z ) = k ∗ } . It is a closed subset of H , as the function p (cid:55)→ lip p ( Z ) is upper semicontin-uous, and it is j (Γ)-invariant (Observation 3.5). We define the stretch locus relative to X to be E := (cid:92) Z ∈Z E Z . We claim that there exists Y ∈ Z such that E = E Y . Indeed, for any p ∈ H (cid:114) E we can find a convex field Z p ∈ Z and a neighborhood V p of p in H such that δ p := k ∗ − lip V p ( Z p ) >
0. Let { V p i } i ∈ N ∗ be a countable set of suchneighborhoods such that H (cid:114) E = (cid:83) i ∈ N ∗ V p i . The equivariant convex field Y = + ∞ (cid:88) i =1 − i Z p i satisfies lip V pi ( Y ) ≤ k ∗ − − i δ p i for all i , hence E Y = E . We next study thestructure of E .By ( ∗∗ ), since k ≤ k ∗ , the set E is contained in U . Moreover, E is notempty: otherwise we could cover U (which is compact modulo j (Γ)) by the j (Γ)-translates of finitely many open sets V p i for which lip V pi ( Y ) < k ∗ . Sincethe local lipschitz constant in N is also bounded by k − ε ≤ k ∗ − ε by ( ∗∗ ),it would then follow by j (Γ)-invariance and subdivision (Observation 3.5)that lip( Y ) < k ∗ , a contradiction. Thus E contains a point of U .In fact, E must contain a point of U . Indeed, let us show that if E contains a point p of ∂U = ∂N , then there exists q ∈ U such that(4.3) d (cid:48) Y ( p, q ) = k ∗ d ( p, q ) . Let B be a small ball centered at p , of radius r >
0, and let A be a thinannulus neighborhood of ∂B in B . We have lip ( B ∩ N ) ∪ A ( Y ) ≤ lip( Y ) = k ∗ .Suppose by contradiction that(4.4) sup q ∈ A d (cid:48) Y ( p, q ) d ( p, q ) < k ∗ . By assumption ( ∗∗∗ ) on X , if r is sufficiently small then X ( B ∩ N ) admitsa vector field extension X (cid:48) to B with lip( X (cid:48) ) < k ≤ k ∗ . If B (cid:48) ⊂ B isanother ball centered at p , of radius r (cid:48) (cid:28) r small enough, then (4.4) and thecontinuity of the vector field X (cid:48) imply that d (cid:48) ( X (cid:48) ( p (cid:48) ) , y q ) < k ∗ d ( p (cid:48) , q ) for all p (cid:48) ∈ B (cid:48) , all q ∈ A , and all y q ∈ Y ( q ). Then the convex field defined over C := ( B ∩ N ) ∪ A ∪ B (cid:48) that agrees with Y over A and B ∩ N and with X (cid:48) over B (cid:48) is k ∗ -lipschitz (see Figure 6). Applying Theorem 4.2 to C , wefind a k ∗ -lipschitz convex field Y (cid:48) on the ball B = Conv( C ) that contains Y ( A ) and X ( B ∩ N ) and satisfies lip p ( Y (cid:48) ) = lip p ( X (cid:48) ) < k ≤ k ∗ ; we canextend it to j (Γ) · B in a ( j, u )-equivariant way, and then to H by taking Y (cid:48) = Y on H (cid:114) j (Γ) · B . Using subdivision at points of j (Γ) · A , we see that OMPLETE LORENTZ SPACETIMES OF CONSTANT CURVATURE 31 lip( Y (cid:48) ) ≤ k ∗ , which contradicts the fact that p ∈ E . Thus (4.4) is false, and(4.3) holds for some q ∈ U , which implies q ∈ E . U BN p B (cid:48) A Figure 6.
Definition of the region C = ( B ∩ N ) ∪ A ∪ B (cid:48) .Let us now prove that E contains the lift of a k ∗ -stretched laminationcontained in the convex core.Assume first that k ∗ >
0. Consider a point p ∈ E ∩ U . Let Y (cid:48) bethe convex field obtained from Y by simply removing all vectors above asmall ball B p ⊂ U centered at p , so that Y (cid:48) is defined over H (cid:114) B p .Proposition 4.4, applied to the restriction of Y (cid:48) to a small neighborhood of ∂B p , implies that p lies on a unique k ∗ -stretched segment [ q, q (cid:48) ] with q, q (cid:48) ∈ ∂B p (or else Y (cid:48) could have been extended with a smaller lipschitz constantat p , and similarly at each j ( γ ) · p in an equivariant fashion, contradictingthe fact that p ∈ E ). This applies to all points p ∈ E ∩ U , and so E ∩ U is a union of geodesic segments. Moreover, the direction of the k ∗ -stretchedsegment [ q, q (cid:48) ] at p is unique for each p , and its length 2 d ( p, q ) (the diameterof the ball B p ) can be taken to be bounded from below by a continuous,positive function of p ∈ U : it follows that any segment of U that containsa k ∗ -stretched subsegment is k ∗ -stretched. Note that one or more of these k ∗ -stretched (partial) geodesics in E may have an endpoint in ∂U ; however,they cannot have two endpoints in ∂U , by the assumption ( ∗ ). Thus any(partial) geodesic (cid:96) ⊂ E descends to a simple (partial) geodesic in j (Γ) \ H which, at least in one direction, remains in j (Γ) \ U and accumulates on ageodesic lamination L . This lamination L must be contained in the convexcore. By closedness of E , its preimage ˜ L ⊂ H is also part of E , and eachleaf of ˜ L is k ∗ -stretched.Next, assume k ∗ = 0. Consider a point p ∈ E ∩ U and let Y (cid:48) be theconvex field obtained from Y by removing all vectors above a small ball B p ⊂ U around p , so that Y (cid:48) is defined over H (cid:114) B p . Proposition 4.4applied to Y (cid:48) implies that either:( i ) p lies on a segment [ q, q (cid:48) ] with q, q (cid:48) ∈ ∂B p and d (cid:48) Y ( q, q (cid:48) ) = 0; or( ii ) p lies in the convex hull of three distinct points q , q , q ∈ ∂B p with d (cid:48) Y ( q i , q i (cid:48) ) = 0 for all 1 ≤ i < i (cid:48) ≤ OMPLETE LORENTZ SPACETIMES OF CONSTANT CURVATURE 32
In case ( i ) the segment [ q, q (cid:48) ] is 0-stretched by Y , and in case ( ii ) the re-striction of Y to the interior of the triangle q q q is a Killing field. Inboth cases there is a 0-stretched segment in E with midpoint p , and we canbound from below the length of this segment by a continuous function of p ∈ U , e.g. half the radius of B p (in case ( ii ), just pick the direction of thesegment to be pq i where q i is at the smallest angle of the triangle q q q ).Now we must conclude that E contains a geodesic lamination. Wheneverthere are two intersecting, 0-stretched, open line segments of E contain-ing p , the 0-lipschitz convex field Y restricts to a Killing field on the unionof these segments and in fact on their convex hull, so that points near p satisfy case ( ii ). So, if case ( ii ) never happens, then the germ of 0-stretchedsegment through p is unique, and we may proceed exactly as when k ∗ > ii ) does occur, then E has an interior point near which Y is a Killingfield, which may be assumed to be 0 up to adjusting u by a coboundary. Theset E := { p ∈ H | p ) ∈ Y } is convex by Proposition 3.8, and containedin E . On the interior of E , the convex field Y coincides with 0. Consider p ∈ ∂E (cid:114) ∂U . Points approaching p from the interior of E are midpoints of0-stretched segments, necessarily included in E , whose lengths are boundedfrom below. This means p cannot be an extremal point of E . Thus, if E isstrictly contained in U , then any p ∈ U ∩ ∂E belongs to a (straight) sideof E which can only terminate, if at all, on ∂U . As in the case k ∗ > j (Γ) \ H to a geodesic lamination (itcannot terminate on ∂U at both ends by the assumption ( ∗ )).Thus, for k ∗ ≥ L in the convex core of j (Γ) \ H whose lift ˜ L to H is k ∗ -stretched by Y . It remains to see that k ∗ = k = k α .We know that k ∗ ≥ k and k ∗ ≥ k α (by (3.6) applied to Y ). In fact, k ∗ = k α by Proposition 3.9, since any minimal component of L can be approximatedby simple closed curves. Choose a smooth, ( j, u )-equivariant vector field W on H and recall the function ν (cid:48) W : T H → R of (3.8). For a long segment[ p, q ] with endpoints in N , spending most of its length near ˜ L , we see asin Proposition 3.9 that d (cid:48) W ( p, q ) = (cid:82) [ p,q ] ν (cid:48) W is roughly k ∗ d ( p, q ) by uniformcontinuity of ν (cid:48) W ; therefore k ≥ k ∗ and finally k = k ∗ . This completes theproof of Proposition 4.5. (cid:3) Existence of a maximally stretched lamination
We now prove Theorem 1.6, which states the existence of a laminationthat is maximally stretched by all equivariant vector fields of minimal lips-chitz constant. First, in Section 5.1, we bring together all strands of Section 4and prove the weaker Theorem 5.1 below, which differs from Theorem 1.6only in that the optimal lipschitz constant k is defined as an infimum overall convex fields (not just vector fields). To prove Theorem 1.6, we thenshow that the infimum over convex fields is the same as the infimum overvector fields: this is done in Section 5.4, after some technical preparationin Sections 5.2 and 5.3 to approximate convex fields by vector fields with OMPLETE LORENTZ SPACETIMES OF CONSTANT CURVATURE 33 almost the same lipschitz constant. Finally, in Section 5.5 we describe asmoothing process that approximates a vector field by smooth vector fields,again with nearly the same lipschiz constant; this will be used in Section 7.5.1.
A weaker version of Theorem 1.6.
We first prove the following.
Theorem 5.1.
Let Γ be a discrete group, j ∈ Hom(Γ , G ) a convex cocompactrepresentation, and u : Γ → g a j -cocycle. Let k ∈ R be the infimumof lipschitz constants of ( j, u ) -equivariant convex fields X defined over H .Suppose k ≥ . Then there exists a geodesic lamination L in the convexcore of j (Γ) \ H that is k -stretched by all ( j, u ) -equivariant, k -lipschitz convexfields X defined over H , meaning that d (cid:48) X ( p, q ) = k d ( p, q ) for any p (cid:54) = q on a common leaf of the lift of L to H ; such convex fields X exist, and can be taken to be standard in the funnels (Definition 3.12).Proof. Recall that by (3.6), k ≥ k α := sup γ with λ ( j ( γ )) > α u ( γ ) λ ( j ( γ )) , and k α is still a lower bound for the lipschitz constant of any locally bounded,( j, u )-equivariant convex field defined over a nonempty subset of H , byProposition 3.13.(1).Let ( X n ) n ∈ N be a sequence of ( j, u )-equivariant convex fields definedover H such that lip( X n ) → k . By Proposition 3.10, the X n are uniformlybounded over any compact subset of the interior U ⊂ H of the convexcore. Therefore, some subsequence of ( X n ( U )) n ∈ N admits a Hausdorff limit X ∞ , which is a k -lipschitz, ( j, u )-equivariant convex field defined over U byProposition 3.7. We now use the extension theory of Section 4 to produce a k -lipschitz, ( j, u )-equivariant convex field, defined over all of H , that agreeswith X ∞ inside (a subset of) U and is standard in the funnels. Let N (resp. N (cid:48) ) be the set of points in H at distance > > /
2) from U . Choose η >
0, small enough so that all the hypercycles H i running inside U atdistance η from the boundary components of U are disjoint. Let U ⊂ U be the closed, connected region bounded by the H i ; the restriction X ∞ ( U )is locally bounded and k -lipschitz. Choose a hypercycle H i parallel to a con-nected component N (cid:48) i of N (cid:48) . Applying Proposition 3.13.(2) with η = − / k -lipschitz extension X of X ∞ ( U ) to U ∪ N (cid:48) i such that • d (cid:48) X ( p, q ) < k α d ( p, q ) for all distinct p ∈ N (cid:48) i and q ∈ H i ∪ N (cid:48) i , • lip p ( X ) is uniformly smaller than k α for p ∈ N (cid:48) i , • X is standard on N (cid:48) i .We then extend X equivariantly to U ∪ j (Γ) · N (cid:48) i , and repeat the procedurefor each hypercycle H i modulo j (Γ): this produces a vector field X defined OMPLETE LORENTZ SPACETIMES OF CONSTANT CURVATURE 34 on U ∪ N (cid:48) . We have k N := lip N ( X ) ≤ k by construction, using a subdivi-sion argument (Observation 3.5) to bound d (cid:48) X ( N ) ( p, q ) /d ( p, q ) for p and q indifferent components of N . Moreover, the vector field X ( N ), with lipschitzconstant k N , satisfies the hypotheses ( ∗ ) , ( ∗∗ ) , ( ∗∗∗ ) of Proposition 4.5 since k N ≥ k α . (The hypothesis ( ∗∗∗ ) is satisfied because X ( N ) is a restrictionof the standard field X ( N (cid:48) ).) Let Y be the ( j, u )-equivariant extension of X ( N ) to H given by Proposition 4.5. If k N <
0, then lip( Y ) <
0, which isimpossible since k ≥ k N ≥ k N = lip( Y ).Since Y is defined over H , this yields k N ≥ k , hence k N = k . Proposition 4.5also gives a k -stretched lamination in the convex core, and k N = k α . (cid:3) By Theorem 5.1, the infimum k of lipschitz constants of ( j, u )-equivariantconvex fields defined on H is achieved by a convex field X that is standardin the funnels. To prove Theorem 1.6, we now only need to establish thefollowing proposition. Proposition 5.2.
Given a ( j, u ) -equivariant convex field X defined on H and standard in the funnels, there exist ( j, u ) -equivariant vector fields X ∗ defined on H with lip( X ∗ ) arbitrarily close to lip( X ) . The main idea (Proposition 5.4 below) is that the backwards flow of alipschitz convex field gives vector field approximates with nearly the samelipschitz constant.5.2.
The flow-back construction.
Let Z be a convex field on H . Forany x, y ∈ Z with x ∈ T p H and y ∈ T q H , we set d (cid:48) d ( x, y ) := d (cid:48) ( x, y ) d ( p, q ) , where d (cid:48) has been defined in (3.3). When Z is a vector field, there is noambiguity about the vectors x ∈ Z ( p ) and y ∈ Z ( q ) and we set(5.1) d (cid:48) Z d ( p, q ) := d (cid:48) d ( x, y ) = d (cid:48) ( x, y ) d ( p, q ) ;then lip( Z ) = sup p (cid:54) = q d (cid:48) Z d ( p, q ). Definition 5.3.
Let ( ϕ t ) t ∈ R be the geodesic flow of H , acting on T H . For t ∈ R , we shall denote by Z t the convex field produced by flowing for time t : Z t := ϕ t ( Z ) . When t <
0, we refer to Z t as the flow-back of Z .The point is that the flow-back of a lipschitz convex field is a (globallydefined) vector field (see Figure 7 for a one-dimensional illustration): Proposition 5.4.
Let Z be an R -lipschitz convex field defined on all of H ,with R > . For any negative t ∈ ( − R , , the set Z t is a lipschitz vector OMPLETE LORENTZ SPACETIMES OF CONSTANT CURVATURE 35 lip = α lip = α − εα lip = − ε Z Z − ε Figure 7.
A one-dimensional illustration: a lipschitz convexfield Z over R identifies with a subset of R (cid:39) T R , namelya curve whose slope is bounded from above (by lip( Z )). Itsflow-back Z − ε is the image of Z under the linear map (cid:0) − ε (cid:1) ,and is a vector field ( i.e. the graph of a continuous function R → R ) with slightly larger lipschitz constant than Z .field defined on H . Moreover, for any ε > there exists t < such thatfor all t ∈ ( t , and all x (cid:54) = y in Z , (5.2) d (cid:48) d ( x t , y t ) ≤ max (cid:26) d (cid:48) d ( x, y ) , − R (cid:27) + ε, where we set x t := ϕ t ( x ) ∈ Z t and y t := ϕ t ( y ) ∈ Z t , and we interpret themaximum in (5.2) to be − R if x and y are based at the same point of H .If Z is j -invariant, then so is Z t .Proof. The set Z t is closed in T H (since Z is), and j -invariant if Z is. Weclaim that if − /R < t <
0, then Z t has at most one vector above eachpoint p ∈ H : indeed, if Z t ( p ) contained two vectors x (cid:54) = y , then we wouldhave ϕ − t ( x ) , ϕ − t ( y ) ∈ Z but d (cid:48) d (cid:0) ϕ − t ( x ) , ϕ − t ( y ) (cid:1) ≥ − t > R by convexity of the distance function, which would contradict lip( Z ) ≤ R .Moreover, Z t = ϕ t ( Z ) also has at least one point above p : indeed, if W denotes the vector field q (cid:55)→ log q ( p ) that flows all points to p in time 1, thenlip( W ) ≤ − x ∈ T H satisfies ϕ t ( x ) ∈ T p H if and only if x ∈ t W . But Z contains a vectorof t W , because Z + − t W has lipschitz constant at most lip( Z ) + − t ( − < Z t isa vector field defined on all of H (necessarily continuous, since Z t is closedin T H ).We now prove (5.2). Consider x (cid:54) = y in Z , with x ∈ T p H and y ∈ T q H forsome p, q ∈ H . Let x t = ϕ t ( x ) and y t = ϕ t ( y ) be the corresponding vectorsof Z t , based at p t := exp p ( tx ) and q t := exp q ( ty ) respectively. If p = q , then d (cid:48) d ( x t , y t ) ≤ /t for all t < OMPLETE LORENTZ SPACETIMES OF CONSTANT CURVATURE 36 (5.2) holds for all − /R < t <
0. We now assume p (cid:54) = q . Fix ε > t := R + ε − R <
0. It is enough to prove that for all t ∈ ( t , d (cid:48) d ( x t , y t ) ≤ f t (cid:18) d (cid:48) d ( x, y ) (cid:19) , where f t ( ξ ) = ξ tξ = ( ξ − + t ) − . Indeed, assume that (5.3) holds. One checks that f t ( ξ ) ≤ ξ + ε for all | ξ | ≤ R , that f t ( ξ ) ≤ − R + ε for all ξ ≤ − R , and that f t ≤ f t for all t ∈ ( t , d (cid:48) d ( x, y ) ≤ R , this implies (5.2). ψ ( τ ) s (0) s ( τ ) t ττ − s ( τ ) 0 ψ (0) Figure 8.
The graph of ψ with two tangents.We now prove (5.3). It is a pure consequence of the convexity of thedistance function ψ : R −→ R + τ (cid:55)−→ d ( p τ , q τ ) . We can rewrite (5.3) as(5.4) ψ (cid:48) ( t ) ψ ( t ) ≤ f t (cid:18) ψ (cid:48) (0) ψ (0) (cid:19) . By convexity, ψ (cid:48) ( t ) ≤ ψ (cid:48) (0). If ψ (cid:48) ( t ) ≤ ≤ ψ (cid:48) (0), then (5.4) holds because f t ( ξ ) has the sign of ξ when t ∈ ( t ,
0) and ξ ≤ R . We now assume that ψ (cid:48) (0) and ψ (cid:48) ( t ) have the same sign. Then we can invert: (5.4) amountsto ψ ( t ) ψ (cid:48) ( t ) ≥ t + ψ (0) ψ (cid:48) (0) . If s ( τ ) denotes the abscissa where the tangent tothe graph of ψ at ( τ, ψ ( τ )) meets the horizontal axis (see Figure 8), then ψ ( τ ) ψ (cid:48) ( τ ) = τ − s ( τ ), hence (5.4) becomes s ( t ) ≤ s (0), which is true by convexityof ψ since t < ψ (cid:48) ( t ) and ψ (cid:48) (0) have the same sign. This completes theproof of Proposition 5.4. (cid:3) One more ingredient.
The following lemma expresses the idea that iftwo points, moving uniformly on straight lines of H , stay at nearly constantdistance, then the line through them stays of nearly constant direction. Lemma 5.5.
For any θ > there exists < δ < θ with the followingproperty: if p, p (cid:48) , q, q (cid:48) ∈ H (with p (cid:54) = q and p (cid:48) (cid:54) = q (cid:48) ) all belong to a ball B of OMPLETE LORENTZ SPACETIMES OF CONSTANT CURVATURE 37 diameter δ and the oriented lines pq and p (cid:48) q (cid:48) intersect (possibly outside B )at an angle ≥ θ , then the midpoints p (cid:48)(cid:48) of pp (cid:48) and q (cid:48)(cid:48) of qq (cid:48) satisfy d ( p (cid:48)(cid:48) , q (cid:48)(cid:48) ) ≤ (1 − δ ) max { d ( p, q ) , d ( p (cid:48) , q (cid:48) ) } . Proof.
Let r be the midpoint of p (cid:48) q . Since H is CAT(0), we have d ( p (cid:48)(cid:48) , q (cid:48)(cid:48) ) ≤ d ( p (cid:48)(cid:48) , r ) + d ( r, q (cid:48)(cid:48) ) ≤ d ( p, q )2 + d ( p (cid:48) , q (cid:48) )2 ≤ max { d ( p, q ) , d ( p (cid:48) , q (cid:48) ) } . We just need to find a spare factor (1 − δ ) between the first and last terms.Such a spare factor exists in the rightmost inequality unless d ( p,q ) d ( p (cid:48) ,q (cid:48) ) is closeto 1. In the latter case, a spare factor exists in the leftmost inequalityprovided we can bound the angle (cid:92) p (cid:48)(cid:48) rq (cid:48)(cid:48) from above by π − θ/ p (cid:48) p (cid:48)(cid:48) rq p q (cid:48) q (cid:48)(cid:48) H Figure 9. If pq and p (cid:48) q (cid:48) form an angle, then so do p (cid:48)(cid:48) r and rq (cid:48)(cid:48) .The main observation is that for any noncolinear a, b, c ∈ H , the line (cid:96) (cid:48) through the midpoints of ab and ac does not intersect the line (cid:96) through b and c . Indeed, suppose ( x t ) t ∈ R is a parameterization of (cid:96) , and let x (cid:48) t be themidpoint of ax t . Then ( x (cid:48) t ) t ∈ R is a convex curve with the same endpointsas (cid:96) : indeed, if we see H as a hyperboloid in R , as in Section 2.2, then x (cid:48) t isjust some positive multiple of a + x t which describes a branch of hyperbola( not contained in a plane through the origin) as t ranges over R . Sincea branch of hyperbola in R looks convex when seen from the origin, x (cid:48) t describes a curve C in H that looks convex (in fact, an arc of conic) in theKlein model. By convexity, (cid:96) (cid:48) ∩ C is then reduced to the two midpoints of ab and ac ; the line (cid:96) (cid:48) cannot cross C a third time, hence does not cross the line (cid:96) .In our situation (see Figure 9), this means the line p (cid:48)(cid:48) r (resp. rq (cid:48)(cid:48) ) is δ -close to, but disjoint from the line pq (resp. p (cid:48) q (cid:48) ). But pq and p (cid:48) q (cid:48) intersectat an angle ≥ θ , at distance at most on the order of δθ from B (hence closeto B if δ is small). If δ is small enough in terms of θ , this means the orientedlines p (cid:48)(cid:48) r and rq (cid:48)(cid:48) cross at an angle > θ/ r , which is what we wanted toprove. (cid:3) Convex fields do no better than vector fields.
Proposition 5.4does not immediately give Proposition 5.2, since the flow-back of a ( j, u )-equivariant convex field is not necessarily ( j, u )-equivariant. However, flow-ing backwards preserves j -invariance. The trick will be to decompose anequivariant convex field into the sum of a smooth, equivariant vector fieldand a rough, but invariant convex field, and apply Proposition 5.4 only tothe latter term; the former term will be controlled by Lemma 5.5. OMPLETE LORENTZ SPACETIMES OF CONSTANT CURVATURE 38
Proposition 5.6.
Let X be a k -lipschitz, ( j, u ) -equivariant convex field,defined on H and standard in the funnels (Definition 3.12). Let Y be asmooth, ( j, u ) -equivariant vector field on H that coincides with X outsidesome cocompact j (Γ) -invariant set, and let Z be the j -invariant convex fielddefined on H such that X = Y + Z . Then Y + Z t is ( j, u ) -equivariant and lim sup t → − lip( Y + Z t ) ≤ lip( X ) as t goes to from below.Proof. By Proposition 5.4 and Observation 3.5.(i), for any t < X t := Y + Z t is a ( j, u )-equivariant lipschitz vector field. Let us checkthat lip( X t ) ≤ lip( X ) + o (1) as t goes to 0 from below.Let U (cid:48) be a cocompact, j (Γ)-invariant set outside of which Z is zero. Notethat X is bounded on a compact fundamental domain for U (cid:48) by Lemma 4.3,and so is Y by smoothness. Therefore the j -invariant convex field Z isbounded: (cid:107) Z (cid:107) < + ∞ .Fix ε >
0. By smoothness and equivariance of Y , there exist θ, R (cid:48) > p (cid:54) = q in U (cid:48) and p (cid:48) (cid:54) = q (cid:48) in H with d ( p, q ) , d ( p, p (cid:48) ) , d ( q, q (cid:48) ) ≤ θ , ifthe oriented lines pq and p (cid:48) q (cid:48) intersect at an angle ≤ θ (or not at all), then(5.5) d (cid:48) Y d ( p, q ) ∈ [ − R (cid:48) , R (cid:48) ] and (cid:12)(cid:12)(cid:12)(cid:12) d (cid:48) Y d ( p, q ) − d (cid:48) Y d ( p (cid:48) , q (cid:48) ) (cid:12)(cid:12)(cid:12)(cid:12) ≤ ε. (The second condition means that θ is a modulus of ε -continuity for thefunction ν (cid:48) Y : T H → R of (3.9), because d (cid:48) Y d ( p, q ) is the average of ν (cid:48) Y overthe unit tangent bundle of the segment [ p, q ].) We set R := | lip( X ) | + R (cid:48) > , so that lip( Z ) ≤ R . Let δ ∈ (0 , θ ) be given by Lemma 5.5, and t = t ( ε ) < X t is ( k + 2 ε )-lipschitz for any(5.6) max (cid:26) t , − δR , − δ (cid:107) Z (cid:107) (cid:27) < t < . Let x, y ∈ Z be vectors based at δ -close points p, q ∈ U (cid:48) . For t as in (5.6),let x t := ϕ t ( x ) and y t := ϕ t ( y ) be the corresponding vectors of Z t , based at p t := exp p ( tx ) and q t := exp q ( ty ) respectively. If d (cid:48) Zt d ( p t , q t ) ≤ − R , then d (cid:48) X t d ( p t , q t ) ≤ R (cid:48) − R = −| lip( X ) | ≤ lip( X ) . Note that this includes the case p = q by Proposition 5.4. We can thereforeassume that p (cid:54) = q and that d (cid:48) Zt d ( p t , q t ) ≥ − R . As in the proof of Proposi-tion 5.4, the distance function ψ : R −→ R + τ (cid:55)−→ d ( p τ , q τ ) OMPLETE LORENTZ SPACETIMES OF CONSTANT CURVATURE 39 is convex and satisfies ψ (cid:48) ( τ ) ψ ( τ ) = d (cid:48) Zτ d ( p τ , q τ ) for all τ . Since Z is R -lipschitz wehave ψ (cid:48) (0) ψ (0) ≤ R , hence(5.7) ψ ( τ ) ≥ (1 + tR ) ψ (0) > (1 − δ ) ψ (0)for all τ ∈ [ t, ψ and choice (5.6) of t . Using convexityagain and the assumption that d (cid:48) Zt d ( p t , q t ) ≥ − R , we have(5.8) ψ ( τ ) ≥ (1 + tR ) ψ ( t ) > (1 − δ ) ψ ( t ) . Since | t | ≤ δ (cid:107) Z (cid:107) , the points p t , q t , p , q are all within δ ≤ θ of each other,and therefore the contrapositive of Lemma 5.5, together with (5.7) and (5.8),implies that the lines p t q t and p q form an angle ≤ θ , or stay disjoint. Then d (cid:48) Y d ( p t , q t ) ≤ d (cid:48) Y d ( p , q ) + ε by (5.5). We also have d (cid:48) Z t d ( p t , q t ) ≤ d (cid:48) Z d ( p , q ) + ε by Proposition 5.4, since t ∈ ( t , d (cid:48) X t d ( p t , q t ) ≤ d (cid:48) X d ( p , q ) + 2 ε. By subdivision (Observation 3.5), we obtain lip( X t ) ≤ lip( X )+2 ε , as wished.This completes the proof of Proposition 5.6. (cid:3) Proposition 5.6 immediately implies Proposition 5.2, and Theorem 1.6follows using Theorem 5.1.5.5.
Smooth vector fields.
As it will be useful in Section 7, we prove thatthe ( j, u )-equivariant vector field X ∗ of Proposition 5.2 can be taken to besmooth (and standard in the funnels). Proposition 5.7.
Given a ( j, u ) -equivariant convex field X defined on H and standard in the funnels, there exist smooth , standard, ( j, u ) -equivariant vector fields X ∗ defined on H with lip( X ∗ ) arbitrarily close to lip( X ) . It is sufficient to smooth out the j -invariant vector field Z t of Proposi-tion 5.6 (without destroying the lipschitz constant). Our first step is to provethat Z t is actually a Lipschitz (uppercase!) section of the tangent bundleof H . First, let us recall the definition.For p, q ∈ H , let ϕ qp be the isometry of H that takes p to q by translatingalong the geodesic ( p, q ). To condense notation, the differential of this mapwill again be denoted by ϕ qp . Note that the action of ϕ qp on T p H is paralleltransport along the geodesic segment [ p, q ]. We also note that ( ϕ qp ) − = ϕ pq .A vector field X on H is said to be Lipschitz if it is a Lipschitz section of T H , in the sense that there exists L ≥ p, q ∈ H , (cid:107) ϕ qp X ( p ) − X ( q ) (cid:107) ≤ L d ( p, q ) OMPLETE LORENTZ SPACETIMES OF CONSTANT CURVATURE 40
This implies in particular lip( X ) ≤ L . Lemma 5.8.
Let Z (cid:48) be a bounded lipschitz vector field. For all t < closeenough to , the flow-back Z (cid:48) t is a Lipschitz vector field.Proof.
By Proposition 5.4, for t close enough to 0 (depending on lip( Z (cid:48) )only), the vector fields Z (cid:48) t for t ≤ t ≤ R -lipschitz for some R ∈ R independent of t . Fix such t and R , and define Z := Z (cid:48) t .Consider distinct points p, q ∈ H . For 0 ≤ t ≤ | t | , set p t := exp p ( tZ ( p ))and q t := exp q ( tZ ( q )). Since Z t = Z (cid:48) t + t , we have lip( Z t ) ≤ R , which byintegrating implies d ( p t , q t ) ≤ e Rt d ( p, q ). Define moreover r t := ϕ pq ( q t ) ∈ H .If (cid:107) Z (cid:107) ≤ N (notation (3.1)), then d ( q t , r t ) ≤ cosh( N t ) d ( p, q ).Define comparison points p ∗ , q ∗ ∈ R , distance d ( p, q ) apart, as well asvectors P, Q ∈ R such that (cid:107) P (cid:107) = (cid:107) Z ( p ) (cid:107) and (cid:107) Q (cid:107) = (cid:107) Z ( q ) (cid:107) , and suchthat [ p ∗ , q ∗ ] forms the same two angles with P and Q as [ p, q ] does with Z ( p ) and Z ( q ) respectively. Define p ∗ t := p ∗ + tP and q ∗ t := q ∗ + tQ and r ∗ t := p ∗ + tQ . Then d ( p ∗ t , q ∗ t ) ≤ d ( p ∗ t , r ∗ t ) + d ( r ∗ t , q ∗ t ) ≤ d ( p t , r t ) + d ( p, q ) ≤ d ( p t , q t ) + d ( q t , r t ) + d ( p, q ) ≤ (cid:0) e Rt + cosh( N t ) + 1 (cid:1) d ( p, q ) . On the other hand, by the triangle inequality, d ( p ∗ t , q ∗ t ) ≥ d ( p ∗ t , r ∗ t ) − d ( r ∗ t , q ∗ t ) = t (cid:107) P − Q (cid:107) − d ( p, q ) . But (cid:107) P − Q (cid:107) = (cid:107) ϕ qp Z ( p ) − Z ( q ) (cid:107) . Combining these estimates, we find (cid:107) ϕ qp Z ( p ) − Z ( q ) (cid:107) ≤ e Rt + cosh( N t ) + 2 t d ( p, q ) . Taking t = | t | gives an upper bound for the Lipschitz constant of Z . (cid:3) It follows from Proposition 5.4 and Lemma 5.8 that the j -invariant com-ponent of the vector field constructed in Proposition 5.6 is Lipschitz. Wenext apply to it a smoothing procedure.For any ∆ >
0, we choose a smooth kernel ψ ∆ : H × H → R + , invariantunder all isometries of H , that vanishes on all pairs of points distance > ∆apart, and such that(5.9) (cid:90) H ψ ∆ ( p, p (cid:48) ) d p (cid:48) = 1for all p ∈ H . For any vector field Z on H , we define a smoothed vectorfield (cid:101) Z ∆ on H by convolution by ψ ∆ : (cid:101) Z ∆ : p (cid:55)−→ (cid:90) H ψ ∆ ( p, p (cid:48) ) ϕ pp (cid:48) Z ( p (cid:48) ) d p (cid:48) . Then (cid:101) Z ∆ inherits the smoothness of ψ . If Z is j -invariant, then so is (cid:101) Z ∆ . OMPLETE LORENTZ SPACETIMES OF CONSTANT CURVATURE 41
Lemma 5.9.
Let X be a lipschitz, ( j, u ) -equivariant vector field on H .Let Y be a smooth, ( j, u ) -equivariant vector field on H that coincides with X outside some cocompact j (Γ) -invariant set, and let Z be the j -invariantvector field on H such that X = Y + Z . If Z is Lipschitz, then for any ε > and any small enough ∆ > , lip (cid:0) Y + (cid:101) Z ∆ (cid:1) ≤ lip( X ) + ε. Proof.
Fix ε > > (cid:101) Z and ψ in place of (cid:101) Z ∆ and ψ ∆ . By subdivision(Observation 3.5), it is enough to prove that if ∆ is small enough, then d (cid:48) Y + (cid:101) Z ( p, q ) ≤ (lip( X ) + ε ) d ( p, q ) for all distinct p, q ∈ H with d ( p, q ) ≤ ∆.Consider such a pair ( p, q ), let u p ∈ T p H be the unit vector at p pointingto q , and set u q := ϕ qp ( u p ) ∈ T q H . By Remark 3.4, d (cid:48) (cid:101) Z ( p, q ) = (cid:104) (cid:101) Z ( q ) | u q (cid:105) − (cid:104) (cid:101) Z ( p ) | u p (cid:105) = (cid:90) H ψ ( q, q (cid:48) ) (cid:104) ϕ qq (cid:48) Z ( q (cid:48) ) | u q (cid:105) d q (cid:48) − (cid:90) H ψ ( p, p (cid:48) ) (cid:104) ϕ pp (cid:48) Z ( p (cid:48) ) | u p (cid:105) d p (cid:48) = (cid:90) H ψ ( q, q (cid:48) ) (cid:104) Z ( q (cid:48) ) | ϕ q (cid:48) q u q (cid:105) d q (cid:48) − (cid:90) H ψ ( p, p (cid:48) ) (cid:104) Z ( p (cid:48) ) | ϕ p (cid:48) p u p (cid:105) d p (cid:48) . In the first integral, we make the substitution q (cid:48) = ϕ qp ( p (cid:48) ): (cid:90) H ψ ( q, q (cid:48) ) (cid:104) Z ( q (cid:48) ) | ϕ q (cid:48) q u q (cid:105) d q (cid:48) = (cid:90) H ψ ( p, p (cid:48) ) (cid:104) ϕ qp Z ( q (cid:48) ) | ϕ p (cid:48) p u p (cid:105) d p (cid:48) , where we use the invariance of ψ under ϕ qp and the fact that ϕ pq takes ϕ q (cid:48) q u q to ϕ p (cid:48) p u p , by the conjugacy relation ϕ p (cid:48) p = ϕ pq ϕ q (cid:48) q ( ϕ pq ) − . Therefore,(5.10) d (cid:48) (cid:101) Z ( p, q ) = (cid:90) H ψ ( p, p (cid:48) ) (cid:104) ϕ pq Z ( q (cid:48) ) − Z ( p (cid:48) ) | ϕ p (cid:48) p u p (cid:105) d p (cid:48) . We now focus on the integrand. We may restrict to p (cid:48) at distance ≤ ∆from p , otherwise ψ ( p, p (cid:48) ) = 0. Let u (cid:48) to be the unit vector at p (cid:48) pointingto q (cid:48) (see Figure 10). Then (cid:104) ϕ pq Z ( q (cid:48) ) − Z ( p (cid:48) ) | ϕ p (cid:48) p u p (cid:105) = (cid:104) ϕ p (cid:48) q (cid:48) Z ( q (cid:48) ) − Z ( p (cid:48) ) | u (cid:48) (cid:105) + (cid:104) ϕ p (cid:48) q (cid:48) Z ( q (cid:48) ) − Z ( p (cid:48) ) | ϕ p (cid:48) p u p − u (cid:48) (cid:105) + (cid:104) ( ϕ pq − ϕ p (cid:48) q (cid:48) ) Z ( q (cid:48) ) | ϕ p (cid:48) p u p (cid:105) . The first term on the right-hand side is d (cid:48) Z ( p (cid:48) , q (cid:48) ) (Remark 3.4), and weshall see that the second and third terms are but small corrections, boundedrespectively by L ∆ d ( p (cid:48) , q (cid:48) ) and (cid:107) Z (cid:107) ∆ d ( p (cid:48) , q (cid:48) ) if ∆ is small enough. Here L is any positive number such that Z is L -Lipschitz; we have (cid:107) Z (cid:107) < + ∞ since Z is continuous, j -invariant, and is zero outside some cocompact set.Let us explain these bounds. OMPLETE LORENTZ SPACETIMES OF CONSTANT CURVATURE 42
In the second term we have (cid:107) ϕ p (cid:48) q (cid:48) Z ( q (cid:48) ) − Z ( p (cid:48) ) (cid:107) ≤ L d ( p (cid:48) , q (cid:48) ) because Z is L -Lipschitz. For ∆ small enough, we also have (cid:107) ϕ p (cid:48) p u p − u (cid:48) (cid:107) ≤ ∆ by the Gauss–Bonnet formula. Indeed, let w = ϕ p (cid:48) p u p , let p (cid:48)(cid:48) and q (cid:48)(cid:48) be the closest pointsto p (cid:48) and q (cid:48) respectively on the line pq , and let r (cid:48) and r (cid:48)(cid:48) be the midpointsof [ p (cid:48) , q (cid:48) ] and [ p (cid:48)(cid:48) , q (cid:48)(cid:48) ] respectively as in Figure 10. Then the angle between u (cid:48) and w is equal to the area of the quadrilateral pr (cid:48)(cid:48) r (cid:48) p (cid:48) , because u (cid:48) maybe obtained by transporting u p along the broken line pr (cid:48)(cid:48) r (cid:48) p (cid:48) while w is thetransport of u p along pp (cid:48) . Since d ( p, q ) and d ( p, p (cid:48) ) = d ( q, q (cid:48) ) are boundedby ∆, we obtain that (cid:107) w − u (cid:48) (cid:107) is bounded by the area 2 π (cosh(2∆) − H . In particular, (cid:107) w − u (cid:48) (cid:107) ≤ ∆ if ∆ is smallenough. We note that as q → p , the angle between w and u (cid:48) approaches thearea of triangle pp (cid:48)(cid:48) p (cid:48) , which in general is nonzero. This negative curvaturephenomenon makes the Lipschitz assumption on Z necessary (at least forthis proof).In the third term, we observe that ϕ pq Z ( q (cid:48) ) is the parallel translation of Z ( q (cid:48) ) along the broken line q (cid:48) qpp (cid:48) and thus differs from ϕ p (cid:48) q (cid:48) Z ( q (cid:48) ) by a rotationof angle equal to the area of the quadrilateral q (cid:48) qpp (cid:48) . This area is at most∆ d ( p (cid:48) , q (cid:48) ), hence (cid:107) ( ϕ pq − ϕ p (cid:48) q (cid:48) ) Z ( q (cid:48) ) (cid:107) ≤ ∆ (cid:107) Z (cid:107) d ( p (cid:48) , q (cid:48) ). u p u q p (cid:48)(cid:48) p q q (cid:48) p (cid:48) q (cid:48)(cid:48) u (cid:48) w r (cid:48) r (cid:48)(cid:48) Figure 10.
In the proof of Lemma 5.9, the angle between u (cid:48) and w = ϕ p (cid:48) p u p is the area of the shaded quadrilateral pr (cid:48)(cid:48) r (cid:48) p (cid:48) .We now turn back to the integrand in (5.10). If ∆ is small enough, with( L + (cid:107) Z (cid:107) )∆ ≤ ε/ ≤ L + ε/ L + ε/ , then (cid:104) ϕ pq Z ( q (cid:48) ) − Z ( p (cid:48) ) | ϕ p (cid:48) p ( u p ) (cid:105) ≤ (cid:18) d (cid:48) Z d ( p (cid:48) , q (cid:48) ) + L ∆ + (cid:107) Z (cid:107) ∆ (cid:19) d ( p (cid:48) , q (cid:48) )= (cid:18) d (cid:48) Z d ( p (cid:48) , q (cid:48) ) + ε (cid:19) d ( p, q ) cosh ∆ ≤ (cid:18) d (cid:48) Z d ( p (cid:48) , q (cid:48) ) + ε (cid:19) d ( p, q ) , (5.11)where the last inequality follows from the bounds cosh ∆ ≤ L + ε/ L + ε/ and | d (cid:48) Z d ( p (cid:48) , q (cid:48) ) | ≤ L , and from the monotonicity of the function t (cid:55)→ t + ε/ t + ε/ . OMPLETE LORENTZ SPACETIMES OF CONSTANT CURVATURE 43
We now bound lip( Y + (cid:101) Z ). The vector field Z is zero outside some j (Γ)-invariant, cocompact set U (cid:48)(cid:48) ⊂ H . Let U (cid:48) be a j (Γ)-invariant, cocompactneighborhood of U (cid:48)(cid:48) . If ∆ is small enough, then the interior of U (cid:48) containsthe closed ∆-neighborhood U ∆ of U (cid:48)(cid:48) . By smoothness of Y , up to taking ∆even smaller, we may assume that for any p (cid:54) = q in U (cid:48) and q (cid:48) = ϕ qp ( p (cid:48) ) with d ( p, q ) , d ( p, p (cid:48) ) ≤ ∆ we have d (cid:48) Y d ( p, q ) ≤ d (cid:48) Y d ( p (cid:48) , q (cid:48) ) + ε/
2. Then (5.9), (5.10),and (5.11) imply that for all p, q ∈ U (cid:48) with 0 < d ( p, q ) ≤ ∆, d (cid:48) Y + (cid:101) Z d ( p, q ) ≤ (cid:90) H ψ ( p, p (cid:48) ) (cid:18) d (cid:48) Y d (cid:0) p (cid:48) , ϕ qp ( p (cid:48) ) (cid:1) + ε d (cid:48) Z d (cid:0) p (cid:48) , ϕ qp ( p (cid:48) ) (cid:1) + ε (cid:19) d p (cid:48) ≤ lip( Y + Z ) + ε = lip( X ) + ε. By subdivision, lip U (cid:48) ( Y + (cid:101) Z ) ≤ lip( X ) + ε . On the other hand, on H (cid:114) U ∆ we have Y + (cid:101) Z = Y = X , hence lip H (cid:114) U ∆ ( Y + (cid:101) Z ) = lip( X ). We concludeusing Observation 3.5.(iv). (cid:3) Proposition 5.7 follows from Proposition 5.6 and Lemmas 5.8 and 5.9.6.
Applications of Theorem 1.6: proper actions and fibrations
What we have proved so far shows that the conditions (1) and (2) ofTheorem 1.1 are equivalent: (1) implies (2) by (3.6), and the negation of(1) implies by Theorem 1.6 the existence of a k -stretched lamination L (with k ≥ L can be approached by simple closed curves).In Section 6.1 (resp. 6.2), we prove that (1) is necessary (resp. sufficient)for the action on R , to be properly discontinuous, thus proving Theo-rem 1.1. The first direction (Proposition 6.2) uses Theorem 1.6; the second(Proposition 6.3, where fibrations by timelike lines appear) can be read in-dependently. Theorem 1.2 is then a byproduct of Proposition 6.3.Note that it is enough to prove Theorems 1.1 and 1.2 for a finite-index,torsion-free subgroup Γ (cid:48) of Γ (such a subgroup exists by the Selberg lemma[Se, Lem. 8]). Indeed, if X is a ( j | Γ (cid:48) , u | Γ (cid:48) )-equivariant convex or vector field,we can always average out the translates γ i • X (notation (2.5)), where thecosets γ i Γ (cid:48) form a partition of Γ, to produce a ( j, ρ )-equivariant field whoselipschitz constant is at most that of X (using Observation 3.5). Thus, weassume Γ to be torsion-free in this section.6.1. A necessary condition for properness.
Let Γ be a torsion-free dis-crete group and j ∈ Hom(Γ , G ) a convex cocompact representation. Re-call that λ ( g ) refers to the translation length of g ∈ G , see (1.1). In the“macroscopic” case, the following was established in [Kas]; see also [GK] forgeneralizations. Proposition 6.1 [Kas, Th. 5.1.1] . Let ρ ∈ Hom(Γ , G ) be an arbitrary rep-resentation such that λ ( ρ ( β )) ≤ λ ( j ( β )) for at least one β ∈ Γ (cid:114) { } . If the OMPLETE LORENTZ SPACETIMES OF CONSTANT CURVATURE 44 group Γ j,ρ := (cid:8) ( j ( γ ) , ρ ( γ )) | γ ∈ Γ (cid:9) ⊂ G × G acts properly discontinuously on AdS , then there exists a ( j, ρ ) -equivariantLipschitz map f : H → H with Lipschitz constant < . Here we prove the following “microscopic” version. Recall that if u is a j -cocycle, then the Margulis invariant α u ( γ ) = dd t (cid:12)(cid:12) t =0 λ ( e tu ( j ( γ )) j ( γ )) is theinfinitesimal rate of change of the length of j ( γ ) under deformation in the u direction (see Section 2.5). Proposition 6.2.
Let u : Γ → g be a j -cocycle such that α u ( β ) ≤ for atleast one β ∈ Γ (cid:114) { } . If the group Γ j,u := (cid:8) ( j ( γ ) , u ( γ )) | γ ∈ Γ (cid:9) ⊂ G (cid:110) g acts properly discontinuously on R , , then there exists a ( j, u ) -equivariantlipschitz vector field on H with lipschitz constant < .Proof. We prove the contrapositive. Let k be the infimum of lipschitz con-stants of ( j, u )-equivariant vector fields on H and assume k ≥
0. Let (cid:96) be ageodesic line in H that projects to a leaf of the maximally stretched lami-nation L given by Theorem 1.6. Let X be the k -lipschitz, globally definedconvex field also given by Theorem 1.6. • The case k = 0 . We first suppose that k = 0. Since X is defined on allof H , Lemma 4.3 implies that (cid:107) X ( C ) (cid:107) < + ∞ for any compact set C ⊂ H .We claim that there is a Killing field Y on H such that X ( p ) contains Y ( p ) for any p in the leaf (cid:96) . Indeed, choose for every p ∈ (cid:96) a vector x p ∈ X ( p ). If p, q ∈ (cid:96) are distinct, then there is a unique Killing field Y such that x p = Y ( p ) and x q = Y ( q ). For any point r on the segment [ p, q ],the vector Y ( r ) belongs to X ( r ) by Proposition 3.8. The Killing field Y containing x p and x q may depend on p and q , but as p and q escape alongthe line (cid:96) in opposite directions, the possible Killing fields Y that arise allbelong to a compact subset of g , because any X ( r ) is bounded. Thereforewe can extract a limit Y such that Y ( p ) ∈ X ( p ) for all p ∈ (cid:96) .Up to modifying u by a coboundary, we may assume that Y is the zerovector field, i.e. p ) ∈ X ( p ) for all p ∈ (cid:96) . Since (cid:96) is contained in theconvex core, which is compact modulo j (Γ), we can find a ball B ⊂ H ,a sequence ( γ n ) n ∈ N of pairwise distinct elements of Γ, and, for any n ∈ N ,two points p n , p (cid:48) n ∈ (cid:96) , distance one apart, such that q n := j ( γ n ) · p n and q (cid:48) n := j ( γ n ) · p (cid:48) n both belong to B . Since 0( p n ) ∈ X ( p n ), the set X ( q n ) = j ( γ n ) ∗ ( X ( p n )) + u ( γ n )( q n ) contains u ( γ n )( q n ); similarly u ( γ n )( q (cid:48) n ) ∈ X ( q (cid:48) n ).Since R := (cid:107) X ( B ) (cid:107) < + ∞ and d ( q n , q (cid:48) n ) = 1, for any n ∈ N the Killing field u ( γ n ) lies in the compact set C = (cid:8) v ∈ g | ∃ q, q (cid:48) ∈ B with d ( q, q (cid:48) ) = 1 and (cid:107) v ( q ) (cid:107) , (cid:107) v ( q (cid:48) ) (cid:107) ≤ R (cid:9) . Thus the action of Γ j,u on g ∼ = R , fails to take the origin 0 ∈ g off C ; inparticular, it cannot be properly discontinuous. OMPLETE LORENTZ SPACETIMES OF CONSTANT CURVATURE 45 • The case k > and the opposite sign lemma. We now suppose that k >
0. By Proposition 3.9, there is an element α ∈ Γ (cid:114) { } , correspondingto a closed curve of j (Γ) \ H nearly carried by L , whose Margulis invariant α u ( α ) is positive. Recall that by assumption in Proposition 6.2, there isalso an element β ∈ Γ (cid:114) { } with α u ( β ) ≤
0. The existence of such α, β isa well-known obstruction to properness, known as Margulis’s opposite signlemma [Ma1, Ma2]. For convenience, we give the idea of a proof here.If α u ( β ) = 0, then ( j ( β ) , u ( β )) ∈ G (cid:110) g has a fixed point in g ∼ = R , ,hence Γ j,u does not act properly discontinuously on R , . We now suppose α u ( α ) > > α u ( β ). Let ( a + , a , a − ) and ( b + , b , b − ) be bases of eigenvectorsfor the adjoint action of j ( α ) and j ( β ) on g , respectively. As in Section 2.4,we assume that a − , a + , b − , b + belong to the positive light cone L of g , that a − , b − (resp. a + , b + ) correspond to eigenvalues < > a (resp. b ) is a positive multiple of a − ∧ a + (resp. of b − ∧ b + ), of norm 1.From a , when looking at L , one sees a + on the left and a − on the right, andsimilarly for b , b + , b − . Let A, B ⊂ g be the affine lines, of directions R a and R b , that are preserved by ( j ( α ) , u ( α )) and ( j ( β ) , u ( β )) respectively(see Section 2.4). For simplicity, we first assume that A and B both containthe origin O .Let A = A + R a + be the unstable lightlike plane containing A , and B = B + R b − the stable lightlike plane containing B . Up to applying alinear transformation in O(2 , and rescaling a + and b − , we may assumethat A ∩ B is the x -axis R (1 , ,
0) and that a + = (0 , − ,
1) and b − = (0 , , a ∈ A and b ∈ B have positive x -coordinates, which we denote by a x and b x respectively. Figure 11 looks down the z -axis. L O p Ox BA A Bβ m ( p ) V b a a + b − Figure 11.
Looking down the z -axis when the axes A and B both contain the origin O . The positive light cone L , trun-cated, appears as a circle.Let V ⊂ A be a compact neighborhood of a point of A , for examplenear the origin. Since α u ( α ) >
0, the action of ( j ( α ) , u ( α )) ∈ G (cid:110) g on A is to translate in the x -direction to the right (preserving the axis A ), whileexpanding exponentially in the a + -direction outwards from A . In particular, OMPLETE LORENTZ SPACETIMES OF CONSTANT CURVATURE 46 for all large enough n , the iterate ( j ( α ) , u ( α )) n · V meets the x -axis at a point p n of large positive abscissa n a x α u ( α ) + O (1).Since α u ( β ) <
0, the action of ( j ( β ) , u ( β )) ∈ G (cid:110) g on B is to translatein the x -direction to the left (preserving the axis B ), while contracting ex-ponentially in the b − -direction towards B . In particular, ( j ( β ) , u ( β )) m · p n is close to the origin O if m = (cid:22) n a x α u ( α ) b x | α u ( β ) | (cid:23) , where (cid:98)·(cid:99) denotes the integral part. For such integers n, m , the element( j ( β ) , u ( β )) m ( j ( α ) , u ( α )) n ∈ Γ j,u does not carry V off some compact set.Thus the action of Γ j,u on R , is not properly continuous.If the axes A and B do not both contain the origin O , then the argument isessentially the same, replacing the x -axis with the intersection line ∆ = A∩B (which we may assume to be parallel to the x -axis): we see that β m · p n isstill close to the point B ∩ ∆ for n, m as above. This completes the proof ofProposition 6.2. (cid:3) Fibrations by timelike geodesics and a sufficient conditionfor properness.
The following proposition, together with Proposition 6.2,completes the proof of Theorems 1.1 and 1.2.
Proposition 6.3.
Let Γ be a torsion-free discrete group and j ∈ Hom(Γ , G ) a discrete and injective representation, with quotient surface S := j (Γ) \ H .Let ρ ∈ Hom(Γ , G ) be an arbitrary representation and u : Γ → g a j -cocycle.(1) Suppose there exists a ( j, ρ ) -equivariant Lipschitz map f : H → H with Lipschitz constant K < . Then the group Γ j,ρ := (cid:8) ( j ( γ ) , ρ ( γ )) | γ ∈ Γ (cid:9) ⊂ G × G acts properly discontinuously on AdS and f induces a fibration F f of the quotient Γ j,ρ \ AdS over S by timelike geodesic circles.(2) Suppose there exists a ( j, u ) -equivariant lipschitz vector field X : H → T H with lipschitz constant k < . Then the group Γ j,u := (cid:8) ( j ( γ ) , u ( γ )) | γ ∈ Γ (cid:9) ⊂ G (cid:110) g acts properly discontinuously on R , and X induces a fibration F X of the quotient Γ j,u \ R , over S by timelike geodesic lines. The fact that the existence of a ( j, ρ )-equivariant contracting Lipschitzmap implies the properness of the action of Γ j,ρ is an easy consequenceof the general properness criterion of Benoist [Be] and Kobayashi [Ko1]:see [Kas]. Our method here gives a different, short proof.Note that we do not require j to be convex cocompact in Proposition 6.3.In particular, using [GK, Th. 1.8], we obtain that any Lorentzian -manifoldwhich is the quotient of AdS by a finitely generated group fibers in circlesover a ( -dimensional) hyperbolic orbifold . OMPLETE LORENTZ SPACETIMES OF CONSTANT CURVATURE 47
Proof of Proposition 6.3. (1) The timelike geodesics in AdS = G are pa-rameterized by H × H : for p, q ∈ H , the corresponding timelike geodesic is L p,q := { g ∈ G | g · p = q } , which is a coset of the stabilizer of p , hence topologically a circle. The map f : H → H determines a natural collection { L p,f ( p ) | p ∈ H } of timelikegeodesics in G . This collection is a fibration of G over H . Indeed, for g ∈ G , the map g − ◦ f is K -Lipschitz because f is; since K <
1, it hasa unique fixed point, which we denote by Π( g ) ∈ H . Thus g belongs to L p,f ( p ) for a unique p ∈ H , namely Π( g ). The surjective map Π : G → H is continuous. Indeed, if g (cid:48) ∈ G is close enough to g in the sense that d ( p, g (cid:48)− ◦ f ( p )) < (1 − K ) δ for p = Π( g ), then g (cid:48)− ◦ f maps the ball ofradius δ around p into itself, hence the unique fixed point Π( g (cid:48) ) of g (cid:48)− ◦ f iswithin δ of p = Π( g ). Finally, the map Π satisfies the following equivarianceproperty: Π (cid:0) ρ ( γ ) gj ( γ ) − (cid:1) = j ( γ ) · Π( g )for all γ ∈ Γ and g ∈ G . Therefore, the properness of the action of Γ j,ρ on G = AdS follows from the properness of the action of j (Γ) on H , and Πdescends to a bundle projection Π : Γ j,ρ \ AdS → S .(2) The timelike geodesics in R , = g are parameterized by the tangentbundle T H : for p ∈ H and x ∈ T p H , the corresponding timelike geodesicis the affine line (cid:96) x := { Y ∈ g | Y ( p ) = x } , which is a translate of the infinitesimal stabilizer of p . The vector field X : H → T H determines a natural collection { (cid:96) X ( p ) | p ∈ H } of timelikegeodesics. This collection is a fibration of g over H . Indeed, for Y ∈ g ,the vector field X − Y is k -lipschitz because X is k -lipschitz and Y is aKilling field; by Proposition 3.6, it has a unique zero, which we denote by (cid:36) ( Y ) ∈ H . Thus Y belongs to (cid:96) X ( p ) for a unique p ∈ H , namely (cid:36) ( Y ).The surjective map (cid:36) : g → H is continuous. Indeed, if Y (cid:48) ∈ g is closeenough to Y in the sense that (cid:107) ( Y − Y (cid:48) )( p ) (cid:107) < | k | δ for p = (cid:36) ( Y ), then the k -lipschitz field X − Y (cid:48) = ( X − Y ) + ( Y − Y (cid:48) ) points inward along the sphereof radius δ centered at p , hence the unique zero (cid:36) ( Y (cid:48) ) of X − Y (cid:48) is within δ of p = (cid:36) ( Y ). Finally, the map (cid:36) satisfies the following equivariance property: (cid:36) (cid:0) j ( γ ) · Y + u ( γ ) (cid:1) = j ( γ ) · (cid:36) ( Y )for all γ ∈ Γ and Y ∈ g . Therefore, the properness of the action of Γ j,u on g = R , follows from the properness of the action of j (Γ) on H , and (cid:36) descends to a bundle projection (cid:36) : Γ j,u \ R , → S . (cid:3) Note that in Proposition 6.3.(2), replacing X with a convex field of neg-ative lipschitz constant would still lead to a fibration of Γ j,u \ R , (distinctvectors in X ( p ) for the same p ∈ H lead to parallel, but distinct linesof R , ). However, using a vector field has the advantage of making the leafspace canonically homeomorphic to S . OMPLETE LORENTZ SPACETIMES OF CONSTANT CURVATURE 48
In fact, in the R , case the quotient manifold is diffeomorphic to S × R as the fibers are oriented by the time direction. Note however that the quo-tient manifold is not globally hyperbolic (see e.g. [Me] for a definition), forCharette–Drumm–Brill [CDB] have shown that Margulis spacetimes containclosed timelike curves (smooth, but not geodesic). Nonetheless, the fibration F X of Proposition 6.3.(2) admits a spacelike section. Indeed, Barbot [Ba]has shown the existence of a convex, future-complete domain Ω + in R , ,invariant under Γ j,u , such that the quotient Γ j,u \ Ω + is globally hyperbolicand Cauchy-complete (this works in the general context where j is convexcocompact and u is any cocycle). In particular, the quotient manifold con-tains a Cauchy surface whose lift to Ω + is a spacelike, convex, complete,embedded disk. This disk intersects all timelike geodesics in R , exactlyonce and so gives a spacelike section of F X .7. Margulis spacetimes are limits of
AdS manifolds
In this section, we make precise the idea that all Margulis spacetimesshould arise by zooming in on collapsing AdS spacetimes. As in the hyperbo-lic-to-AdS transition described in [Da], one natural framework to describethis geometric transition is that of real projective geometry (Section 7.1);this approach eliminates Lorentzian metrics from the analysis. The proof ofTheorem 1.4 is given in Section 7.3; the main technical step (Section 7.2) isto control the geometry of collapsing AdS manifolds by producing geodesicfibrations as in Proposition 6.3 and controlling how they degenerate. Fi-nally, in Section 7.4 we prove Corollary 1.5, which describes the geometrictransition in the language of Lorentzian metrics.Background material on locally homogeneous geometric structures, par-ticularly relevant for Section 7.3, may be found in [T1, Go2].7.1. AdS and R , as projective geometries. Both AdS and R , canbe realized as domains in projective space. Indeed, the map I : (cid:18) y + y y − y y + y − y + y (cid:19) (cid:55)−→ [ y : y : y : y ]defines an embedding of AdS = G = PSL ( R ) into RP whose image is theopen set { [ y ] ∈ RP | y + y − y − y < } (the interior of a projectivequadric); it induces an injective group homomorphism I ∗ : Isom(AdS ) = G × G (cid:44) → PGL ( R ), and I is I ∗ -equivariant:(7.1) I ( A · x ) = I ∗ ( A ) · I ( x )for all A ∈ Isom(AdS ) and x ∈ AdS . The map i : (cid:18) z z − z z + z − z (cid:19) (cid:55)−→ [ z : z : z : 1]defines an embedding of R , = g = sl ( R ) into RP whose image is the affinechart { [ z ] ∈ RP | z (cid:54) = 0 } ; it induces an injective group homomorphism OMPLETE LORENTZ SPACETIMES OF CONSTANT CURVATURE 49 i ∗ : Isom( R , ) = G (cid:110) g (cid:44) → PGL ( R ), and i is i ∗ -equivariant:(7.2) i (cid:0) B · w ) = i ∗ ( B ) · i ( w )for all B ∈ Isom( R , ) and w ∈ R , . We see flat Lorentzian geometryas a limit of AdS geometry by inflating an infinitesimal neighborhood ofthe identity matrix. More precisely, we rescale by applying the family ofprojective transformations (7.3) r t := t − t − t − ∈ PGL ( R ) . Note that r t · I (AdS ) ⊂ r t (cid:48) · I (AdS ) for 0 < t (cid:48) < t and that (cid:91) t> r t · I (AdS ) = i ( R , ) ∪ H ∞ , where H ∞ := { [ y ] ∈ RP | y + y − y < y } is a copy of the hyperbolicplane. We observe that the limit as t → r t is differentiation: Proposition 7.1. (1) For any smooth path t (cid:55)→ g t ∈ G = AdS with g = 1 , r t · I ( g t ) −−→ t → i (cid:18) dd t (cid:12)(cid:12)(cid:12) t =0 g t (cid:19) ∈ RP . (2) For any smooth path t (cid:55)→ ( h t , k t ) ∈ G × G = Isom(AdS ) with h = k , r t I ∗ ( h t , k t ) r − t −−→ t → i ∗ (cid:18) h , dd t (cid:12)(cid:12)(cid:12) t =0 h t k − t (cid:19) ∈ PGL ( R ) . Proof.
Statement (1) is an immediate consequence of the definitions. State-ment (2) follows from (1) by using the equivariance relations (7.1) and (7.2):given X ∈ R , , for small enough t > i ( X ) = r t · I ( g t ) forsome g t ∈ G , and g t converges to 1; by (1) we have dd t (cid:12)(cid:12) t =0 g t = X and (cid:0) r t I ∗ ( h t , k t ) r − t (cid:1) · i ( X ) = r t · I ( k t g t h − t )converges to i (cid:18) dd t (cid:12)(cid:12)(cid:12) t =0 k t g t h − t (cid:19) = i (cid:18) dd t (cid:12)(cid:12)(cid:12) t =0 k t g t k − t k t h − t (cid:19) = i (cid:18) Ad( k ) (cid:18) dd t (cid:12)(cid:12)(cid:12) t =0 g t (cid:19) + dd t (cid:12)(cid:12)(cid:12) t =0 k t h − t (cid:19) = i ∗ (cid:18) k , dd t (cid:12)(cid:12)(cid:12) t =0 k t h − t (cid:19) · i ( X ) . We conclude using the fact that an element of PGL ( R ) is determined byits action on i ( R , ). (cid:3) Now, let Γ be a discrete group, j ∈ Hom(Γ , G ) a convex cocompact rep-resentation, and u : Γ → g a j -cocycle. Proposition 7.1 has the followingimmediate consequence. OMPLETE LORENTZ SPACETIMES OF CONSTANT CURVATURE 50
Corollary 7.2.
Let t (cid:55)→ j t and t (cid:55)→ ρ t be two smooth paths in Hom(Γ , G ) with j = ρ = j and dd t (cid:12)(cid:12) t =0 ρ t j − t = u . For all γ ∈ Γ , r t I ∗ (cid:0) j t ( γ ) , ρ t ( γ ) (cid:1) r − t −−→ t → i ∗ (cid:0) j ( γ ) , u ( γ ) (cid:1) ∈ PGL ( R ) . Corollary 7.2 states that r t ( I ∗ Γ j t ,ρ t ) r − t converges to i ∗ Γ j,u as groups act-ing on RP . When Γ j,u and Γ j t ,ρ t act properly discontinuously on R , andAdS respectively (which will be the case below), we thus obtain a path ofproper actions on the subspaces r t · I (AdS ) that converges (algebraically)to a proper action on i ( R , ). However, this is not enough to construct ageometric transition, for algebraic convergence may not in general give awell-defined continuous path of quotient manifolds. It is even possible thatthe topology of the quotient manifolds could change: see [AC] or [Ca] forexamples of “bumping” in the context of hyperbolic 3-manifolds. We thusneed a more careful geometric investigation of the situation.7.2. Collapsing fibered
AdS manifolds.
In this section, we prove twotechnical statements needed for Theorem 1.4. The first gives the proper dis-continuity of Theorem 1.4.(1) and, using Proposition 6.3, produces geodesicfibrations that are well controlled as the quotient AdS manifolds collapse.The second statement gives sections of these fibrations with suitable behav-ior under the collapse.Let Γ be a discrete group, j ∈ Hom(Γ , G ) a convex cocompact representa-tion, and u : Γ → g a proper deformation of j in the sense of Definition 2.1.Fix two smooth paths t (cid:55)→ j t ∈ Hom(Γ , G ) and t (cid:55)→ ρ t ∈ Hom(Γ , G ) with j = ρ = j and dd t (cid:12)(cid:12) t =0 ρ t j − t = u , as well as a smooth, ( j, u )-equivariantvector field X defined on H , standard in the funnels (Definition 3.12), with k := lip( X ) < X exists by Proposition 5.7). The following twopropositions are the main tools we will need for Theorem 1.4. Proposition 7.3.
There is a smooth family of ( j t , ρ t ) -equivariant diffeo-morphisms f t : H → H , defined for small enough t ≥ , such that(1) f t is Lipschitz with Lip( f t ) = 1 + kt + O ( t ) ;(2) f = Id H ;(3) dd t (cid:12)(cid:12) t =0 f t = X . In particular, by Proposition 6.3, the group Γ j t ,ρ t acts properly discontin-uously on AdS for all small enough t > Proposition 7.4.
Given a smooth family ( f t ) of diffeomorphisms as inProposition 7.3, there is a smooth family of smooth maps σ t : H → G ,defined for small enough t ≥ , such that(1) σ t is equivariant with respect to j t and ( j t , ρ t ) , meaning that for all γ ∈ Γ and p ∈ H , σ t ( j t ( γ ) · p ) = ρ t ( γ ) σ t ( p ) j t ( γ ) − ; (2) σ t is an embedding for all t > , while σ ( H ) = { } ⊂ G ; OMPLETE LORENTZ SPACETIMES OF CONSTANT CURVATURE 51 (3) σ t ( p ) · p = f t ( p ) for all p ∈ H ; in other words, σ t defines a sectionof the fibration F f t of Γ j t ,ρ t \ G from Proposition 6.3;(4) the derivative σ (cid:48) := ( dd t (cid:12)(cid:12) t =0 σ t ) : H → g = R , is an embedding,equivariant with respect to j and ( j, u ) , meaning that for all γ ∈ Γ and p ∈ H , σ (cid:48) ( j ( γ ) · p ) = j ( γ ) · σ (cid:48) ( p ) + u ( γ ); (5) σ (cid:48) ( p )( p ) = X ( p ) for all p ∈ H ; in other words, σ (cid:48) defines a sectionof the fibration F X of Γ j,u \ g from Proposition 6.3. We first prove Proposition 7.4. For a fixed t , there is a lot of flexibilityin defining a section σ t of F f t , since the bundle is trivial. One naturalconstruction with the desired first-order behavior is to take the osculatingisometry map to f t . Proof of Proposition 7.4.
Let σ t : H → G be the osculating isometry mapto the diffeomorphism f t : H → H : by definition, for any p ∈ H , theelement σ t ( p ) ∈ G is the orientation-preserving isometry of H that coincideswith f t at p and maps the (mutually orthogonal) principal directions of f t in T p H to their (mutually orthogonal) images in T f t ( p ) H . (The differentialmap d p f t has two principal values close to 1; if they happen to be equal,then all pairs of mutually perpendicular directions in T p H yield the samedefinition.) Then σ t is smooth for any t , and varies smoothly with t . For t = 0, we obtain the constant map with image 1 ∈ G (the isometry osculatingthe identity map is the identity). The ( j t , ( j t , ρ t ))-equivariance of σ t followsfrom the ( j t , ρ t )-equivariance of f t (using the fact that the osculating mapis unique).The derivative σ (cid:48) = ( dd t (cid:12)(cid:12) t =0 σ t ) : H → g maps any p ∈ H to the infini-tesimal isometry osculating X at p in the sense that the Killing field σ (cid:48) ( p )agrees with X at p and has the same curl. (This means that the vector field σ (cid:48) ( p ) − X vanishes at p and that its linearization x (cid:55)→ ∇ x ( σ (cid:48) ( p ) − X ) is a sym-metric endomorphism of T p H ; note by contrast that the linearization of aKilling field near a zero is anti symmetric.) Finally, σ (cid:48) inherits both smooth-ness and equivariance from X (using the uniqueness of the construction). (cid:3) We now turn to the proof of Proposition 7.3. Suppose ( f t ) is any smoothfamily of smooth maps H → H with f = Id H and dd t (cid:12)(cid:12) t =0 f t = X . For any p (cid:54) = q in H , dd t (cid:12)(cid:12)(cid:12) t =0 d (cid:0) f t ( p ) , f t ( q ) (cid:1) = d (cid:48) (cid:0) X ( p ) , X ( q ) (cid:1) , hence d (cid:0) f t ( p ) , f t ( q ) (cid:1) d ( p, q ) = 1 + t d (cid:48) X d ( p, q ) + O ( t ) . Since ( f t ) is a smooth family of smooth maps, the constant in the O ( t ) canbe taken uniform for p, q in a compact set, including for q → p , henceLip( f t ) ≤ kt + O ( t ) OMPLETE LORENTZ SPACETIMES OF CONSTANT CURVATURE 52 over any compact set. Thus we just need to find a smooth family ( f t ) forwhich the Lipschitz constant can be controlled uniformly above the funnels of the surface j (Γ) \ H . We now explain how this can be done. Proof of Proposition 7.3.
Let ( s t ) and ( s t ) be smooth families of diffeo-morphisms of H with s i = Id H , such that s t is ( j, j t )-equivariant and s t is( j, ρ t )-equivariant. For i ∈ { , } , we assume that the family ( s it ) is standardin the funnels , by which we mean the following. Let E ⊂ H be a connectedcomponent of the exterior of the convex core for j , whose boundary is thetranslation axis A j ( γ ) for some peripheral γ ∈ Γ (cid:114) { } . In Fermi coordinates F ( ξ, η ) (see Section 3.4) relative to A j ( γ ) , such that E = F ( R × R ∗− ), we askthat there exist a smooth family ( a it ) of isometries of H with a i = Id H ,and smooth families ( κ it ) , ( r it ) of reals with κ i = r i = 1, such that s it (cid:0) F ( ξ, η ) (cid:1) = a it ◦ F ( κ it ξ, r it η )for all η < ξ ∈ R . Note that, byequivariance, the isometry a t takes the axis A j ( γ ) to A j t ( γ ) , and a t takes A j ( γ ) to A ρ t ( γ ) ; these axes vary smoothly. We ask that this hold for anycomponent E of the complement of the convex core. Then the smooth,( j, u )-equivariant vector field Y := dd t (cid:12)(cid:12) t =0 s t ◦ ( s t ) − on H is standard in thefunnels in the sense of Definition 3.12. By choosing the first-order behaviorof a it , κ it , r it appropriately in each component E , we can arrange for Y toagree with X outside some cocompact j (Γ)-invariant neighborhood U (cid:48) ofthe convex core (because X itself is standard in the funnels).We claim that the ( j t , ρ t )-equivariant diffeomorphisms s t := s t ◦ ( s t ) − have the desired Lipschitz properties in the funnels. Indeed, in each compo-nent E of the complement of the convex core, direct computation givessup p ∈ s t ( E (cid:114) U (cid:48) ) (cid:107) d s t ( p ) (cid:107) ≤ max (cid:26) r t r t , κ t κ t (cid:27) ≤ kt + O ( t ) , where the right-hand inequality follows from the fact that dd t (cid:12)(cid:12) t =0 ( r t /r t ) ≤ k and dd t (cid:12)(cid:12) t =0 ( κ t /κ t ) ≤ k since X is k -lipschitz. Therefore, for each component E there is a constant R > d ( s t ( p ) , s t ( q )) d ( p, q ) ≤ kt + Rt (7.4)for all small enough t > p (cid:54) = q in E (cid:114) s t ( U (cid:48) ). We can actually takethe same R for all components E by equivariance (there are only finitelymany funnels in the quotient surface).We now modify ( s t ) into a smooth family of ( j t , ρ t )-equivariant diffeo-morphisms whose derivative at t = 0 is X instead of Y , keeping the goodLipschitz properties in the funnels. The vector field Z := X − Y is smooth, j -invariant, and zero outside U (cid:48) . Let ϕ Zt : H → H be the time- t flow of Z :it is a j (Γ)-invariant map, equal to the identity outside the 1-neighborhood OMPLETE LORENTZ SPACETIMES OF CONSTANT CURVATURE 53 U (cid:48)(cid:48) of U (cid:48) for small enough t ≥
0. The map f t := s t ◦ ϕ Zt ◦ ( s t ) − is ( j t , ρ t )-equivariant, coincides with s t outside s t ( U (cid:48)(cid:48) ), and satisfiesdd t (cid:12)(cid:12)(cid:12) t =0 f t = Y + Z = X. Let C be a compact neighborhood of a fundamental domain of U (cid:48)(cid:48) for theaction of j (Γ). Then C also contains fundamental domains of s t ( U (cid:48)(cid:48) ) for theaction of j t (Γ) for small enough t ≥
0. By smoothness and cocompactness,as explained before the proof, we have Lip( f t ) ≤ kt + O ( t ) on C , hence on j t (Γ) · C by subdivision. Since j t (Γ) · C is a neighborhood of s t ( U (cid:48)(cid:48) ) for small t ,by subdivision and (7.4) we have Lip( f t ) ≤ kt + O ( t ) on all of H . (cid:3) Convergence of projective structures.
In this section we proveTheorem 1.4, using Propositions 7.3 and 7.4. We have already seen that(1) for all t > j t ,ρ t acts properly on AdS (Proposi-tions 6.3 and 7.3). We now aim to prove that: (2) there is a smoothfamily of ( j × Id S , ( j t , ρ t ))-equivariant diffeomorphisms (developing maps)Dev t : H × S → AdS determining complete AdS structures A t on S × S ;(3) the RP structures P t underlying A t have a well-defined limit P as t →
0, and the Margulis spacetime M is obtained by removing S × { π } from P .Note that as AdS structures, the A t do not converge as t →
0. In fact, weshall see in the proof that the central surface S ×{ } collapses to a point (thedeveloping maps Dev t satisfy Dev t ( p, → ∈ G for all p ∈ H as t → S = R / π Z . The normalization of themetric of AdS chosen in Section 2.1 makes the length of any timelike geo-desic circle 2 π . Proof of Theorem 1.4.
Let X be a smooth, ( j, u )-equivariant vector fielddefined on H , standard in the funnels, with lip( X ) <
0, and let ( f t ) and( σ t ) be as in Propositions 7.3 and 7.4. By Proposition 6.3, for any t > f t induces a principal fiber bundle structure S (cid:44) −→ G Π t −→ H with timelike geodesic fibers. The maps Π t vary smoothly because the f t do.Using ( σ t ), which is a smooth family of smooth sections, and the S -actionon the fibers, we obtain a smooth family of global trivializationsΦ t : H × S −→ G. Explicitly, Φ t ( p, θ ) is the point at distance θ ∈ S from σ t ( p ) along the fiberΠ − t ( p ) in the future direction:Φ t ( p, θ ) = σ t ( p ) Rot( p, θ ) , where Rot( p, θ ) ∈ G is the rotation of angle θ centered at p ∈ H , and theproduct is for the group structure of G . By construction, Φ t is equivariant OMPLETE LORENTZ SPACETIMES OF CONSTANT CURVATURE 54 with respect to j t × Id S and ( j t , ρ t ); it is therefore a developing map defininga complete AdS structure on the manifold j t (Γ) \ H × S , which is diffeo-morphic to S × S . To prove (2), we precompose the maps Φ t with a smoothfamily of (lifts of) diffeomorphisms identifying j t (Γ) \ H × S with S × S .Recall the ( j, j t )-equivariant diffeomorphisms s t : H → H from the proofof Proposition 7.3. For t >
0, the map Dev t : H × S → G defined byDev t ( p, θ ) := Φ t (cid:0) s t ( p ) , θ (cid:1) is equivariant with respect to j × Id S and ( j t , ρ t ), hence is a developing mapfor a complete AdS structure A t on the fixed manifold S × S , as desired.Note that only the smooth structure of S (and not the hyperbolic structuredetermined by j ) is important for this definition: indeed, if Σ is a surfacediffeomorphic to S , then the maps s t could be replaced by any smoothfamily of diffeomorphisms taking the action of π (Σ) on (cid:101) Σ to the j t -actionof Γ on H .We now prove (3). By Proposition 6.3, the vector field X induces aprincipal fiber bundle structure R (cid:44) −→ g (cid:36) −→ H with timelike geodesic fibers. The derivative σ (cid:48) := ( dd t (cid:12)(cid:12) t =0 σ t ) : H → g is asmooth section; as above, we obtain a global trivializationdev : H × R −→ g . Explicitly, dev( p, θ (cid:48) ) is the point at signed distance θ (cid:48) ∈ R in the future of σ (cid:48) ( p ) along the fiber (cid:36) − ( p ):dev( p, θ (cid:48) ) = σ (cid:48) ( p ) + rot( p, θ (cid:48) ) , where rot( p, θ (cid:48) ) ∈ g is the infinitesimal rotation by amount θ (cid:48) around p .By construction, dev is equivariant with respect to j × Id R and ( j, u ); it istherefore a developing map for the Margulis spacetime M = Γ j,u \ g .In order to obtain the convergence of projective structures as in (3),we precompose the developing maps Dev t and dev with diffeomorphims,changing coordinates in the fiber direction. Let ψ : S → RP be anydiffeomorphism with ψ ( θ ) ∼ θ for θ near 0 and ψ ( π ) = ∞ , for instance ψ ( θ ) = 2 tan( θ/ t >
0, let ξ t : S → S be the diffeomorphism ξ t ( θ ) := ψ − (cid:0) t ψ ( θ ) (cid:1) . We precompose Dev t by this change of coordinates in the S factor, yieldinga new developing map (cid:100) Dev t : H × S → G for the same AdS structure:(7.5) (cid:100) Dev t ( p, θ ) := Dev t (cid:0) p, ξ t ( θ ) (cid:1) . The map (cid:100) dev : H × ( − π, π ) → g given by (cid:100) dev( p, θ ) := dev (cid:0) p, ψ ( θ ) (cid:1) is a developing map for a complete flat Lorentzian structure on S × ( − π, π )isometric to M . OMPLETE LORENTZ SPACETIMES OF CONSTANT CURVATURE 55
Claim 7.5.
For any ( p, θ ) ∈ H × ( − π, π ) , (cid:100) Dev t ( p, θ ) −−→ t → ∈ G and dd t (cid:12)(cid:12)(cid:12) t =0 (cid:100) Dev t ( p, θ ) = (cid:100) dev( p, θ ) . Proof.
Fix ( p, θ ) ∈ H × ( − π, π ). Recall that ( s t ) and ( σ t ) are smoothfamilies, that s = Id H , and that σ : H → G is the constant map withimage 1 ∈ G , whose differential is zero everywhere. Moreover, ξ t ( θ ) → t → ψ (0) = 0, hence (cid:100) Dev t ( p, θ ) = σ t (cid:0) s t ( p ) (cid:1) Rot (cid:0) s t ( p ) , ξ t ( θ ) (cid:1) −−→ t → ∈ G and dd t (cid:12)(cid:12)(cid:12) t =0 (cid:100) Dev t ( p, θ ) = σ (cid:48) ( p ) + dd t (cid:12)(cid:12)(cid:12) t =0 Rot (cid:0) s t ( p ) , ξ t ( θ ) (cid:1) . Since ψ ( x ) ∼ x near 0, we have dd t (cid:12)(cid:12) t =0 ξ t ( θ ) = ψ ( θ ), hencedd t (cid:12)(cid:12)(cid:12) t =0 Rot (cid:0) s t ( p ) , ξ t ( θ ) (cid:1) = rot (cid:18) s ( p ) , dd t (cid:12)(cid:12)(cid:12) t =0 ξ t ( θ ) (cid:19) = rot (cid:0) p, ψ ( θ ) (cid:1) . We conclude that dd t (cid:12)(cid:12) t =0 (cid:100) Dev t ( p, θ ) = dev( p, ψ ( θ )) = (cid:100) dev( p, θ ). (cid:3) Note that the right-hand equality in Claim 7.5 can also be written as1 t log ◦ (cid:100) Dev t −−→ t → (cid:100) devon H × ( − π, π ), where log is the inverse of the exponential map, definedfrom a neighborhood of 1 in G to a neighborhood of 0 in g , and it followsfrom the proof that this convergence is uniform on compact sets. UsingProposition 7.1, we conclude that on H × ( − π, π ),(7.6) r t I ◦ (cid:100) Dev t −−→ t → i ◦ (cid:100) devuniformly on compact sets. This shows that, when restricted to S × ( − π, π ),the real projective structure P t underlying the AdS structures A t convergesto the real projective structure underlying the Margulis spacetime. Whatremains to be shown is that the projective structures P t on the full manifold S × S converge.The embedding i : g (cid:44) → RP of Section 7.1 extends to a diffeomorphism i : g ∼ → RP , where g := P ( g ⊕ R ) = g ∪ P ( g )is the visual compactification of g . The map (cid:100) dev : H × ( − π, π ) → g extendsto a map dev : H × S → g , withdev( p, π ) = [rot( p, ∈ P ( g ) ⊂ g for all p ∈ H . The restriction of i ◦ dev to H × { π } is a diffeomorphismonto the set H ∞ := (cid:8) [ y ] ∈ RP | y + y − y < y (cid:9) of timelike directions of i ( R , ), which is a copy of the hyperbolic plane. Theextended map i ◦ dev is a diffeomorphism onto i ( R , ) ∪ H ∞ ⊂ RP . OMPLETE LORENTZ SPACETIMES OF CONSTANT CURVATURE 56
Note that the action of Γ j,u on i ( R , ) via i ∗ (see Section 7.1) extendsto an action of Γ j,u on i ( R , ) ∪ H ∞ . This action is properly discontinuousbecause i ◦ dev is an equivariant diffeomorphism. Thus, i ◦ dev : H × S ∼ → i ( R , ) ∪ H ∞ is a developing map identifying S × S with the real projectivemanifold M := Γ j,u \ (cid:0) i ( R , ) ∪ H ∞ (cid:1) . We conclude the proof of (3) by extending Formula (7.6) in this context.
Claim 7.6. On H × S , we have r t I ◦ (cid:100) Dev t −→ t → i ◦ dev .Proof. Given (7.6), we can restrict to H × { π } . Note (to be compared withProposition 7.1.(1)) that for any smooth path t (cid:55)→ g t ∈ G with g (cid:54) = 1,(7.7) r t · I ( g t ) −−→ t → i (cid:0) [log( g )] (cid:1) ∈ H ∞ , where log( g ) is any preimage of g under the exponential map exp : g → G (the projective class [log( g )] does not depend on the choice of the preimage).Indeed, (7.7) can be checked using the explicit coordinates of Section 7.1:just note that i (cid:20) log (cid:18) y + y y − y y + y − y + y (cid:19)(cid:21) = i (cid:20)(cid:18) y y − y y + y − y (cid:19)(cid:21) = [ y : y : y : 0] . We then apply (7.7) to g t = (cid:100) Dev t ( p, π ), which satisfies g = Rot( p, π ) and[log( g )] = [rot( p, π )] = dev( p, π ). (cid:3) Thus the limit P of the projective structures P t exists and is naturallyidentified with M (which might reasonably be called the timelike completion of M ). This completes the proof of Theorem 1.4. (cid:3) Here is a consequence of the proof.
Corollary 7.7.
The S fibers in Theorem 1.4 can always be assumed to betimelike geodesics in the manifolds M t . These geodesic fibers converge totimelike geodesic fibers in the limiting Margulis spacetime. Convergence of metrics.
Finally, we prove Corollary 1.5 by showingthat the AdS metrics on S × S determined by the developing maps (cid:100) Dev t that we constructed in Section 7.3 converge under appropriate rescaling tothe complete flat metric on the limiting Margulis spacetime. Proof of Corollary 1.5.
Recall from Section 2.1 that we equip both R , = g and AdS = G with Lorentzian metrics induced by half the Killing form of g .Let (cid:37) Min and (cid:37)
AdS denote these metrics, of respective curvature 0 and − / ς Min be the parallel flat metric on V := i ( R , ) ⊂ RP obtained bypushing forward (cid:37) Min by i , and let ς AdS be the constant-curvature metricon I (AdS ) ⊂ RP obtained by pushing forward (cid:37) AdS by I . If we identify V with the tangent space to RP at x := i (0) = I (1), then i : g → V coincideswith the differential of I at 1 ∈ G ; therefore, ς Min x = ς AdS x . OMPLETE LORENTZ SPACETIMES OF CONSTANT CURVATURE 57
For t >
0, consider the AdS metric ς t defined on r t · I (AdS ) ⊂ RP by ς t := ( r t ) ∗ ς AdS , where ( r t ) ∗ is the pushforward by r t . Let us check that t − ς t converges to ς Min uniformly on compact subsets of V as t →
0, where for a given compactset C we only consider t small enough so that ς t is defined on C (recall fromSection 7.1 that the union of the sets r t · I (AdS ) for t > V ). In what follows, we use the trivialization of T V (which ispreserved under affine transformations), denoting the parallel transport ofa vector v ∈ T x V to T y V again by v . First, note that for any tangent vector v ∈ T x V we have ( r − t ) ∗ v = tv ∈ T r − t ( x ) . Thus, for v, w ∈ T x V , t − ς tx ( v, w ) = t − (cid:0) ( r t ) ∗ ς AdS (cid:1) x ( v, w )= t − ς AdS r − t ( x ) (cid:0) ( r − t ) ∗ v, ( r − t ) ∗ w (cid:1) = ς AdS r − t ( x ) ( v, w ) . Given a compact set
C ⊂ V , the projective transformation r − t maps C intoarbitrarily small neighborhoods of the basepoint x as t →
0. Therefore, bycontinuity of ς AdS , t − ς tx = ς AdS r − t ( x ) −−→ t → ς AdS x = ς Min x = ς Min x uniformly for x ∈ C (where we use again the trivialization of T V ).Now, recall the developing maps (cid:100)
Dev t : H × S → AdS defined for t > t >
0, we consider the AdS metric (cid:37) t := ( (cid:100) Dev t ) ∗ (cid:37) AdS on H × S , where ( (cid:100) Dev t ) ∗ is the pullback by (cid:100) Dev t . The rescaled metrics t − (cid:37) t determine complete metrics of curvature − t / S × S . We alsoconsider the flat metric (cid:37) := ( (cid:100) dev) ∗ (cid:37) Min on H × ( − π, π ). By (7.6) and the convergence of t − ς t proved above, t − (cid:37) t = (cid:0) r t I ◦ (cid:100) Dev t (cid:1) ∗ t − ς t −−→ t → ( i ◦ (cid:100) dev) ∗ ς Min = ( (cid:100) dev) ∗ (cid:37) Min = (cid:37) , on H × ( − π, π ), and the convergence is uniform on compact sets. Theinduced rescaled metrics t − (cid:37) t on S × ( − π, π ) converge to the induced flatmetric (cid:37) on S × ( − π, π ). The metric (cid:37) makes S × ( − π, π ) isometric to M since (cid:100) dev is a developing map for M . (cid:3) OMPLETE LORENTZ SPACETIMES OF CONSTANT CURVATURE 58
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Department of Mathematics, The University of Texas at Austin, 1 Univer-sity Station C1200, Austin, TX 78712, USA
E-mail address : [email protected] CNRS and Universit´e Lille 1, Laboratoire Paul Painlev´e, 59655 Villeneuved’Ascq Cedex, France
E-mail address : [email protected] CNRS and Universit´e Lille 1, Laboratoire Paul Painlev´e, 59655 Villeneuved’Ascq Cedex, France
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