Foliations between crooked planes in 3-dimensional Minkowski space
FFOLIATIONS BETWEEN CROOKED PLANES IN -DIMENSIONAL MINKOWSKI SPACE JEAN-PHILIPPE BURELLE AND DOMINIK FRANCOEUR
Abstract.
We show that any two disjoint crooked planes in R are leaves ofa crooked foliation. This answers a question asked by Charette and Kim [5]. Introduction
In 1983, answering a question of Milnor [10], Margulis constructed the first exam-ples of nonabelian free groups which act freely and properly discontinuously on R by affine transformations [9]. In order to better understand these examples, ToddDrumm defined piecewise linear surfaces called crooked planes which can boundfundamental domains for such actions [7].Crooked planes have proven to be very useful in the study of affine actions.Charette-Drumm-Goldman have used them in order to obtain a complete classifi-cation for free groups of rank two [2–4]. In particular, they show that every freeand properly discontinuous affine action of a rank two free group on R admits afundamental domain bounded by finitely many crooked planes (the crooked planeconjecture ). A consequence of this is the tameness conjecture , that the quotient of R by one of these actions is homeomorphic to the interior of a compact manifoldwith boundary.Building on this work, Danciger-Gu´eritaud-Kassel showed in [6] that crookedplanes have a natural interpretation in terms of the deformation theory of hyper-bolic surfaces, and used this fact in order to prove the crooked plane conjecture inarbitrary rank, assuming that the linear part is convex cocompact in O (2 , Drumm-Goldman inequality provides a necessary and sufficient criterion for twocrooked planes to be disjoint [8]. This criterion was later expanded upon in [1] andreinterpreted in terms of hyperbolic geometry in [6].As an application of the disjointness criterion, the first example of a crookedfoliation , a smooth 1-parameter family of pairwise disjoint crooked planes, was givenin [1]. Charette-Kim [5] investigated these foliations further and gave necessary andsufficient criteria for a one-parameter family of crooked planes to foliate a subsetof R . They ask the following question : given a pair of disjoint crooked planes in Mathematics Subject Classification. a r X i v : . [ m a t h . DG ] M a r JEAN-PHILIPPE BURELLE AND DOMINIK FRANCOEUR R , can the region between them be foliated by crooked planes? We answer thisquestion in the affirmative. Theorem 1.
Let
C, C (cid:48) be a pair of disjoint crooked planes in R . Then, there isa crooked foliation , that is, a smooth family of pairwise disjoint crooked planes C t , ≤ t ≤ with C = C and C = C (cid:48) . After recalling some definitions from the theory of crooked planes in Minkowski3-space in Section 2, we will prove the main theorem in Section 3.We are thankful to the referee for insightful comments and for suggesting anelegant way to shorten the proof of the main theorem.2.
Definitions
Definition 2.1.
Lorentzian 3-space R , is the real three dimensional vector space R endowed with the following symmetric bilinear form of signature (2 , · : R × R → R ( u , v ) (cid:55)→ u v + u v − u v . We fix the orientation given by the standard basis e , e , e and we define the Lorentzian cross product u × v = ( u v − u v , u v − u v , u v − u v ) ∈ R , , for u , v ∈ R , .A null frame of R , is a positively oriented basis u , u (cid:48) , u (cid:48)(cid:48) such that u · u = 1, u (cid:48) · u (cid:48)(cid:48) = − Notation 2.1.
Any unit spacelike vector u can be extended to a null frame. Thisframe is unique up to scaling u (cid:48) and u (cid:48)(cid:48) by inverse scalars. As normalization wewill choose u (cid:48) and u (cid:48)(cid:48) so that their third coordinates are positive and equal. Given u , we will denote these two null vectors by u − and u + , respectively.We will denote by Min the pseudo-Euclidean affine space which is modeled onthe vector space R , . In other words, Min is a topological space on which R , actssimply transitively by homeomorphisms. For v ∈ R , and p ∈ Min, we denote thisaction by v ( p ) = p + v . If q = p + v , we will also write q − p = v . A choice of origin o ∈ Min identifies Min with R , via the map v (cid:55)→ o + v .We now recall the definition of a crooked plane. First, we define a stem , whichwill be one of the three linear pieces of a crooked plane. Definition 2.2.
Let u ∈ R , be a unit spacelike vector. The stem S ( u ) is the setof causal vectors orthogonal to u : S ( u ) = { v ∈ R , | u · v = 0 and v · v ≤ } . A stem is the union of two opposite closed quadrants (see Figure 1).
Definition 2.3.
Let u ∈ R , be a unit spacelike vector. The linear crooked plane C ( u ) is the piecewise linear surface defined by: C ( u ) := { v ∈ R , | v × w = k w for some w ∈ S ( u ) and k ∈ R ≥ } . OLIATIONS BETWEEN CROOKED PLANES IN 3-DIMENSIONAL MINKOWSKI SPACE 3
From this definition, we see that S ( u ) ⊂ C ( u ) since v × v = 0 for all v ∈ R , .The complement of the stem C ( u ) − S ( u ) has two connected components whichare called the wings of the crooked plane. Each wing is a half-plane on which theLorentzian bilinear form is degenerate, attached to the stem along its boundary(See Fig. 1). Note that C ( u ) = C ( − u ). Definition 2.4.
Let p ∈ Min and u ∈ R , unit spacelike. The crooked plane C ( p, u ) is the set p + C ( u ) ⊂ Min. The vector u is called a directing vector of thecrooked plane, and p its vertex .In order to formally state the disjointness criteria from [5, 8], we need a normal-ization for pairs of unit spacelike vectors. Definition 2.5.
Two unit spacelike vectors u , u ∈ R , are consistently oriented if • u · u ≤ −
1, and • u i · u ± j ≤ ≤ i, j ≤ u , u (cid:48) are called ultraparallel if u · u (cid:48) < −
1. They are called asymptotic if u · u (cid:48) = − u (cid:48) (cid:54) = − u . Intersect-ing u ⊥ and u (cid:48)⊥ with the hyperboloid model of the hyperbolic plane defines a pairof hyperbolic geodesics, and the terminology comes from the relative position ofthese geodesics. Choosing one of the unit vectors ± u endows the geodesic in thehyperboloid model of H defined by u ⊥ with a transverse orientation. Two unitspacelike vectors are consistently oriented when the corresponding transversely ori-ented geodesics are disjoint with transverse orientations pointing away from eachother (see Figure 1).Whenever there exists a choice of directing vectors u , u (cid:48) which are consistentlyoriented, we will also call a pair of crooked planes C ( p, u ) , C ( p (cid:48) , u (cid:48) ) ultraparallel orasymptotic accordingly.We will use two disjointness criteria for crooked planes, one for pairs of crookedplanes and one for foliations. Both depend on the following notion: Definition 2.6.
The stem quadrant associated to a unit spacelike vector u is theset V ( u ) := { a u − − b u + : a, b ≥ }\{ } . Note that V ( − u ) = − V ( u ).The following disjointness criterion is a restatement of the Drumm-Goldmaninequality [8].
Theorem 2 (Burelle-Charette-Goldman [1]) . Let C = C ( p, u ) , C (cid:48) = C ( p (cid:48) , u (cid:48) ) becrooked planes and assume that u , u (cid:48) are consistently oriented. Then, C and C (cid:48) aredisjoint if and only if p (cid:48) − p ∈ A ( u , u (cid:48) ) := int( V ( u (cid:48) ) − V ( u )) . Remark 2.1.
It is also shown in [8] that if there is no choice of sign for u , u (cid:48) making them consistently oriented, then C ( p, u ) and C ( p (cid:48) , u (cid:48) ) necessarily intersect.Therefore, the above theorem is a characterization of disjoint crooked planes.We will use the following straightforward consequence of the Charette-Kim cri-terion for crooked foliations (foliations of R , by crooked planes) : JEAN-PHILIPPE BURELLE AND DOMINIK FRANCOEUR (a)
A pair of consistently oriented unit spacelike vectors u , u and the corresponding stems. (b) The linear crooked plane C ( u ). Figure 1.
Consistent orientations, stems and crooked planes.
Theorem 3 (Charette-Kim [5]) . Let ( u t ) t ∈ R be a path of pairwise ultraparallel orasymptotic unit spacelike vectors such that − u t , u s are consistently oriented for all t < s . Suppose ( p t ) t ∈ R , is a regular curve such that for every t ∈ R , ˙ p t ∈ int( V ( u t )) . Then, C ( p t , u t ) is a crooked foliation. Foliations between crooked planes
We now prove Theorem 1 : there exists a crooked foliation containing any pairof disjoint crooked planes. The theorem is a consequence of the following strongerresult :
OLIATIONS BETWEEN CROOKED PLANES IN 3-DIMENSIONAL MINKOWSKI SPACE 5
Proposition 1.
Let ( u t ) t ∈ [0 , be a smooth path of unit spacelike vectors whichare pairwise ultraparallel or asymptotic. Let p , p ∈ Min such that C ( p , u ) and C ( p , u ) are disjoint crooked planes. Then, there exists a path ( p t ) t ∈ [0 , startingat p and ending at p such that C ( p t , u t ) is a smooth crooked foliation.Proof. Since we assume that u s are pairwise ultraparallel or asymptotic, we havethat u t · u s ≥ t ≤ s . Changing the path u s to − u s if needed (bothpaths define the same linear crooked planes) we may also assume that − u t , u s areconsistently oriented for all t < s .For any pair of smooth functions f, g : [0 , → R > , define v f,g ( s ) := f ( s ) u − s − g ( s ) u + s . Then, the path of vertices p f,g ( t ) := p + (cid:82) t v f,g ( s ) d s satisfies the hypotheses ofTheorem 3 since its derivative ˙ p f,g ( t ) = v f,g ( t )lies in the interior of V ( u t ) by definition.Let D denote collection of displacement vectors p f,g (1) − p : D = (cid:26) (cid:90) v f,g ( s ) d s (cid:12)(cid:12)(cid:12)(cid:12) f, g : [0 , → R > (cid:27) . Then D is a convex cone since k v f,g = v kf,kg for k ∈ R > and v f ,g + v f ,g = v f + f ,g + g . Moreover, since by Theorem 3 the crooked planes C ( p f,g ( t ) , u t ) definecrooked foliations, the initial and final crooked planes are disjoint and so D ⊂ A ( − u , u ). Since the cone A ( − u , u ) is the interior of the convex hull of the fourrays generated by u − , − u +0 , u − , − u +1 , to show equality of the cones it suffices toshow that these rays can be approximated by vectors in D .Consider the sequences f n ( s ) = ne − ns and g n ( s ) = e − n . Integrating by parts weget (cid:90) f n ( s ) u − s d s = u − − e − n u − + (cid:90) e − ns ˙ u − s d s. Therefore, as u s is smooth and so u + s and ˙ u − s are bounded on [0 , n →∞ (cid:90) v f n ,g n ( s ) d s = u − . We conclude that D contains vectors arbitrarily close to the ray R > u − .Similarly, if f is concentrated near s = 1 and g is small we can approximate theray u − , and exchanging the roles of f and g we approximate the other two rays onthe boundary of the convex cone A ( u , u ). (cid:3) The previous proposition has the following interpretation : given any geodesicfoliation F of the region between two geodesics (cid:96) , (cid:96) of H and basepoints p , p ∈ Min such that the crooked planes with vertices p i and stems corresponding to (cid:96) i are disjoint, F can be lifted to a foliation by crooked planes of the region betweenthe crooked planes. JEAN-PHILIPPE BURELLE AND DOMINIK FRANCOEUR
References [1] Jean-Philippe Burelle, Virginie Charette, Todd A. Drumm, and William M. Goldman.Crooked halfspaces.
Enseign. Math. , 60(1-2):43–78, 2014.[2] Virginie Charette, Todd A. Drumm, and William M. Goldman. Affine deformations of athree-holed sphere.
Geom. Topol. , 14(3):1355–1382, 2010.[3] Virginie Charette, Todd A. Drumm, and William M. Goldman. Finite-sided deformationspaces of complete affine 3-manifolds.
J. Topol. , 7(1):225–246, 2014.[4] Virginie Charette, Todd A. Drumm, and William M. Goldman. Proper affine deformationsof the one-holed torus.
Transform. Groups , 21(4):953–1002, 2016.[5] Virginie Charette and Youngju Kim. Foliations of Minkowski 2 + 1 spacetime by crookedplanes.
Internat. J. Math. , 25(9):1450088, 25, 2014.[6] Jeffrey Danciger, Fran¸cois Gu´eritaud, and Fanny Kassel. Margulis spacetimes via the arccomplex.
Invent. Math. , 204(1):133–193, 2016.[7] Todd A. Drumm.
Fundamental polyhedra for Margulis space-times . ProQuest LLC, AnnArbor, MI, 1990. Thesis (Ph.D.)–University of Maryland, College Park.[8] Todd A Drumm and William M Goldman. The geometry of crooked planes.
Topology ,38(2):323 – 351, 1999.[9] G. A. Margulis. Free completely discontinuous groups of affine transformations.
Dokl. Akad.Nauk SSSR , 272(4):785–788, 1983.[10] John Milnor. On fundamental groups of complete affinely flat manifolds.
Advances in Math. ,25(2):178–187, 1977.
Jean-Philippe Burelle, CNRS and Institut des Hautes ´Etudes Scientifiques, 35 routede Chartres, 91440 Bures-sur-Yvette, France
E-mail address : [email protected] Dominik Francoeur, Universit´e de Gen`eve, 1205 Geneva, Switzerland
E-mail address ::