Foliations of Minkowski 2 + 1 spacetime by crooked planes

Charette, Virginie; Kim, Young‐Ju

doi:10.1142/s0129167x14500888

Cited by 4 publications

(8 citation statements)

References 9 publications

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“…

We show that any two disjoint crooked planes in R 3 are leaves of a crooked foliation. This answers a question asked by Charette and Kim [5].

…”

supporting

confidence: 51%

“…Note that C(u) = C(−u). In order to formally state the disjointness criteria from [5,8], we need a normalization for pairs of unit spacelike vectors.…”

Section: Definitionsmentioning

confidence: 99%

“…We will use the following straightforward consequence of the Charette-Kim criterion for crooked foliations (foliations of R 2,1 by crooked planes) : Theorem 3 (Charette-Kim [5]). Let (u t ) t∈R be a path of pairwise ultraparallel or asymptotic 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,…”

Section: Definitionsmentioning

confidence: 99%

“…Theorem 3 (Charette-Kim [5]). Let (u t ) t∈R be a path of pairwise ultraparallel or asymptotic 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,…”

Section: Foliations Between Crooked Planesmentioning

confidence: 99%

“…As an application of the disjointness criterion, the first example of a crooked foliation, a smooth 1-parameter family of pairwise disjoint crooked planes, was given in [1]. Charette-Kim [5] investigated these foliations further and gave necessary and sufficient criteria for a one-parameter family of crooked planes to foliate a subset of R 3 . They ask the following question : given a pair of disjoint crooked planes in R 3 , can the region between them be foliated by crooked planes?…”

Section: Introductionmentioning

confidence: 99%

See 4 more Smart Citations

Foliations between crooked planes in 3-dimensional Minkowski space

Burelle

Francoeur

2019

Int. J. Math.

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“…

We show that any two disjoint crooked planes in R 3 are leaves of a crooked foliation. This answers a question asked by Charette and Kim [5].

…”

supporting

confidence: 51%

“…Note that C(u) = C(−u). In order to formally state the disjointness criteria from [5,8], we need a normalization for pairs of unit spacelike vectors.…”

Section: Definitionsmentioning

confidence: 99%

Section: Definitionsmentioning

confidence: 99%

Section: Foliations Between Crooked Planesmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

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Foliations between crooked planes in 3-dimensional Minkowski space

Burelle

Francoeur

2019

Int. J. Math.

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Tameness of Margulis space-times with parabolics

2022

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Let 𝖤 {\mathsf{E}} be a flat Lorentzian space of signature ( 2 , 1 ) {(2,1)} . A Margulis space-time is a noncompact complete Lorentz flat 3-manifold 𝖤 / Γ {\mathsf{E}/\Gamma} with a free holonomy group Γ of rank 𝗀 {\mathsf{g}} , 𝗀 ≥ 2 {\mathsf{g}\geq 2} . We consider the case when Γ contains a parabolic element. We obtain a characterization of proper Γ-actions in terms of Margulis and Charette–Drumm invariants. We show that 𝖤 / Γ {\mathsf{E}/\Gamma} is homeomorphic to the interior of a compact handlebody of genus 𝗀 {\mathsf{g}} generalizing our earlier result. Also, we obtain a bordification of the Margulis space-time with parabolics by adding a real projective surface at infinity giving us a compactification as a manifold relative to parabolic end neighborhoods. Our method is to estimate the translational parts of the affine transformation group and use some 3-manifold topology.

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Tameness of Margulis space-times with parabolics

Choi¹,

Drumm²,

Goldman³

2017

Preprint

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Let E be a flat Lorentzian space of signature (2, 1). A Margulis space-time is a noncompact complete Lorentz flat 3-manifold E/Γ with a free holonomy group Γ of rank g, g ≥ 2. We consider the case when Γ contains a parabolic element. We show that E/Γ is homeomorphic to the interior of a compact handlebody of genus g generalizing our earlier result. Also, we obtain a bordification of the Margulis space-time with parabolics by adding a real projective surface at infinity giving us a compactification as a manifold relative to parabolic end neighborhoods. Our method is to find the estimation of the translational parts of the affine transformation group and to use some 3-manifold topology. Contents1. Introduction 2. Preliminary 2.1. The projective geometry of the Margulis space-time 2.2. Thin parts of hyperbolic surfaces 2.3. Hausdorff limits 2.4. The Poincaré polyhedron theorem 3. Margulis invariants and Charette-Drumm invariants 3.1. Parabolic action 3.2. Proper affine deformations and Margulis and Charette-Drumm invariants 4. Orbits of proper affine deformations and translation vectors 4.1. Convergence sequences 4.2. The bundles E over US 4.3. The Anosov property of the geodesic flow 4.4. Computing translation vectors 4.5. Translation vectors have direction limits in S 0 4.6. Accumulation points of Γ-orbits 4.7. A consequence 5. The topology of Margulis space-times with parabolics Date: May 7, 2019. 2010 Mathematics Subject Classification. 57M50 (primary) 83A99 (secondary).

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Foliations of Minkowski 2 + 1 spacetime by crooked planes

Cited by 4 publications

References 9 publications

Foliations between crooked planes in 3-dimensional Minkowski space

Foliations between crooked planes in 3-dimensional Minkowski space

Tameness of Margulis space-times with parabolics

Tameness of Margulis space-times with parabolics

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