Cauchy-compact flat spacetimes with extreme BTZ

Brunswic, Léo

doi:10.1007/s10711-021-00629-8

Cited by 4 publications

(7 citation statements)

References 38 publications

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“…We now state and prove a structure Theorem for radiant spacetimes. This result is in the line of Mess Theorem [Mes07] and is akin to previous results by Bonsante and Seppi [SB15] or the author [Bru20a] though in a much simpler context. To the author's knowledge, while this result is expected and "folkoric", there is no existing reference to point to.…”

Section: Radiant Spacetimessupporting

confidence: 82%

“…C-morphism. Since M is Cauchymaximal and Cauchy-compact by Proposition 4 in [Bru20a], the map ι is surjective thus an isomorphism.…”

Section: Radiant Spacetimesmentioning

confidence: 93%

“…A manifold M 1 is Cauchy-maximal if for any Cauchy-embedding M 1 ϕ − → M 2 , the map ϕ is an isomorphism. One can prove [Bru17,Bru20a] a version of Choquet-Bruhat-Geroch Theorem for C ≥0 -manifolds following the lines of [Sbi15] which states that any C ≥0 -manifold admits a unique Cauchy-maximal Cauchy-extension.…”

Section: Causal Structurementioning

confidence: 99%

“…Let M be a C ≥0 -manifold, Reg(M ) admits a natural causal geodesic foliation. We notice that in the model spaces C α the foliation can be extended to the whole C α , furthermore by Propsition 1 of [Bru20a] if ϕ : U α → U β is an a.e. SO 0 (1, 2)-isomorphism between neighborhoods of singular points in C α and C β respectively then α = β and ϕ is induced by an element of SO 0 (1, 2); hence the extended foliation to the whole C α induces a causal foliation on M .…”

Section: H 2 -Structure Of the Space Of Leaves And Suspensionsmentioning

confidence: 99%

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Alexandrov Theorem for 2+1 flat radiant spacetimes

Brunswic

2020

Preprint

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A classical Theorem of Alexandrov states that the map associating its boundary to a convex polyhdedron of the 3-dimensional Euclidean space is a bijection from the set of convex polyhdedron up to congruence to the set of isometry classes of locally Euclidean metric on the 2-sphere with conical singularities smaller that 2π. Fillastre proved a similar statement for locally Euclidean metric on higher genus surfaces with conical singularities bigger than 2π by embedding their universal covering in 3-dimensional Minkowski space as the boundary of Fuchsian polyhedra. The original proofs of Alexandrov and Fillastre both rely on invariance of domain Theorem hence are not effective. Volkov, in his thesis, provided a variational, hence effective, proof of Alexandrov Theorem which has then been generalised by Bobenko, Izmestiev and Fillastre. The present work goes further by adapting Volkov's variational method to provide an effective version of Fillastre Theorem and extend Fillastre's result: we show that for any closed locally Euclidean surface Σ with conical singularities of arbitrary angles (θ i ) i∈ [[1,n]] and any choice of Lorentzian angles (κ i ) i∈ [[1,n]] such that κ i < θ i and κ i ≤ 2π, there exists a locally Minkoswki 3-manifold M of linear holonomy with conical singularities (κ i ) i∈ [[1,n]] and a convex polyedron P in M whose boundary is isometric to Σ; furthermore such a couple (M, P ) is unique.

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Section: Radiant Spacetimessupporting

confidence: 82%

“…C-morphism. Since M is Cauchymaximal and Cauchy-compact by Proposition 4 in [Bru20a], the map ι is surjective thus an isomorphism.…”

Section: Radiant Spacetimesmentioning

confidence: 93%

Section: Causal Structurementioning

confidence: 99%

Section: H 2 -Structure Of the Space Of Leaves And Suspensionsmentioning

confidence: 99%

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Alexandrov Theorem for 2+1 flat radiant spacetimes

Brunswic

2020

Preprint

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“…In [5], the author introduced the BTZ model space E 1,2 0 which is defined as R 3 endowed with the singular E 1,2 -structure induced by the flat lorentzian metric ds 2 = −2dτ dr + dr 2 + r 2 dθ in cylindrical coordinates. The singular locus of E 1,2 0 is then Sing(E 1,2 0 ) = {r = 0} and the regular locus is Reg(E 1,2 0 ) := {r > 0}.…”

Section: Manifoldsmentioning

confidence: 99%

On branched coverings of singular $(G,X)$-manifolds

Brunswic

2020

Preprint

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Branched coverings have a long history from ramification of Riemann surfaces to realization of 3 manifolds as coverings ramified over a knots; from geometrical topology to algebraic geometry. The present work investigates a notion of branched covering "à la Fox" which is particularly natural for (G, X)-manifolds. The work is two fold. First, we recall and enrich the current state of the art (based upon Montesinos) and develop a Galois theory for such branched coverings together with a description of the fiber above branching points. Second, we present a theory of singular (G, X)-manifolds and apply the theory of branched coverings we develop to extend the usual framework of (G, X)-manifolds to singular (G, X)-manifolds; in particular, we construct a developing map for such singular manifolds. An application to a singular locally Minkowski manifolds is given.

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On branched coverings of singular (G, X)-manifolds

Brunswic

2024

Geom Dedicata

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Cauchy-compact flat spacetimes with extreme BTZ

Cited by 4 publications

References 38 publications

Alexandrov Theorem for 2+1 flat radiant spacetimes

Alexandrov Theorem for 2+1 flat radiant spacetimes

On branched coverings of singular $(G,X)$-manifolds

On branched coverings of singular (G, X)-manifolds

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