Surfaces convexes dans des espaces lorentziens à courbure constante

Schlenker, Jean-Marc

doi:10.4310/cag.1996.v4.n2.a4

Cited by 33 publications

(48 citation statements)

References 9 publications

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“…Following the construction made in step (1), γ ′ can be considered as a closed curve in the boundary of C. Since γ is simple, this curve is homotopically non-trivial in ∂C, while it is homotopically trivial in C. But ∂C lifts to a complete, convex, polyhedral surface in H 3 . It is known that, for any non-trivial closed curve on such a surface, the sum of the exterior dihedral angles of the edges crossed by the closed curve is strictly larger than 2π, see [RH93,CD95,Sch96,Sch98]. Condition (2) follows.…”

Section: A Theorem Of Otalmentioning

confidence: 88%

Hyperideal circle patterns

Schlenker¹

2005

Mathematical Research Letters

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Abstract. A "hyperideal circle pattern" in S 2 is a finite family of oriented circles, similar to an "usual" circle pattern but such that the closed disks bounded by the circles do not cover the whole sphere. Hyperideal circle patterns are directly related to hyperideal hyperbolic polyhedra, and also to circle packings.To each hyperideal circle pattern, one can associate an incidence graph and a set of intersection angles. We characterize the possible incidence graphs and intersection angles of hyperideal circle patterns in the sphere, the torus, and in higher genus surfaces. It is a consequence of a more general result, describing the hyperideal circle patterns in the boundaries of geometrically finite hyperbolic 3-manifolds (for the corresponding CP 1 -structures). This more general statement is obtained as a consequence of a theorem of Otal [Ota94, BO01] on the pleating laminations of the convex cores of geometrically finite hyperbolic manifolds.Résumé Un "motif de cercles hyperidéal" sur S 2 est une famille finie de cercles orientés, similaireà un motif de cercles "usuel" mais tel que la réunion des disques fermés bordés par les cercles ne couvre pas la sphère. Les motifs de cercles hyperidéaux sont directement liés aux polyèdres hyperboliques hyperidéaux, et aussi aux empilements de cercles.A chaque motif de cercles hyperidéal, on associe un graphe d'incidence et un ensemble d'angles d'intersection. On caractérise les graphes d'incidence et les angles d'intersection possibles dans la sphère, le tore, et sur les surfaces de genre supérieur. C'est une conséquence d'un résultat plus général décrivant les motifs de cercles hyperidéaux dans les bords de variétés hyperboliques géométriquement finies (pour les CP 1 -structures correspondantes). Ce résultat plus général est obtenu comme conséquence d'un théorème d'Otal [Ota94, BO01] sur les laminations de plissage des coeurs convexes de variétés hyperboliques géométriquement finies.

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Section: A Theorem Of Otalmentioning

confidence: 88%

Hyperideal circle patterns

Schlenker¹

2005

Mathematical Research Letters

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“…A fortiori, the sequence of embeddings ( f n ) n∈N does not converge in the C ∞ topology. So we are under the assumption of Schlenkers result: Theorem 8.5 (see [56], Théorème 5.6). -Let ( f n ) n ∈ N : D → X be a sequence of uniformly elliptic immersions (10) of a disc D in a simply connected Lorentzian spacetime of constant curvature (X, g).…”

Section: Annales De L'institut Fouriermentioning

confidence: 99%

Prescribing Gauss curvature of surfaces in 3-dimensional spacetimes Application to the Minkowski problem in the Minkowski space

Barbot¹,

Béguin²,

Zeghib³

2011

Ann. inst. Fourier

103

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We study the existence of surfaces with constant or prescribed Gauss curvature in certain Lorentzian spacetimes. We prove in particular that every (non-elementary) 3-dimensional maximal globally hyperbolic spatially compact spacetime with constant non-negative curvature is foliated by compact spacelike surfaces with constant Gauss curvature. In the constant negative curvature case, such a foliation exists outside the convex core. The existence of these foliations, together with a theorem of C. Gerhardt, yield several corollaries. For example, they allow to solve the Minkowski problem in Min3 for datas that are invariant under the action of a co-compact Fuchsian group.-the Minkowski problem in the 3-dimensional Minkowski space.The results concerning the last two items are essentially application of the first one. STATEMENTS OF RESULTS K-slicings of MGHC spacetimes with constant curvature.The following is our main result: Theorem 2.1. Let M be a 3-dimensional non-elementary MGHC spacetime with constant curvature Λ. If Λ ≥ 0, reversing the time orientation if necessary, we assume that M is future complete.• If Λ ≥ 0 (flat case or locally de Sitter case), then M admits a unique K-slicing. The leaves of this slicing are the level sets of a K-time ranging over (−∞, −Λ). • If Λ < 0 (locally anti-de Sitter case), then M does not admit any global K-slicing, but each of the two connected component of the complement of the convex core 2 of M admits a unique K-slicing. The leaves of the K-slicing of the past of the convex core are the level sets of a K-time ranging over (−∞, 0). The leaves of the K-slicing of the future of the convex core are the level sets of a reverse K-time 3 ranging over (−∞, 0).Let us make a few comments on this result.Gaussian curvature. Since the Gaussian curvature of a surface is R = Λ + κ, the Gaussian curvature of the leaves of the K-slicings provided by Theorem 2.1 varies in (−∞, 0) when Λ ≥ 0, and in (−∞, Λ) when Λ < 0.CMC times. It is our interest on CMC-times that led us to extend our attention to more general geometric times. Existence of CMC times on MGHC spacetimes of constant non-positive curvature and any dimension was proved in [3,4]. For spacetimes locally modelled on the de Sitter space, there are some restrictions but not in dimension 3.Regularity. The slicings provided by Theorem 2.1 are continuous. It follows from their uniqueness (i.e. they are canonical). Extra smoothness is not automatic (e.g. the cosmological time is C 1,1 , but not C 2 ).Here, we can hope that our K-slicings are (real) analytic, and even more, they depend analytically on the spacetime (within the space of MGHC spacetimes of curvature Λ and fixed topology). All this depends on consideration of Einstein equations in a K-gauge.Non-standard isometric immersions of H 2 in Min 3 . It was observed by Hano and Nomizu [38], that the hyperbolic plane H 2 admits non standard isometric immersions in the Minkowski space Min 3 (i.e. different from the hyperbola, up to a Lorentz motion).

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“…On peut alors se restreindre à un disque D (pour an) centré en x du revêtement universel È de E, et appliquer le théorème 5.6 (p. 287) de [Sch96], qui affirme que, si ((j)n) ne converge pas C°° vers un plongement isométrique d'une surface à courbure constante, alors il existe une soussuite (^) de (<^n), une géodésique maximale 7 de D, et une géodésique Y de S^ telle que (<^n|-y) converge vers une isométrie sur r.…”

Section: Propretéunclassified

Réalisations de surfaces hyperboliques complètes dans $H^3$

Schlenker¹

1998

Annales De L’institut Fourier

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Surfaces convexes dans des espaces lorentziens à courbure constante

Cited by 33 publications

References 9 publications

Hyperideal circle patterns

Hyperideal circle patterns

Prescribing Gauss curvature of surfaces in 3-dimensional spacetimes Application to the Minkowski problem in the Minkowski space

Réalisations de surfaces hyperboliques complètes dans $H^3$

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