Ads Manifolds With Particles and Earthquakes on Singular Surfaces

Bonsante, Francesco; Schlenker, Jean-Marc

doi:10.1007/s00039-009-0716-9

Cited by 39 publications

(89 citation statements)

References 13 publications

(22 reference statements)

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“…-The condition of convexity in the definition will allow us to use a convex core. As pointed out by the authors in [6], we do not know if every AdS GHM manifold with particles is convex GHM. Many results known in the non-singular case extend to the singular case (that is with particles of angles less than π).…”

Section: Extension Of Mess' Parametrizationmentioning

confidence: 98%

“…The natural candidates for these barriers are equidistant surfaces from the boundary of the convex core of (M, g). It is proved in [6,Section 5] that the future (respectively past) boundary component ∂ + (respectively ∂ − ) of the convex core is a future-convex (respectively past-convex) spacelike pleated surface orthogonal to the particles. Moreover, each point of the boundary components is either contained in the interior of a geodesic segment (a pleating locus) or of a totally geodesic disk contained in the boundary components.…”

Section: Annales De L'institut Fouriermentioning

confidence: 99%

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Maximal surfaces in anti-de Sitter 3-manifolds with particles

Toulisse¹

2016

Annales De l'Institut Fourier

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We prove the existence of a unique maximal surface in each anti-de Sitter (AdS) Globally Hyperbolic Maximal (GHM) manifold with particles (that is, with conical singularities along time-like lines) for cone angles less than π. We interpret this result in terms of Teichmüller theory, and prove the existence of a unique minimal Lagrangian diffeomorphism isotopic to the identity between two hyperbolic surfaces with cone singularities when the cone angles are the same for both surfaces and are less than π.

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Section: Extension Of Mess' Parametrizationmentioning

confidence: 98%

Section: Annales De L'institut Fouriermentioning

confidence: 99%

Maximal surfaces in anti-de Sitter 3-manifolds with particles

Toulisse¹

2016

Annales De l'Institut Fourier

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“…The definition can be naturally extended to certain hyperbolic conemanifolds with singularities along infinite lines; see eg Krasnov and Schlenker [19], Moroianu and Schlenker [23] and Bonsante and Schlenker [7]. By analogy, we have adopted the same terminology for our "cusps with particles".…”

Section: Manifolds With Particlesmentioning

confidence: 99%

“…Since P i Ä i D 0, we may assume Ä i < 0. Due to (7), it suffices to show that the particle length h i can be decreased; in other words, that there exists a cusp M 0 2 M.T ; g/ with truncated particle lengths…”

Section: Proof Of Theorem Amentioning

confidence: 99%

Hyperbolic cusps with convex polyhedral boundary

Fillastre

Izmestiev²

2009

Geom. Topol.

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We prove that a 3-dimensional hyperbolic cusp with convex polyhedral boundary is uniquely determined by the metric induced on its boundary. Furthermore, any hyperbolic metric on the torus with cone singularities of positive curvature can be realized as the induced metric on the boundary of a convex polyhedral cusp.The proof uses the discrete total curvature functional on the space of "cusps with particles", which are hyperbolic cone-manifolds with the singular locus a union of half-lines. We prove, in addition, that convex polyhedral cusps with particles are rigid with respect to the induced metric on the boundary and the curvatures of the singular locus.Our main theorem is equivalent to a part of a general statement about isometric immersions of compact surfaces.

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“…Other examples of conformally flat Lorentzian manifolds have recently been studied by Frances [61], Zeghib [176], and Bonsante-Schlenker [22], also closely relating to hyperbolic geometry.…”

Section: Complete Affine 3-manifoldsmentioning

confidence: 99%

Locally Homogeneous Geometric Manifolds

Goldman

2011

Proceedings of the International Congress of Mathematicians 2010 (ICM 2010)

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Abstract.Motivated by Felix Klein's notion that geometry is governed by its group of symmetry transformations, Charles Ehresmann initiated the study of geometric structures on topological spaces locally modeled on a homogeneous space of a Lie group. These locally homogeneous spaces later formed the context of Thurston's 3-dimensional geometrization program. The basic problem is for a given topology Σ and a geometry X = G/H, to classify all the possible ways of introducing the local geometry of X into Σ. For example, a sphere admits no local Euclidean geometry: there is no metrically accurate Euclidean atlas of the earth. One develops a space whose points are equivalence classes of geometric structures on Σ, which itself exhibits a rich geometry and symmetries arising from the topological symmetries of Σ.We survey several examples of the classification of locally homogeneous geometric structures on manifolds in low dimension, and how it leads to a general study of surface group representations. In particular geometric structures are a useful tool in understanding local and global properties of deformation spaces of representations of fundamental groups.Mathematics Subject Classification (2000). Primary 57M50; Secondary 57N16.Keywords. connection, curvature, fiber bundle, homogeneous space, Thurston geometrization of 3-manifolds, uniformization, crystallographic group, discrete group, proper action, Lie group, fundamental group, holonomy, completeness, development, geodesic, symplectic structure, Teichmüller space, Fricke space, hypebolic structure, Riemannian metric, Riemann surface, affine structure, projective structure, conformal structure, spherical CR structure, complex hyperbolic structure, deformation space, mapping class group, ergodic action. Historical backgroundWhile geometry involves quantitative measurements and rigid metric relations, topology deals with the loose quantitative organization of points. Felix Klein proposed in his 1872 Erlangen Program that the classical geometries be considered as the properties of a space invariant under a transitive Lie group action. Therefore one may ask which topologies support a system of local coordinates modeled on a fixed homogeneous space X = G/H such that on overlapping coordinate patches, the coordinate changes are locally restrictions of transformations from G. W. GoldmanIn this generality this question was first asked by Charles Ehresmann [55] at the conference "Quelques questions de Geométrie et de Topologie," in Geneva in 1935. Forty years later, the subject of such locally homogeneous geometric structures experienced a resurgence when W. Thurston placed his 3-dimensional geometrization program [158] in the context of locally homogeneous (Riemannian) structures. The rich diversity of geometries on homogeneous spaces brings in a wide range of techniques, and the field has thrived through their interaction.Before Ehresmann, the subject may be traced to several independent threads in the 19th century:• The theory of monodromy of Schwarzian differential equ...

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Ads Manifolds With Particles and Earthquakes on Singular Surfaces

Cited by 39 publications

References 13 publications

Maximal surfaces in anti-de Sitter 3-manifolds with particles

Maximal surfaces in anti-de Sitter 3-manifolds with particles

Hyperbolic cusps with convex polyhedral boundary

Locally Homogeneous Geometric Manifolds

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