Fuchsian polyhedra in Lorentzian space-forms
type: Item
instance of: scholarly article
title: Fuchsian polyhedra in Lorentzian space-forms
zbMATH Open document ID: 1229.52017
author: François Fillastre
published in: Mathematische Annalen
publication date: 2011-06-03T00:00:00Z
full work available at URL: https://arxiv.org/abs/math/0702532
review text: A. D. Aleksandrov proved that a hyperbolic cone metric on the sphere \(\mathbb{S}^2\) with positively curved singularities can be realized (uniquely, up to congruence) by the induced metric of a convex polyhedron in \(\mathbb{H}^3 .\) Much later, the characterization of compact convex polyhedral surfaces in the de Sitter space \(\mathbb{S}^2_1 \) was obtained by \textit{C. D. Hodgson} and \textit{I. Rivin} in [Invent. Math. 111, No.~1, 77--111 (1993; Zbl 0784.52013)]. The paper under review greatly generalizes the above results to Lorentzian space forms of dimension 3 with constant curvature \(K, K\in \{-1,0,1\}\): Let \(S\) be a compact surface of genus \(>1\) and \(g\) a metric of constant curvature \(K\) with conical singularities of negative singular curvature (in the spherical case the lengths of the contractible geodesics are assumed to be \(>2\pi\)). It is shown that there exists a convex polyhedral surface \(P\) in the Minkowski space (\(K=0\)), in the de Sitter space (\(K=1\)) and in the anti-de Sitter space (\(K=-1\)) and a group \(G\) of isometries of this space such that the induced metric on \(P/G\) is isometric to \((S,g).\) Moreover, uniqueness (up to global isometries) is shown within a particular class of convex polyhedra.
Mathematics Subject Classification ID: 52B70
Mathematics Subject Classification ID: 52A15
Mathematics Subject Classification ID: 53C24
Mathematics Subject Classification ID: 53C45
zbMATH DE Number: 5903695
zbMATH Keywords: Lorentzian space forms
zbMATH Keywords: convex polyhedral surface
zbMATH Keywords: Minkowski space
zbMATH Keywords: de Sitter space
zbMATH Keywords: anti-de Sitter space
MaRDI profile type: Publication
OpenAlex ID: W2592280817
arXiv ID: math/0702532
cites work: Q5837689
cites work: Q5589820
cites work: Geometry and spectra of compact Riemann surfaces
cites work: The polar dual of a convex polyhedral set in hyperbolic space
cites work: A Geometrical Background for De Sitter's World
cites work: Metrics of positive curvature with conic singularities on the sphere
cites work: Hyperbolic cusps with convex polyhedral boundary
cites work: Polyhedral realisation of hyperbolic metrics with conical singularities on compact surfaces
cites work: Polyhedral hyperbolic metrics on surfaces
cites work: Q4028058
cites work: Voronoi diagrams on piecewise flat surfaces and an application to biological growth
cites work: Projective background of the infinitesimal rigidity of frameworks
cites work: Convex Fuchs surfaces in Lorentz spaces of constant curvature
cites work: Liouville Equation and Spherical Convex Polytopes
cites work: Point Singularities and Conformal Metrics on Riemann Surfaces
cites work: Lorentz spacetimes of constant curvature
cites work: Q4069454
cites work: A characterization of compact convex polyhedra in hyperbolic 3-space
cites work: Andreev's theorem on hyperbolic polyhedra
cites work: On geometry of convex ideal polyhedra in hyperbolic 3-space
cites work: Sur la rigidité de polyèdres hyperboliques en dimension $3$ : cas de volume fini, cas hyperidéal, cas fuchsien
cites work: Convex surfaces in Lorentzian spaces of constant curvature
cites work: Metrics on hyperbolic convex polyhedra
cites work: Réalisations de surfaces hyperboliques complètes dans \(H^3\). (Realizations of complete hyperbolic surfaces in \(H^3\))
cites work: Convex polyhedra in Lorentzian space-forms
cites work: A rigidity criterion for non-convex polyhedra
cites work: Hyperbolic manifolds with convex boundary
cites work: Small deformations of polygons and polyhedra
cites work: Q4337982
cites work: Q4231188
cites work: Les surfaces euclidiennes à singularités coniques. (Euclidean surfaces with cone singularities)
cites work: Prescribing Curvature on Compact Surfaces with Conical Singularities
cites work: Metrics of constant curvature 1 with three conical singularities on the 2-sphere
cites work: Surfaces and planar discontinuous groups. Revised and expanded transl. from the German by J. Stillwell
DOI: 10.1007/S00208-010-0563-X
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description: scientific article