Hyperbolic cusps with convex polyhedral boundary

Article, Preprint, Other literature type English OPEN
Fillastre, François; Izmestiev, Ivan;
(2007)
  • Publisher: Mathematical Sciences Publishers
  • Journal: issn: 1465-3060
  • Publisher copyright policies & self-archiving
  • Related identifiers: doi: 10.2140/gt.2009.13.457
  • Subject: infinitesimal rigidity | 57M50 | 57M50 (Geometric structures on low-dimensional manifolds), secondary: 53C24 | 53C24 | convex polyhedral boundary | hyperbolic cone-manifold | discrete total curvature | Mathematics - Differential Geometry | Hyperbolic cusp | 57M50 (Primary) 53C24 (Secondary) | Alexandrov's theorem | [ MATH.MATH-DG ] Mathematics [math]/Differential Geometry [math.DG] | discrete total scalar curvature
    arxiv: Mathematics::Differential Geometry | Mathematics::Geometric Topology

International audience; We prove that a 3-dimensional hyperbolic cusp with convex polyhedral boundary is uniquely determined by the metric induced on its boundary. Furthemore, any hyperbolic metric on the torus with cone singularities of positive curvature can be realiz... View more
  • References (30)
    30 references, page 1 of 3

    [Ale42] A. D. Alexandroff. Existence of a convex polyhedron and of a convex surface with a given metric. Rec. Math. [Mat. Sbornik] N.S., 11(53):15-65, 1942. (Russian).

    [Ale05] A. D. Alexandrov. Convex polyhedra. Springer Monographs in Mathematics. SpringerVerlag, Berlin, 2005.

    [And70] E. M. Andreev. Convex polyhedra in Lobaˇcevski˘ı spaces. Mat. Sb. (N.S.), 81 (123):445- 478, 1970.

    [And02] M. T. Anderson. Scalar curvature and the existence of geometric structures on 3- manifolds. I. J. Reine Angew. Math., 553:125-182, 2002.

    [BH37] W. Blaschke and G. Herglotz. U¨ ber die Verwirklichung einer geschlossenen Fla¨che mit vorgeschriebenem Bogenelement im Euklidischen Raum. Sitzungsber. Bayer. Akad. Wiss., Math.-Naturwiss. Abt., No.2:229-230, 1937.

    [BI07] A. Bobenko and I. Izmestiev. Alexandrov's theorem, weighted Delaunay triangulations, and mixed volumes. math.DG/0609447. To appear in Annales de l'institut Fourier, 2007.

    [BS06] F. Bonsante and J.-M. Schlenker. AdS manifolds with particles and earthquakes on singular surfaces, 2006. arXiv.org:math/0609116.

    [Cau05] A.-L. Cauchy. Sur les polygones et poly`edres (Second M´emoire). In Œvres compl`etes, volume 1 of Seconde s´erie, pages 26-38. 1905.

    [CEG06] R. D. Canary, D. B. A. Epstein, and P. L. Green. Notes on notes of Thurston. In Fundamentals of hyperbolic geometry: selected expositions, volume 328 of London Math. Soc. Lecture Note Ser., pages 1-115. Cambridge Univ. Press, Cambridge, 2006.

    [CS06] R. Connelly and J.-M. Schlenker. On the infinitesimal rigidity of weakly convex polyhedra, 2006. arXiv.org:math/0606681.

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