Geometry & Topology

Hyperbolic cusps with convex polyhedral boundary

François Fillastre and Ivan Izmestiev

Full-text: Open access

Abstract

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.

Article information

Source
Geom. Topol., Volume 13, Number 1 (2009), 457-492.

Dates
Received: 14 December 2007
Revised: 7 October 2008
Accepted: 17 September 2008
First available in Project Euclid: 20 December 2017

Permanent link to this document
https://projecteuclid.org/euclid.gt/1513800186

Digital Object Identifier
doi:10.2140/gt.2009.13.457

Mathematical Reviews number (MathSciNet)
MR2469522

Zentralblatt MATH identifier
1179.57026

Subjects
Primary: 57M50: Geometric structures on low-dimensional manifolds
Secondary: 53C24: Rigidity results

Keywords
Alexandrov's theorem convex polyhedral boundary hyperbolic cone-manifold discrete total curvature

Citation

Fillastre, François; Izmestiev, Ivan. Hyperbolic cusps with convex polyhedral boundary. Geom. Topol. 13 (2009), no. 1, 457--492. doi:10.2140/gt.2009.13.457. https://projecteuclid.org/euclid.gt/1513800186


Export citation

References

  • A Alexandroff, Existence of a convex polyhedron and of a convex surface with a given metric, Rec. Math. [Mat. Sbornik] N.S. 11(53) (1942) 15–65
  • A D Alexandrov, Convex polyhedra, Springer Monographs in Math., Springer, Berlin (2005) Translated from the 1950 Russian edition by N S Dairbekov, S S Kutateladze and A B Sossinsky, With comments and bibliography by V A Zalgaller and appendices by L A Shor and Yu A Volkov
  • M T Anderson, Scalar curvature and the existence of geometric structures on $3$–manifolds. I, J. Reine Angew. Math. 553 (2002) 125–182
  • E M Andreev, Convex polyhedra in Lobačevskiĭ spaces, Mat. Sb. $($N.S.$)$ 81 (123) (1970) 445–478
  • W Blaschke, G Herglotz, Über die Verwirklichung einer geschlossenen Fläche mit vorgeschriebenem Bogenelement im Euklidischen Raum, Sitzungsber. Bayer. Akad. Wiss., Math.-Naturwiss. Abt. 2 (1937) 229–230
  • A I Bobenko, I Izmestiev, Alexandrov's theorem, weighted Delaunay triangulations, and mixed volumes, Ann. Inst. Fourier $($Grenoble$)$ 58 (2008) 447–505
  • F Bonsante, J-M Schlenker, AdS manifolds with particles and earthquakes on singular surfaces, To appear in Geom. Funct. Anal.
  • R D Canary, D B A Epstein, P L Green, Notes on notes of Thurston [MR0903850], from: “Fundamentals of hyperbolic geometry: selected expositions”, London Math. Soc. Lecture Note Ser. 328, Cambridge Univ. Press (2006) 1–115 With a new foreword by Canary
  • A-L Cauchy, Sur les polygones et polyèdres (Second Mémoire), from: “Œvres complètes, Seconde série 1” (1905) 26–38
  • R Connelly, J-M Schlenker, On the infinitesimal rigidity of weakly convex polyhedra
  • F Fillastre, Fuchsian polyhedra in Lorentzian space-forms
  • F Fillastre, Polyhedral realisation of hyperbolic metrics with conical singularities on compact surfaces, Ann. Inst. Fourier $($Grenoble$)$ 57 (2007) 163–195
  • F Fillastre, Polyhedral hyperbolic metrics on surfaces, Geom. Dedicata 134 (2008) 177–196
  • F Fillastre, I Izmestiev, Dual metrics of hyperbolic cusps with convex polyhedral boundary, In preparation (2008)
  • C D Hodgson, Deduction of Andreev's theorem from Rivin's characterization of convex hyperbolic polyhedra, from: “Topology '90 (Columbus, OH, 1990)”, Ohio State Univ. Math. Res. Inst. Publ. 1, de Gruyter, Berlin (1992) 185–193
  • C D Hodgson, I Rivin, A characterization of compact convex polyhedra in hyperbolic $3$–space, Invent. Math. 111 (1993) 77–111
  • C Indermitte, T M Liebling, M Troyanov, H Clémençon, Voronoi diagrams on piecewise flat surfaces and an application to biological growth, Theoret. Comput. Sci. 263 (2001) 263–274 Combinatorics and computer science (Palaiseau, 1997)
  • I Izmestiev, A variational proof of Alexandrov's convex cap theorem, To appear in Discrete Comput. Geom.
  • K Krasnov, J-M Schlenker, Minimal surfaces and particles in $3$–manifolds, Geom. Dedicata 126 (2007) 187–254
  • F Labourie, J-M Schlenker, Surfaces convexes fuchsiennes dans les espaces lorentziens à courbure constante, Math. Ann. 316 (2000) 465–483
  • F Luo, Rigidity of polyhedral surfaces
  • J Milnor, Collected papers. Vol. 1. Geometry, Publish or Perish, Houston, TX (1994)
  • S Moroianu, J-M Schlenker, Quasi-Fuchsian manifolds with particles
  • I Rivin, On geometry of convex polyhedra in hyperbolic $3$–space, PhD thesis, Princeton University (1986)
  • I Rivin, Euclidean structures on simplicial surfaces and hyperbolic volume, Ann. of Math. $(2)$ 139 (1994) 553–580
  • J-M Schlenker, Hyperbolic manifolds with polyhedral boundary, Available online at http://www.picard.ups-tlse.fr/\char'176schlenker/texts/ideal.pdf
  • J-M Schlenker, On weakly convex star-shaped polyhedra
  • J-M Schlenker, Surfaces convexes dans des espaces lorentziens à courbure constante, Comm. Anal. Geom. 4 (1996) 285–331
  • J-M Schlenker, Convex polyhedra in Lorentzian space-forms, Asian J. Math. 5 (2001) 327–363
  • J-M Schlenker, Hyperbolic manifolds with convex boundary, Invent. Math. 163 (2006) 109–169
  • S Sechelmann, Alexandrov polyhedron editor http://www.math.tu-berlin.de/geometrie/ps/software.shtml
  • W P Thurston, Shapes of polyhedra and triangulations of the sphere, from: “The Epstein birthday schrift”, Geom. Topol. Monogr. 1, Geom. Topol. Publ., Coventry (1998) 511–549
  • Y A Volkov, Existence of a polyhedron with a given development, PhD thesis, Leningrad State University (1955) In Russian
  • Y A Volkov, Existence of convex polyhedra with prescribed development I, Vestn. Leningr. Univ. 15 (1960) 75–86