Geometry & Topology

Characterizing the Delaunay decompositions of compact hyperbolic surfaces

Gregory Leibon

Full-text: Open access

Abstract

Given a Delaunay decomposition of a compact hyperbolic surface, one may record the topological data of the decomposition, together with the intersection angles between the “empty disks” circumscribing the regions of the decomposition. The main result of this paper is a characterization of when a given topological decomposition and angle assignment can be realized as the data of an actual Delaunay decomposition of a hyperbolic surface.

Article information

Source
Geom. Topol., Volume 6, Number 1 (2002), 361-391.

Dates
Received: 28 March 2001
Revised: 8 July 2002
Accepted: 9 July 2002
First available in Project Euclid: 21 December 2017

Permanent link to this document
https://projecteuclid.org/euclid.gt/1513882798

Digital Object Identifier
doi:10.2140/gt.2002.6.361

Mathematical Reviews number (MathSciNet)
MR1914573

Zentralblatt MATH identifier
1028.52014

Subjects
Primary: 52C26: Circle packings and discrete conformal geometry
Secondary: 30F10: Compact Riemann surfaces and uniformization [See also 14H15, 32G15]

Keywords
Delaunay triangulation hyperbolic polyhedra disk pattern

Citation

Leibon, Gregory. Characterizing the Delaunay decompositions of compact hyperbolic surfaces. Geom. Topol. 6 (2002), no. 1, 361--391. doi:10.2140/gt.2002.6.361. https://projecteuclid.org/euclid.gt/1513882798


Export citation

References

  • B H Bowditch, Singular Euclidean structures on surfaces, J. London Math. Soc. 44 (1991) 553–565
  • W Brägger, Kreispackungen und Triangulation, Ens. Math. 38 (1992) 201–217
  • George B Dantzig, Linear programming and extensions, Princeton University Press, Princeton, NJ (1963)
  • Y Colin de Verdiére, Un principe variationnel pour les empilements de cercles, Prépublication de l'Institut Fourier, Grenoble 147 (1990) 0–16
  • B Delaunay, Sur la sphere vide, Proc. Int. Congr. Mth. 1 (1928) 695–700
  • D B A Epstein, R C Penner, Euclidean decompostions of non-compact hyperbolic manifolds, Journal of Differnetial Geometry, 27 (1988) 67–80
  • Marvin Jay Greenberg, Euclidean and non-euclidean geometries, second edition, W H Freeman and Co. San Francisco (1972)
  • J L Harer, The virtual cohomological dimension of the mapping class group of an orientable surface, Invent. Math. 84 (1986) 157–176
  • Z He, Rigidity of infinite disk patterns, Ann. of Math. 149 (1999) 1–33
  • Zheng-Xu He, Disk patterns and 3–dimensional convex hyperbolic polyhedra, in preparation
  • Gregory Leibon, Random Delaunay triangulations and metric uniformization, IMRN, 25 (2002) 1331–1345
  • Gregory Leibon, Random Delaunay triangulations, the Thurston–Andreev Theorem and metric uniformization, PhD thesis, UCSD (1999)
  • R C Penner, The decorated Teichmüller space of a punctured surface, Comm. Math. Phys. 113 (1987) 299–339
  • I Rivin, Combinatorial optimization in geometry.
  • I Rivin, Euclidean structures on simplicial surfaces and hyperbolic volume, Ann. of Math. 139 (1994) 553–580
  • William P Thurston, Three–dimensional geometry and topology, draft edition, The Geometry Center, University of Minnesota (1991)
  • M Troyanov, Les surfaces euclidiennes a singularites coniques, Ens. Math. 32 (1986) 74–94
  • W A Veech, Delaunay partitions, Topology, 36 (1997) 1–28