Algebraic & Geometric Topology

The moduli space of hex spheres

Aldo-Hilario Cruz-Cota

Full-text: Open access

PDF File (345 KB)

Abstract

A hex sphere is a singular Euclidean sphere with four cone points whose cone angles are (integer) multiples of 2π3 but less than 2π. We prove that the Moduli space of hex spheres of unit area is homeomorphic to the the space of similarity classes of Voronoi polygons in the Euclidean plane. This result gives us as a corollary that each unit-area hex sphere M satisfies the following properties:

(1) it has an embedded (open Euclidean) annulus that is disjoint from the singular locus of M;

(2) it embeds isometrically in the 3–dimensional Euclidean space as the boundary of a tetrahedron; and

(3) there is a simple closed geodesic γ in M such that a fractional Dehn twist along γ converts M to the double of a parallelogram.

Article information

Source
Algebr. Geom. Topol., Volume 11, Number 3 (2011), 1323-1343.

Dates
Received: 31 October 2010
Revised: 26 January 2011
Accepted: 15 February 2011
First available in Project Euclid: 19 December 2017

Permanent link to this document
https://projecteuclid.org/euclid.agt/1513715229

Digital Object Identifier
doi:10.2140/agt.2011.11.1323

Mathematical Reviews number (MathSciNet)
MR2801420

Zentralblatt MATH identifier
1233.57010

Subjects
Primary: 57M50: Geometric structures on low-dimensional manifolds
Secondary: 57M15: Relations with graph theory [See also 05Cxx]

Keywords
singular Euclidean surfaces moduli spaces

Citation

Cruz-Cota, Aldo-Hilario. The moduli space of hex spheres. Algebr. Geom. Topol. 11 (2011), no. 3, 1323--1343. doi:10.2140/agt.2011.11.1323. https://projecteuclid.org/euclid.agt/1513715229


Export citation

References

  • M Boileau, J Porti, Geometrization of 3–orbifolds of cyclic type, Astérisque (2001) 208 Appendix A by Michael Heusener and Porti
    Mathematical Reviews (MathSciNet): MR1844891
    Zentralblatt MATH: 0971.57004
  • D Burago, Y Burago, S Ivanov, A course in metric geometry, Graduate Studies in Mathematics 33, American Mathematical Society, Providence, RI (2001)
    Mathematical Reviews (MathSciNet): MR1835418
    Zentralblatt MATH: 0981.51016
  • S Choi, W M Goldman, Convex real projective structures on closed surfaces are closed, Proc. Amer. Math. Soc. 118 (1993) 657–661
    Mathematical Reviews (MathSciNet): MR1145415
    Zentralblatt MATH: 0810.57005
    Digital Object Identifier: doi:10.1090/S0002-9939-1993-1145415-8
  • D Cooper, C D Hodgson, S P Kerckhoff, Three-dimensional orbifolds and cone-manifolds, MSJ Memoirs 5, Mathematical Society of Japan, Tokyo (2000) With a postface by Sadayoshi Kojima
    Mathematical Reviews (MathSciNet): MR1778789
    Zentralblatt MATH: 0955.57014
  • A-H Cruz-Cota, Classifying Voronoi graphs of hex sphere, preprint
    Mathematical Reviews (MathSciNet): MR2876727
    Zentralblatt MATH: 1243.57010
    Digital Object Identifier: doi:10.1016/j.topol.2011.11.008
  • W M Goldman, Convex real projective structures on compact surfaces, J. Differential Geom. 31 (1990) 791–845
    Mathematical Reviews (MathSciNet): MR1053346
    Zentralblatt MATH: 0711.53033
    Digital Object Identifier: doi:10.4310/jdg/1214444635
    Project Euclid: euclid.jdg/1214444635
  • N Hitchin, Lie groups and Teichmüller space, Topology 31 (1992) 449–473
    Mathematical Reviews (MathSciNet): MR1174252
    Digital Object Identifier: doi:10.1016/0040-9383(92)90044-I
  • F Labourie, Flat projective structures on surfaces and cubic holomorphic differentials, Pure Appl. Math. Q. 3 (2007) 1057–1099
    Mathematical Reviews (MathSciNet): MR2402597
    Zentralblatt MATH: 1158.32006
    Digital Object Identifier: doi:10.4310/PAMQ.2007.v3.n4.a10
  • J Loftin, Flat metrics, cubic differentials and limits of projective holonomies, Geom. Dedicata 128 (2007) 97–106
    Mathematical Reviews (MathSciNet): MR2350148
    Zentralblatt MATH: 1145.57014
    Digital Object Identifier: doi:10.1007/s10711-007-9184-2
  • M Troyanov, On the moduli space of singular Euclidean surfaces, from: “Handbook of Teichmüller theory Vol I”, IRMA Lect. Math. Theor. Phys. 11, Eur. Math. Soc., Zürich (2007) 507–540
    Mathematical Reviews (MathSciNet): MR2349679
    Zentralblatt MATH: 1127.32009

MSP

Algebraic & Geometric Topology

New content alerts

Email RSS ToC RSS Article
  • You have access to this content.
  • You have partial access to this content.
  • You do not have access to this content.
Turn Off MathJax

What is MathJax?

More like this

+ See more