Open Access
2011 The moduli space of hex spheres
Aldo-Hilario Cruz-Cota
Algebr. Geom. Topol. 11(3): 1323-1343 (2011). DOI: 10.2140/agt.2011.11.1323


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.


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Aldo-Hilario Cruz-Cota. "The moduli space of hex spheres." Algebr. Geom. Topol. 11 (3) 1323 - 1343, 2011.


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

zbMATH: 1233.57010
MathSciNet: MR2801420
Digital Object Identifier: 10.2140/agt.2011.11.1323

Primary: 57M50
Secondary: 57M15

Keywords: moduli spaces , singular Euclidean surfaces

Rights: Copyright © 2011 Mathematical Sciences Publishers

Vol.11 • No. 3 • 2011
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