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The moduli space of hex spheres

Aldo-Hilario Cruz-Cota

Algebraic & Geometric Topology 11 (2011) 1323–1343

A hex sphere is a singular Euclidean sphere with four cone points whose cone angles are (integer) multiples of 2π 3 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.

singular Euclidean surfaces, moduli spaces
Mathematical Subject Classification 2000
Primary: 57M50
Secondary: 57M15
Received: 31 October 2010
Revised: 26 January 2011
Accepted: 15 February 2011
Published: 11 May 2011
Aldo-Hilario Cruz-Cota
Department of Mathematics
Grand Valley State University
Allendale MI 49401-9401