Geodesic
The moduli space of hex spheres
Cruz-Cota, Aldo-Hilario1
1Department of Mathematics, Grand Valley State University, Allendale MI 49401-9401, USA
Algebraic and Geometric Topology, Tome 11 (2011) no. 3, pp. 1323-1343
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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.

DOI : 10.2140/agt.2011.11.1323
Keywords: singular Euclidean surfaces, moduli spaces

Cruz-Cota, Aldo-Hilario  1

1 Department of Mathematics, Grand Valley State University, Allendale MI 49401-9401, USA 
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Cruz-Cota, Aldo-Hilario. The moduli space of hex spheres. Algebraic and Geometric Topology, Tome 11 (2011) no. 3, pp. 1323-1343. doi: 10.2140/agt.2011.11.1323

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