Volume 11, issue 3 (2011)

Download this article
Download this article For screen
For printing
Recent Issues

Volume 17
Issue 6, 3213–3852
Issue 5, 2565–3212
Issue 4, 1917–2564
Issue 3, 1283–1916
Issue 2, 645–1281
Issue 1, 1–643

Volume 16, 6 issues

Volume 15, 6 issues

Volume 14, 6 issues

Volume 13, 6 issues

Volume 12, 4 issues

Volume 11, 5 issues

Volume 10, 4 issues

Volume 9, 4 issues

Volume 8, 4 issues

Volume 7, 4 issues

Volume 6, 5 issues

Volume 5, 4 issues

Volume 4, 2 issues

Volume 3, 2 issues

Volume 2, 2 issues

Volume 1, 2 issues

The Journal
About the Journal
Subscriptions
Editorial Board
Editorial Interests
Editorial Procedure
Submission Guidelines
Submission Page
Author Index
To Appear
ISSN (electronic): 1472-2739
ISSN (print): 1472-2747
The moduli space of hex spheres

Aldo-Hilario Cruz-Cota

Algebraic & Geometric Topology 11 (2011) 1323–1343
Abstract

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.

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