Shapes of polyhedra and triangulations of the sphere

Details Shapes of polyhedra and triangulations of the sphere Shapes of polyhedra and triangulations of the sphere

1998-01-19

arxiv.org/abs/math/9801088v2

Geom. Topol. Monogr. 1 (1998), 511-549

Mathematics

Geometric Topology

39 pages. Published copy, also available at http://www.maths.warwick.ac.uk/gt/GTMon1/paper25.abs.html

Scientific paper

The space of shapes of a polyhedron with given total angles less than 2\pi at each of its n vertices has a Kaehler metric, locally isometric to complex hyperbolic space CH^{n-3}. The metric is not complete: collisions between vertices take place a finite distance from a nonsingular point. The metric completion is a complex hyperbolic cone-manifold. In some interesting special cases, the metric completion is an orbifold. The concrete description of these spaces of shapes gives information about the combinatorial classification of triangulations of the sphere with no more than 6 triangles at a vertex.

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