Shapes of polyhedra and triangulations of the sphere

TL;DR

This paper explores the geometric structure of polyhedron shapes with fixed angles, revealing a complex hyperbolic metric space and its completion, which informs the classification of sphere triangulations with limited vertex degrees.

Contribution

It characterizes the shape space of polyhedra with given angles as a complex hyperbolic cone-manifold and links this to sphere triangulation classifications.

Findings

The shape space has a Kaehler metric locally isometric to complex hyperbolic space.

The metric completion forms a complex hyperbolic cone-manifold or orbifold in special cases.

Provides a classification of sphere triangulations with up to 6 triangles per vertex.

Abstract

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.