Shapes of Polyhedra and Triangulations of the Sphere

The space of shapes of a polyhedron with given total angles less than 2π at each of its n vertices has a Kähler metric, locally isometric to complex hyperbolic space CH. The metric is not complete: collisions between vertices take place a finite distance from a nonsingular point. The metric completion is a complex hyperbolic conemanifold. 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.