Shapes of polyhedra and triangulations of the sphere

Abstract

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