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Shapes of polyhedra and triangulations of the sphere
William P Thurston
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Geometry & Topology Monographs 1
(1998) 511–549
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Abstract
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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 CHn-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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Keywords
polyhedra, triangulations, configuration
spaces, braid groups, complex hyperbolic orbifolds
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Mathematical Subject Classification
Primary: 51M20
Secondary: 20H15, 51F15, 57M50
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Publication
Received: 15 November 1997
Revised: 27 November 1998
Published: 30 November 1998
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