Existence and uniqueness of packings with specified combinatorics

Generalizations of the Andreev-Thurston circle packing theorem are proved. One such result is the following. Let G=G(V) be a planar graph, and for each vertex v ∈ V, let ℱ _v be a proper 3-manifold of smooth topological disks in S ²,with the property that the pattern of intersection of any two sets A, B ∈ ℱ _v is topologically the pattern of intersection of two circles (i.e., there is a homeomorphism h:S ²→S ² taking A and B to circles). Then there is a packing P=(P _v :v ∈V)whose nerve is G, and which satisfies P _v ∈ ℱ _ν for v ∈ V. (‘The nerve is G’ means that two sets, P _v ,P _u ,touch, if, and only if, u ↔ v is an edge in G.) In the case whereG is the 1-skeleton of a triangulation, we also give a precise uniqueness statement. Various examples and applications are discussed.