Conformal tetrahedra and global rigidity of sphere packings in constant curvature background geometry

RESUMO

In this paper, we give the dual characterization of conformal tetrahedra in constant curvature spaces, which develops a geometric technique to obtain the global rigidity of (ideal) hyperbolic, Euclidean, spherical sphere packings on a tetrahedron. Moreover, we introduce ideal hyperbolic sphere packings and extend the variational principle to obtain the global rigidity of (ideal) hyperbolic sphere packings on 3-manifolds. We also provide a characterization of conformal $n(\ge3)$-simplices in constant curvature spaces, which is useful for studying higher-dimensional sphere packings.