In this paper, we prove the global rigidity of sphere packings on $3$-dimensional manifolds. This is a $3$-dimensional analogue of the rigidity theorem of Andreev–Thurston and was conjectured by Cooper and Rivin in . We also prove a global rigidity result using a combinatorial scalar curvature introduced by Ge and the author in .
"On the global rigidity of sphere packings on $3$-dimensional manifolds." J. Differential Geom. 115 (1) 175 - 193, May 2020. https://doi.org/10.4310/jdg/1586224843