On the global rigidity of sphere packings on $3$-dimensional manifolds

Abstract

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 [5]. We also prove a global rigidity result using a combinatorial scalar curvature introduced by Ge and the author in [13].