2025/11/12 by Huabin Ge, Longsong Jia, Ge, Huabin +5
Engineering · Mathematics · #Advanced Materials and Mechanics #Complex Variables (math.CV) #Differential Geometry (math.DG) #FOS: Mathematics #Geometric Topology (math.GT) #Geometric and Algebraic Topology #Metric Geometry (math.MG) #Structural Analysis and Optimization
paper · pdf · doi:10.48550/arxiv.2511.09368
openalex publication_date 2025/11/12 · openalex created_date 2025/11/14 · openalex updated_date 2026/07/28
Since Thurston pioneered the connection between circle packing (abbr. CP) and three-dimensional geometric topology, the characterization of CPs and hyperbolic polyhedra has become increasingly profound. Some milestones have been achieved, for example, Rodin-Sullivan \citeRodin-Sullivan and Schramm \citeschramm91 proved the rigidity of infinite CPs with the intersection angle Θ=0. Rivin-Hodgson \citeRH93 fully characterized the existence and rigidity of compact convex polyhedra in ℍ3. He \citeHe proved the rigidity and uniformization theorem for infinite CPs with 0≤Θ≤ π/2. Therefore, the remaining unresolved issues are the rigidity and uniformization theorems for infinite CPs with 0≤Θ<π, as well as for infinite hyperbolic polyhedra. In fact, He specifically claimed in the abstract of \citeHe that ``in a future paper, the techniques of this paper will be extended to the case when 0≤Θ<π. In particular, we will show a rigidity property for a class of infinite convex polyhedra in the 3-dimensional hyperbolic space". The objective of the article is to accomplish the work claimed in \citeHe by proving the rigidity and uniformization theorem for infinite CPs with 0≤Θ<π, as well as infinite trivalent hyperbolic polyhedra. We will pay special attention to CPs whose contact graphs are disk triangulation graphs. Such CPs are called regular because they exclude some singular configurations and correspond well to hyperbolic polyhedra. We will establish the existence and rigidity of infinite regular CPs. Moreover, we will prove a uniformization theorem for regular CPs, which solves the classification problem for regular CPs. Thereby, the existence and rigidity of infinite convex trivalent polyhedra are obtained.