Characterizations of infinite circle patterns and convex polyhedra in hyperbolic 3-space

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 \cite{Rodin-Sullivan} and Schramm \cite{schramm91} proved the rigidity of infinite CPs with the intersection angle $\Theta=0$. Rivin-Hodgson \cite{RH93} fully characterized the existence and rigidity of compact convex polyhedra in $\mathbb{H}^3$. He \cite{He} proved the rigidity and uniformization theorem for infinite CPs with $0\leq\Theta\leq \pi/2$. Therefore, the remaining unresolved issues are the rigidity and uniformization theorems for infinite CPs with $0\leq\Theta<\pi$, as well as for infinite hyperbolic polyhedra. In fact, He specifically claimed in the abstract of \cite{He} that ``in a future paper, the techniques of this paper will be extended to the case when $0\leq\Theta<\pi$. 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 \cite{He} by proving the rigidity and uniformization theorem for infinite CPs with $0\leq\Theta<\pi$, 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.