Characterization of Infinite Ideal Polyhedra in Hyperbolic 3-Space via Combinatorial Ricci Flow
Geometric Topology
2025-06-06 v1 Differential Geometry
Abstract
In his seminal work \cite{Ri96}, Rivin characterized finite ideal polyhedra in three-dimensional hyperbolic space. However, the characterization of infinite ideal polyhedra, as proposed by Rivin, has remained a long-standing open problem. In this paper, we introduce the combinatorial Ricci flow for infinite ideal circle patterns, a discrete analogue of Ricci flow on non-compact Riemannian manifolds, and prove a characterization of such circle patterns under certain combinatorial conditions. Our results provide affirmative solutions to Rivin's problem.
Keywords
Cite
@article{arxiv.2506.05036,
title = {Characterization of Infinite Ideal Polyhedra in Hyperbolic 3-Space via Combinatorial Ricci Flow},
author = {Huabin Ge and Bobo Hua and Hao Yu and Puchun Zhou},
journal= {arXiv preprint arXiv:2506.05036},
year = {2025}
}