Characterization of Infinite Ideal Polyhedra in Hyperbolic 3-Space via Combinatorial Ricci Flow

TL;DR

This paper introduces a combinatorial Ricci flow approach to characterize infinite ideal polyhedra in hyperbolic 3-space, solving a long-standing open problem and extending Rivin's finite polyhedra characterization.

Contribution

It develops a novel combinatorial Ricci flow method for infinite ideal circle patterns, providing a new characterization of infinite ideal polyhedra in hyperbolic space.

Findings

Proves a new characterization of infinite ideal polyhedra.

Introduces combinatorial Ricci flow for non-compact cases.

Provides affirmative solutions to Rivin's open problem.

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.