https://neckpinch.com/authors/huabin-ge/New arXiv papers by Huabin Ge, from Neckpinch Tue, 03 Nov 2026 07:15:30 GMT https://arxiv.org/abs/2506.05036arxiv:2506.05036Thu, 05 Jun 2025 00:00:00 GMTHuabin Ge, Bobo Hua, Hao Yu, Puchun Zhou — In his seminal work, 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.Ricci flow3-manifoldsBoundaries & non-compactDiscrete & graph Ricci https://arxiv.org/abs/2504.05817arxiv:2504.05817Tue, 08 Apr 2025 00:00:00 GMTHuabin Ge, Bobo Hua, Puchun Zhou — In this paper, we introduce combinatorial Ricci flows (CRFs in short) in Euclidean and hyperbolic background geometries on infinite triangulations of the open disk, which are discrete analogs of Ricci flows on simply connected open surfaces. We establish well-posedness results, the existence and the uniqueness, of CRFs in both Euclidean and hyperbolic background geometries. Moreover, we prove convergence results of CRFs, which indicate a uniformization theorem for CRFs on infinite disk triangulations. As an application, we prove an existence result of circle-packing metrics with infinite prescribed cone angles in hyperbolic background geometry. To our knowledge, these are the first results of CRFs on infinite triangulations.Ricci flowSurfacesStability & uniquenessDiscrete & graph Ricci https://arxiv.org/abs/2503.07421arxiv:2503.07421Mon, 10 Mar 2025 00:00:00 GMTKe Feng, Huabin Ge, Yunpeng Meng — Thurston's triangulation conjecture asserts that every hyperbolic 3-manifold admits a geometric triangulation into hyper-ideal hyperbolic tetrahedra. So far, this conjecture had only been proven for a few special 3-manifolds. In this article, we confirm this conjecture for a class of 3-manifolds. To be precise, let M be an oriented compact 3-manifold with boundary, no component of which is a 2-sphere, and 𝓣 is an ideal triangulation of M. If 𝓣 satisfies properly gluing condition, and the valence is at least 6 at each ideal edge and 11 at each hyper-ideal edge, then M admits an unique complete hyperbolic metric with totally geodesic boundary, so that 𝓣 is isotopic to a geometric ideal triangulation of M. We use analytical tools such as combinatorial Ricci flow (CRF, abbr.) to derive the conclusions. There are intrinsic difficulties in dealing with CRF. First, the CRF may collapse in a finite time, second, most of the smooth curvature flow methods are no longer applicable since there is no local coordinates in 𝓣, and third, the evolution of CRF is affected by certain combinatorial obstacles in addition to topology. To this end, we introduce the ideas as "extending CRF", "tetrahedral comparison principles", and "control CRF with edge valence" to solve the above difficulties. In addition, the presence of torus boundary adds substantial difficulties in this article, which we have solved by introducing the properly gluing conditions on 𝓣 and reducing the ECRF to a flow relatively easy to handle.Ricci flowOther curvature flowsSingularitiesGeometrization3-manifoldsBoundaries & non-compactDiscrete & graph Ricci https://arxiv.org/abs/2502.06497arxiv:2502.06497Mon, 10 Feb 2025 00:00:00 GMTFeng Ke, Ge Huabin — Thurston's triangulation conjecture asserts that every hyperbolic 3-manifold admits a geometric decomposition into ideal hyperbolic tetrahedra, a result proven only for certain special 3-manifolds. This paper presents combinatorial Ricci flow as a systematic and general approach to addressing Thurston's triangulation conjecture, showing that the flow converges if and only if the triangulation is geometric. First, we prove the rigidity of the most general hyperbolic polyhedral 3-manifolds constructed by isometrically gluing partially truncated and decorated hyperbolic tetrahedra, demonstrating that the metrics are uniquely determined by cone angles modulo isometry and decoration changes. Then, we demonstrate that combinatorial Ricci flow evolves polyhedral metrics toward complete hyperbolic structures with geometric decompositions when convergent. Conversely, the existence of a geometric triangulation guarantees flow convergence.Ricci flowGeometrization3-manifoldsStability & uniquenessDiscrete & graph Ricci https://arxiv.org/abs/2303.09191arxiv:2303.09191Thu, 16 Mar 2023 00:00:00 GMTHuabin Ge, Bobo Hua, Puchun Zhou — In [12], the existence of ideal circle patterns in Euclidean or hyperbolic background geometry under the combinatorial conditions was proved using flow approaches. It remains as an open problem for the spherical case. In this paper, we introduce a combinatorial geodesic curvature flow in spherical background geometry, which is analogous to the combinatorial Ricci flow of Chow and Luo in [4]. We characterize the sufficient and necessary condition for the convergence of the flow. That is, the prescribed geodesic curvature satisfies certain geometric and combinatorial condition if and only if for any initial data the flow converges exponentially fast to a circle pattern with given total geodesic curvature on each circle. Our result could be regarded as a resolution of the problem in the spherical case. As far as we know, this is the first combinatorial curvature flow in spherical background geometry with fine properties, and it provides an algorithm to find the desired ideal circle pattern.Ricci flowOther curvature flowsLong-time existence & convergenceDiscrete & graph Ricci