https://neckpinch.com/authors/ke-feng/New arXiv papers by Ke Feng, from Neckpinch Tue, 03 Nov 2026 08:41:12 GMT https://neckpinch.com/papers/2503.07421/arxiv: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 First, the CRF may collapse in a finite time, second, most of the smooth curvature flow methods are no longer applicable since there is no local coordinates in 𝓣, and third, the evolution of CRF is affected by certain combinatorial obstacles in addition to topology.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://neckpinch.com/papers/2502.06497/arxiv: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