Hyperbolization and geometric decomposition of a class of 3-manifolds
Ke Feng, Huabin Ge, Yunpeng Meng
TL;DR
The paper proves Thurston's triangulation conjecture for a class of 3-manifolds by coupling combinatorial triangulations with extended hyperbolic geometry through an extended combinatorial Ricci flow (ECRF). It introduces decorated 3-1 and 4-0 hyperbolic tetrahedra, the cov and co-volume framework, and a reduced ECRF that evolves only along hyper-ideal edges under proper gluing constraints. Under edge-valence conditions $d_e\ge6$ for ideal edges and $d_e\ge11$ for hyper-ideal edges, the reduced flow produces a zero-curvature decorated metric, which corresponds to a complete hyperbolic structure on $M$ with totally geodesic boundary and a geometric triangulation isotopic to $\\mathcal{T}$. The approach provides a constructive analytic bridge from triangulation data to hyperbolic geometry in manifolds with torus boundaries and extends prior results by incorporating torus boundaries and degeneration phenomena into a tractable geometric-flow framework.
Abstract
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 $\mathcal{T}$ is an ideal triangulation of $M$. If $\mathcal{T}$ 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 $\mathcal{T}$ 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 First, the CRF may collapse in a finite time, second, most of the smooth curvature flow methods are no longer applicable since there is no local coordinates in $\mathcal{T}$, and third, the evolution of CRF is affected by certain combinatorial obstacles in addition to topology.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 $\mathcal{T}$ and reducing the ECRF to a flow relatively easy to handle.