Combinatorial Ricci Flows and Hyperbolic Structures on a Class of Compact $3$-Manifolds with Boundary
Xinrong Zhao
TL;DR
The work develops a flow-based method to realize hyperbolic structures on a class of compact 3-manifolds with boundary by employing generalized hyper-ideal tetrahedra and an extended combinatorial Ricci flow. Under a valence condition $v(e)\ge 9$ on all edges, the extended Ricci flow converges exponentially to a unique zero-curvature hyper-ideal metric, yielding a complete hyperbolic metric with totally geodesic boundary on the manifold; the triangulation then aligns with a geometric decomposition. The authors derive explicit lower and upper bounds for edge lengths and dihedral angles through a careful analysis of the co-volume functional and its gradient, including numerical verifications for parameter regimes. Altogether, the paper provides a constructive, flow-driven realization of Thurston’s geometric ideal triangulation conjecture for this class of manifolds and sharpens existing results by lowering the valence threshold.
Abstract
In this paper, we study a combinatorial Ricci flow on closed pseudo $3$-manifolds $(M,\mathcal{T})$. We prove that if every edge in the triangulation $\mathcal{T}$ has valence at least $9$, then the combinatorial Ricci flow converges exponentially fast to a hyperbolic metric. As a consequence, for any compact $3$-manifold $N$ with boundary admitting an ideal triangulation $\mathcal{T}_N$ whose edges all have valence at least $9$, there exists a unique complete hyperbolic metric with totally geodesic boundary on $N$ such that $\mathcal{T}_N$ is isotopic to a geometric decomposition of $N$. This provides a partial solution to the conjecture of Costantino, Frigerio, Martelli and Petronio, and hence an affirmative answer of Thurston's geometric ideal triangulation conjecture for such manifolds. Moreover, we obtain explicit upper and lower bounds for the resulting hyperbolic metric.