2020/09/11 by Xu Xu, Xu, Xu
Computer Science · Mathematics · #FOS: Mathematics #Geometric Analysis and Curvature Flows #Geometric Topology (math.GT) #Geometry and complex manifolds #Topological and Geometric Data Analysis
paper · pdf · doi:10.48550/arxiv.2009.05477
openalex publication_date 2020/09/11 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
Combinatorial Ricci flow on a cusped 3-manifold is an analogue of Chow-Luo's combinatorial Ricci flow on surfaces and Luo's combinatorial Ricci flow on compact 3-manifolds with boundary for finding complete hyperbolic metrics on cusped 3-manifolds. Dual to Casson and Rivin's program of maximizing the volume of angle structures, combinatorial Ricci flow finds the complete hyperbolic metric on a cusped 3-manifold by minimizing the co-volume of decorated hyperbolic polyhedral metrics. The combinatorial Ricci flow may develop singularities. We overcome this difficulty by extending the flow through the potential singularities using Luo-Yang's extension. It is shown that the existence of a complete hyperbolic metric on a cusped 3-manifold is equivalent to the convergence of the extended combinatorial Ricci flow, which gives a new characterization of existence of a complete hyperbolic metric on a cusped 3-manifold dual to Casson and Rivin's program. The extended combinatorial Ricci flow also provides an effective algorithm for finding complete hyperbolic metrics on cusped 3-manifolds.