Combinatorial Ricci flow on cusped 3-manifolds

Combinatorial Ricci flow on a cusped $3$-manifold is an analogue of Chow-Luos combinatorial Ricci flow on surfaces and Luos combinatorial Ricci flow on compact $3$-manifolds with boundary for finding complete hyperbolic metrics on cusped $3$-manifolds. Dual to Casson and Rivins 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-Yangs 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 Rivins program. The extended combinatorial Ricci flow also provides an effective algorithm for finding complete hyperbolic metrics on cusped $3$-manifolds.