Combinatorial Calabi flow on 3-manifolds with toroidal boundary

Abstract

Motivated by Chow-Luo's combinatorial Ricci flow on surfaces and Luo's combinatorial Ricci flow on compact 3-manifolds with boundary, we introduce combinatorial Calabi flow for decorated hyperbolic polyhedral metrics on 3-manifolds with toroidal boundary to find complete hyperbolic metrics. Dual to Casson and Rivin's approach to solve Thurston's gluing equation by maximizing the volume function for angle structures, the combinatorial Calabi flow on 3-manifolds with toroidal boundary works on decorated hyperbolic polyhedral metrics to ensure that there is no shearing around the edges and finds the complete hyperbolic metric by minimizing the combinatorial Calabi energy. Basic properties of combinatorial Calabi flow on 3-manifolds with toroidal boundary are established. The most important ones include that the equilibrium points of the combinatorial Calabi flow correspond to complete hyperbolic metrics on 3-manifolds with toroidal boundary and the local convergence of the combinatorial Calabi flow. We also study the combinatorial Calabi flow on an ideal tetrahedron. It is shown that for any prescribed admissible dihedral angles, the solution of combinatorial Calabi flow on an ideal tetrahedron exists for all time and converges exponentially fast to a complete hyperbolic polyhedral metric with the prescribed dihedral angles.