Combinatorial Ricci flows and the hyperbolization of a class of compact 3-manifolds

Abstract

We prove that for a compact 3-manifold M with boundary admitting an ideal triangulation T with valence at least 10 at all edges, there exists a unique complete hyperbolic metric with totally geodesic boundary, so that T is isotopic to a geometric decomposition of M . Our approach is to use a variant of the combinatorial Ricci flow introduced by Luo [Luo05] for pseudo 3-manifolds. In this case, we prove that the extended Ricci flow converges to the hyperbolic metric exponentially fast. CONTENTS1. Introduction 1 2. Preliminaries 7 2.1. Generalized hyper-ideal tetrahedra 8 2.2. Co-volume of a generalized hyper-ideal tetrahedron 11 3. Geometric properties for a generalized hyper-ideal tetrahedron 12 4. Extended Ricci flows 18 5. Proof of Main Theorems 24 References 30

Rigidity and deformation of discrete conformal structures on polyhedral surfaces

“…For the combinatorial Yamabe flow for vertex scaling on polyhedral surfaces, please refer to [22,43,44,57,88,93] and others. There are also some research activities for combinatorial Ricci flow and combinatorial Yamabe flow on 3-dimensional manifolds, please refer to [11,14,15,20,26,27,35,36,58,89,90,94] and others.…”