Volume 26, issue 3 (2022)

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

Ke Feng, Huabin Ge and Bobo Hua

Geometry & Topology 26 (2022) 1349–1384
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 (Electron. Res. Announc. Amer. Math. Soc. 11 (2005) 12–20) for pseudo-3–manifolds. In this case, we prove that the extended Ricci flow converges to the hyperbolic metric exponentially fast.

Keywords
Ricci flow, hyperbolization, 3–manifold, ideal triangulation, hyperbolic metric
Mathematical Subject Classification
Primary: 05E45, 53E20, 57K32, 57M50, 57Q15
References
Publication
Received: 6 October 2020
Revised: 7 March 2021
Accepted: 23 April 2021
Published: 3 August 2022
Proposed: David Gabai
Seconded: John Lott, Tobias H Colding
Authors
Ke Feng
School of Mathematical Sciences
University of Electronic Science and Technology of China
Chengdu
Sichuan
China
Huabin Ge
School of Mathematics
Renmin University of China
Beijing
China
Bobo Hua
School of Mathematical Sciences, LMNS
Fudan University
Shanghai
China
Shanghai Center for Mathematical Sciences
Fudan University
Shanghai
China