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

Feng, Ke¹ ; Ge, Huabin² ; Hua, Bobo³

¹ School of Mathematical Sciences, University of Electronic Science and Technology of China, Chengdu, Sichuan, China
² School of Mathematics, Renmin University of China, Beijing, China
³ School of Mathematical Sciences, LMNS, Fudan University, Shanghai, China, Shanghai Center for Mathematical Sciences, Fudan University, Shanghai, China

Geometry & topology, Tome 26 (2022) no. 3, pp. 1349-1384.

Résumé

We prove that for a compact $3$–manifold $M$ with boundary admitting an ideal triangulation $\mathcal{𝒯}$ with valence at least 10 at all edges, there exists a unique complete hyperbolic metric with totally geodesic boundary, so that $\mathcal{𝒯}$ 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.

Détail

DOI : 10.2140/gt.2022.26.1349

Keywords: Ricci flow, hyperbolization, 3–manifold, ideal triangulation, hyperbolic metric

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