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Rigidity and deformation of generalized sphere packings on 3-dimensional manifolds with boundary

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Abstract

Motivated by Guo–Luo’s generalized circle packings on surfaces with boundary (Guo–Luo in Geom Topol 13(3):1265–1312, 2009), we introduce the generalized sphere packings on 3-dimensional manifolds with boundary. Then we investigate the rigidity of the generalized sphere packing metrics. We prove that the generalized sphere packing metric is determined by the combinatorial scalar curvature. To find the hyper-ideal polyhedral metrics on 3-dimensional manifolds with prescribed combinatorial scalar curvature, we introduce the combinatorial Ricci flow and combinatorial Calabi flow for the generalized sphere packings on 3-dimensional manifolds with boundary. Then we study the longtime existence and convergence for the solutions of these combinatorial curvature flows.

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Acknowledgements

The first author thanks Professor Tian Yang at Texas A &M University for helpful communications.

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  1. School of Mathematics and Statistics, Wuhan University, Wuhan, 430072, People’s Republic of China

    Xu Xu & Chao Zheng

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  1. Xu Xu
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  2. Chao Zheng
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Correspondence to Chao Zheng.

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Communicated by Andrea Mondino.

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Xu, X., Zheng, C. Rigidity and deformation of generalized sphere packings on 3-dimensional manifolds with boundary. Calc. Var. 62, 232 (2023). https://doi.org/10.1007/s00526-023-02572-w

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