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Combinatorial Calabi flow with surgery on surfaces

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Abstract

Computing uniformization maps for surfaces has been a challenging problem and has many practical applications. In this paper, we provide a theoretically rigorous algorithm to compute such maps via combinatorial Calabi flow for vertex scaling of polyhedral metrics on surfaces, which is an analogue of the combinatorial Yamabe flow introduced by Luo (Commun Contemp Math 6(5):765–780, 2004). To handle the singularies along the combinatorial Calabi flow, we do surgery on the flow by flipping. Using the discrete conformal theory established in Gu et al. (J Differ Geom 109(3):431–466, 2018; J Differ Geom 109(2):223–256, 2018), we prove that for any initial Euclidean or hyperbolic polyhedral metric on a closed surface, the combinatorial Calabi flow with surgery exists for all time and converges exponentially fast after finite number of surgeries. The convergence is independent of the combinatorial structure of the initial triangulation on the surface.

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Acknowledgements

The first author thanks Professor Jian Sun for introduction and guidance on discrete geometry. Part of this work was done when the first author was visiting the School of Mathematics and Statistics, Wuhan University. He would like to thank Wuhan University for its hospitality. The second author also thanks Professor Kai Zheng for communications on Calabi flow. The authors thank Dr. Tianqi Wu for helpful communications. The research of the second author is supported by Hubei Provincial Natural Science Foundation of China under Grant No. 2017CFB681, National Natural Science Foundation of China under Grant Nos. 61772379 and 11301402 and Fundamental Research Funds for the Central Universities.

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Authors and Affiliations

  1. Yau Mathematical Sciences Center, Tsinghua University, Beijing, 100084, People’s Republic of China

    Xiang Zhu

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

    Xu Xu

  3. Hubei Key Laboratory of Computational Science, Wuhan University, Wuhan, 430072, China

    Xu Xu

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  1. Xiang Zhu
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  2. Xu Xu
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Correspondence to Xu Xu.

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Communicated by J. Jost.

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Zhu, X., Xu, X. Combinatorial Calabi flow with surgery on surfaces. Calc. Var. 58, 195 (2019). https://doi.org/10.1007/s00526-019-1654-5

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