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The character of Thurston’s circle packings

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Abstract

We introduce the character of Thurston’s circle packings in the hyperbolic background geometry Consequently, some quite simple criteria are obtained for the existence of hyperbolic circle packings. For example, if a closed surface X admits a circle packing with all the vertex degrees di ⩾ 7, then it admits a unique complete hyperbolic metric so that the triangulation graph of the circle packing is isotopic to a geometric decomposition of X. This criterion is sharp due to the fact that any closed hyperbolic surface admits no triangulations with all di ⩽ 6. As a corollary, we obtain a new proof of the uniformization theorem for closed surfaces with genus g ⩾ 2, and moreover, any hyperbolic closed surface has a geometric decomposition. To obtain our results, we use Chow-Luo’s combinatorial Ricci flow as a fundamental tool.

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Acknowledgements

Huabin Ge was supported by National Natural Science Foundation of China (Grant Nos. 11871094 and 12122119). Aijin Lin was supported by National Natural Science Foundation of China (Grant No. 12171480), Hunan Provincial Natural Science Foundation of China (Grant Nos. 2020JJ4658 and 2022JJ10059), and Scientific Research Program Funds of National University of Defense Technology (Grant No. 22-ZZCX-016). The second author thanks Ke Feng and Ze Zhou for communication on related topics. The authors thank Liangming Shen for many useful conversations. The authors are very grateful to the referees for carefully reading the original manuscript and pointing out some typos.

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Authors and Affiliations

  1. School of Mathematics, Renmin University of China, Beijing, 100872, China

    Huabin Ge

  2. Department of Mathematics, National University of Defense Technology, Changsha, 410073, China

    Aijin Lin

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  1. Huabin Ge
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  2. Aijin Lin
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Correspondence to Aijin Lin.

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Ge, H., Lin, A. The character of Thurston’s circle packings. Sci. China Math. (2024). https://doi.org/10.1007/s11425-023-2182-2

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