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<p>In this paper, we introduce discrete conformal structures on surfaces with boundary through an axiomatic approach, ensuring that the Poincaré dual of an ideally triangulated surface with boundary possesses a well-defined geometric structure. We then classify discrete conformal structures on surfaces with boundary, thereby unifying and generalizing Guo–Luo’s generalized circle packings (Geom. Topol. <b>13</b>(3), 1265–1312 (2009)), Guo’s vertex scalings (Commun. Contemp. Math. <b>13</b>(5), 827–842 (2011)), and Xu’s partial discrete conformal structures (Calc. Var. Partial Differ. Equ. <b>61</b>(4), Paper No. 141, 23 pp. (2022)) on surfaces with boundary. These results extend the work of Glickenstein–Thomas (Adv. Math. <b>320</b>, 250–278 (2017)) from closed surfaces to surfaces with boundary. Based on the works of Bobenko–Pinkall–Springborn (Geom. Topol. <b>19</b>(4), 2155–2215 (2015)) and Zhang–Guo–Zeng–Luo–Yau–Gu (Graph. Models <b>76</b>(5), 321–339 (2014)), we establish the relationships between discrete conformal structures on surfaces with boundary and hyperbolic trigonometry. Surprisingly, we discover that certain subclasses of discrete conformal structures on surfaces with boundary are closely related to the twisted generalized hyperbolic triangles introduced by Roger–Yang (J. Differ. Geom. <b>96</b>(1), 95–140 (2014)), a phenomenon absent in the case of closed surfaces (see [Geom. Topol. <b>19</b>(4), 2155–2215 (2015)] and [Graph. Models <b>76</b>(5), 321–339 (2014)]). Finally, we investigate the relationships between discrete conformal structures on surfaces with boundary and 3-dimensional hyperbolic geometry by constructing ten types of generalized hyperbolic tetrahedra. Some recently introduced generalized hyperbolic tetrahedra by Belletti–Yang (arXiv:2308.13864 (2023)) naturally emerge within these constructions.</p>

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Discrete Conformal Structures on Surfaces with Boundary (I)—Classification

  • 期刊论文

  • 发表日期：

    2025年5月6日

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  • Xu Xu,
  • Chao Zheng

摘要

In this paper, we introduce discrete conformal structures on surfaces with boundary through an axiomatic approach, ensuring that the Poincaré dual of an ideally triangulated surface with boundary possesses a well-defined geometric structure. We then classify discrete conformal structures on surfaces with boundary, thereby unifying and generalizing Guo–Luo’s generalized circle packings (Geom. Topol. 13(3), 1265–1312 (2009)), Guo’s vertex scalings (Commun. Contemp. Math. 13(5), 827–842 (2011)), and Xu’s partial discrete conformal structures (Calc. Var. Partial Differ. Equ. 61(4), Paper No. 141, 23 pp. (2022)) on surfaces with boundary. These results extend the work of Glickenstein–Thomas (Adv. Math. 320, 250–278 (2017)) from closed surfaces to surfaces with boundary. Based on the works of Bobenko–Pinkall–Springborn (Geom. Topol. 19(4), 2155–2215 (2015)) and Zhang–Guo–Zeng–Luo–Yau–Gu (Graph. Models 76(5), 321–339 (2014)), we establish the relationships between discrete conformal structures on surfaces with boundary and hyperbolic trigonometry. Surprisingly, we discover that certain subclasses of discrete conformal structures on surfaces with boundary are closely related to the twisted generalized hyperbolic triangles introduced by Roger–Yang (J. Differ. Geom. 96(1), 95–140 (2014)), a phenomenon absent in the case of closed surfaces (see [Geom. Topol. 19(4), 2155–2215 (2015)] and [Graph. Models 76(5), 321–339 (2014)]). Finally, we investigate the relationships between discrete conformal structures on surfaces with boundary and 3-dimensional hyperbolic geometry by constructing ten types of generalized hyperbolic tetrahedra. Some recently introduced generalized hyperbolic tetrahedra by Belletti–Yang (arXiv:2308.13864 (2023)) naturally emerge within these constructions.