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<p>In (II. Geom. Topol. 13, 1265 -1312, 2009), Guo and Luo introduced generalized circle patterns on surfaces and proved their rigidity. In this paper, we prove the existence of Guo-Luo’s generalized circle patterns with prescribed generalized intersection angles on surfaces with cusps, which partially answers a question raised by Guo-Luo in (II. Geom. Topol. 13, 1265 -1312, 2009) and generalizes Bobenko-Springborn’s hyperbolic circle patterns on closed surfaces to generalized hyperbolic circle patterns on surfaces with cusps. We further introduce the combinatorial Ricci flow and combinatorial Calabi flow for generalized circle patterns on surfaces with cusps, and prove the longtime existence and convergence of the solutions for these combinatorial curvature flows.</p>

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Generalized circle patterns on surfaces with cusps

  • 期刊论文

  • 发表日期：

    2025年9月8日

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  • Zhiwen Xiong,
  • Xu Xu

摘要

In (II. Geom. Topol. 13, 1265 -1312, 2009), Guo and Luo introduced generalized circle patterns on surfaces and proved their rigidity. In this paper, we prove the existence of Guo-Luo’s generalized circle patterns with prescribed generalized intersection angles on surfaces with cusps, which partially answers a question raised by Guo-Luo in (II. Geom. Topol. 13, 1265 -1312, 2009) and generalizes Bobenko-Springborn’s hyperbolic circle patterns on closed surfaces to generalized hyperbolic circle patterns on surfaces with cusps. We further introduce the combinatorial Ricci flow and combinatorial Calabi flow for generalized circle patterns on surfaces with cusps, and prove the longtime existence and convergence of the solutions for these combinatorial curvature flows.