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<p>In this paper, we investigate the deformation of hyperbolic inversive distance circle packings with the inversive distance greater than 1 on surfaces. We develop three distinct combinatorial curvature flows—the combinatorial Ricci flow, the combinatorial Calabi flow and the fractional combinatorial Calabi flow for hyperbolic inversive distance circle packings—as tools to find the hyperbolic metrics on surfaces with prescribed combinatorial curvatures. To handle the potential singularities along these combinatorial curvature flows, we do surgery along these flows by edge flipping under the weighted Delaunay condition. Then we prove the longtime existence and global convergence for the solutions of these combinatorial curvature flows with surgery. These combinatorial curvature flows with surgery provide effective algorithms for finding hyperbolic metrics with prescribed combinatorial curvatures on surfaces.</p>

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Combinatorial curvature flows with surgery for hyperbolic inversive distance circle packings on surfaces

  • 期刊论文

  • 发表日期：

    2025年8月20日

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  • Xu Xu,
  • Chao Zheng

摘要

In this paper, we investigate the deformation of hyperbolic inversive distance circle packings with the inversive distance greater than 1 on surfaces. We develop three distinct combinatorial curvature flows—the combinatorial Ricci flow, the combinatorial Calabi flow and the fractional combinatorial Calabi flow for hyperbolic inversive distance circle packings—as tools to find the hyperbolic metrics on surfaces with prescribed combinatorial curvatures. To handle the potential singularities along these combinatorial curvature flows, we do surgery along these flows by edge flipping under the weighted Delaunay condition. Then we prove the longtime existence and global convergence for the solutions of these combinatorial curvature flows with surgery. These combinatorial curvature flows with surgery provide effective algorithms for finding hyperbolic metrics with prescribed combinatorial curvatures on surfaces.