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<p>For triangulated surfaces and any <InlineEquation ID="IEq1"> <EquationSource Format="TEX">\(p&gt;1\)</EquationSource> </InlineEquation>, we introduce the branched combinatorial <i>p</i>-th Calabi flows in Euclidean (hyperbolic resp.) background geometry. Then using an important estimate established by the second author and his collaborators, we show that the solutions to the branched combinatorial <i>p</i>-th Calabi flow exists for all time and converges if and only if there exists a branched circle packing metric. Our results generalize the work of Ge (Trans Amer Math Soc 370(2):1377–1391, 2018), Ge and Hua (Adv Math 333:523–538, 2018), Ge and Xu (Diff Geom Appl 47:86–98, 2016), Gao and Lin (Adv Appl Math 11(10):7451–7463, 2022) and Lin and Zhang (Adv Math 346:1067–1090, 2019) on the combinatorial Calabi flows.</p>

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Branched combinatorial p-th Calabi flows on surfaces

  • 期刊论文

  • 发表日期：

    2025年11月8日

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  • Kaicheng Gao,
  • Aijin Lin,
  • Rongyuan Liu,
  • Longxiang Wu

摘要

For triangulated surfaces and any \(p>1\) , we introduce the branched combinatorial p-th Calabi flows in Euclidean (hyperbolic resp.) background geometry. Then using an important estimate established by the second author and his collaborators, we show that the solutions to the branched combinatorial p-th Calabi flow exists for all time and converges if and only if there exists a branched circle packing metric. Our results generalize the work of Ge (Trans Amer Math Soc 370(2):1377–1391, 2018), Ge and Hua (Adv Math 333:523–538, 2018), Ge and Xu (Diff Geom Appl 47:86–98, 2016), Gao and Lin (Adv Appl Math 11(10):7451–7463, 2022) and Lin and Zhang (Adv Math 346:1067–1090, 2019) on the combinatorial Calabi flows.