Abstract
For triangulated surfaces, we introduce the combinatorial p-th (\(p>1\)) Calabi flow for Euclidean and hyperbolic polyhedral metrics on surfaces which precisely equals the combinatorial Calabi flows for discrete conformal factors first introduced in Ge’s Ph.D. thesis (Combinatorial methods and geometric equations, Peking University, Beijing, 2012) and then followed by Zhu and Xu (Combinatorial Calabi flow with surgery on surfaces, arXiv:1806.02166v1, 2018) when \(p=2\). Adopting different approaches, we show that the solution to the combinatorial p-th Calabi flow for Euclidean (hyperbolic resp.) polyhedral metric exists for all time and converges to a piecewise linear (hyperbolic resp.) polyhedral metric with constant (zero resp.) combinatorial curvature. Our results generalize the work of Ge (2012) and Zhu and Xu (2018) on the combinatorial Calabi flow with surgery from \(p=2\) to any \(p>1\).
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Acknowledgements
The first and second authors would like to thank Professor Gang Tian for constant guidance and encouragement. The second author would like to thank Professor Huijun Fan for constant encouragement. The third author would like to thank Professor Yanxun Chang for constant guidance and encouragement. They authors would also like to thank Professor Huabin Ge for many helpful conversations. The second author is supported by NNSF of China under Grant No. 11401578. The third author is supported by NNSF of China under Grant No. 11871094.
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Feng, K., Lin, A. & Zhang, X. Combinatorial p-th Calabi Flows for Discrete Conformal Factors on Surfaces. J Geom Anal 30, 3979–3994 (2020). https://doi.org/10.1007/s12220-019-00224-0
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DOI: https://doi.org/10.1007/s12220-019-00224-0