Abstract
For two dimensional surfaces (smooth) Ricci and Yamabe flows are equivalent. In 2003, Chow and Luo developed the theory of combinatorial Ricci flow for circle packing metrics on closed triangulated surfaces. In 2004, Luo developed a theory of discrete Yamabe flow for closed triangulated surfaces. He investigated the formation of singularities and convergence to a metric of constant curvature.
In this note we develop the theory of a naïve discrete Ricci flow and its modification — the so-called weighted Ricci flow. We prove that this flow has a rich family of first integrals and is equivalent to a certain modification of Luo’s discrete Yamabe flow. We investigate the types of singularities of solutions for these flows and discuss convergence to a metric of weighted constant curvature.
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ACKNOWLEDGMENTS
The author thanks the anonymous referee for a thorough reading of the paper and valuable comments and suggestions.
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This work was funded by Russian Science Foundation (grant 22-11-00272).
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MSC2010
52C26
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Popelensky, T.Y. A Note on the Weighted Yamabe Flow. Regul. Chaot. Dyn. 28, 309–320 (2023). https://doi.org/10.1134/S1560354723030048
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DOI: https://doi.org/10.1134/S1560354723030048