{"title":"A Note on the Weighted Yamabe Flow","authors":"Theodore Yu. Popelensky","doi":"10.1134/S1560354723030048","DOIUrl":null,"url":null,"abstract":"<div><p>For two dimensional surfaces (smooth) Ricci and Yamabe flows are equivalent.\nIn 2003, Chow and Luo developed the theory of combinatorial Ricci flow for circle packing metrics on closed triangulated surfaces.\nIn 2004, Luo developed a theory of discrete Yamabe flow for closed triangulated surfaces.\nHe investigated the formation of singularities and convergence to a metric of constant curvature.</p><p>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.\nWe investigate the types of singularities of solutions for these flows and discuss convergence to a metric of weighted\nconstant curvature.</p></div>","PeriodicalId":752,"journal":{"name":"Regular and Chaotic Dynamics","volume":"28 3","pages":"309 - 320"},"PeriodicalIF":0.8000,"publicationDate":"2023-06-02","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"0","resultStr":null,"platform":"Semanticscholar","paperid":null,"PeriodicalName":"Regular and Chaotic Dynamics","FirstCategoryId":"4","ListUrlMain":"https://link.springer.com/article/10.1134/S1560354723030048","RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"Q3","JCRName":"MATHEMATICS, APPLIED","Score":null,"Total":0}
引用次数: 0
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.
期刊介绍：
Regular and Chaotic Dynamics (RCD) is an international journal publishing original research papers in dynamical systems theory and its applications. Rooted in the Moscow school of mathematics and mechanics, the journal successfully combines classical problems, modern mathematical techniques and breakthroughs in the field. Regular and Chaotic Dynamics welcomes papers that establish original results, characterized by rigorous mathematical settings and proofs, and that also address practical problems. In addition to research papers, the journal publishes review articles, historical and polemical essays, and translations of works by influential scientists of past centuries, previously unavailable in English. Along with regular issues, RCD also publishes special issues devoted to particular topics and events in the world of dynamical systems.
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