{"title":"Ancient solutions to the Ricci flow in dimension $3$","authors":"S. Brendle","doi":"10.4310/acta.2020.v225.n1.a1","DOIUrl":null,"url":null,"abstract":"It is known from work of Perelman that any finite-time singularity of the Ricci flow on a compact three-manifold is modeled on an ancient $\\kappa$-solution. \nWe prove that the every noncompact ancient $\\kappa$-solution in dimension $3$ is isometric to either the shrinking cylinders (or a quotient thereof), or the Bryant soliton.","PeriodicalId":50895,"journal":{"name":"Acta Mathematica","volume":" ","pages":""},"PeriodicalIF":4.9000,"publicationDate":"2018-11-06","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"60","resultStr":null,"platform":"Semanticscholar","paperid":null,"PeriodicalName":"Acta Mathematica","FirstCategoryId":"100","ListUrlMain":"https://doi.org/10.4310/acta.2020.v225.n1.a1","RegionNum":1,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"Q1","JCRName":"MATHEMATICS","Score":null,"Total":0}
引用次数: 60
Abstract
It is known from work of Perelman that any finite-time singularity of the Ricci flow on a compact three-manifold is modeled on an ancient $\kappa$-solution.
We prove that the every noncompact ancient $\kappa$-solution in dimension $3$ is isometric to either the shrinking cylinders (or a quotient thereof), or the Bryant soliton.
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