{"title":"Stability of Llarull's theorem in all dimensions","authors":"Sven Hirsch , Yiyue Zhang","doi":"10.1016/j.aim.2024.109980","DOIUrl":null,"url":null,"abstract":"<div><div>Llarull's theorem characterizes the round sphere <span><math><msup><mrow><mi>S</mi></mrow><mrow><mi>n</mi></mrow></msup></math></span> among all spin manifolds whose scalar curvature is bounded from below by <span><math><mi>n</mi><mo>(</mo><mi>n</mi><mo>−</mo><mn>1</mn><mo>)</mo></math></span>. In this paper we show that if the scalar curvature is bounded from below by <span><math><mi>n</mi><mo>(</mo><mi>n</mi><mo>−</mo><mn>1</mn><mo>)</mo><mo>−</mo><mi>ε</mi></math></span>, the underlying manifold is <span><math><msup><mrow><mi>C</mi></mrow><mrow><mn>0</mn></mrow></msup></math></span>-close to a finite number of spheres outside a small bad set. This completely solves Gromov's spherical stability problem and is the first instance of a scalar curvature stability result that both holds in all dimensions and is stated without any additional geometrical or topological assumptions.</div></div>","PeriodicalId":1,"journal":{"name":"Accounts of Chemical Research","volume":null,"pages":null},"PeriodicalIF":16.4000,"publicationDate":"2024-10-18","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"0","resultStr":null,"platform":"Semanticscholar","paperid":null,"PeriodicalName":"Accounts of Chemical Research","FirstCategoryId":"100","ListUrlMain":"https://www.sciencedirect.com/science/article/pii/S0001870824004961","RegionNum":1,"RegionCategory":"化学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"Q1","JCRName":"CHEMISTRY, MULTIDISCIPLINARY","Score":null,"Total":0}
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Abstract
Llarull's theorem characterizes the round sphere among all spin manifolds whose scalar curvature is bounded from below by . In this paper we show that if the scalar curvature is bounded from below by , the underlying manifold is -close to a finite number of spheres outside a small bad set. This completely solves Gromov's spherical stability problem and is the first instance of a scalar curvature stability result that both holds in all dimensions and is stated without any additional geometrical or topological assumptions.
期刊介绍：
Accounts of Chemical Research presents short, concise and critical articles offering easy-to-read overviews of basic research and applications in all areas of chemistry and biochemistry. These short reviews focus on research from the author’s own laboratory and are designed to teach the reader about a research project. In addition, Accounts of Chemical Research publishes commentaries that give an informed opinion on a current research problem. Special Issues online are devoted to a single topic of unusual activity and significance.
Accounts of Chemical Research replaces the traditional article abstract with an article "Conspectus." These entries synopsize the research affording the reader a closer look at the content and significance of an article. Through this provision of a more detailed description of the article contents, the Conspectus enhances the article's discoverability by search engines and the exposure for the research.
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