拉鲁尔定理在所有维度上的稳定性

IF 16.4 1区化学 Q1 CHEMISTRY, MULTIDISCIPLINARY

Accounts of Chemical Research Pub Date : 2024-10-18 DOI:10.1016/j.aim.2024.109980

Sven Hirsch , Yiyue Zhang

{"title":"拉鲁尔定理在所有维度上的稳定性","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":"{\"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}","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}

引用次数: 0

摘要

拉鲁尔定理描述了所有标量曲率自下而上受 n(n-1) 约束的自旋流形中圆球 Sn 的特征。在本文中，我们证明了如果标量曲率自下而上受 n(n-1)-ε 约束，则底层流形在一个小的坏集之外与有限数量的球面是 C0-接近的。这完全解决了格罗莫夫的球面稳定性问题，是标量曲率稳定性结果的第一个实例，它既在所有维度上都成立，又无需任何额外的几何或拓扑假设。

本文章由计算机程序翻译，如有差异，请以英文原文为准。

查看原文本刊更多论文

Stability of Llarull's theorem in all dimensions

Llarull's theorem characterizes the round sphere

S^{n}

among all spin manifolds whose scalar curvature is bounded from below by

n (n - 1)

. In this paper we show that if the scalar curvature is bounded from below by

n (n - 1) - ε

, the underlying manifold is

C^{0}

-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.

求助全文

通过发布文献求助，成功后即可免费获取论文全文。去求助

来源期刊

Accounts of Chemical Research 化学-化学综合

CiteScore

31.40

自引率

1.10%

发文量

312

审稿时长

2 months

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.