里奇和积分标量曲率约束下的环的格罗莫夫-豪斯多夫稳定性

IF 16.4 1区化学 Q1 CHEMISTRY, MULTIDISCIPLINARY

Accounts of Chemical Research Pub Date : 2024-08-12 DOI:10.1016/j.na.2024.113629

Shouhei Honda , Christian Ketterer , Ilaria Mondello , Raquel Perales , Chiara Rigoni

{"title":"里奇和积分标量曲率约束下的环的格罗莫夫-豪斯多夫稳定性","authors":"Shouhei Honda , Christian Ketterer , Ilaria Mondello , Raquel Perales , Chiara Rigoni","doi":"10.1016/j.na.2024.113629","DOIUrl":null,"url":null,"abstract":"<div><p>We establish a nonlinear analogue of a splitting map into a Euclidean space, as a harmonic map into a flat torus. We prove that the existence of such a map implies Gromov–Hausdorff closeness to a flat torus in any dimension. Furthermore, Gromov–Hausdorff closeness to a flat torus and an integral bound on <span><math><mrow><msub><mrow><mi>r</mi></mrow><mrow><mi>M</mi></mrow></msub><mrow><mo>(</mo><mi>x</mi><mo>)</mo></mrow></mrow></math></span>, the smallest eigenvalue of the Ricci tensor <span><math><msub><mrow><mo>ric</mo></mrow><mrow><mi>x</mi></mrow></msub></math></span> in <span><math><mi>x</mi></math></span>, imply the existence of a harmonic splitting map. Combining these results with Stern’s inequality, we provide a new Gromov–Hausdorff stability theorem for flat 3-tori. The main tools we employ include the harmonic map heat flow, Ricci flow, and both Ricci limits and RCD theories.</p></div>","PeriodicalId":1,"journal":{"name":"Accounts of Chemical Research","volume":null,"pages":null},"PeriodicalIF":16.4000,"publicationDate":"2024-08-12","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://www.sciencedirect.com/science/article/pii/S0362546X24001482/pdfft?md5=ba09939bdd60c2c66bc8258ccc472db4&pid=1-s2.0-S0362546X24001482-main.pdf","citationCount":"0","resultStr":"{\"title\":\"Gromov–Hausdorff stability of tori under Ricci and integral scalar curvature bounds\",\"authors\":\"Shouhei Honda , Christian Ketterer , Ilaria Mondello , Raquel Perales , Chiara Rigoni\",\"doi\":\"10.1016/j.na.2024.113629\",\"DOIUrl\":null,\"url\":null,\"abstract\":\"<div><p>We establish a nonlinear analogue of a splitting map into a Euclidean space, as a harmonic map into a flat torus. We prove that the existence of such a map implies Gromov–Hausdorff closeness to a flat torus in any dimension. Furthermore, Gromov–Hausdorff closeness to a flat torus and an integral bound on <span><math><mrow><msub><mrow><mi>r</mi></mrow><mrow><mi>M</mi></mrow></msub><mrow><mo>(</mo><mi>x</mi><mo>)</mo></mrow></mrow></math></span>, the smallest eigenvalue of the Ricci tensor <span><math><msub><mrow><mo>ric</mo></mrow><mrow><mi>x</mi></mrow></msub></math></span> in <span><math><mi>x</mi></math></span>, imply the existence of a harmonic splitting map. Combining these results with Stern’s inequality, we provide a new Gromov–Hausdorff stability theorem for flat 3-tori. The main tools we employ include the harmonic map heat flow, Ricci flow, and both Ricci limits and RCD theories.</p></div>\",\"PeriodicalId\":1,\"journal\":{\"name\":\"Accounts of Chemical Research\",\"volume\":null,\"pages\":null},\"PeriodicalIF\":16.4000,\"publicationDate\":\"2024-08-12\",\"publicationTypes\":\"Journal Article\",\"fieldsOfStudy\":null,\"isOpenAccess\":false,\"openAccessPdf\":\"https://www.sciencedirect.com/science/article/pii/S0362546X24001482/pdfft?md5=ba09939bdd60c2c66bc8258ccc472db4&pid=1-s2.0-S0362546X24001482-main.pdf\",\"citationCount\":\"0\",\"resultStr\":null,\"platform\":\"Semanticscholar\",\"paperid\":null,\"PeriodicalName\":\"Accounts of Chemical Research\",\"FirstCategoryId\":\"100\",\"ListUrlMain\":\"https://www.sciencedirect.com/science/article/pii/S0362546X24001482\",\"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/S0362546X24001482","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

我们建立了欧几里得空间分裂映射的非线性类比，即平面环的谐波映射。我们证明了这种映射的存在意味着在任何维度上都与平环面的格罗莫夫-豪斯多夫接近。此外，Gromov-Hausdorff 与平坦环面的接近性和 rM(x) 的积分约束（即 x 中里奇张量 ricx 的最小特征值）意味着谐波分裂映射的存在。将这些结果与斯特恩不等式相结合，我们为平面 3 蝶形提供了一个新的格罗莫夫-豪斯多夫稳定性定理。我们使用的主要工具包括谐波图热流、利玛窦流以及利玛窦极限和 RCD 理论。

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

Gromov–Hausdorff stability of tori under Ricci and integral scalar curvature bounds

We establish a nonlinear analogue of a splitting map into a Euclidean space, as a harmonic map into a flat torus. We prove that the existence of such a map implies Gromov–Hausdorff closeness to a flat torus in any dimension. Furthermore, Gromov–Hausdorff closeness to a flat torus and an integral bound on $r_{M} (x)$ , the smallest eigenvalue of the Ricci tensor ${ric}_{x}$ in $x$ , imply the existence of a harmonic splitting map. Combining these results with Stern’s inequality, we provide a new Gromov–Hausdorff stability theorem for flat 3-tori. The main tools we employ include the harmonic map heat flow, Ricci flow, and both Ricci limits and RCD theories.

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.