{"title":"几乎Ricci平坦5流形的Kummer型构造","authors":"Chanyoung Sung","doi":"10.1007/s10455-023-09900-5","DOIUrl":null,"url":null,"abstract":"<div><p>A smooth closed manifold <i>M</i> is called almost Ricci-flat if </p><div><div><span>$$\\begin{aligned} \\inf _g||\\text {Ric}_g||_\\infty \\cdot \\text {diam}_g(M)^2=0 \\end{aligned}$$</span></div></div><p>where <span>\\(\\text {Ric}_g\\)</span> and <span>\\(\\text {diam}_g\\)</span>, respectively, denote the Ricci tensor and the diameter of <i>g</i> and <i>g</i> runs over all Riemannian metrics on <i>M</i>. By using Kummer-type method, we construct a smooth closed almost Ricci-flat nonspin 5-manifold <i>M</i> which is simply connected. It is minimal volume vanishes; namely, it collapses with sectional curvature bounded.</p></div>","PeriodicalId":0,"journal":{"name":"","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2023-04-13","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://link.springer.com/content/pdf/10.1007/s10455-023-09900-5.pdf","citationCount":"0","resultStr":"{\"title\":\"Kummer-type constructions of almost Ricci-flat 5-manifolds\",\"authors\":\"Chanyoung Sung\",\"doi\":\"10.1007/s10455-023-09900-5\",\"DOIUrl\":null,\"url\":null,\"abstract\":\"<div><p>A smooth closed manifold <i>M</i> is called almost Ricci-flat if </p><div><div><span>$$\\\\begin{aligned} \\\\inf _g||\\\\text {Ric}_g||_\\\\infty \\\\cdot \\\\text {diam}_g(M)^2=0 \\\\end{aligned}$$</span></div></div><p>where <span>\\\\(\\\\text {Ric}_g\\\\)</span> and <span>\\\\(\\\\text {diam}_g\\\\)</span>, respectively, denote the Ricci tensor and the diameter of <i>g</i> and <i>g</i> runs over all Riemannian metrics on <i>M</i>. By using Kummer-type method, we construct a smooth closed almost Ricci-flat nonspin 5-manifold <i>M</i> which is simply connected. It is minimal volume vanishes; namely, it collapses with sectional curvature bounded.</p></div>\",\"PeriodicalId\":0,\"journal\":{\"name\":\"\",\"volume\":null,\"pages\":null},\"PeriodicalIF\":0.0,\"publicationDate\":\"2023-04-13\",\"publicationTypes\":\"Journal Article\",\"fieldsOfStudy\":null,\"isOpenAccess\":false,\"openAccessPdf\":\"https://link.springer.com/content/pdf/10.1007/s10455-023-09900-5.pdf\",\"citationCount\":\"0\",\"resultStr\":null,\"platform\":\"Semanticscholar\",\"paperid\":null,\"PeriodicalName\":\"\",\"FirstCategoryId\":\"100\",\"ListUrlMain\":\"https://link.springer.com/article/10.1007/s10455-023-09900-5\",\"RegionNum\":0,\"RegionCategory\":null,\"ArticlePicture\":[],\"TitleCN\":null,\"AbstractTextCN\":null,\"PMCID\":null,\"EPubDate\":\"\",\"PubModel\":\"\",\"JCR\":\"\",\"JCRName\":\"\",\"Score\":null,\"Total\":0}","platform":"Semanticscholar","paperid":null,"PeriodicalName":"","FirstCategoryId":"100","ListUrlMain":"https://link.springer.com/article/10.1007/s10455-023-09900-5","RegionNum":0,"RegionCategory":null,"ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"","JCRName":"","Score":null,"Total":0}
where \(\text {Ric}_g\) and \(\text {diam}_g\), respectively, denote the Ricci tensor and the diameter of g and g runs over all Riemannian metrics on M. By using Kummer-type method, we construct a smooth closed almost Ricci-flat nonspin 5-manifold M which is simply connected. It is minimal volume vanishes; namely, it collapses with sectional curvature bounded.
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