{"title":"开放的问题","authors":"Kefeng Liu","doi":"10.4310/iccm.2019.v7.n2.a10","DOIUrl":null,"url":null,"abstract":"Recently Neves and Marques [1] proved there are infinite number of minimal surfaces in a compact Riemannian manifold with positive Ricci curvature and dimension at most seven. It will be interesting to know the Euler number of such surfaces. Are they bounded by the index linearly? In [2, 3], Grigor’yan, Netrusov and I proved this if the three manifold has positive Ricci curvature. One can ask similar question for codimension one minimal hypersurface in higher dimensions. Can one bound the sum of Betti number in terms of the index in a linear manner?","PeriodicalId":415664,"journal":{"name":"Notices of the International Congress of Chinese Mathematicians","volume":"22 1","pages":"0"},"PeriodicalIF":0.0000,"publicationDate":"1900-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"0","resultStr":"{\"title\":\"Open Problems\",\"authors\":\"Kefeng Liu\",\"doi\":\"10.4310/iccm.2019.v7.n2.a10\",\"DOIUrl\":null,\"url\":null,\"abstract\":\"Recently Neves and Marques [1] proved there are infinite number of minimal surfaces in a compact Riemannian manifold with positive Ricci curvature and dimension at most seven. It will be interesting to know the Euler number of such surfaces. Are they bounded by the index linearly? In [2, 3], Grigor’yan, Netrusov and I proved this if the three manifold has positive Ricci curvature. One can ask similar question for codimension one minimal hypersurface in higher dimensions. Can one bound the sum of Betti number in terms of the index in a linear manner?\",\"PeriodicalId\":415664,\"journal\":{\"name\":\"Notices of the International Congress of Chinese Mathematicians\",\"volume\":\"22 1\",\"pages\":\"0\"},\"PeriodicalIF\":0.0000,\"publicationDate\":\"1900-01-01\",\"publicationTypes\":\"Journal Article\",\"fieldsOfStudy\":null,\"isOpenAccess\":false,\"openAccessPdf\":\"\",\"citationCount\":\"0\",\"resultStr\":null,\"platform\":\"Semanticscholar\",\"paperid\":null,\"PeriodicalName\":\"Notices of the International Congress of Chinese Mathematicians\",\"FirstCategoryId\":\"1085\",\"ListUrlMain\":\"https://doi.org/10.4310/iccm.2019.v7.n2.a10\",\"RegionNum\":0,\"RegionCategory\":null,\"ArticlePicture\":[],\"TitleCN\":null,\"AbstractTextCN\":null,\"PMCID\":null,\"EPubDate\":\"\",\"PubModel\":\"\",\"JCR\":\"\",\"JCRName\":\"\",\"Score\":null,\"Total\":0}","platform":"Semanticscholar","paperid":null,"PeriodicalName":"Notices of the International Congress of Chinese Mathematicians","FirstCategoryId":"1085","ListUrlMain":"https://doi.org/10.4310/iccm.2019.v7.n2.a10","RegionNum":0,"RegionCategory":null,"ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"","JCRName":"","Score":null,"Total":0}
Recently Neves and Marques [1] proved there are infinite number of minimal surfaces in a compact Riemannian manifold with positive Ricci curvature and dimension at most seven. It will be interesting to know the Euler number of such surfaces. Are they bounded by the index linearly? In [2, 3], Grigor’yan, Netrusov and I proved this if the three manifold has positive Ricci curvature. One can ask similar question for codimension one minimal hypersurface in higher dimensions. Can one bound the sum of Betti number in terms of the index in a linear manner?
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