{"title":"关于具有两个临界值的平稳函数","authors":"Antonio Lerario, Chiara Meroni, Daniele Zuddas","doi":"10.1007/s13163-023-00484-z","DOIUrl":null,"url":null,"abstract":"<p>We prove that every smooth closed connected manifold admits a smooth real-valued function with only two critical values such that the set of minima (or maxima) can be arbitrarily prescribed, as soon as this set is a finite subcomplex of the manifold (we call a function of this type a <i>Reeb function</i>). In analogy with Reeb’s Sphere Theorem, we use such functions to study the topology of the underlying manifold. In dimension 3, we give a characterization of manifolds having a Heegaard splitting of genus <i>g</i> in terms of the existence of certain Reeb functions. Similar results are proved in dimension <span>\\(n\\ge 5\\)</span>.</p>","PeriodicalId":501429,"journal":{"name":"Revista Matemática Complutense","volume":null,"pages":null},"PeriodicalIF":0.0000,"publicationDate":"2024-01-24","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"0","resultStr":"{\"title\":\"On smooth functions with two critical values\",\"authors\":\"Antonio Lerario, Chiara Meroni, Daniele Zuddas\",\"doi\":\"10.1007/s13163-023-00484-z\",\"DOIUrl\":null,\"url\":null,\"abstract\":\"<p>We prove that every smooth closed connected manifold admits a smooth real-valued function with only two critical values such that the set of minima (or maxima) can be arbitrarily prescribed, as soon as this set is a finite subcomplex of the manifold (we call a function of this type a <i>Reeb function</i>). In analogy with Reeb’s Sphere Theorem, we use such functions to study the topology of the underlying manifold. In dimension 3, we give a characterization of manifolds having a Heegaard splitting of genus <i>g</i> in terms of the existence of certain Reeb functions. Similar results are proved in dimension <span>\\\\(n\\\\ge 5\\\\)</span>.</p>\",\"PeriodicalId\":501429,\"journal\":{\"name\":\"Revista Matemática Complutense\",\"volume\":null,\"pages\":null},\"PeriodicalIF\":0.0000,\"publicationDate\":\"2024-01-24\",\"publicationTypes\":\"Journal Article\",\"fieldsOfStudy\":null,\"isOpenAccess\":false,\"openAccessPdf\":\"\",\"citationCount\":\"0\",\"resultStr\":null,\"platform\":\"Semanticscholar\",\"paperid\":null,\"PeriodicalName\":\"Revista Matemática Complutense\",\"FirstCategoryId\":\"1085\",\"ListUrlMain\":\"https://doi.org/10.1007/s13163-023-00484-z\",\"RegionNum\":0,\"RegionCategory\":null,\"ArticlePicture\":[],\"TitleCN\":null,\"AbstractTextCN\":null,\"PMCID\":null,\"EPubDate\":\"\",\"PubModel\":\"\",\"JCR\":\"\",\"JCRName\":\"\",\"Score\":null,\"Total\":0}","platform":"Semanticscholar","paperid":null,"PeriodicalName":"Revista Matemática Complutense","FirstCategoryId":"1085","ListUrlMain":"https://doi.org/10.1007/s13163-023-00484-z","RegionNum":0,"RegionCategory":null,"ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"","JCRName":"","Score":null,"Total":0}
引用次数: 0
摘要
我们证明,每个光滑闭合连通流形都有一个光滑实值函数,它只有两个临界值,只要这个临界值集是流形的一个有限子复数,那么它的最小值(或最大值)集就可以任意规定(我们称这类函数为里布函数)。与里布球定理类似,我们利用这类函数来研究底层流形的拓扑结构。在维度 3 中,我们根据某些里布函数的存在性,给出了具有属 g 的希嘉分裂的流形的特征。在维数 \(n\ge 5\) 中也证明了类似的结果。
We prove that every smooth closed connected manifold admits a smooth real-valued function with only two critical values such that the set of minima (or maxima) can be arbitrarily prescribed, as soon as this set is a finite subcomplex of the manifold (we call a function of this type a Reeb function). In analogy with Reeb’s Sphere Theorem, we use such functions to study the topology of the underlying manifold. In dimension 3, we give a characterization of manifolds having a Heegaard splitting of genus g in terms of the existence of certain Reeb functions. Similar results are proved in dimension \(n\ge 5\).