{"title":"Geroch猜想的一个推广","authors":"Simon Brendle, Sven Hirsch, Florian Johne","doi":"10.1002/cpa.22137","DOIUrl":null,"url":null,"abstract":"<p>The Theorem of Bonnet–Myers implies that manifolds with topology <math>\n <semantics>\n <mrow>\n <msup>\n <mi>M</mi>\n <mrow>\n <mi>n</mi>\n <mo>−</mo>\n <mn>1</mn>\n </mrow>\n </msup>\n <mo>×</mo>\n <msup>\n <mi>S</mi>\n <mn>1</mn>\n </msup>\n </mrow>\n <annotation>$M^{n-1} \\times \\mathbb {S}^1$</annotation>\n </semantics></math> do not admit a metric of positive Ricci curvature, while the resolution of Geroch's conjecture implies that the torus <math>\n <semantics>\n <msup>\n <mi>T</mi>\n <mi>n</mi>\n </msup>\n <annotation>$\\mathbb {T}^n$</annotation>\n </semantics></math> does not admit a metric of positive scalar curvature. In this work we introduce a new notion of curvature interpolating between Ricci and scalar curvature (so-called <i>m</i>-intermediate curvature), and use stable weighted slicings to show that for <math>\n <semantics>\n <mrow>\n <mi>n</mi>\n <mo>≤</mo>\n <mn>7</mn>\n </mrow>\n <annotation>$n \\le 7$</annotation>\n </semantics></math> and <math>\n <semantics>\n <mrow>\n <mn>1</mn>\n <mo>≤</mo>\n <mi>m</mi>\n <mo>≤</mo>\n <mi>n</mi>\n <mo>−</mo>\n <mn>1</mn>\n </mrow>\n <annotation>$1 \\le m \\le n-1$</annotation>\n </semantics></math> the manifolds <math>\n <semantics>\n <mrow>\n <msup>\n <mi>N</mi>\n <mi>n</mi>\n </msup>\n <mo>=</mo>\n <msup>\n <mi>M</mi>\n <mrow>\n <mi>n</mi>\n <mo>−</mo>\n <mi>m</mi>\n </mrow>\n </msup>\n <mo>×</mo>\n <msup>\n <mi>T</mi>\n <mi>m</mi>\n </msup>\n </mrow>\n <annotation>$N^n = M^{n-m} \\times \\mathbb {T}^m$</annotation>\n </semantics></math> do not admit a metric of positive <i>m</i>-intermediate curvature.</p>","PeriodicalId":10601,"journal":{"name":"Communications on Pure and Applied Mathematics","volume":null,"pages":null},"PeriodicalIF":3.1000,"publicationDate":"2023-08-12","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"7","resultStr":"{\"title\":\"A generalization of Geroch's conjecture\",\"authors\":\"Simon Brendle, Sven Hirsch, Florian Johne\",\"doi\":\"10.1002/cpa.22137\",\"DOIUrl\":null,\"url\":null,\"abstract\":\"<p>The Theorem of Bonnet–Myers implies that manifolds with topology <math>\\n <semantics>\\n <mrow>\\n <msup>\\n <mi>M</mi>\\n <mrow>\\n <mi>n</mi>\\n <mo>−</mo>\\n <mn>1</mn>\\n </mrow>\\n </msup>\\n <mo>×</mo>\\n <msup>\\n <mi>S</mi>\\n <mn>1</mn>\\n </msup>\\n </mrow>\\n <annotation>$M^{n-1} \\\\times \\\\mathbb {S}^1$</annotation>\\n </semantics></math> do not admit a metric of positive Ricci curvature, while the resolution of Geroch's conjecture implies that the torus <math>\\n <semantics>\\n <msup>\\n <mi>T</mi>\\n <mi>n</mi>\\n </msup>\\n <annotation>$\\\\mathbb {T}^n$</annotation>\\n </semantics></math> does not admit a metric of positive scalar curvature. In this work we introduce a new notion of curvature interpolating between Ricci and scalar curvature (so-called <i>m</i>-intermediate curvature), and use stable weighted slicings to show that for <math>\\n <semantics>\\n <mrow>\\n <mi>n</mi>\\n <mo>≤</mo>\\n <mn>7</mn>\\n </mrow>\\n <annotation>$n \\\\le 7$</annotation>\\n </semantics></math> and <math>\\n <semantics>\\n <mrow>\\n <mn>1</mn>\\n <mo>≤</mo>\\n <mi>m</mi>\\n <mo>≤</mo>\\n <mi>n</mi>\\n <mo>−</mo>\\n <mn>1</mn>\\n </mrow>\\n <annotation>$1 \\\\le m \\\\le n-1$</annotation>\\n </semantics></math> the manifolds <math>\\n <semantics>\\n <mrow>\\n <msup>\\n <mi>N</mi>\\n <mi>n</mi>\\n </msup>\\n <mo>=</mo>\\n <msup>\\n <mi>M</mi>\\n <mrow>\\n <mi>n</mi>\\n <mo>−</mo>\\n <mi>m</mi>\\n </mrow>\\n </msup>\\n <mo>×</mo>\\n <msup>\\n <mi>T</mi>\\n <mi>m</mi>\\n </msup>\\n </mrow>\\n <annotation>$N^n = M^{n-m} \\\\times \\\\mathbb {T}^m$</annotation>\\n </semantics></math> do not admit a metric of positive <i>m</i>-intermediate curvature.</p>\",\"PeriodicalId\":10601,\"journal\":{\"name\":\"Communications on Pure and Applied Mathematics\",\"volume\":null,\"pages\":null},\"PeriodicalIF\":3.1000,\"publicationDate\":\"2023-08-12\",\"publicationTypes\":\"Journal Article\",\"fieldsOfStudy\":null,\"isOpenAccess\":false,\"openAccessPdf\":\"\",\"citationCount\":\"7\",\"resultStr\":null,\"platform\":\"Semanticscholar\",\"paperid\":null,\"PeriodicalName\":\"Communications on Pure and Applied Mathematics\",\"FirstCategoryId\":\"100\",\"ListUrlMain\":\"https://onlinelibrary.wiley.com/doi/10.1002/cpa.22137\",\"RegionNum\":1,\"RegionCategory\":\"数学\",\"ArticlePicture\":[],\"TitleCN\":null,\"AbstractTextCN\":null,\"PMCID\":null,\"EPubDate\":\"\",\"PubModel\":\"\",\"JCR\":\"Q1\",\"JCRName\":\"MATHEMATICS\",\"Score\":null,\"Total\":0}","platform":"Semanticscholar","paperid":null,"PeriodicalName":"Communications on Pure and Applied Mathematics","FirstCategoryId":"100","ListUrlMain":"https://onlinelibrary.wiley.com/doi/10.1002/cpa.22137","RegionNum":1,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"Q1","JCRName":"MATHEMATICS","Score":null,"Total":0}
The Theorem of Bonnet–Myers implies that manifolds with topology do not admit a metric of positive Ricci curvature, while the resolution of Geroch's conjecture implies that the torus does not admit a metric of positive scalar curvature. In this work we introduce a new notion of curvature interpolating between Ricci and scalar curvature (so-called m-intermediate curvature), and use stable weighted slicings to show that for and the manifolds do not admit a metric of positive m-intermediate curvature.
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