A generalization of Geroch's conjecture

IF 3.1 1区数学 Q1 MATHEMATICS

Communications on Pure and Applied Mathematics Pub Date : 2023-08-12 DOI:10.1002/cpa.22137

Simon Brendle, Sven Hirsch, Florian Johne

{"title":"A generalization of Geroch's conjecture","authors":"Simon Brendle, Sven Hirsch, Florian Johne","doi":"10.1002/cpa.22137","DOIUrl":null,"url":null,"abstract":"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 m-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 m-intermediate curvature.","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}

引用次数: 7

Abstract

The Theorem of Bonnet–Myers implies that manifolds with topology $M^{n - 1} \times S^{1}$ do not admit a metric of positive Ricci curvature, while the resolution of Geroch's conjecture implies that the torus $T^{n}$ 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 $n \leq 7$ and $1 \leq m \leq n - 1$ the manifolds $N^{n} = M^{n - m} \times T^{m}$ do not admit a metric of positive m-intermediate curvature.

查看原文本刊更多论文

Geroch猜想的一个推广

Bonnet–Myers定理暗示了具有拓扑的流形不允许正Ricci曲率的度量，而Geroch猜想的分辨率暗示了环面不允许正标量曲率的度量。在这项工作中，我们引入了Ricci和标量曲率（称为m-中间曲率）之间的曲率插值的新概念，并使用稳定的加权切片来表明对于和流形不允许正m-中间弯曲的度量。

本文章由计算机程序翻译，如有差异，请以英文原文为准。

求助全文

约1分钟内获得全文求助全文

来源期刊

Communications on Pure and Applied Mathematics 数学-数学

CiteScore

6.70

自引率

3.30%

发文量

审稿时长

>12 weeks

期刊介绍： Communications on Pure and Applied Mathematics (ISSN 0010-3640) is published monthly, one volume per year, by John Wiley & Sons, Inc. © 2019. The journal primarily publishes papers originating at or solicited by the Courant Institute of Mathematical Sciences. It features recent developments in applied mathematics, mathematical physics, and mathematical analysis. The topics include partial differential equations, computer science, and applied mathematics. CPAM is devoted to mathematical contributions to the sciences; both theoretical and applied papers, of original or expository type, are included.