{"title":"Non-compact Einstein manifolds with symmetry","authors":"Christoph Böhm, Ramiro A. Lafuente","doi":"10.1090/jams/1022","DOIUrl":null,"url":null,"abstract":"For Einstein manifolds with negative scalar curvature admitting an isometric action of a Lie group <inline-formula content-type=\"math/mathml\"> <mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" alttext=\"sans-serif upper G\"> <mml:semantics> <mml:mrow class=\"MJX-TeXAtom-ORD\"> <mml:mi mathvariant=\"sans-serif\">G</mml:mi> </mml:mrow> <mml:annotation encoding=\"application/x-tex\">\\mathsf {G}</mml:annotation> </mml:semantics> </mml:math> </inline-formula> with compact, smooth orbit space, we show that the nilradical <inline-formula content-type=\"math/mathml\"> <mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" alttext=\"sans-serif upper N\"> <mml:semantics> <mml:mrow class=\"MJX-TeXAtom-ORD\"> <mml:mi mathvariant=\"sans-serif\">N</mml:mi> </mml:mrow> <mml:annotation encoding=\"application/x-tex\">\\mathsf {N}</mml:annotation> </mml:semantics> </mml:math> </inline-formula> of <inline-formula content-type=\"math/mathml\"> <mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" alttext=\"sans-serif upper G\"> <mml:semantics> <mml:mrow class=\"MJX-TeXAtom-ORD\"> <mml:mi mathvariant=\"sans-serif\">G</mml:mi> </mml:mrow> <mml:annotation encoding=\"application/x-tex\">\\mathsf {G}</mml:annotation> </mml:semantics> </mml:math> </inline-formula> acts polarly and that the <inline-formula content-type=\"math/mathml\"> <mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" alttext=\"sans-serif upper N\"> <mml:semantics> <mml:mrow class=\"MJX-TeXAtom-ORD\"> <mml:mi mathvariant=\"sans-serif\">N</mml:mi> </mml:mrow> <mml:annotation encoding=\"application/x-tex\">\\mathsf {N}</mml:annotation> </mml:semantics> </mml:math> </inline-formula>-orbits can be extended to minimal Einstein submanifolds. As an application, we prove the Alekseevskii conjecture: Any homogeneous Einstein manifold with negative scalar curvature is diffeomorphic to a Euclidean space.","PeriodicalId":54764,"journal":{"name":"Journal of the American Mathematical Society","volume":"23 1","pages":"0"},"PeriodicalIF":3.5000,"publicationDate":"2023-02-28","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"4","resultStr":null,"platform":"Semanticscholar","paperid":null,"PeriodicalName":"Journal of the American Mathematical Society","FirstCategoryId":"1085","ListUrlMain":"https://doi.org/10.1090/jams/1022","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: 4
Abstract
For Einstein manifolds with negative scalar curvature admitting an isometric action of a Lie group G\mathsf {G} with compact, smooth orbit space, we show that the nilradical N\mathsf {N} of G\mathsf {G} acts polarly and that the N\mathsf {N}-orbits can be extended to minimal Einstein submanifolds. As an application, we prove the Alekseevskii conjecture: Any homogeneous Einstein manifold with negative scalar curvature is diffeomorphic to a Euclidean space.
期刊介绍：
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This journal is devoted to research articles of the highest quality in all areas of pure and applied mathematics.
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