{"title":"翘曲积度量的谱常数刚性","authors":"Xiaoxiang Chai, Juncheol Pyo, Xueyuan Wan","doi":"10.1112/jlms.12958","DOIUrl":null,"url":null,"abstract":"<p>A theorem of Llarull says that if a smooth metric <span></span><math>\n <semantics>\n <mi>g</mi>\n <annotation>$g$</annotation>\n </semantics></math> on the <span></span><math>\n <semantics>\n <mi>n</mi>\n <annotation>$n$</annotation>\n </semantics></math>-sphere <span></span><math>\n <semantics>\n <msup>\n <mi>S</mi>\n <mi>n</mi>\n </msup>\n <annotation>$\\mathbb {S}^n$</annotation>\n </semantics></math> is bounded below by the standard round metric and the scalar curvature <span></span><math>\n <semantics>\n <msub>\n <mi>R</mi>\n <mi>g</mi>\n </msub>\n <annotation>$R_g$</annotation>\n </semantics></math> of <span></span><math>\n <semantics>\n <mi>g</mi>\n <annotation>$g$</annotation>\n </semantics></math> is bounded below by <span></span><math>\n <semantics>\n <mrow>\n <mi>n</mi>\n <mo>(</mo>\n <mi>n</mi>\n <mo>−</mo>\n <mn>1</mn>\n <mo>)</mo>\n </mrow>\n <annotation>$n (n - 1)$</annotation>\n </semantics></math>, then the metric <span></span><math>\n <semantics>\n <mi>g</mi>\n <annotation>$g$</annotation>\n </semantics></math> must be the standard round metric. We prove a spectral Llarull theorem by replacing the bound <span></span><math>\n <semantics>\n <mrow>\n <msub>\n <mi>R</mi>\n <mi>g</mi>\n </msub>\n <mo>⩾</mo>\n <mi>n</mi>\n <mrow>\n <mo>(</mo>\n <mi>n</mi>\n <mo>−</mo>\n <mn>1</mn>\n <mo>)</mo>\n </mrow>\n </mrow>\n <annotation>$R_g \\geqslant n (n - 1)$</annotation>\n </semantics></math> by a lower bound on the first eigenvalue of an elliptic operator involving the Laplacian and the scalar curvature <span></span><math>\n <semantics>\n <msub>\n <mi>R</mi>\n <mi>g</mi>\n </msub>\n <annotation>$R_g$</annotation>\n </semantics></math>. We utilize two methods: spinor and spacetime harmonic function.</p>","PeriodicalId":49989,"journal":{"name":"Journal of the London Mathematical Society-Second Series","volume":null,"pages":null},"PeriodicalIF":1.0000,"publicationDate":"2024-06-30","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://onlinelibrary.wiley.com/doi/epdf/10.1112/jlms.12958","citationCount":"0","resultStr":"{\"title\":\"Spectral constant rigidity of warped product metrics\",\"authors\":\"Xiaoxiang Chai, Juncheol Pyo, Xueyuan Wan\",\"doi\":\"10.1112/jlms.12958\",\"DOIUrl\":null,\"url\":null,\"abstract\":\"<p>A theorem of Llarull says that if a smooth metric <span></span><math>\\n <semantics>\\n <mi>g</mi>\\n <annotation>$g$</annotation>\\n </semantics></math> on the <span></span><math>\\n <semantics>\\n <mi>n</mi>\\n <annotation>$n$</annotation>\\n </semantics></math>-sphere <span></span><math>\\n <semantics>\\n <msup>\\n <mi>S</mi>\\n <mi>n</mi>\\n </msup>\\n <annotation>$\\\\mathbb {S}^n$</annotation>\\n </semantics></math> is bounded below by the standard round metric and the scalar curvature <span></span><math>\\n <semantics>\\n <msub>\\n <mi>R</mi>\\n <mi>g</mi>\\n </msub>\\n <annotation>$R_g$</annotation>\\n </semantics></math> of <span></span><math>\\n <semantics>\\n <mi>g</mi>\\n <annotation>$g$</annotation>\\n </semantics></math> is bounded below by <span></span><math>\\n <semantics>\\n <mrow>\\n <mi>n</mi>\\n <mo>(</mo>\\n <mi>n</mi>\\n <mo>−</mo>\\n <mn>1</mn>\\n <mo>)</mo>\\n </mrow>\\n <annotation>$n (n - 1)$</annotation>\\n </semantics></math>, then the metric <span></span><math>\\n <semantics>\\n <mi>g</mi>\\n <annotation>$g$</annotation>\\n </semantics></math> must be the standard round metric. We prove a spectral Llarull theorem by replacing the bound <span></span><math>\\n <semantics>\\n <mrow>\\n <msub>\\n <mi>R</mi>\\n <mi>g</mi>\\n </msub>\\n <mo>⩾</mo>\\n <mi>n</mi>\\n <mrow>\\n <mo>(</mo>\\n <mi>n</mi>\\n <mo>−</mo>\\n <mn>1</mn>\\n <mo>)</mo>\\n </mrow>\\n </mrow>\\n <annotation>$R_g \\\\geqslant n (n - 1)$</annotation>\\n </semantics></math> by a lower bound on the first eigenvalue of an elliptic operator involving the Laplacian and the scalar curvature <span></span><math>\\n <semantics>\\n <msub>\\n <mi>R</mi>\\n <mi>g</mi>\\n </msub>\\n <annotation>$R_g$</annotation>\\n </semantics></math>. We utilize two methods: spinor and spacetime harmonic function.</p>\",\"PeriodicalId\":49989,\"journal\":{\"name\":\"Journal of the London Mathematical Society-Second Series\",\"volume\":null,\"pages\":null},\"PeriodicalIF\":1.0000,\"publicationDate\":\"2024-06-30\",\"publicationTypes\":\"Journal Article\",\"fieldsOfStudy\":null,\"isOpenAccess\":false,\"openAccessPdf\":\"https://onlinelibrary.wiley.com/doi/epdf/10.1112/jlms.12958\",\"citationCount\":\"0\",\"resultStr\":null,\"platform\":\"Semanticscholar\",\"paperid\":null,\"PeriodicalName\":\"Journal of the London Mathematical Society-Second Series\",\"FirstCategoryId\":\"100\",\"ListUrlMain\":\"https://onlinelibrary.wiley.com/doi/10.1112/jlms.12958\",\"RegionNum\":2,\"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":"Journal of the London Mathematical Society-Second Series","FirstCategoryId":"100","ListUrlMain":"https://onlinelibrary.wiley.com/doi/10.1112/jlms.12958","RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"Q1","JCRName":"MATHEMATICS","Score":null,"Total":0}
引用次数: 0
摘要
拉鲁尔定理指出，如果 n $n$ -球面 S n $\mathbb {S}^n$ 上的光滑度量 g $g$ 的下界是标准圆度量，并且 g $g$ 的标量曲率 R g $R_g$ 的下界是 n ( n - 1 ) $n (n - 1)$ ，那么度量 g $g$ 一定是标准圆度量。我们将 R g ⩾ n ( n - 1 ) $R_g \geqslant n (n - 1)$ 约束替换为涉及拉普拉卡和标量曲率 R g $R_g$ 的椭圆算子的第一个特征值的下限，从而证明了谱拉鲁尔定理。我们采用两种方法：旋量和时空谐函数。
Spectral constant rigidity of warped product metrics
A theorem of Llarull says that if a smooth metric on the -sphere is bounded below by the standard round metric and the scalar curvature of is bounded below by , then the metric must be the standard round metric. We prove a spectral Llarull theorem by replacing the bound by a lower bound on the first eigenvalue of an elliptic operator involving the Laplacian and the scalar curvature . We utilize two methods: spinor and spacetime harmonic function.
期刊介绍：
The Journal of the London Mathematical Society has been publishing leading research in a broad range of mathematical subject areas since 1926. The Journal welcomes papers on subjects of general interest that represent a significant advance in mathematical knowledge, as well as submissions that are deemed to stimulate new interest and research activity.
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