{"title":"共形拉普拉斯的负特征值","authors":"Guillermo Henry, Jimmy Petean","doi":"10.1090/proc/16798","DOIUrl":null,"url":null,"abstract":"<p>Let <inline-formula content-type=\"math/mathml\"> <mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" alttext=\"upper M\"> <mml:semantics> <mml:mi>M</mml:mi> <mml:annotation encoding=\"application/x-tex\">M</mml:annotation> </mml:semantics> </mml:math> </inline-formula> be a closed differentiable manifold of dimension at least <inline-formula content-type=\"math/mathml\"> <mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" alttext=\"3\"> <mml:semantics> <mml:mn>3</mml:mn> <mml:annotation encoding=\"application/x-tex\">3</mml:annotation> </mml:semantics> </mml:math> </inline-formula>. Let <inline-formula content-type=\"math/mathml\"> <mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" alttext=\"normal upper Lamda 0 left-parenthesis upper M right-parenthesis\"> <mml:semantics> <mml:mrow> <mml:msub> <mml:mi mathvariant=\"normal\">Λ</mml:mi> <mml:mn>0</mml:mn> </mml:msub> <mml:mo stretchy=\"false\">(</mml:mo> <mml:mi>M</mml:mi> <mml:mo stretchy=\"false\">)</mml:mo> </mml:mrow> <mml:annotation encoding=\"application/x-tex\">\\Lambda _0 (M)</mml:annotation> </mml:semantics> </mml:math> </inline-formula> be the minimum number of non-positive eigenvalues that the conformal Laplacian of a metric on <inline-formula content-type=\"math/mathml\"> <mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" alttext=\"upper M\"> <mml:semantics> <mml:mi>M</mml:mi> <mml:annotation encoding=\"application/x-tex\">M</mml:annotation> </mml:semantics> </mml:math> </inline-formula> can have. We prove that for any <inline-formula content-type=\"math/mathml\"> <mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" alttext=\"k\"> <mml:semantics> <mml:mi>k</mml:mi> <mml:annotation encoding=\"application/x-tex\">k</mml:annotation> </mml:semantics> </mml:math> </inline-formula> greater than or equal to <inline-formula content-type=\"math/mathml\"> <mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" alttext=\"normal upper Lamda 0 left-parenthesis upper M right-parenthesis\"> <mml:semantics> <mml:mrow> <mml:msub> <mml:mi mathvariant=\"normal\">Λ</mml:mi> <mml:mn>0</mml:mn> </mml:msub> <mml:mo stretchy=\"false\">(</mml:mo> <mml:mi>M</mml:mi> <mml:mo stretchy=\"false\">)</mml:mo> </mml:mrow> <mml:annotation encoding=\"application/x-tex\">\\Lambda _0 (M)</mml:annotation> </mml:semantics> </mml:math> </inline-formula>, there exists a Riemannian metric on <inline-formula content-type=\"math/mathml\"> <mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" alttext=\"upper M\"> <mml:semantics> <mml:mi>M</mml:mi> <mml:annotation encoding=\"application/x-tex\">M</mml:annotation> </mml:semantics> </mml:math> </inline-formula> such that its conformal Laplacian has exactly <inline-formula content-type=\"math/mathml\"> <mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" alttext=\"k\"> <mml:semantics> <mml:mi>k</mml:mi> <mml:annotation encoding=\"application/x-tex\">k</mml:annotation> </mml:semantics> </mml:math> </inline-formula> negative eigenvalues. Also, we discuss upper bounds for <inline-formula content-type=\"math/mathml\"> <mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" alttext=\"normal upper Lamda 0 left-parenthesis upper M right-parenthesis\"> <mml:semantics> <mml:mrow> <mml:msub> <mml:mi mathvariant=\"normal\">Λ</mml:mi> <mml:mn>0</mml:mn> </mml:msub> <mml:mo stretchy=\"false\">(</mml:mo> <mml:mi>M</mml:mi> <mml:mo stretchy=\"false\">)</mml:mo> </mml:mrow> <mml:annotation encoding=\"application/x-tex\">\\Lambda _0 (M)</mml:annotation> </mml:semantics> </mml:math> </inline-formula>.</p>","PeriodicalId":20696,"journal":{"name":"Proceedings of the American Mathematical Society","volume":"33 1","pages":""},"PeriodicalIF":0.8000,"publicationDate":"2024-02-07","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"0","resultStr":"{\"title\":\"Negative eigenvalues of the conformal Laplacian\",\"authors\":\"Guillermo Henry, Jimmy Petean\",\"doi\":\"10.1090/proc/16798\",\"DOIUrl\":null,\"url\":null,\"abstract\":\"<p>Let <inline-formula content-type=\\\"math/mathml\\\"> <mml:math xmlns:mml=\\\"http://www.w3.org/1998/Math/MathML\\\" alttext=\\\"upper M\\\"> <mml:semantics> <mml:mi>M</mml:mi> <mml:annotation encoding=\\\"application/x-tex\\\">M</mml:annotation> </mml:semantics> </mml:math> </inline-formula> be a closed differentiable manifold of dimension at least <inline-formula content-type=\\\"math/mathml\\\"> <mml:math xmlns:mml=\\\"http://www.w3.org/1998/Math/MathML\\\" alttext=\\\"3\\\"> <mml:semantics> <mml:mn>3</mml:mn> <mml:annotation encoding=\\\"application/x-tex\\\">3</mml:annotation> </mml:semantics> </mml:math> </inline-formula>. Let <inline-formula content-type=\\\"math/mathml\\\"> <mml:math xmlns:mml=\\\"http://www.w3.org/1998/Math/MathML\\\" alttext=\\\"normal upper Lamda 0 left-parenthesis upper M right-parenthesis\\\"> <mml:semantics> <mml:mrow> <mml:msub> <mml:mi mathvariant=\\\"normal\\\">Λ</mml:mi> <mml:mn>0</mml:mn> </mml:msub> <mml:mo stretchy=\\\"false\\\">(</mml:mo> <mml:mi>M</mml:mi> <mml:mo stretchy=\\\"false\\\">)</mml:mo> </mml:mrow> <mml:annotation encoding=\\\"application/x-tex\\\">\\\\Lambda _0 (M)</mml:annotation> </mml:semantics> </mml:math> </inline-formula> be the minimum number of non-positive eigenvalues that the conformal Laplacian of a metric on <inline-formula content-type=\\\"math/mathml\\\"> <mml:math xmlns:mml=\\\"http://www.w3.org/1998/Math/MathML\\\" alttext=\\\"upper M\\\"> <mml:semantics> <mml:mi>M</mml:mi> <mml:annotation encoding=\\\"application/x-tex\\\">M</mml:annotation> </mml:semantics> </mml:math> </inline-formula> can have. We prove that for any <inline-formula content-type=\\\"math/mathml\\\"> <mml:math xmlns:mml=\\\"http://www.w3.org/1998/Math/MathML\\\" alttext=\\\"k\\\"> <mml:semantics> <mml:mi>k</mml:mi> <mml:annotation encoding=\\\"application/x-tex\\\">k</mml:annotation> </mml:semantics> </mml:math> </inline-formula> greater than or equal to <inline-formula content-type=\\\"math/mathml\\\"> <mml:math xmlns:mml=\\\"http://www.w3.org/1998/Math/MathML\\\" alttext=\\\"normal upper Lamda 0 left-parenthesis upper M right-parenthesis\\\"> <mml:semantics> <mml:mrow> <mml:msub> <mml:mi mathvariant=\\\"normal\\\">Λ</mml:mi> <mml:mn>0</mml:mn> </mml:msub> <mml:mo stretchy=\\\"false\\\">(</mml:mo> <mml:mi>M</mml:mi> <mml:mo stretchy=\\\"false\\\">)</mml:mo> </mml:mrow> <mml:annotation encoding=\\\"application/x-tex\\\">\\\\Lambda _0 (M)</mml:annotation> </mml:semantics> </mml:math> </inline-formula>, there exists a Riemannian metric on <inline-formula content-type=\\\"math/mathml\\\"> <mml:math xmlns:mml=\\\"http://www.w3.org/1998/Math/MathML\\\" alttext=\\\"upper M\\\"> <mml:semantics> <mml:mi>M</mml:mi> <mml:annotation encoding=\\\"application/x-tex\\\">M</mml:annotation> </mml:semantics> </mml:math> </inline-formula> such that its conformal Laplacian has exactly <inline-formula content-type=\\\"math/mathml\\\"> <mml:math xmlns:mml=\\\"http://www.w3.org/1998/Math/MathML\\\" alttext=\\\"k\\\"> <mml:semantics> <mml:mi>k</mml:mi> <mml:annotation encoding=\\\"application/x-tex\\\">k</mml:annotation> </mml:semantics> </mml:math> </inline-formula> negative eigenvalues. Also, we discuss upper bounds for <inline-formula content-type=\\\"math/mathml\\\"> <mml:math xmlns:mml=\\\"http://www.w3.org/1998/Math/MathML\\\" alttext=\\\"normal upper Lamda 0 left-parenthesis upper M right-parenthesis\\\"> <mml:semantics> <mml:mrow> <mml:msub> <mml:mi mathvariant=\\\"normal\\\">Λ</mml:mi> <mml:mn>0</mml:mn> </mml:msub> <mml:mo stretchy=\\\"false\\\">(</mml:mo> <mml:mi>M</mml:mi> <mml:mo stretchy=\\\"false\\\">)</mml:mo> </mml:mrow> <mml:annotation encoding=\\\"application/x-tex\\\">\\\\Lambda _0 (M)</mml:annotation> </mml:semantics> </mml:math> </inline-formula>.</p>\",\"PeriodicalId\":20696,\"journal\":{\"name\":\"Proceedings of the American Mathematical Society\",\"volume\":\"33 1\",\"pages\":\"\"},\"PeriodicalIF\":0.8000,\"publicationDate\":\"2024-02-07\",\"publicationTypes\":\"Journal Article\",\"fieldsOfStudy\":null,\"isOpenAccess\":false,\"openAccessPdf\":\"\",\"citationCount\":\"0\",\"resultStr\":null,\"platform\":\"Semanticscholar\",\"paperid\":null,\"PeriodicalName\":\"Proceedings of the American Mathematical Society\",\"FirstCategoryId\":\"100\",\"ListUrlMain\":\"https://doi.org/10.1090/proc/16798\",\"RegionNum\":3,\"RegionCategory\":\"数学\",\"ArticlePicture\":[],\"TitleCN\":null,\"AbstractTextCN\":null,\"PMCID\":null,\"EPubDate\":\"\",\"PubModel\":\"\",\"JCR\":\"Q2\",\"JCRName\":\"MATHEMATICS\",\"Score\":null,\"Total\":0}","platform":"Semanticscholar","paperid":null,"PeriodicalName":"Proceedings of the American Mathematical Society","FirstCategoryId":"100","ListUrlMain":"https://doi.org/10.1090/proc/16798","RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"Q2","JCRName":"MATHEMATICS","Score":null,"Total":0}
引用次数: 0
摘要
设 M M 是维数至少为 3 3 的闭可微分流形。设 Λ 0 ( M ) \Lambda _0 (M) 是 M M 上度量的保角拉普拉卡矩的最小非正特征值个数。我们证明，对于大于或等于 Λ 0 ( M ) \Lambda _0 (M) 的任何 k k，存在一个 M M 上的黎曼度量，使得它的共形拉普拉卡恰好有 k k 个负特征值。此外，我们还讨论了Λ 0 ( M ) \Lambda _0 (M) 的上限。
Let MM be a closed differentiable manifold of dimension at least 33. Let Λ0(M)\Lambda _0 (M) be the minimum number of non-positive eigenvalues that the conformal Laplacian of a metric on MM can have. We prove that for any kk greater than or equal to Λ0(M)\Lambda _0 (M), there exists a Riemannian metric on MM such that its conformal Laplacian has exactly kk negative eigenvalues. Also, we discuss upper bounds for Λ0(M)\Lambda _0 (M).
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