{"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}
引用次数: 0
Abstract
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).
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。
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