{"title":"特殊爱因斯坦积的 GJMS 算子因式分解及其应用","authors":"Jeffrey S. Case, Andrea Malchiodi","doi":"10.1112/jlms.70023","DOIUrl":null,"url":null,"abstract":"<p>We show that the GJMS operators of a special Einstein product factor as a composition of second- and fourth-order differential operators. In particular, our formula applies to the Riemannian product <span></span><math>\n <semantics>\n <mrow>\n <msup>\n <mi>H</mi>\n <mi>ℓ</mi>\n </msup>\n <mo>×</mo>\n <msup>\n <mi>S</mi>\n <mrow>\n <mi>d</mi>\n <mo>−</mo>\n <mi>ℓ</mi>\n </mrow>\n </msup>\n </mrow>\n <annotation>$H^{\\ell } \\times S^{d-\\ell }$</annotation>\n </semantics></math>. We also show that there is an integer <span></span><math>\n <semantics>\n <mrow>\n <mi>D</mi>\n <mo>=</mo>\n <mi>D</mi>\n <mo>(</mo>\n <mi>k</mi>\n <mo>,</mo>\n <mi>ℓ</mi>\n <mo>)</mo>\n </mrow>\n <annotation>$D = D(k,\\ell)$</annotation>\n </semantics></math> such that if <span></span><math>\n <semantics>\n <mrow>\n <mi>d</mi>\n <mo>⩾</mo>\n <mi>D</mi>\n </mrow>\n <annotation>$d \\geqslant D$</annotation>\n </semantics></math>, then for any special Einstein product <span></span><math>\n <semantics>\n <mrow>\n <msup>\n <mi>N</mi>\n <mi>ℓ</mi>\n </msup>\n <mo>×</mo>\n <msup>\n <mi>M</mi>\n <mrow>\n <mi>d</mi>\n <mo>−</mo>\n <mi>ℓ</mi>\n </mrow>\n </msup>\n </mrow>\n <annotation>$N^\\ell \\times M^{d-\\ell }$</annotation>\n </semantics></math>, the Green's function for the GJMS operator of order <span></span><math>\n <semantics>\n <mrow>\n <mn>2</mn>\n <mi>k</mi>\n </mrow>\n <annotation>$2k$</annotation>\n </semantics></math> is positive. As a result, these products give new examples of closed Riemannian manifolds for which the <span></span><math>\n <semantics>\n <msub>\n <mi>Q</mi>\n <mrow>\n <mn>2</mn>\n <mi>k</mi>\n </mrow>\n </msub>\n <annotation>$Q_{2k}$</annotation>\n </semantics></math>-Yamabe problem is solvable.</p>","PeriodicalId":49989,"journal":{"name":"Journal of the London Mathematical Society-Second Series","volume":null,"pages":null},"PeriodicalIF":1.0000,"publicationDate":"2024-11-03","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://onlinelibrary.wiley.com/doi/epdf/10.1112/jlms.70023","citationCount":"0","resultStr":"{\"title\":\"A factorization of the GJMS operators of special Einstein products and applications\",\"authors\":\"Jeffrey S. Case, Andrea Malchiodi\",\"doi\":\"10.1112/jlms.70023\",\"DOIUrl\":null,\"url\":null,\"abstract\":\"<p>We show that the GJMS operators of a special Einstein product factor as a composition of second- and fourth-order differential operators. In particular, our formula applies to the Riemannian product <span></span><math>\\n <semantics>\\n <mrow>\\n <msup>\\n <mi>H</mi>\\n <mi>ℓ</mi>\\n </msup>\\n <mo>×</mo>\\n <msup>\\n <mi>S</mi>\\n <mrow>\\n <mi>d</mi>\\n <mo>−</mo>\\n <mi>ℓ</mi>\\n </mrow>\\n </msup>\\n </mrow>\\n <annotation>$H^{\\\\ell } \\\\times S^{d-\\\\ell }$</annotation>\\n </semantics></math>. We also show that there is an integer <span></span><math>\\n <semantics>\\n <mrow>\\n <mi>D</mi>\\n <mo>=</mo>\\n <mi>D</mi>\\n <mo>(</mo>\\n <mi>k</mi>\\n <mo>,</mo>\\n <mi>ℓ</mi>\\n <mo>)</mo>\\n </mrow>\\n <annotation>$D = D(k,\\\\ell)$</annotation>\\n </semantics></math> such that if <span></span><math>\\n <semantics>\\n <mrow>\\n <mi>d</mi>\\n <mo>⩾</mo>\\n <mi>D</mi>\\n </mrow>\\n <annotation>$d \\\\geqslant D$</annotation>\\n </semantics></math>, then for any special Einstein product <span></span><math>\\n <semantics>\\n <mrow>\\n <msup>\\n <mi>N</mi>\\n <mi>ℓ</mi>\\n </msup>\\n <mo>×</mo>\\n <msup>\\n <mi>M</mi>\\n <mrow>\\n <mi>d</mi>\\n <mo>−</mo>\\n <mi>ℓ</mi>\\n </mrow>\\n </msup>\\n </mrow>\\n <annotation>$N^\\\\ell \\\\times M^{d-\\\\ell }$</annotation>\\n </semantics></math>, the Green's function for the GJMS operator of order <span></span><math>\\n <semantics>\\n <mrow>\\n <mn>2</mn>\\n <mi>k</mi>\\n </mrow>\\n <annotation>$2k$</annotation>\\n </semantics></math> is positive. As a result, these products give new examples of closed Riemannian manifolds for which the <span></span><math>\\n <semantics>\\n <msub>\\n <mi>Q</mi>\\n <mrow>\\n <mn>2</mn>\\n <mi>k</mi>\\n </mrow>\\n </msub>\\n <annotation>$Q_{2k}$</annotation>\\n </semantics></math>-Yamabe problem is solvable.</p>\",\"PeriodicalId\":49989,\"journal\":{\"name\":\"Journal of the London Mathematical Society-Second Series\",\"volume\":null,\"pages\":null},\"PeriodicalIF\":1.0000,\"publicationDate\":\"2024-11-03\",\"publicationTypes\":\"Journal Article\",\"fieldsOfStudy\":null,\"isOpenAccess\":false,\"openAccessPdf\":\"https://onlinelibrary.wiley.com/doi/epdf/10.1112/jlms.70023\",\"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.70023\",\"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.70023","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
我们证明，特殊爱因斯坦积的 GJMS 算子因子是二阶和四阶微分算子的组合。特别是，我们的公式适用于黎曼积 H ℓ × S d - ℓ $H^{\ell }。\times S^{d-\ell }$ 。我们还证明，存在一个整数 D = D ( k , ℓ ) $D = D(k,\ell)$ ，如果 d ⩾ D $d \geqslant D$，那么对于任何特殊的爱因斯坦积 N ℓ × M d - ℓ $N^\ell \times M^{d-\ell }$，阶数为 2 k $2k$ 的 GJMS 算子的格林函数为正。因此，这些乘积给出了Q 2 k $Q_{2k}$ -Yamabe问题可解的封闭黎曼流形的新例子。
A factorization of the GJMS operators of special Einstein products and applications
We show that the GJMS operators of a special Einstein product factor as a composition of second- and fourth-order differential operators. In particular, our formula applies to the Riemannian product . We also show that there is an integer such that if , then for any special Einstein product , the Green's function for the GJMS operator of order is positive. As a result, these products give new examples of closed Riemannian manifolds for which the -Yamabe problem is solvable.
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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