{"title":"爱因斯坦求解漫流的非均质变形","authors":"Adam Thompson","doi":"10.1112/jlms.12904","DOIUrl":null,"url":null,"abstract":"<p>For each non-flat, unimodular Ricci soliton solvmanifold <span></span><math>\n <semantics>\n <mrow>\n <mo>(</mo>\n <msub>\n <mi>S</mi>\n <mn>0</mn>\n </msub>\n <mo>,</mo>\n <msub>\n <mi>g</mi>\n <mn>0</mn>\n </msub>\n <mo>)</mo>\n </mrow>\n <annotation>$(\\mathsf {S}_0,g_0)$</annotation>\n </semantics></math>, we construct a one-parameter family of complete, expanding, gradient Ricci solitons that admit a cohomogeneity one isometric action by <span></span><math>\n <semantics>\n <msub>\n <mi>S</mi>\n <mn>0</mn>\n </msub>\n <annotation>$\\mathsf {S}_0$</annotation>\n </semantics></math>. The orbits of this action are hypersurfaces homothetic to <span></span><math>\n <semantics>\n <mrow>\n <mo>(</mo>\n <msub>\n <mi>S</mi>\n <mn>0</mn>\n </msub>\n <mo>,</mo>\n <msub>\n <mi>g</mi>\n <mn>0</mn>\n </msub>\n <mo>)</mo>\n </mrow>\n <annotation>$(\\mathsf {S}_0,g_0)$</annotation>\n </semantics></math>. These metrics are asymptotic at one end to an Einstein solvmanifold. In the one-parameter family, exactly one metric is Einstein, and exactly one has orbits that are isometric to <span></span><math>\n <semantics>\n <mrow>\n <mo>(</mo>\n <msub>\n <mi>S</mi>\n <mn>0</mn>\n </msub>\n <mo>,</mo>\n <msub>\n <mi>g</mi>\n <mn>0</mn>\n </msub>\n <mo>)</mo>\n </mrow>\n <annotation>$(\\mathsf {S}_0,g_0)$</annotation>\n </semantics></math>.</p>","PeriodicalId":49989,"journal":{"name":"Journal of the London Mathematical Society-Second Series","volume":null,"pages":null},"PeriodicalIF":1.0000,"publicationDate":"2024-04-30","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://onlinelibrary.wiley.com/doi/epdf/10.1112/jlms.12904","citationCount":"0","resultStr":"{\"title\":\"Inhomogeneous deformations of Einstein solvmanifolds\",\"authors\":\"Adam Thompson\",\"doi\":\"10.1112/jlms.12904\",\"DOIUrl\":null,\"url\":null,\"abstract\":\"<p>For each non-flat, unimodular Ricci soliton solvmanifold <span></span><math>\\n <semantics>\\n <mrow>\\n <mo>(</mo>\\n <msub>\\n <mi>S</mi>\\n <mn>0</mn>\\n </msub>\\n <mo>,</mo>\\n <msub>\\n <mi>g</mi>\\n <mn>0</mn>\\n </msub>\\n <mo>)</mo>\\n </mrow>\\n <annotation>$(\\\\mathsf {S}_0,g_0)$</annotation>\\n </semantics></math>, we construct a one-parameter family of complete, expanding, gradient Ricci solitons that admit a cohomogeneity one isometric action by <span></span><math>\\n <semantics>\\n <msub>\\n <mi>S</mi>\\n <mn>0</mn>\\n </msub>\\n <annotation>$\\\\mathsf {S}_0$</annotation>\\n </semantics></math>. The orbits of this action are hypersurfaces homothetic to <span></span><math>\\n <semantics>\\n <mrow>\\n <mo>(</mo>\\n <msub>\\n <mi>S</mi>\\n <mn>0</mn>\\n </msub>\\n <mo>,</mo>\\n <msub>\\n <mi>g</mi>\\n <mn>0</mn>\\n </msub>\\n <mo>)</mo>\\n </mrow>\\n <annotation>$(\\\\mathsf {S}_0,g_0)$</annotation>\\n </semantics></math>. These metrics are asymptotic at one end to an Einstein solvmanifold. In the one-parameter family, exactly one metric is Einstein, and exactly one has orbits that are isometric to <span></span><math>\\n <semantics>\\n <mrow>\\n <mo>(</mo>\\n <msub>\\n <mi>S</mi>\\n <mn>0</mn>\\n </msub>\\n <mo>,</mo>\\n <msub>\\n <mi>g</mi>\\n <mn>0</mn>\\n </msub>\\n <mo>)</mo>\\n </mrow>\\n <annotation>$(\\\\mathsf {S}_0,g_0)$</annotation>\\n </semantics></math>.</p>\",\"PeriodicalId\":49989,\"journal\":{\"name\":\"Journal of the London Mathematical Society-Second Series\",\"volume\":null,\"pages\":null},\"PeriodicalIF\":1.0000,\"publicationDate\":\"2024-04-30\",\"publicationTypes\":\"Journal Article\",\"fieldsOfStudy\":null,\"isOpenAccess\":false,\"openAccessPdf\":\"https://onlinelibrary.wiley.com/doi/epdf/10.1112/jlms.12904\",\"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.12904\",\"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.12904","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
对于每一个非平坦、单模态的利玛窦孤素解旋体( S 0 , g 0 ) $ (\mathsf {S}_0,g_0)$ ,我们构建了一个完整的、扩展的、梯度的利玛窦孤素的单参数族,该族通过 S 0 $\mathsf {S}_0$ 接受同构一等轴作用。该作用的轨道是与 ( S 0 , g 0 ) $(\mathsf {S}_0,g_0)$ 同调的超曲面。这些度量在一端渐近于爱因斯坦溶域。在单参数族中,正好有一个度量是爱因斯坦度量,正好有一个度量的轨道与 ( S 0 , g 0 ) $(\mathsf {S}_0,g_0)$ 等距。
Inhomogeneous deformations of Einstein solvmanifolds
For each non-flat, unimodular Ricci soliton solvmanifold , we construct a one-parameter family of complete, expanding, gradient Ricci solitons that admit a cohomogeneity one isometric action by . The orbits of this action are hypersurfaces homothetic to . These metrics are asymptotic at one end to an Einstein solvmanifold. In the one-parameter family, exactly one metric is Einstein, and exactly one has orbits that are isometric to .
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.