{"title":"Grushin半球作为曲率≥1的Ricci极限空间","authors":"Jiayin Pan","doi":"10.1090/bproc/160","DOIUrl":null,"url":null,"abstract":"<p>The Grushin sphere is an almost-Riemannian manifold that degenerates along its equator. We construct a sequence of Riemannian metrics on a sphere <inline-formula content-type=\"math/mathml\">\n<mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" alttext=\"upper S Superscript m plus n\">\n <mml:semantics>\n <mml:msup>\n <mml:mi>S</mml:mi>\n <mml:mrow class=\"MJX-TeXAtom-ORD\">\n <mml:mi>m</mml:mi>\n <mml:mo>+</mml:mo>\n <mml:mi>n</mml:mi>\n </mml:mrow>\n </mml:msup>\n <mml:annotation encoding=\"application/x-tex\">S^{m+n}</mml:annotation>\n </mml:semantics>\n</mml:math>\n</inline-formula> with <inline-formula content-type=\"math/mathml\">\n<mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" alttext=\"upper R i c greater-than-or-equal-to 1\">\n <mml:semantics>\n <mml:mrow>\n <mml:mi>R</mml:mi>\n <mml:mi>i</mml:mi>\n <mml:mi>c</mml:mi>\n <mml:mo>≥<!-- ≥ --></mml:mo>\n <mml:mn>1</mml:mn>\n </mml:mrow>\n <mml:annotation encoding=\"application/x-tex\">Ric\\ge 1</mml:annotation>\n </mml:semantics>\n</mml:math>\n</inline-formula> such that its Gromov-Hausdorff limit is the <inline-formula content-type=\"math/mathml\">\n<mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" alttext=\"n\">\n <mml:semantics>\n <mml:mi>n</mml:mi>\n <mml:annotation encoding=\"application/x-tex\">n</mml:annotation>\n </mml:semantics>\n</mml:math>\n</inline-formula>-dimensional Grushin hemisphere.</p>","PeriodicalId":106316,"journal":{"name":"Proceedings of the American Mathematical Society, Series B","volume":"22 1","pages":"0"},"PeriodicalIF":0.0000,"publicationDate":"2023-03-24","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"3","resultStr":"{\"title\":\"The Grushin hemisphere as a Ricci limit space with curvature ≥1\",\"authors\":\"Jiayin Pan\",\"doi\":\"10.1090/bproc/160\",\"DOIUrl\":null,\"url\":null,\"abstract\":\"<p>The Grushin sphere is an almost-Riemannian manifold that degenerates along its equator. We construct a sequence of Riemannian metrics on a sphere <inline-formula content-type=\\\"math/mathml\\\">\\n<mml:math xmlns:mml=\\\"http://www.w3.org/1998/Math/MathML\\\" alttext=\\\"upper S Superscript m plus n\\\">\\n <mml:semantics>\\n <mml:msup>\\n <mml:mi>S</mml:mi>\\n <mml:mrow class=\\\"MJX-TeXAtom-ORD\\\">\\n <mml:mi>m</mml:mi>\\n <mml:mo>+</mml:mo>\\n <mml:mi>n</mml:mi>\\n </mml:mrow>\\n </mml:msup>\\n <mml:annotation encoding=\\\"application/x-tex\\\">S^{m+n}</mml:annotation>\\n </mml:semantics>\\n</mml:math>\\n</inline-formula> with <inline-formula content-type=\\\"math/mathml\\\">\\n<mml:math xmlns:mml=\\\"http://www.w3.org/1998/Math/MathML\\\" alttext=\\\"upper R i c greater-than-or-equal-to 1\\\">\\n <mml:semantics>\\n <mml:mrow>\\n <mml:mi>R</mml:mi>\\n <mml:mi>i</mml:mi>\\n <mml:mi>c</mml:mi>\\n <mml:mo>≥<!-- ≥ --></mml:mo>\\n <mml:mn>1</mml:mn>\\n </mml:mrow>\\n <mml:annotation encoding=\\\"application/x-tex\\\">Ric\\\\ge 1</mml:annotation>\\n </mml:semantics>\\n</mml:math>\\n</inline-formula> such that its Gromov-Hausdorff limit is the <inline-formula content-type=\\\"math/mathml\\\">\\n<mml:math xmlns:mml=\\\"http://www.w3.org/1998/Math/MathML\\\" alttext=\\\"n\\\">\\n <mml:semantics>\\n <mml:mi>n</mml:mi>\\n <mml:annotation encoding=\\\"application/x-tex\\\">n</mml:annotation>\\n </mml:semantics>\\n</mml:math>\\n</inline-formula>-dimensional Grushin hemisphere.</p>\",\"PeriodicalId\":106316,\"journal\":{\"name\":\"Proceedings of the American Mathematical Society, Series B\",\"volume\":\"22 1\",\"pages\":\"0\"},\"PeriodicalIF\":0.0000,\"publicationDate\":\"2023-03-24\",\"publicationTypes\":\"Journal Article\",\"fieldsOfStudy\":null,\"isOpenAccess\":false,\"openAccessPdf\":\"\",\"citationCount\":\"3\",\"resultStr\":null,\"platform\":\"Semanticscholar\",\"paperid\":null,\"PeriodicalName\":\"Proceedings of the American Mathematical Society, Series B\",\"FirstCategoryId\":\"1085\",\"ListUrlMain\":\"https://doi.org/10.1090/bproc/160\",\"RegionNum\":0,\"RegionCategory\":null,\"ArticlePicture\":[],\"TitleCN\":null,\"AbstractTextCN\":null,\"PMCID\":null,\"EPubDate\":\"\",\"PubModel\":\"\",\"JCR\":\"\",\"JCRName\":\"\",\"Score\":null,\"Total\":0}","platform":"Semanticscholar","paperid":null,"PeriodicalName":"Proceedings of the American Mathematical Society, Series B","FirstCategoryId":"1085","ListUrlMain":"https://doi.org/10.1090/bproc/160","RegionNum":0,"RegionCategory":null,"ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"","JCRName":"","Score":null,"Total":0}
The Grushin hemisphere as a Ricci limit space with curvature ≥1
The Grushin sphere is an almost-Riemannian manifold that degenerates along its equator. We construct a sequence of Riemannian metrics on a sphere Sm+nS^{m+n} with Ric≥1Ric\ge 1 such that its Gromov-Hausdorff limit is the nn-dimensional Grushin hemisphere.
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