{"title":"任何萨萨基结构都可以通过嵌入球面来近似","authors":"Andrea Loi, Giovanni Placini","doi":"10.1515/forum-2023-0364","DOIUrl":null,"url":null,"abstract":"We show that, for any given <jats:inline-formula> <jats:alternatives> <m:math xmlns:m=\"http://www.w3.org/1998/Math/MathML\"> <m:mrow> <m:mi>q</m:mi> <m:mo>≥</m:mo> <m:mn>0</m:mn> </m:mrow> </m:math> <jats:inline-graphic xmlns:xlink=\"http://www.w3.org/1999/xlink\" xlink:href=\"graphic/j_forum-2023-0364_eq_0372.png\"/> <jats:tex-math>{q\\geq 0}</jats:tex-math> </jats:alternatives> </jats:inline-formula>, any Sasakian structure on a closed manifold <jats:italic>M</jats:italic> is approximated in the <jats:inline-formula> <jats:alternatives> <m:math xmlns:m=\"http://www.w3.org/1998/Math/MathML\"> <m:msup> <m:mi>C</m:mi> <m:mi>q</m:mi> </m:msup> </m:math> <jats:inline-graphic xmlns:xlink=\"http://www.w3.org/1999/xlink\" xlink:href=\"graphic/j_forum-2023-0364_eq_0146.png\"/> <jats:tex-math>{C^{{q}}}</jats:tex-math> </jats:alternatives> </jats:inline-formula>-norm by structures induced by CR embeddings into weighted Sasakian spheres. In order to obtain this result, we also strengthen the approximation of an orbifold Kähler form by projectively induced ones given in [J. Ross and R. Thomas, Weighted projective embeddings, stability of orbifolds, and constant scalar curvature Kähler metrics, J. Differential Geom. 88 2011, 1, 109–159] in the <jats:inline-formula> <jats:alternatives> <m:math xmlns:m=\"http://www.w3.org/1998/Math/MathML\"> <m:msup> <m:mi>C</m:mi> <m:mn>0</m:mn> </m:msup> </m:math> <jats:inline-graphic xmlns:xlink=\"http://www.w3.org/1999/xlink\" xlink:href=\"graphic/j_forum-2023-0364_eq_0138.png\"/> <jats:tex-math>{C^{0}}</jats:tex-math> </jats:alternatives> </jats:inline-formula>-norm to a <jats:inline-formula> <jats:alternatives> <m:math xmlns:m=\"http://www.w3.org/1998/Math/MathML\"> <m:msup> <m:mi>C</m:mi> <m:mi>q</m:mi> </m:msup> </m:math> <jats:inline-graphic xmlns:xlink=\"http://www.w3.org/1999/xlink\" xlink:href=\"graphic/j_forum-2023-0364_eq_0146.png\"/> <jats:tex-math>{C^{{q}}}</jats:tex-math> </jats:alternatives> </jats:inline-formula>-approximation.","PeriodicalId":12433,"journal":{"name":"Forum Mathematicum","volume":null,"pages":null},"PeriodicalIF":1.0000,"publicationDate":"2024-08-07","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"0","resultStr":"{\"title\":\"Any Sasakian structure is approximated by embeddings into spheres\",\"authors\":\"Andrea Loi, Giovanni Placini\",\"doi\":\"10.1515/forum-2023-0364\",\"DOIUrl\":null,\"url\":null,\"abstract\":\"We show that, for any given <jats:inline-formula> <jats:alternatives> <m:math xmlns:m=\\\"http://www.w3.org/1998/Math/MathML\\\"> <m:mrow> <m:mi>q</m:mi> <m:mo>≥</m:mo> <m:mn>0</m:mn> </m:mrow> </m:math> <jats:inline-graphic xmlns:xlink=\\\"http://www.w3.org/1999/xlink\\\" xlink:href=\\\"graphic/j_forum-2023-0364_eq_0372.png\\\"/> <jats:tex-math>{q\\\\geq 0}</jats:tex-math> </jats:alternatives> </jats:inline-formula>, any Sasakian structure on a closed manifold <jats:italic>M</jats:italic> is approximated in the <jats:inline-formula> <jats:alternatives> <m:math xmlns:m=\\\"http://www.w3.org/1998/Math/MathML\\\"> <m:msup> <m:mi>C</m:mi> <m:mi>q</m:mi> </m:msup> </m:math> <jats:inline-graphic xmlns:xlink=\\\"http://www.w3.org/1999/xlink\\\" xlink:href=\\\"graphic/j_forum-2023-0364_eq_0146.png\\\"/> <jats:tex-math>{C^{{q}}}</jats:tex-math> </jats:alternatives> </jats:inline-formula>-norm by structures induced by CR embeddings into weighted Sasakian spheres. In order to obtain this result, we also strengthen the approximation of an orbifold Kähler form by projectively induced ones given in [J. Ross and R. Thomas, Weighted projective embeddings, stability of orbifolds, and constant scalar curvature Kähler metrics, J. Differential Geom. 88 2011, 1, 109–159] in the <jats:inline-formula> <jats:alternatives> <m:math xmlns:m=\\\"http://www.w3.org/1998/Math/MathML\\\"> <m:msup> <m:mi>C</m:mi> <m:mn>0</m:mn> </m:msup> </m:math> <jats:inline-graphic xmlns:xlink=\\\"http://www.w3.org/1999/xlink\\\" xlink:href=\\\"graphic/j_forum-2023-0364_eq_0138.png\\\"/> <jats:tex-math>{C^{0}}</jats:tex-math> </jats:alternatives> </jats:inline-formula>-norm to a <jats:inline-formula> <jats:alternatives> <m:math xmlns:m=\\\"http://www.w3.org/1998/Math/MathML\\\"> <m:msup> <m:mi>C</m:mi> <m:mi>q</m:mi> </m:msup> </m:math> <jats:inline-graphic xmlns:xlink=\\\"http://www.w3.org/1999/xlink\\\" xlink:href=\\\"graphic/j_forum-2023-0364_eq_0146.png\\\"/> <jats:tex-math>{C^{{q}}}</jats:tex-math> </jats:alternatives> </jats:inline-formula>-approximation.\",\"PeriodicalId\":12433,\"journal\":{\"name\":\"Forum Mathematicum\",\"volume\":null,\"pages\":null},\"PeriodicalIF\":1.0000,\"publicationDate\":\"2024-08-07\",\"publicationTypes\":\"Journal Article\",\"fieldsOfStudy\":null,\"isOpenAccess\":false,\"openAccessPdf\":\"\",\"citationCount\":\"0\",\"resultStr\":null,\"platform\":\"Semanticscholar\",\"paperid\":null,\"PeriodicalName\":\"Forum Mathematicum\",\"FirstCategoryId\":\"100\",\"ListUrlMain\":\"https://doi.org/10.1515/forum-2023-0364\",\"RegionNum\":3,\"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":"Forum Mathematicum","FirstCategoryId":"100","ListUrlMain":"https://doi.org/10.1515/forum-2023-0364","RegionNum":3,"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，对于任何给定的 q ≥ 0 {q\geq 0}，闭流形 M 上的任何萨萨基结构在 C q {C^{{q}} 中都是近似的。｝ -的结构近似。为了得到这个结果，我们还加强了 [J. Ross and R. Thomas, Weighted Sasakian spheres] 中给出的用投影诱导的球面凯勒形式对球面凯勒形式的逼近。Ross and R. Thomas, Weighted projective embeddings, stability of orbifolds, and constant scalar curvature Kähler metrics, J. Differential Geom.88 2011, 1, 109-159] 中的 C 0 {C^{0}} -norm。 -C q {C^{{q}} 的近似。｝ -的近似。
Any Sasakian structure is approximated by embeddings into spheres
We show that, for any given q≥0{q\geq 0}, any Sasakian structure on a closed manifold M is approximated in the Cq{C^{{q}}}-norm by structures induced by CR embeddings into weighted Sasakian spheres. In order to obtain this result, we also strengthen the approximation of an orbifold Kähler form by projectively induced ones given in [J. Ross and R. Thomas, Weighted projective embeddings, stability of orbifolds, and constant scalar curvature Kähler metrics, J. Differential Geom. 88 2011, 1, 109–159] in the C0{C^{0}}-norm to a Cq{C^{{q}}}-approximation.
期刊介绍：
Forum Mathematicum is a general mathematics journal, which is devoted to the publication of research articles in all fields of pure and applied mathematics, including mathematical physics. Forum Mathematicum belongs to the top 50 journals in pure and applied mathematics, as measured by citation impact.
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