{"title":"关于ℝⁿ中超曲面的新加权几何不等式的说明","authors":"Jie Wu","doi":"10.1090/proc/16875","DOIUrl":null,"url":null,"abstract":"<p>In this note, we prove a family of sharp weighed inequalities which involve weighted <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>-th mean curvature integral and two distinct quermassintegrals for closed hypersurfaces in <inline-formula content-type=\"math/mathml\"> <mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" alttext=\"double-struck upper R Superscript n\"> <mml:semantics> <mml:msup> <mml:mrow> <mml:mi mathvariant=\"double-struck\">R</mml:mi> </mml:mrow> <mml:mi>n</mml:mi> </mml:msup> <mml:annotation encoding=\"application/x-tex\">\\mathbb {R}^n</mml:annotation> </mml:semantics> </mml:math> </inline-formula>. This inequality generalizes the corresponding result of Wei and Zhou [Bull. Lond. Math. Soc. 55 (2023), pp. 263–281] where their proof is based on earlier results of Kwong-Miao [Pacific J. Math. 267 (2014), pp. 417–422; Commun. Contemp. Math. 17 (2015), p. 1550014]. Here we present a proof which does not rely on Kwong-Miao’s results.</p>","PeriodicalId":20696,"journal":{"name":"Proceedings of the American Mathematical Society","volume":"10 1","pages":""},"PeriodicalIF":0.8000,"publicationDate":"2024-04-10","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"0","resultStr":"{\"title\":\"A note on new weighted geometric inequalities for hypersurfaces in ℝⁿ\",\"authors\":\"Jie Wu\",\"doi\":\"10.1090/proc/16875\",\"DOIUrl\":null,\"url\":null,\"abstract\":\"<p>In this note, we prove a family of sharp weighed inequalities which involve weighted <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>-th mean curvature integral and two distinct quermassintegrals for closed hypersurfaces in <inline-formula content-type=\\\"math/mathml\\\"> <mml:math xmlns:mml=\\\"http://www.w3.org/1998/Math/MathML\\\" alttext=\\\"double-struck upper R Superscript n\\\"> <mml:semantics> <mml:msup> <mml:mrow> <mml:mi mathvariant=\\\"double-struck\\\">R</mml:mi> </mml:mrow> <mml:mi>n</mml:mi> </mml:msup> <mml:annotation encoding=\\\"application/x-tex\\\">\\\\mathbb {R}^n</mml:annotation> </mml:semantics> </mml:math> </inline-formula>. This inequality generalizes the corresponding result of Wei and Zhou [Bull. Lond. Math. Soc. 55 (2023), pp. 263–281] where their proof is based on earlier results of Kwong-Miao [Pacific J. Math. 267 (2014), pp. 417–422; Commun. Contemp. Math. 17 (2015), p. 1550014]. Here we present a proof which does not rely on Kwong-Miao’s results.</p>\",\"PeriodicalId\":20696,\"journal\":{\"name\":\"Proceedings of the American Mathematical Society\",\"volume\":\"10 1\",\"pages\":\"\"},\"PeriodicalIF\":0.8000,\"publicationDate\":\"2024-04-10\",\"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/16875\",\"RegionNum\":3,\"RegionCategory\":\"数学\",\"ArticlePicture\":[],\"TitleCN\":null,\"AbstractTextCN\":null,\"PMCID\":null,\"EPubDate\":\"\",\"PubModel\":\"\",\"JCR\":\"Q2\",\"JCRName\":\"MATHEMATICS\",\"Score\":null,\"Total\":0}","platform":"Semanticscholar","paperid":null,"PeriodicalName":"Proceedings of the American Mathematical Society","FirstCategoryId":"100","ListUrlMain":"https://doi.org/10.1090/proc/16875","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
摘要
在本注释中，我们证明了一系列尖锐的权重不等式，它们涉及 R n \mathbb {R}^n 中封闭超曲面的加权 k k -th 平均曲率积分和两个不同的质点积分。这个不等式概括了 Wei 和 Zhou [Bull. Lond. Math. Soc. 55 (2023), pp.267 (2014), pp.Contemp.Math.17 (2015), p. 1550014].这里我们提出一个不依赖邝淼结果的证明。
A note on new weighted geometric inequalities for hypersurfaces in ℝⁿ
In this note, we prove a family of sharp weighed inequalities which involve weighted kk-th mean curvature integral and two distinct quermassintegrals for closed hypersurfaces in Rn\mathbb {R}^n. This inequality generalizes the corresponding result of Wei and Zhou [Bull. Lond. Math. Soc. 55 (2023), pp. 263–281] where their proof is based on earlier results of Kwong-Miao [Pacific J. Math. 267 (2014), pp. 417–422; Commun. Contemp. Math. 17 (2015), p. 1550014]. Here we present a proof which does not rely on Kwong-Miao’s results.
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