{"title":"里奇曲率积分，局部泛函，和里奇流","authors":"Yuanqing Ma, Bing Wang","doi":"10.1090/btran/155","DOIUrl":null,"url":null,"abstract":"<p>Consider a Riemannian manifold <inline-formula content-type=\"math/mathml\"> <mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" alttext=\"left-parenthesis upper M Superscript m Baseline comma g right-parenthesis\"> <mml:semantics> <mml:mrow> <mml:mo stretchy=\"false\">(</mml:mo> <mml:msup> <mml:mi>M</mml:mi> <mml:mrow class=\"MJX-TeXAtom-ORD\"> <mml:mi>m</mml:mi> </mml:mrow> </mml:msup> <mml:mo>,</mml:mo> <mml:mi>g</mml:mi> <mml:mo stretchy=\"false\">)</mml:mo> </mml:mrow> <mml:annotation encoding=\"application/x-tex\">(M^{m}, g)</mml:annotation> </mml:semantics> </mml:math> </inline-formula> whose volume is the same as the standard sphere <inline-formula content-type=\"math/mathml\"> <mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" alttext=\"left-parenthesis upper S Superscript m Baseline comma g Subscript r o u n d Baseline right-parenthesis\"> <mml:semantics> <mml:mrow> <mml:mo stretchy=\"false\">(</mml:mo> <mml:msup> <mml:mi>S</mml:mi> <mml:mrow class=\"MJX-TeXAtom-ORD\"> <mml:mi>m</mml:mi> </mml:mrow> </mml:msup> <mml:mo>,</mml:mo> <mml:msub> <mml:mi>g</mml:mi> <mml:mrow class=\"MJX-TeXAtom-ORD\"> <mml:mi>r</mml:mi> <mml:mi>o</mml:mi> <mml:mi>u</mml:mi> <mml:mi>n</mml:mi> <mml:mi>d</mml:mi> </mml:mrow> </mml:msub> <mml:mo stretchy=\"false\">)</mml:mo> </mml:mrow> <mml:annotation encoding=\"application/x-tex\">(S^{m}, g_{round})</mml:annotation> </mml:semantics> </mml:math> </inline-formula>. If <inline-formula content-type=\"math/mathml\"> <mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" alttext=\"p greater-than StartFraction m Over 2 EndFraction\"> <mml:semantics> <mml:mrow> <mml:mi>p</mml:mi> <mml:mspace width=\"negativethinmathspace\" /> <mml:mo>></mml:mo> <mml:mspace width=\"negativethinmathspace\" /> <mml:mfrac> <mml:mi>m</mml:mi> <mml:mn>2</mml:mn> </mml:mfrac> </mml:mrow> <mml:annotation encoding=\"application/x-tex\">p\\!>\\!\\frac {m}{2}</mml:annotation> </mml:semantics> </mml:math> </inline-formula> and <inline-formula content-type=\"math/mathml\"> <mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" alttext=\"integral Underscript upper M Endscripts left-brace upper R c minus left-parenthesis m minus 1 right-parenthesis g right-brace Subscript minus Superscript p Baseline d v\"> <mml:semantics> <mml:mrow> <mml:msub> <mml:mo>∫<!-- ∫ --></mml:mo> <mml:mrow class=\"MJX-TeXAtom-ORD\"> <mml:mi>M</mml:mi> </mml:mrow> </mml:msub> <mml:mspace width=\"negativethinmathspace\" /> <mml:msubsup> <mml:mrow> <mml:mo>{</mml:mo> <mml:mi>R</mml:mi> <mml:mi>c</mml:mi> <mml:mspace width=\"negativethinmathspace\" /> <mml:mo>−<!-- − --></mml:mo> <mml:mspace width=\"negativethinmathspace\" /> <mml:mo stretchy=\"false\">(</mml:mo> <mml:mi>m</mml:mi> <mml:mspace width=\"negativethinmathspace\" /> <mml:mo>−<!-- − --></mml:mo> <mml:mspace width=\"negativethinmathspace\" /> <mml:mn>1</mml:mn> <mml:mo stretchy=\"false\">)</mml:mo> <mml:mi>g</mml:mi> <mml:mo>}</mml:mo> </mml:mrow> <mml:mrow class=\"MJX-TeXAtom-ORD\"> <mml:mo>−<!-- − --></mml:mo> </mml:mrow> <mml:mrow class=\"MJX-TeXAtom-ORD\"> <mml:mi>p</mml:mi> </mml:mrow> </mml:msubsup> <mml:mi>d</mml:mi> <mml:mi>v</mml:mi> </mml:mrow> <mml:annotation encoding=\"application/x-tex\">\\int _{M}\\! \\left \\{ Rc\\!-\\!(m\\!-\\!1)g\\right \\}_{-}^{p} dv</mml:annotation> </mml:semantics> </mml:math> </inline-formula> is sufficiently small, we show that the normalized Ricci flow initiated from <inline-formula content-type=\"math/mathml\"> <mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" alttext=\"left-parenthesis upper M Superscript m Baseline comma g right-parenthesis\"> <mml:semantics> <mml:mrow> <mml:mo stretchy=\"false\">(</mml:mo> <mml:msup> <mml:mi>M</mml:mi> <mml:mrow class=\"MJX-TeXAtom-ORD\"> <mml:mi>m</mml:mi> </mml:mrow> </mml:msup> <mml:mo>,</mml:mo> <mml:mi>g</mml:mi> <mml:mo stretchy=\"false\">)</mml:mo> </mml:mrow> <mml:annotation encoding=\"application/x-tex\">(M^{m}, g)</mml:annotation> </mml:semantics> </mml:math> </inline-formula> will exist immortally and converge to the standard sphere. The choice of <inline-formula content-type=\"math/mathml\"> <mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" alttext=\"p\"> <mml:semantics> <mml:mi>p</mml:mi> <mml:annotation encoding=\"application/x-tex\">p</mml:annotation> </mml:semantics> </mml:math> </inline-formula> is optimal.</p>","PeriodicalId":377306,"journal":{"name":"Transactions of the American Mathematical Society, Series B","volume":"1 1","pages":"0"},"PeriodicalIF":0.0000,"publicationDate":"2021-09-06","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"1","resultStr":"{\"title\":\"Ricci curvature integrals, local functionals, and the Ricci flow\",\"authors\":\"Yuanqing Ma, Bing Wang\",\"doi\":\"10.1090/btran/155\",\"DOIUrl\":null,\"url\":null,\"abstract\":\"<p>Consider a Riemannian manifold <inline-formula content-type=\\\"math/mathml\\\"> <mml:math xmlns:mml=\\\"http://www.w3.org/1998/Math/MathML\\\" alttext=\\\"left-parenthesis upper M Superscript m Baseline comma g right-parenthesis\\\"> <mml:semantics> <mml:mrow> <mml:mo stretchy=\\\"false\\\">(</mml:mo> <mml:msup> <mml:mi>M</mml:mi> <mml:mrow class=\\\"MJX-TeXAtom-ORD\\\"> <mml:mi>m</mml:mi> </mml:mrow> </mml:msup> <mml:mo>,</mml:mo> <mml:mi>g</mml:mi> <mml:mo stretchy=\\\"false\\\">)</mml:mo> </mml:mrow> <mml:annotation encoding=\\\"application/x-tex\\\">(M^{m}, g)</mml:annotation> </mml:semantics> </mml:math> </inline-formula> whose volume is the same as the standard sphere <inline-formula content-type=\\\"math/mathml\\\"> <mml:math xmlns:mml=\\\"http://www.w3.org/1998/Math/MathML\\\" alttext=\\\"left-parenthesis upper S Superscript m Baseline comma g Subscript r o u n d Baseline right-parenthesis\\\"> <mml:semantics> <mml:mrow> <mml:mo stretchy=\\\"false\\\">(</mml:mo> <mml:msup> <mml:mi>S</mml:mi> <mml:mrow class=\\\"MJX-TeXAtom-ORD\\\"> <mml:mi>m</mml:mi> </mml:mrow> </mml:msup> <mml:mo>,</mml:mo> <mml:msub> <mml:mi>g</mml:mi> <mml:mrow class=\\\"MJX-TeXAtom-ORD\\\"> <mml:mi>r</mml:mi> <mml:mi>o</mml:mi> <mml:mi>u</mml:mi> <mml:mi>n</mml:mi> <mml:mi>d</mml:mi> </mml:mrow> </mml:msub> <mml:mo stretchy=\\\"false\\\">)</mml:mo> </mml:mrow> <mml:annotation encoding=\\\"application/x-tex\\\">(S^{m}, g_{round})</mml:annotation> </mml:semantics> </mml:math> </inline-formula>. If <inline-formula content-type=\\\"math/mathml\\\"> <mml:math xmlns:mml=\\\"http://www.w3.org/1998/Math/MathML\\\" alttext=\\\"p greater-than StartFraction m Over 2 EndFraction\\\"> <mml:semantics> <mml:mrow> <mml:mi>p</mml:mi> <mml:mspace width=\\\"negativethinmathspace\\\" /> <mml:mo>></mml:mo> <mml:mspace width=\\\"negativethinmathspace\\\" /> <mml:mfrac> <mml:mi>m</mml:mi> <mml:mn>2</mml:mn> </mml:mfrac> </mml:mrow> <mml:annotation encoding=\\\"application/x-tex\\\">p\\\\!>\\\\!\\\\frac {m}{2}</mml:annotation> </mml:semantics> </mml:math> </inline-formula> and <inline-formula content-type=\\\"math/mathml\\\"> <mml:math xmlns:mml=\\\"http://www.w3.org/1998/Math/MathML\\\" alttext=\\\"integral Underscript upper M Endscripts left-brace upper R c minus left-parenthesis m minus 1 right-parenthesis g right-brace Subscript minus Superscript p Baseline d v\\\"> <mml:semantics> <mml:mrow> <mml:msub> <mml:mo>∫<!-- ∫ --></mml:mo> <mml:mrow class=\\\"MJX-TeXAtom-ORD\\\"> <mml:mi>M</mml:mi> </mml:mrow> </mml:msub> <mml:mspace width=\\\"negativethinmathspace\\\" /> <mml:msubsup> <mml:mrow> <mml:mo>{</mml:mo> <mml:mi>R</mml:mi> <mml:mi>c</mml:mi> <mml:mspace width=\\\"negativethinmathspace\\\" /> <mml:mo>−<!-- − --></mml:mo> <mml:mspace width=\\\"negativethinmathspace\\\" /> <mml:mo stretchy=\\\"false\\\">(</mml:mo> <mml:mi>m</mml:mi> <mml:mspace width=\\\"negativethinmathspace\\\" /> <mml:mo>−<!-- − --></mml:mo> <mml:mspace width=\\\"negativethinmathspace\\\" /> <mml:mn>1</mml:mn> <mml:mo stretchy=\\\"false\\\">)</mml:mo> <mml:mi>g</mml:mi> <mml:mo>}</mml:mo> </mml:mrow> <mml:mrow class=\\\"MJX-TeXAtom-ORD\\\"> <mml:mo>−<!-- − --></mml:mo> </mml:mrow> <mml:mrow class=\\\"MJX-TeXAtom-ORD\\\"> <mml:mi>p</mml:mi> </mml:mrow> </mml:msubsup> <mml:mi>d</mml:mi> <mml:mi>v</mml:mi> </mml:mrow> <mml:annotation encoding=\\\"application/x-tex\\\">\\\\int _{M}\\\\! \\\\left \\\\{ Rc\\\\!-\\\\!(m\\\\!-\\\\!1)g\\\\right \\\\}_{-}^{p} dv</mml:annotation> </mml:semantics> </mml:math> </inline-formula> is sufficiently small, we show that the normalized Ricci flow initiated from <inline-formula content-type=\\\"math/mathml\\\"> <mml:math xmlns:mml=\\\"http://www.w3.org/1998/Math/MathML\\\" alttext=\\\"left-parenthesis upper M Superscript m Baseline comma g right-parenthesis\\\"> <mml:semantics> <mml:mrow> <mml:mo stretchy=\\\"false\\\">(</mml:mo> <mml:msup> <mml:mi>M</mml:mi> <mml:mrow class=\\\"MJX-TeXAtom-ORD\\\"> <mml:mi>m</mml:mi> </mml:mrow> </mml:msup> <mml:mo>,</mml:mo> <mml:mi>g</mml:mi> <mml:mo stretchy=\\\"false\\\">)</mml:mo> </mml:mrow> <mml:annotation encoding=\\\"application/x-tex\\\">(M^{m}, g)</mml:annotation> </mml:semantics> </mml:math> </inline-formula> will exist immortally and converge to the standard sphere. The choice of <inline-formula content-type=\\\"math/mathml\\\"> <mml:math xmlns:mml=\\\"http://www.w3.org/1998/Math/MathML\\\" alttext=\\\"p\\\"> <mml:semantics> <mml:mi>p</mml:mi> <mml:annotation encoding=\\\"application/x-tex\\\">p</mml:annotation> </mml:semantics> </mml:math> </inline-formula> is optimal.</p>\",\"PeriodicalId\":377306,\"journal\":{\"name\":\"Transactions of the American Mathematical Society, Series B\",\"volume\":\"1 1\",\"pages\":\"0\"},\"PeriodicalIF\":0.0000,\"publicationDate\":\"2021-09-06\",\"publicationTypes\":\"Journal Article\",\"fieldsOfStudy\":null,\"isOpenAccess\":false,\"openAccessPdf\":\"\",\"citationCount\":\"1\",\"resultStr\":null,\"platform\":\"Semanticscholar\",\"paperid\":null,\"PeriodicalName\":\"Transactions of the American Mathematical Society, Series B\",\"FirstCategoryId\":\"1085\",\"ListUrlMain\":\"https://doi.org/10.1090/btran/155\",\"RegionNum\":0,\"RegionCategory\":null,\"ArticlePicture\":[],\"TitleCN\":null,\"AbstractTextCN\":null,\"PMCID\":null,\"EPubDate\":\"\",\"PubModel\":\"\",\"JCR\":\"\",\"JCRName\":\"\",\"Score\":null,\"Total\":0}","platform":"Semanticscholar","paperid":null,"PeriodicalName":"Transactions of the American Mathematical Society, Series B","FirstCategoryId":"1085","ListUrlMain":"https://doi.org/10.1090/btran/155","RegionNum":0,"RegionCategory":null,"ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"","JCRName":"","Score":null,"Total":0}
引用次数: 1
摘要
考虑到riemannefold (M M, g)， g)它的体积和标准球形差不多。如果p > m 2 p\ > >\公元\ frac{}{2}和 ∫ M { R c − ( m − 1 ) g } − p d v的int {M} \ !我们向他指出，从mm g (m)到g)，普通的Ricci flow的主动性质将永远存在，并将其转化为标准圈子。p - p选择是最佳的。
Ricci curvature integrals, local functionals, and the Ricci flow
Consider a Riemannian manifold (Mm,g)(M^{m}, g) whose volume is the same as the standard sphere (Sm,ground)(S^{m}, g_{round}). If p>m2p\!>\!\frac {m}{2} and ∫M{Rc−(m−1)g}−pdv\int _{M}\! \left \{ Rc\!-\!(m\!-\!1)g\right \}_{-}^{p} dv is sufficiently small, we show that the normalized Ricci flow initiated from (Mm,g)(M^{m}, g) will exist immortally and converge to the standard sphere. The choice of pp is optimal.
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