具有大量收缩的双曲流形

IF 1.2 2区数学 Q1 MATHEMATICS

Transactions of the American Mathematical Society Pub Date : 2023-10-11 DOI:10.1090/tran/9049

Cayo Dória, Emanoel Freire, Plinio Murillo

{"title":"具有大量收缩的双曲流形","authors":"Cayo Dória, Emanoel Freire, Plinio Murillo","doi":"10.1090/tran/9049","DOIUrl":null,"url":null,"abstract":"In this article, for any <inline-formula content-type=\"math/mathml\"> <mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" alttext=\"n greater-than-or-equal-to 4\"> <mml:semantics> <mml:mrow> <mml:mi>n</mml:mi> <mml:mo>≥</mml:mo> <mml:mn>4</mml:mn> </mml:mrow> <mml:annotation encoding=\"application/x-tex\">n\\geq 4</mml:annotation> </mml:semantics> </mml:math> </inline-formula> we construct a sequence of compact hyperbolic <inline-formula content-type=\"math/mathml\"> <mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" alttext=\"n\"> <mml:semantics> <mml:mi>n</mml:mi> <mml:annotation encoding=\"application/x-tex\">n</mml:annotation> </mml:semantics> </mml:math> </inline-formula>-manifolds <inline-formula content-type=\"math/mathml\"> <mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" alttext=\"left-brace upper M Subscript i Baseline right-brace\"> <mml:semantics> <mml:mrow> <mml:mo fence=\"false\" stretchy=\"false\">{</mml:mo> <mml:msub> <mml:mi>M</mml:mi> <mml:mi>i</mml:mi> </mml:msub> <mml:mo fence=\"false\" stretchy=\"false\">}</mml:mo> </mml:mrow> <mml:annotation encoding=\"application/x-tex\">\\{M_i\\}</mml:annotation> </mml:semantics> </mml:math> </inline-formula> with number of systoles at least as <inline-formula content-type=\"math/mathml\"> <mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" alttext=\"normal v normal o normal l left-parenthesis upper M Subscript i Baseline right-parenthesis Superscript 1 plus StartFraction 1 Over 3 n left-parenthesis n plus 1 right-parenthesis EndFraction minus epsilon\"> <mml:semantics> <mml:mrow> <mml:mrow class=\"MJX-TeXAtom-ORD\"> <mml:mi mathvariant=\"normal\">v</mml:mi> <mml:mi mathvariant=\"normal\">o</mml:mi> <mml:mi mathvariant=\"normal\">l</mml:mi> </mml:mrow> <mml:mo stretchy=\"false\">(</mml:mo> <mml:msub> <mml:mi>M</mml:mi> <mml:mi>i</mml:mi> </mml:msub> <mml:msup> <mml:mo stretchy=\"false\">)</mml:mo> <mml:mrow class=\"MJX-TeXAtom-ORD\"> <mml:mn>1</mml:mn> <mml:mo>+</mml:mo> <mml:mfrac> <mml:mn>1</mml:mn> <mml:mrow> <mml:mn>3</mml:mn> <mml:mi>n</mml:mi> <mml:mo stretchy=\"false\">(</mml:mo> <mml:mi>n</mml:mi> <mml:mo>+</mml:mo> <mml:mn>1</mml:mn> <mml:mo stretchy=\"false\">)</mml:mo> </mml:mrow> </mml:mfrac> <mml:mo>−</mml:mo> <mml:mi>ϵ</mml:mi> </mml:mrow> </mml:msup> </mml:mrow> <mml:annotation encoding=\"application/x-tex\">\\mathrm {vol}(M_i)^{1+\\frac {1}{3n(n+1)}-\\epsilon }</mml:annotation> </mml:semantics> </mml:math> </inline-formula> for any <inline-formula content-type=\"math/mathml\"> <mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" alttext=\"epsilon greater-than 0\"> <mml:semantics> <mml:mrow> <mml:mi>ϵ</mml:mi> <mml:mo>></mml:mo> <mml:mn>0</mml:mn> </mml:mrow> <mml:annotation encoding=\"application/x-tex\">\\epsilon >0</mml:annotation> </mml:semantics> </mml:math> </inline-formula>. In dimension 3, the bound is improved to <inline-formula content-type=\"math/mathml\"> <mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" alttext=\"normal v normal o normal l left-parenthesis upper M Subscript i Baseline right-parenthesis Superscript four thirds minus epsilon\"> <mml:semantics> <mml:mrow> <mml:mrow class=\"MJX-TeXAtom-ORD\"> <mml:mi mathvariant=\"normal\">v</mml:mi> <mml:mi mathvariant=\"normal\">o</mml:mi> <mml:mi mathvariant=\"normal\">l</mml:mi> </mml:mrow> <mml:mo stretchy=\"false\">(</mml:mo> <mml:msub> <mml:mi>M</mml:mi> <mml:mi>i</mml:mi> </mml:msub> <mml:msup> <mml:mo stretchy=\"false\">)</mml:mo> <mml:mrow class=\"MJX-TeXAtom-ORD\"> <mml:mfrac> <mml:mn>4</mml:mn> <mml:mn>3</mml:mn> </mml:mfrac> <mml:mo>−</mml:mo> <mml:mi>ϵ</mml:mi> </mml:mrow> </mml:msup> </mml:mrow> <mml:annotation encoding=\"application/x-tex\">\\mathrm {vol}(M_i)^{\\frac {4}{3}-\\epsilon }</mml:annotation> </mml:semantics> </mml:math> </inline-formula>. These results generalize previous work of Schmutz for <inline-formula content-type=\"math/mathml\"> <mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" alttext=\"n equals 2\"> <mml:semantics> <mml:mrow> <mml:mi>n</mml:mi> <mml:mo>=</mml:mo> <mml:mn>2</mml:mn> </mml:mrow> <mml:annotation encoding=\"application/x-tex\">n=2</mml:annotation> </mml:semantics> </mml:math> </inline-formula>, and Dória-Murillo for <inline-formula content-type=\"math/mathml\"> <mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" alttext=\"n equals 3\"> <mml:semantics> <mml:mrow> <mml:mi>n</mml:mi> <mml:mo>=</mml:mo> <mml:mn>3</mml:mn> </mml:mrow> <mml:annotation encoding=\"application/x-tex\">n=3</mml:annotation> </mml:semantics> </mml:math> </inline-formula> to higher dimensions.","PeriodicalId":23209,"journal":{"name":"Transactions of the American Mathematical Society","volume":null,"pages":null},"PeriodicalIF":1.2000,"publicationDate":"2023-10-11","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"1","resultStr":"{\"title\":\"Hyperbolic manifolds with a large number of systoles\",\"authors\":\"Cayo Dória, Emanoel Freire, Plinio Murillo\",\"doi\":\"10.1090/tran/9049\",\"DOIUrl\":null,\"url\":null,\"abstract\":\"In this article, for any <inline-formula content-type=\\\"math/mathml\\\"> <mml:math xmlns:mml=\\\"http://www.w3.org/1998/Math/MathML\\\" alttext=\\\"n greater-than-or-equal-to 4\\\"> <mml:semantics> <mml:mrow> <mml:mi>n</mml:mi> <mml:mo>≥</mml:mo> <mml:mn>4</mml:mn> </mml:mrow> <mml:annotation encoding=\\\"application/x-tex\\\">n\\\\geq 4</mml:annotation> </mml:semantics> </mml:math> </inline-formula> we construct a sequence of compact hyperbolic <inline-formula content-type=\\\"math/mathml\\\"> <mml:math xmlns:mml=\\\"http://www.w3.org/1998/Math/MathML\\\" alttext=\\\"n\\\"> <mml:semantics> <mml:mi>n</mml:mi> <mml:annotation encoding=\\\"application/x-tex\\\">n</mml:annotation> </mml:semantics> </mml:math> </inline-formula>-manifolds <inline-formula content-type=\\\"math/mathml\\\"> <mml:math xmlns:mml=\\\"http://www.w3.org/1998/Math/MathML\\\" alttext=\\\"left-brace upper M Subscript i Baseline right-brace\\\"> <mml:semantics> <mml:mrow> <mml:mo fence=\\\"false\\\" stretchy=\\\"false\\\">{</mml:mo> <mml:msub> <mml:mi>M</mml:mi> <mml:mi>i</mml:mi> </mml:msub> <mml:mo fence=\\\"false\\\" stretchy=\\\"false\\\">}</mml:mo> </mml:mrow> <mml:annotation encoding=\\\"application/x-tex\\\">\\\\{M_i\\\\}</mml:annotation> </mml:semantics> </mml:math> </inline-formula> with number of systoles at least as <inline-formula content-type=\\\"math/mathml\\\"> <mml:math xmlns:mml=\\\"http://www.w3.org/1998/Math/MathML\\\" alttext=\\\"normal v normal o normal l left-parenthesis upper M Subscript i Baseline right-parenthesis Superscript 1 plus StartFraction 1 Over 3 n left-parenthesis n plus 1 right-parenthesis EndFraction minus epsilon\\\"> <mml:semantics> <mml:mrow> <mml:mrow class=\\\"MJX-TeXAtom-ORD\\\"> <mml:mi mathvariant=\\\"normal\\\">v</mml:mi> <mml:mi mathvariant=\\\"normal\\\">o</mml:mi> <mml:mi mathvariant=\\\"normal\\\">l</mml:mi> </mml:mrow> <mml:mo stretchy=\\\"false\\\">(</mml:mo> <mml:msub> <mml:mi>M</mml:mi> <mml:mi>i</mml:mi> </mml:msub> <mml:msup> <mml:mo stretchy=\\\"false\\\">)</mml:mo> <mml:mrow class=\\\"MJX-TeXAtom-ORD\\\"> <mml:mn>1</mml:mn> <mml:mo>+</mml:mo> <mml:mfrac> <mml:mn>1</mml:mn> <mml:mrow> <mml:mn>3</mml:mn> <mml:mi>n</mml:mi> <mml:mo stretchy=\\\"false\\\">(</mml:mo> <mml:mi>n</mml:mi> <mml:mo>+</mml:mo> <mml:mn>1</mml:mn> <mml:mo stretchy=\\\"false\\\">)</mml:mo> </mml:mrow> </mml:mfrac> <mml:mo>−</mml:mo> <mml:mi>ϵ</mml:mi> </mml:mrow> </mml:msup> </mml:mrow> <mml:annotation encoding=\\\"application/x-tex\\\">\\\\mathrm {vol}(M_i)^{1+\\\\frac {1}{3n(n+1)}-\\\\epsilon }</mml:annotation> </mml:semantics> </mml:math> </inline-formula> for any <inline-formula content-type=\\\"math/mathml\\\"> <mml:math xmlns:mml=\\\"http://www.w3.org/1998/Math/MathML\\\" alttext=\\\"epsilon greater-than 0\\\"> <mml:semantics> <mml:mrow> <mml:mi>ϵ</mml:mi> <mml:mo>></mml:mo> <mml:mn>0</mml:mn> </mml:mrow> <mml:annotation encoding=\\\"application/x-tex\\\">\\\\epsilon >0</mml:annotation> </mml:semantics> </mml:math> </inline-formula>. In dimension 3, the bound is improved to <inline-formula content-type=\\\"math/mathml\\\"> <mml:math xmlns:mml=\\\"http://www.w3.org/1998/Math/MathML\\\" alttext=\\\"normal v normal o normal l left-parenthesis upper M Subscript i Baseline right-parenthesis Superscript four thirds minus epsilon\\\"> <mml:semantics> <mml:mrow> <mml:mrow class=\\\"MJX-TeXAtom-ORD\\\"> <mml:mi mathvariant=\\\"normal\\\">v</mml:mi> <mml:mi mathvariant=\\\"normal\\\">o</mml:mi> <mml:mi mathvariant=\\\"normal\\\">l</mml:mi> </mml:mrow> <mml:mo stretchy=\\\"false\\\">(</mml:mo> <mml:msub> <mml:mi>M</mml:mi> <mml:mi>i</mml:mi> </mml:msub> <mml:msup> <mml:mo stretchy=\\\"false\\\">)</mml:mo> <mml:mrow class=\\\"MJX-TeXAtom-ORD\\\"> <mml:mfrac> <mml:mn>4</mml:mn> <mml:mn>3</mml:mn> </mml:mfrac> <mml:mo>−</mml:mo> <mml:mi>ϵ</mml:mi> </mml:mrow> </mml:msup> </mml:mrow> <mml:annotation encoding=\\\"application/x-tex\\\">\\\\mathrm {vol}(M_i)^{\\\\frac {4}{3}-\\\\epsilon }</mml:annotation> </mml:semantics> </mml:math> </inline-formula>. These results generalize previous work of Schmutz for <inline-formula content-type=\\\"math/mathml\\\"> <mml:math xmlns:mml=\\\"http://www.w3.org/1998/Math/MathML\\\" alttext=\\\"n equals 2\\\"> <mml:semantics> <mml:mrow> <mml:mi>n</mml:mi> <mml:mo>=</mml:mo> <mml:mn>2</mml:mn> </mml:mrow> <mml:annotation encoding=\\\"application/x-tex\\\">n=2</mml:annotation> </mml:semantics> </mml:math> </inline-formula>, and Dória-Murillo for <inline-formula content-type=\\\"math/mathml\\\"> <mml:math xmlns:mml=\\\"http://www.w3.org/1998/Math/MathML\\\" alttext=\\\"n equals 3\\\"> <mml:semantics> <mml:mrow> <mml:mi>n</mml:mi> <mml:mo>=</mml:mo> <mml:mn>3</mml:mn> </mml:mrow> <mml:annotation encoding=\\\"application/x-tex\\\">n=3</mml:annotation> </mml:semantics> </mml:math> </inline-formula> to higher dimensions.\",\"PeriodicalId\":23209,\"journal\":{\"name\":\"Transactions of the American Mathematical Society\",\"volume\":null,\"pages\":null},\"PeriodicalIF\":1.2000,\"publicationDate\":\"2023-10-11\",\"publicationTypes\":\"Journal Article\",\"fieldsOfStudy\":null,\"isOpenAccess\":false,\"openAccessPdf\":\"\",\"citationCount\":\"1\",\"resultStr\":null,\"platform\":\"Semanticscholar\",\"paperid\":null,\"PeriodicalName\":\"Transactions of the American Mathematical Society\",\"FirstCategoryId\":\"1085\",\"ListUrlMain\":\"https://doi.org/10.1090/tran/9049\",\"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":"Transactions of the American Mathematical Society","FirstCategoryId":"1085","ListUrlMain":"https://doi.org/10.1090/tran/9049","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: 1

摘要

在这个文章,对于任何n≥4 \ geq 4我们构造a序列的紧凑型轿车hyperbolic n n -manifolds {M i} {M_i \}用systoles至少美国的当家v o l (M) 1 + 1 3 n (n + 1)−ϵ\ mathrm{卷}(M_i) ^ {1 + \ frac {1} {3n (n + 1)} - \ epsilon} for任何ϵ>0 \epsilon > 0。在第三维度,被束缚的是改良到v o l (M) 4 3−ϵ\ mathrm{卷}(M_i) ^ {\ frac {4} {3} - epsilon的。这些重复一般用于n=2 =2的Schmutz工作，而Doria-Murillo为n=3 =3，以达到更高的维度。

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Hyperbolic manifolds with a large number of systoles

In this article, for any n ≥ 4 n\geq 4 we construct a sequence of compact hyperbolic n n -manifolds { M i } \{M_i\} with number of systoles at least as v o l ( M i ) 1 + 1 3 n ( n + 1 ) − ϵ \mathrm {vol}(M_i)^{1+\frac {1}{3n(n+1)}-\epsilon } for any ϵ > 0 \epsilon >0 . In dimension 3, the bound is improved to v o l ( M i ) 4 3 − ϵ \mathrm {vol}(M_i)^{\frac {4}{3}-\epsilon } . These results generalize previous work of Schmutz for n = 2 n=2 , and Dória-Murillo for n = 3 n=3 to higher dimensions.

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来源期刊

Transactions of the American Mathematical Society 数学-数学

CiteScore

2.30

自引率

7.70%

发文量

171

审稿时长

3-6 weeks

期刊介绍： All articles submitted to this journal are peer-reviewed. The AMS has a single blind peer-review process in which the reviewers know who the authors of the manuscript are, but the authors do not have access to the information on who the peer reviewers are. This journal is devoted to research articles in all areas of pure and applied mathematics. To be published in the Transactions, a paper must be correct, new, and significant. Further, it must be well written and of interest to a substantial number of mathematicians. Piecemeal results, such as an inconclusive step toward an unproved major theorem or a minor variation on a known result, are in general not acceptable for publication. Papers of less than 15 printed pages that meet the above criteria should be submitted to the Proceedings of the American Mathematical Society. Published pages are the same size as those generated in the style files provided for AMS-LaTeX.