Journal für die reine und angewandte Mathematik (Crelles Journal)最新文献

{"title":"The distribution of Manin’s iterated integrals of modular forms","authors":"Nils Matthes, Morten S. Risager","doi":"10.1515/crelle-2024-0024","DOIUrl":"https://doi.org/10.1515/crelle-2024-0024","url":null,"abstract":"\u0000 We determine the asymptotic distribution of Manin’s iterated integrals of length at most 2.\u0000For all lengths, we compute all the asymptotic moments.\u0000We show that if the length is at least 3, these moments do in general not determine a unique distribution.","PeriodicalId":508691,"journal":{"name":"Journal für die reine und angewandte Mathematik (Crelles Journal)","volume":"24 10","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-05-07","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141005656","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

{"title":"On κ-solutions andbreak canonical neighborhoods in 4d Ricci flow","authors":"Robert Haslhofer","doi":"10.1515/crelle-2024-0022","DOIUrl":"https://doi.org/10.1515/crelle-2024-0022","url":null,"abstract":"\u0000 We introduce a classification conjecture for κ-solutions in 4d Ricci flow. Our conjectured list\u0000includes known examples from the literature, but also a new one-parameter family of \u0000 \u0000 \u0000 \u0000 \u0000 ℤ\u0000 2\u0000 2\u0000 \u0000 ×\u0000 \u0000 O\u0000 3\u0000 \u0000 \u0000 \u0000 \u0000 {mathbb{Z}_{2}^{2}timesmathrm{O}_{3}}\u0000 \u0000 -symmetric bubble-sheet ovals that we construct. We observe that some special cases of the conjecture follow from recent results in the literature. We also introduce a stronger variant of the classification conjecture for ancient asymptotically cylindrical 4d Ricci flows, which does not assume smoothness and nonnegative curvature operator a priori. Assuming this stronger variant holds true, we establish a canonical neighborhood theorem for 4d Ricci flow through cylindrical singularities, which shares some elements in common with Perelman’s canonical neighborhood theorem for 3d Ricci flow as well as the mean-convex neighborhood theorem for mean curvature flow through neck-singularities. Finally, we argue that quotient-necks lead to new phenomena, and sketch an example of non-uniqueness for 4d Ricci flow through singularities.","PeriodicalId":508691,"journal":{"name":"Journal für die reine und angewandte Mathematik (Crelles Journal)","volume":"50 10","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-04-24","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"140663008","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

{"title":"Jacobian determinants for nonlinear gradient of planar ∞-harmonic functions and applications","authors":"Hongjie Dong, Fa Peng, Y. Zhang, Yuan Zhou","doi":"10.1515/crelle-2024-0016","DOIUrl":"https://doi.org/10.1515/crelle-2024-0016","url":null,"abstract":"\u0000 <jats:p>We introduce a distributional Jacobian determinant <jats:inline-formula>\u0000 <jats:alternatives>\u0000 <m:math xmlns:m=\"http://www.w3.org/1998/Math/MathML\">\u0000 <m:mrow>\u0000 <m:mo rspace=\"0.167em\">det</m:mo>\u0000 <m:mrow>\u0000 <m:mi>D</m:mi>\u0000 <m:mo>⁢</m:mo>\u0000 <m:msub>\u0000 <m:mi>V</m:mi>\u0000 <m:mi>β</m:mi>\u0000 </m:msub>\u0000 <m:mo>⁢</m:mo>\u0000 <m:mrow>\u0000 <m:mo stretchy=\"false\">(</m:mo>\u0000 <m:mrow>\u0000 <m:mi>D</m:mi>\u0000 <m:mo>⁢</m:mo>\u0000 <m:mi>v</m:mi>\u0000 </m:mrow>\u0000 <m:mo stretchy=\"false\">)</m:mo>\u0000 </m:mrow>\u0000 </m:mrow>\u0000 </m:mrow>\u0000 </m:math>\u0000 <jats:inline-graphic xmlns:xlink=\"http://www.w3.org/1999/xlink\" xlink:href=\"graphic/j_crelle-2024-0016_ineq_0001.png\" />\u0000 <jats:tex-math>det DV_{beta}(Dv)</jats:tex-math>\u0000 </jats:alternatives>\u0000 </jats:inline-formula> in dimension two for the nonlinear complex gradient <jats:inline-formula>\u0000 <jats:alternatives>\u0000 <m:math xmlns:m=\"http://www.w3.org/1998/Math/MathML\">\u0000 <m:mrow>\u0000 <m:mrow>\u0000 <m:msub>\u0000 <m:mi>V</m:mi>\u0000 <m:mi>β</m:mi>\u0000 </m:msub>\u0000 <m:mo>⁢</m:mo>\u0000 <m:mrow>\u0000 <m:mo stretchy=\"false\">(</m:mo>\u0000 <m:mrow>\u0000 <m:mi>D</m:mi>\u0000 <m:mo>⁢</m:mo>\u0000 <m:mi>v</m:mi>\u0000 </m:mrow>\u0000 <m:mo stretchy=\"false\">)</m:mo>\u0000 </m:mrow>\u0000 </m:mrow>\u0000 <m:mo>=</m:mo>\u0000 <m:mrow>\u0000 <m:msup>\u0000 <m:mrow>\u0000 <m:mo stretchy=\"false\">|</m:mo>\u0000 <m:mrow>\u0000 <m:mi>D</m:mi>\u0000 <m:mo>⁢</m:mo>\u0000 ","PeriodicalId":508691,"journal":{"name":"Journal für die reine und angewandte Mathematik (Crelles Journal)","volume":"15 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-04-11","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"140714514","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

{"title":"Optimal regularity and fine asymptotics for the porous medium equation in bounded domains","authors":"Tianling Jin, Xavier Ros-Oton, Jingang Xiong","doi":"10.1515/crelle-2024-0014","DOIUrl":"https://doi.org/10.1515/crelle-2024-0014","url":null,"abstract":"\u0000 <jats:p>We prove the optimal global regularity of nonnegative solutions to the porous medium equation in smooth bounded domains with the zero Dirichlet boundary condition after certain waiting time <jats:inline-formula id=\"j_crelle-2024-0014_ineq_9999\">\u0000 <jats:alternatives>\u0000 <m:math xmlns:m=\"http://www.w3.org/1998/Math/MathML\">\u0000 <m:msup>\u0000 <m:mi>T</m:mi>\u0000 <m:mo>*</m:mo>\u0000 </m:msup>\u0000 </m:math>\u0000 <jats:inline-graphic xmlns:xlink=\"http://www.w3.org/1999/xlink\" xlink:href=\"graphic/j_crelle-2024-0014_eq_0436.png\" />\u0000 <jats:tex-math>{T^{*}}</jats:tex-math>\u0000 </jats:alternatives>\u0000 </jats:inline-formula>.\u0000More precisely, we show that solutions are <jats:inline-formula id=\"j_crelle-2024-0014_ineq_9998\">\u0000 <jats:alternatives>\u0000 <m:math xmlns:m=\"http://www.w3.org/1998/Math/MathML\">\u0000 <m:mrow>\u0000 <m:msup>\u0000 <m:mi>C</m:mi>\u0000 <m:mrow>\u0000 <m:mn>2</m:mn>\u0000 <m:mo>,</m:mo>\u0000 <m:mi>α</m:mi>\u0000 </m:mrow>\u0000 </m:msup>\u0000 <m:mo>⁢</m:mo>\u0000 <m:mrow>\u0000 <m:mo stretchy=\"false\">(</m:mo>\u0000 <m:mover accent=\"true\">\u0000 <m:mi mathvariant=\"normal\">Ω</m:mi>\u0000 <m:mo>¯</m:mo>\u0000 </m:mover>\u0000 <m:mo stretchy=\"false\">)</m:mo>\u0000 </m:mrow>\u0000 </m:mrow>\u0000 </m:math>\u0000 <jats:inline-graphic xmlns:xlink=\"http://www.w3.org/1999/xlink\" xlink:href=\"graphic/j_crelle-2024-0014_eq_0398.png\" />\u0000 <jats:tex-math>{C^{2,alpha}(overline{Omega})}</jats:tex-math>\u0000 </jats:alternatives>\u0000 </jats:inline-formula> in space, with <jats:inline-formula id=\"j_crelle-2024-0014_ineq_9997\">\u0000 <jats:alternatives>\u0000 <m:math xmlns:m=\"http://www.w3.org/1998/Math/MathML\">\u0000 <m:mrow>\u0000 <m:mi>α</m:mi>\u0000 <m:mo>=</m:mo>\u0000 <m:mfrac>\u0000 <m:mn>1</m:mn>\u0000 <m:mi>m</m:mi>\u0000 </m:mfrac>\u0000 </m:mrow>\u0000 </m:math>\u0000 <jats:inline-graphic xmlns:xlink=\"http://www.w3.org/1999","PeriodicalId":508691,"journal":{"name":"Journal für die reine und angewandte Mathematik (Crelles Journal)","volume":"39 8","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-03-26","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"140378376","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

{"title":"A matrix version of the Steinitz lemma","authors":"Imre Bárány","doi":"10.1515/crelle-2024-0008","DOIUrl":"https://doi.org/10.1515/crelle-2024-0008","url":null,"abstract":"\u0000 <jats:p>The Steinitz lemma, a classic from 1913, states that <jats:inline-formula id=\"j_crelle-2024-0008_ineq_9999\">\u0000 <jats:alternatives>\u0000 <m:math xmlns:m=\"http://www.w3.org/1998/Math/MathML\">\u0000 <m:mrow>\u0000 <m:msub>\u0000 <m:mi>a</m:mi>\u0000 <m:mn>1</m:mn>\u0000 </m:msub>\u0000 <m:mo>,</m:mo>\u0000 <m:mi mathvariant=\"normal\">…</m:mi>\u0000 <m:mo>,</m:mo>\u0000 <m:msub>\u0000 <m:mi>a</m:mi>\u0000 <m:mi>n</m:mi>\u0000 </m:msub>\u0000 </m:mrow>\u0000 </m:math>\u0000 <jats:inline-graphic xmlns:xlink=\"http://www.w3.org/1999/xlink\" xlink:href=\"graphic/j_crelle-2024-0008_eq_0141.png\" />\u0000 <jats:tex-math>{a_{1},ldots,a_{n}}</jats:tex-math>\u0000 </jats:alternatives>\u0000 </jats:inline-formula>, a sequence of vectors in <jats:inline-formula id=\"j_crelle-2024-0008_ineq_9998\">\u0000 <jats:alternatives>\u0000 <m:math xmlns:m=\"http://www.w3.org/1998/Math/MathML\">\u0000 <m:msup>\u0000 <m:mi>ℝ</m:mi>\u0000 <m:mi>d</m:mi>\u0000 </m:msup>\u0000 </m:math>\u0000 <jats:inline-graphic xmlns:xlink=\"http://www.w3.org/1999/xlink\" xlink:href=\"graphic/j_crelle-2024-0008_eq_0091.png\" />\u0000 <jats:tex-math>{mathbb{R}^{d}}</jats:tex-math>\u0000 </jats:alternatives>\u0000 </jats:inline-formula> with <jats:inline-formula id=\"j_crelle-2024-0008_ineq_9997\">\u0000 <jats:alternatives>\u0000 <m:math xmlns:m=\"http://www.w3.org/1998/Math/MathML\">\u0000 <m:mrow>\u0000 <m:mrow>\u0000 <m:msubsup>\u0000 <m:mo largeop=\"true\" symmetric=\"true\">∑</m:mo>\u0000 <m:mrow>\u0000 <m:mi>i</m:mi>\u0000 <m:mo>=</m:mo>\u0000 <m:mn>1</m:mn>\u0000 </m:mrow>\u0000 <m:mi>n</m:mi>\u0000 </m:msubsup>\u0000 <m:msub>\u0000 <m:mi>a</m:mi>\u0000 <m:mi>i</m:mi>\u0000 </m:msub>\u0000 </m:mrow>\u0000 <m:mo>=</m:mo>\u0000 <m:mn>0</m:mn>\u0000 </m:mrow>\u0000 </m:math>\u0000 ","PeriodicalId":508691,"journal":{"name":"Journal für die reine und angewandte Mathematik (Crelles Journal)","volume":"23 8","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-03-26","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"140378491","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

{"title":"The parametric Willmore flow","authors":"Francesco Palmurella, Tristan Rivière","doi":"10.1515/crelle-2024-0011","DOIUrl":"https://doi.org/10.1515/crelle-2024-0011","url":null,"abstract":"\u0000 <jats:p>We establish a minimal positive existence time of the parametric Willmore flow for any smooth initial data (smooth immersion of a closed oriented surface).\u0000The minimal existence time is a function exclusively of geometric data which in particular are all well defined for general weak Lipschitz <jats:inline-formula>\u0000 <jats:alternatives>\u0000 <m:math xmlns:m=\"http://www.w3.org/1998/Math/MathML\">\u0000 <m:msup>\u0000 <m:mi>W</m:mi>\u0000 <m:mrow>\u0000 <m:mn>2</m:mn>\u0000 <m:mo>,</m:mo>\u0000 <m:mn>2</m:mn>\u0000 </m:mrow>\u0000 </m:msup>\u0000 </m:math>\u0000 <jats:inline-graphic xmlns:xlink=\"http://www.w3.org/1999/xlink\" xlink:href=\"graphic/j_crelle-2024-0011_ineq_0001.png\" />\u0000 <jats:tex-math>W^{2,2}</jats:tex-math>\u0000 </jats:alternatives>\u0000 </jats:inline-formula> immersions.\u0000This fact opens in particular the possibility for defining the Willmore flow for weak Lipschitz <jats:inline-formula>\u0000 <jats:alternatives>\u0000 <m:math xmlns:m=\"http://www.w3.org/1998/Math/MathML\">\u0000 <m:msup>\u0000 <m:mi>W</m:mi>\u0000 <m:mrow>\u0000 <m:mn>2</m:mn>\u0000 <m:mo>,</m:mo>\u0000 <m:mn>2</m:mn>\u0000 </m:mrow>\u0000 </m:msup>\u0000 </m:math>\u0000 <jats:inline-graphic xmlns:xlink=\"http://www.w3.org/1999/xlink\" xlink:href=\"graphic/j_crelle-2024-0011_ineq_0001.png\" />\u0000 <jats:tex-math>W^{2,2}</jats:tex-math>\u0000 </jats:alternatives>\u0000 </jats:inline-formula> initial data.</jats:p>","PeriodicalId":508691,"journal":{"name":"Journal für die reine und angewandte Mathematik (Crelles Journal)","volume":"128 4","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-03-21","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"140223343","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

{"title":"Typical Lipschitz images of rectifiable metric spaces","authors":"David Bate, Jakub Takáč","doi":"10.1515/crelle-2024-0004","DOIUrl":"https://doi.org/10.1515/crelle-2024-0004","url":null,"abstract":"\u0000 <jats:p>This article studies typical 1-Lipschitz images of 𝑛-rectifiable metric spaces 𝐸 into <jats:inline-formula>\u0000 <jats:alternatives>\u0000 <m:math xmlns:m=\"http://www.w3.org/1998/Math/MathML\">\u0000 <m:msup>\u0000 <m:mi mathvariant=\"double-struck\">R</m:mi>\u0000 <m:mi>m</m:mi>\u0000 </m:msup>\u0000 </m:math>\u0000 <jats:inline-graphic xmlns:xlink=\"http://www.w3.org/1999/xlink\" xlink:href=\"graphic/j_crelle-2024-0004_ineq_0001.png\" />\u0000 <jats:tex-math>mathbb{R}^{m}</jats:tex-math>\u0000 </jats:alternatives>\u0000 </jats:inline-formula> for <jats:inline-formula>\u0000 <jats:alternatives>\u0000 <m:math xmlns:m=\"http://www.w3.org/1998/Math/MathML\">\u0000 <m:mrow>\u0000 <m:mi>m</m:mi>\u0000 <m:mo>≥</m:mo>\u0000 <m:mi>n</m:mi>\u0000 </m:mrow>\u0000 </m:math>\u0000 <jats:inline-graphic xmlns:xlink=\"http://www.w3.org/1999/xlink\" xlink:href=\"graphic/j_crelle-2024-0004_ineq_0002.png\" />\u0000 <jats:tex-math>mgeq n</jats:tex-math>\u0000 </jats:alternatives>\u0000 </jats:inline-formula>.\u0000For example, if <jats:inline-formula>\u0000 <jats:alternatives>\u0000 <m:math xmlns:m=\"http://www.w3.org/1998/Math/MathML\">\u0000 <m:mrow>\u0000 <m:mi>E</m:mi>\u0000 <m:mo>⊂</m:mo>\u0000 <m:msup>\u0000 <m:mi mathvariant=\"double-struck\">R</m:mi>\u0000 <m:mi>k</m:mi>\u0000 </m:msup>\u0000 </m:mrow>\u0000 </m:math>\u0000 <jats:inline-graphic xmlns:xlink=\"http://www.w3.org/1999/xlink\" xlink:href=\"graphic/j_crelle-2024-0004_ineq_0003.png\" />\u0000 <jats:tex-math>Esubsetmathbb{R}^{k}</jats:tex-math>\u0000 </jats:alternatives>\u0000 </jats:inline-formula>, we show that the Jacobian of such a typical 1-Lipschitz map equals 1 <jats:inline-formula>\u0000 <jats:alternatives>\u0000 <m:math xmlns:m=\"http://www.w3.org/1998/Math/MathML\">\u0000 <m:msup>\u0000 <m:mi mathvariant=\"script\">H</m:mi>\u0000 <m:mi>n</m:mi>\u0000 </m:msup>\u0000 </m:math>\u0000 <jats:inline-graphic xmlns:xlink=\"http://www.w3.org/1999/xlink\" xlink:href=\"graphic/j_crelle-2024-0004_ineq_0004.png\" />\u0000 <jats:tex-math>mathcal{H}^{n}</jats:tex-math>\u0000 </jats:alternatives>\u0000","PeriodicalId":508691,"journal":{"name":"Journal für die reine und angewandte Mathematik (Crelles Journal)","volume":"12 6","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-02-21","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"140442083","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

{"title":"Periodic orbits in the thin part of strata","authors":"Ursula Hamenstädt","doi":"10.1515/crelle-2023-0102","DOIUrl":"https://doi.org/10.1515/crelle-2023-0102","url":null,"abstract":"\u0000 <jats:p>Let <jats:italic>S</jats:italic> be a closed oriented surface of\u0000genus <jats:inline-formula id=\"j_crelle-2023-0102_ineq_9999\">\u0000 <jats:alternatives>\u0000 <m:math xmlns:m=\"http://www.w3.org/1998/Math/MathML\">\u0000 <m:mrow>\u0000 <m:mi>g</m:mi>\u0000 <m:mo>≥</m:mo>\u0000 <m:mn>0</m:mn>\u0000 </m:mrow>\u0000 </m:math>\u0000 <jats:inline-graphic xmlns:xlink=\"http://www.w3.org/1999/xlink\" xlink:href=\"graphic/j_crelle-2023-0102_eq_0733.png\" />\u0000 <jats:tex-math>{ggeq 0}</jats:tex-math>\u0000 </jats:alternatives>\u0000 </jats:inline-formula> with <jats:inline-formula id=\"j_crelle-2023-0102_ineq_9998\">\u0000 <jats:alternatives>\u0000 <m:math xmlns:m=\"http://www.w3.org/1998/Math/MathML\">\u0000 <m:mrow>\u0000 <m:mi>n</m:mi>\u0000 <m:mo>≥</m:mo>\u0000 <m:mn>0</m:mn>\u0000 </m:mrow>\u0000 </m:math>\u0000 <jats:inline-graphic xmlns:xlink=\"http://www.w3.org/1999/xlink\" xlink:href=\"graphic/j_crelle-2023-0102_eq_0798.png\" />\u0000 <jats:tex-math>{ngeq 0}</jats:tex-math>\u0000 </jats:alternatives>\u0000 </jats:inline-formula> punctures and\u0000<jats:inline-formula id=\"j_crelle-2023-0102_ineq_9997\">\u0000 <jats:alternatives>\u0000 <m:math xmlns:m=\"http://www.w3.org/1998/Math/MathML\">\u0000 <m:mrow>\u0000 <m:mrow>\u0000 <m:mrow>\u0000 <m:mrow>\u0000 <m:mn>3</m:mn>\u0000 <m:mo>⁢</m:mo>\u0000 <m:mi>g</m:mi>\u0000 </m:mrow>\u0000 <m:mo>-</m:mo>\u0000 <m:mn>3</m:mn>\u0000 </m:mrow>\u0000 <m:mo>+</m:mo>\u0000 <m:mi>n</m:mi>\u0000 </m:mrow>\u0000 <m:mo>≥</m:mo>\u0000 <m:mn>5</m:mn>\u0000 </m:mrow>\u0000 </m:math>\u0000 <jats:inline-graphic xmlns:xlink=\"http://www.w3.org/1999/xlink\" xlink:href=\"graphic/j_crelle-2023-0102_eq_0143.png\" />\u0000 <jats:tex-math>{3g-3+ngeq 5}</jats:tex-math>\u0000 </jats:alternatives>\u0000 </jats:inline-formula>.\u0000Let <jats:inline-formula id=\"j_crelle-2023-0102_ineq_9996\">\u0000 <jats:alternatives>\u0000 <m:math xmlns:m=\"http://www.w3.org/1998/Math/MathML\">\u0000 ","PeriodicalId":508691,"journal":{"name":"Journal für die reine und angewandte Mathematik (Crelles Journal)","volume":"37 27","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-02-20","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"140449483","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

{"title":"A quantitative stability result for the sphere packing problem in dimensions 8 and 24","authors":"K. Böröczky, Danylo Radchenko, João P. G. Ramos","doi":"10.1515/crelle-2024-0002","DOIUrl":"https://doi.org/10.1515/crelle-2024-0002","url":null,"abstract":"\u0000 <jats:p>We prove explicit stability estimates for the sphere packing problem in dimensions 8 and 24, showing that, in the lattice case, if a lattice is <jats:inline-formula id=\"j_crelle-2024-0002_ineq_9999\">\u0000 <jats:alternatives>\u0000 <m:math xmlns:m=\"http://www.w3.org/1998/Math/MathML\">\u0000 <m:mrow>\u0000 <m:mi />\u0000 <m:mo>∼</m:mo>\u0000 <m:mi>ε</m:mi>\u0000 </m:mrow>\u0000 </m:math>\u0000 <jats:inline-graphic xmlns:xlink=\"http://www.w3.org/1999/xlink\" xlink:href=\"graphic/j_crelle-2024-0002_eq_0559.png\" />\u0000 <jats:tex-math>{simvarepsilon}</jats:tex-math>\u0000 </jats:alternatives>\u0000 </jats:inline-formula> close to satisfying the optimal density, then it is, in a suitable sense, close to the <jats:inline-formula id=\"j_crelle-2024-0002_ineq_9998\">\u0000 <jats:alternatives>\u0000 <m:math xmlns:m=\"http://www.w3.org/1998/Math/MathML\">\u0000 <m:msub>\u0000 <m:mi>E</m:mi>\u0000 <m:mn>8</m:mn>\u0000 </m:msub>\u0000 </m:math>\u0000 <jats:inline-graphic xmlns:xlink=\"http://www.w3.org/1999/xlink\" xlink:href=\"graphic/j_crelle-2024-0002_eq_0349.png\" />\u0000 <jats:tex-math>{E_{8}}</jats:tex-math>\u0000 </jats:alternatives>\u0000 </jats:inline-formula> and Leech lattices, respectively. In the periodic setting, we prove that, under the same assumptions, we may take a large “frame” through which our packing locally looks like <jats:inline-formula id=\"j_crelle-2024-0002_ineq_9997\">\u0000 <jats:alternatives>\u0000 <m:math xmlns:m=\"http://www.w3.org/1998/Math/MathML\">\u0000 <m:msub>\u0000 <m:mi>E</m:mi>\u0000 <m:mn>8</m:mn>\u0000 </m:msub>\u0000 </m:math>\u0000 <jats:inline-graphic xmlns:xlink=\"http://www.w3.org/1999/xlink\" xlink:href=\"graphic/j_crelle-2024-0002_eq_0349.png\" />\u0000 <jats:tex-math>{E_{8}}</jats:tex-math>\u0000 </jats:alternatives>\u0000 </jats:inline-formula> or <jats:inline-formula id=\"j_crelle-2024-0002_ineq_9996\">\u0000 <jats:alternatives>\u0000 <m:math xmlns:m=\"http://www.w3.org/1998/Math/MathML\">\u0000 <m:msub>\u0000 <m:mi mathvariant=\"normal\">Λ</m:mi>\u0000 <m:mn>24</m:mn>\u0000 </m:msub>\u0000 </m:math>\u0000 <jats:inline-graphic xmlns:xlink=\"http://www.w3.org/1999/xlink\" xlink:href=\"graphic/j_crelle-2024-0002_eq_0432.png\" />\u0000","PeriodicalId":508691,"journal":{"name":"Journal für die reine und angewandte Mathematik (Crelles Journal)","volume":"109 ","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-01-30","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"140484391","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

{"title":"Erratum to The level four braid group (J. reine angew. Math. 735 (2018), 249–264))","authors":"Tara E. Brendle, Dan Margalit","doi":"10.1515/crelle-2023-0093","DOIUrl":"https://doi.org/10.1515/crelle-2023-0093","url":null,"abstract":"Abstract The proof of the first statement of Theorem 5.1 of the paper referenced in the title is correct for k = 1 {k=1} and incorrect for k ≥ 2 {kgeq 2} and should be considered an open problem. As such, the proof of the second statement is not correct for k ≥ 2 {kgeq 2} .","PeriodicalId":508691,"journal":{"name":"Journal für die reine und angewandte Mathematik (Crelles Journal)","volume":"13 3","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-01-02","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"139124410","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}