Pub Date : 2024-05-07DOI: 10.1515/crelle-2024-0024
Nils Matthes, Morten S. Risager
We determine the asymptotic distribution of Manin’s iterated integrals of length at most 2. For all lengths, we compute all the asymptotic moments. We show that if the length is at least 3, these moments do in general not determine a unique distribution.
{"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}
Pub Date : 2024-04-24DOI: 10.1515/crelle-2024-0022
Robert Haslhofer
We introduce a classification conjecture for κ-solutions in 4d Ricci flow. Our conjectured list includes known examples from the literature, but also a new one-parameter family of ℤ 2 2 × O 3 {mathbb{Z}_{2}^{2}timesmathrm{O}_{3}} -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.
{"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}
Pub Date : 2024-04-11DOI: 10.1515/crelle-2024-0016
Hongjie Dong, Fa Peng, Y. Zhang, Yuan Zhou
We introduce a distributional Jacobian determinant detDVβ(Dv)det DV_{beta}(Dv) in dimension two for the nonlinear complex gradient Vβ(Dv)=|D
我们引入一个分布雅各布行列式 det D V β ( D v ) det DV_{beta}(Dv) 为非线性复梯度 V β ( D v ) = | D v | β ( v x 1 , - v x 2 ) 的二维 DV_{beta}(Dv) V_{beta}(Dv)=lvert Dvrvert^{beta}(v_{x_{1}},-v_{x_{2}}) for any β > - 1 beta>-1 , whenever v ∈ W loc 1 , 2 vin W^{1smash{,}2}_{mathrm{loc}} and β | D v | 1
{"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}
Pub Date : 2024-03-26DOI: 10.1515/crelle-2024-0014
Tianling Jin, Xavier Ros-Oton, Jingang Xiong
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 T*{T^{*}}. More precisely, we show that solutions are C2,α(Ω¯){C^{2,alpha}(overline{Omega})} in space, with α=1m
我们证明了多孔介质方程在光滑有界域中的非负解的最优全局正则性，该方程具有零 Dirichlet 边界条件，经过一定的等待时间 T * {T^{*}} 。 更确切地说，我们证明解是 C 2 , α ( Ω¯ ) {C^{2,alpha}(overline{Omega})} 在空间中，α = 1 m {alpha=frac{1}{m}} 在时间上，C ∞ {C^{infty}} （在 x∈ Ω ¯ {xinoverline{Omega}} 中均匀分布）。 ），对于 t > T * {t>T^{*}} 。 此外，这使我们能够完善大时间解的渐近性，从两个方面改进了迄今已知的最佳结果：我们建立了一个更快的收敛速率 O ( t - 1 - γ ) {O(t^{-1-gamma})}。 我们证明收敛在 C 1 , α ( Ω ¯ ) 中成立 {C^{1,alpha}(overline{Omega})} 拓扑中收敛成立。
{"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}
Pub Date : 2024-03-26DOI: 10.1515/crelle-2024-0008
Imre Bárány
The Steinitz lemma, a classic from 1913, states that a1,…,an{a_{1},ldots,a_{n}}, a sequence of vectors in ℝd{mathbb{R}^{d}} with ∑i=1nai=0
Steinitz Lemma 是 1913 年的一个经典定理，它指出 a 1 , ... , a n {a_{1},ldots,a_{n}} ，∑ i = 1 n a i = 0 {sum_{i=1}^{n}a_{i}=0} 的ℝ d {mathbb{R}^{d} 中的向量序列，可以重新排列，使之与∑ i = 1 n a i = 0 {sum_{i=1}^{n}a_{i}=0} 中的向量序列相等。 可以重新排列，使得重新排列序列的每个部分和的规范最多为 2 d max ∥ a i ∥ {2dmax|a_{i}|} 。在矩阵版本中，A 是一个 k × n {ktimes n} 矩阵，其条目为 a i j∈ ℝ d {a_{i}^{j}inmathbb{R}^{d}} ，∑ j = 1 k ∑ i = 1 n a i j = 0 {sum_{j=1}^{k}sum_{i=1}^{n}a_{i}^{j}=0} 。这在 [T. Oertel, J. Paat] 中得到证明。Oertel、J. Paat 和 R.
{"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}
Pub Date : 2024-03-21DOI: 10.1515/crelle-2024-0011
Francesco Palmurella, Tristan Rivière
We establish a minimal positive existence time of the parametric Willmore flow for any smooth initial data (smooth immersion of a closed oriented surface). The minimal existence time is a function exclusively of geometric data which in particular are all well defined for general weak Lipschitz W2,2W^{2,2} immersions. This fact opens in particular the possibility for defining the Willmore flow for weak Lipschitz W2,2W^{2,2} initial data.
我们为任何光滑初始数据（闭合定向曲面的光滑浸入）建立了参数威尔莫尔流的最小正存在时间。最小存在时间是几何数据的唯一函数，而几何数据对于一般的弱李普齐兹 W 2 , 2 W^{2,2} 浸入都是定义良好的。
{"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}
Pub Date : 2024-02-21DOI: 10.1515/crelle-2024-0004
David Bate, Jakub Takáč
This article studies typical 1-Lipschitz images of 𝑛-rectifiable metric spaces 𝐸 into Rmmathbb{R}^{m} for m≥nmgeq n. For example, if E⊂RkEsubsetmathbb{R}^{k}, we show that the Jacobian of such a typical 1-Lipschitz map equals 1 Hnmathcal{H}^{n}
例如，如果 E ⊂ R k Esubsetmathbb{R}^{k} ，我们会发现这样一个典型的 1-Lipschitz 映射的 Jacobian 几乎在所有地方都等于 1 H n mathcal{H}^{n} -，并且如果 m > n m>n ，则 1 Lipschitz 映射的 Jacobian 几乎在所有地方都等于 1 H n mathcal{H}^{n} -。 我们证明，这样一个典型的 1-Lipschitz 映射的 Jacobian 几乎在所有地方都等于 1 H n mathcal{H}^{n} ，并且，如果 m > n m>n ，会保留𝐸的 Hausdorff 度量。一般来说，我们从𝐸的切线规范出发，提供了典型的1-Lipschitz映射保留𝐸的Hausdorff度量的充分条件，直至某个常数倍数。
{"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}
Pub Date : 2024-02-20DOI: 10.1515/crelle-2023-0102
Ursula Hamenstädt
Let S be a closed oriented surface of genus g≥0{ggeq 0} with n≥0{ngeq 0} punctures and 3g-3+n≥5{3g-3+ngeq 5}. Let
Let S be a closed oriented surface ofgenus g ≥ 0 {ggeq 0} with n ≥ 0 {ngeq 0} punctures and 3 g - 3 + n ≥ 5 {3g-3+ngeq 5} .Let 𝒬 {{mathcal{Q}}} be a connected componentof a stratum in the moduli space 𝒬 ( S ) {{mathcal{Q}}(S)} of area onemeromorphic quadratic differentials on S with nsimple poles at the puncturesor in the moduli space ℋ ( S ) {{mathcal{H}}(S)} of abelian differentials on S if n = 0 {n=0} .For a compact subset K of 𝒬 ( S ) {{mathcal{Q}}(S)} or of ℋ ( S ) {{mathcal{H}}(S)} ,we show that the asymptotic growth rate of the number of periodic orbits for theTeichmüller flow Φ t {Phi^{t}} on 𝒬 {{mathcal{Q}}} which are entirely contained in 𝒬 - K
Pub Date : 2024-01-30DOI: 10.1515/crelle-2024-0002
K. Böröczky, Danylo Radchenko, João P. G. Ramos
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 ∼ε{simvarepsilon} close to satisfying the optimal density, then it is, in a suitable sense, close to the E8{E_{8}} 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 E8{E_{8}} or Λ24
我们证明了维数为8和24的球体堆积问题的显式稳定性估计，表明在晶格情况下，如果一个晶格接近于满足最优密度，那么在合适的意义上，它分别接近于E 8 {E_{8}} 和Leech晶格。在周期设置中，我们证明，在同样的假设条件下，我们可以取一个大的 "框架"，通过这个框架，我们的堆积局部看起来像 E 8 {E_{8}} 或 Λ 24 {Lambda_{24}} 。 我们的方法明确使用了 [M. S.] 中构建的魔法函数。S.Viazovska,The sphere packing problem in dimension 8,Ann. of Math. (2) 185 2017, 3, 991-1015]中在维度 8 和[H.Cohn, A. Kumar, S. D.Miller, D. Radchenko and M. Viazovska,The sphere packing problem in dimension 24,Ann. of Math. (2) 185 2017, 3, 1017-1033] 中的维度 24，以及关于网格 E 8 {E_{8}} 和Λ 24 {Lambda_{24}} 的抽象稳定性的独立结果。 .
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Pub Date : 2024-01-02DOI: 10.1515/crelle-2023-0093
Tara E. Brendle, Dan Margalit
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} .
摘要 标题中提到的论文中的定理 5.1 的第一个陈述的证明对于 k = 1 {k=1} 是正确的，而对于 k ≥ 2 {kgeq 2} 则是不正确的，应被视为一个未决问题。因此，对于 k ≥ 2 {kgeq 2} 而言，第二个陈述的证明是不正确的。
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