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Surface groups in uniform lattices of some semi-simple groups 一些半简单群均匀网格中的面群
IF 3.7 1区 数学
Acta Mathematica Pub Date : 2024-05-10 DOI: 10.4310/acta.2024.v232.n1.a2
Jeremy Kahn, François Labourie, Mozes Shahar
{"title":"Surface groups in uniform lattices of some semi-simple groups","authors":"Jeremy Kahn, François Labourie, Mozes Shahar","doi":"10.4310/acta.2024.v232.n1.a2","DOIUrl":"https://doi.org/10.4310/acta.2024.v232.n1.a2","url":null,"abstract":"We show that uniform lattices in some semi-simple groups (notably complex ones) admit Anosov surface subgroups. This result has a quantitative version: we introduce a notion, called $K$-Sullivan maps, which generalizes the notion of $K$-quasi-circles in hyperbolic geometry, and show in particular that Sullivan maps are Hölder. Using this notion, we show a quantitative version of our surface subgroup theorem, and in particular that one can obtain $K$-Sullivan limit maps, as close as one wants to smooth round circles. All these results use the coarse geometry of “path of triangles” in a certain flag manifold, and we prove an analogue to the Morse Lemma for quasi-geodesics in that context.","PeriodicalId":50895,"journal":{"name":"Acta Mathematica","volume":"36 1","pages":""},"PeriodicalIF":3.7,"publicationDate":"2024-05-10","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"140939276","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":1,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
引用次数: 0
The dynamical Kirchberg–Phillips theorem 动力基希贝格-菲利普斯定理
IF 3.7 1区 数学
Acta Mathematica Pub Date : 2024-05-10 DOI: 10.4310/acta.2024.v232.n1.a1
James Gabe, Gábor Szabó
{"title":"The dynamical Kirchberg–Phillips theorem","authors":"James Gabe, Gábor Szabó","doi":"10.4310/acta.2024.v232.n1.a1","DOIUrl":"https://doi.org/10.4310/acta.2024.v232.n1.a1","url":null,"abstract":"Let $G$ be a second-countable, locally compact group. In this article we study amenable $G$-actions on Kirchberg algebras that admit an approximately central embedding of a canonical quasi-free action on the Cuntz algebra $mathcal{O}_{^infty}$. If $G$ is discrete, this coincides with the class of amenable and outer $G-$actions on Kirchberg algebras. We show that the resulting $G-C^ast$-dynamical systems are classified by equivariant Kasparov theory, up to cocycle conjugacy. This is the first classification theory of its kind applicable to actions of arbitrary locally compact groups. Among various applications, our main result solves a conjecture of Izumi for actions of discrete amenable torsion-free groups, and recovers the main results of recent work by Izumi–Matui for actions of poly-$mathbb{Z}$ groups.","PeriodicalId":50895,"journal":{"name":"Acta Mathematica","volume":"2 1","pages":""},"PeriodicalIF":3.7,"publicationDate":"2024-05-10","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"140939220","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":1,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
引用次数: 0
Correction to “On the geometry of metric measure spaces. I” 对 "论度量空间的几何。I"
IF 3.7 1区 数学
Acta Mathematica Pub Date : 2023-12-19 DOI: 10.4310/acta.2023.v231.n2.a3
Karl-Theodor Sturm
{"title":"Correction to “On the geometry of metric measure spaces. I”","authors":"Karl-Theodor Sturm","doi":"10.4310/acta.2023.v231.n2.a3","DOIUrl":"https://doi.org/10.4310/acta.2023.v231.n2.a3","url":null,"abstract":"This is a correction to $href{https://dx.doi.org/10.1007/s11511-006-0002-8}{[11]}$ (<i>Acta Math.</i>), as well as to the follow-up publications $href{https://doi.org/10.1016/j.jfa.2010.03.024}{[3]}$ and $href{https://doi.org/10.1016/j.jfa.2011.02.026}{[5]}$ (both in <i>J. Funct. Anal.</i>).","PeriodicalId":50895,"journal":{"name":"Acta Mathematica","volume":"10 1","pages":""},"PeriodicalIF":3.7,"publicationDate":"2023-12-19","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"138745202","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":1,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
引用次数: 0
On the boundaries of highly connected, almost closed manifolds 关于高度连接的几乎封闭流形的边界
IF 3.7 1区 数学
Acta Mathematica Pub Date : 2023-12-19 DOI: 10.4310/acta.2023.v231.n2.a1
Robert Burklund, Jeremy Hahn, Andrew Senger
{"title":"On the boundaries of highly connected, almost closed manifolds","authors":"Robert Burklund, Jeremy Hahn, Andrew Senger","doi":"10.4310/acta.2023.v231.n2.a1","DOIUrl":"https://doi.org/10.4310/acta.2023.v231.n2.a1","url":null,"abstract":"Building on work of Stolz, we prove for integers $0 leqslant d leqslant 3$ and $k gt 232$ that the boundaries of $(k-1)$-connected, almost closed $(2k+d)$-manifolds also bound parallelizable manifolds. Away from finitely many dimensions, this settles longstanding questions of C.T.C. Wall, determines all Stein fillable homotopy spheres, and proves a conjecture of Galatius and Randal–Williams. Implications are drawn for both the classification of highly connected manifolds and, via work of Kreck and Krannich, the calculation of their mapping class groups. Our technique is to recast the Galatius and Randal–Williams conjecture in terms of the vanishing of a certain Toda bracket, and then to analyze this Toda bracket by bounding its $mathrm{H}mathbb{F}_p$-Adams filtrations for all primes $p$. We additionally prove new vanishing lines in the $mathrm{H}mathbb{F}_p$-Adams spectral sequences of spheres and Moore spectra, which are likely to be of independent interest. Several of these vanishing lines rely on an Appendix by Robert Burklund, which answers a question of Mathew about vanishing curves in $mathrm{BP}{langle n rangle}$-based Adams spectral sequences.","PeriodicalId":50895,"journal":{"name":"Acta Mathematica","volume":"27 1","pages":""},"PeriodicalIF":3.7,"publicationDate":"2023-12-19","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"138745199","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":1,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
引用次数: 0
Every complete Pick space satisfies the column-row property 每个完整的 Pick 空间都满足列-行属性
IF 3.7 1区 数学
Acta Mathematica Pub Date : 2023-12-19 DOI: 10.4310/acta.2023.v231.n2.a2
Michael Hartz
{"title":"Every complete Pick space satisfies the column-row property","authors":"Michael Hartz","doi":"10.4310/acta.2023.v231.n2.a2","DOIUrl":"https://doi.org/10.4310/acta.2023.v231.n2.a2","url":null,"abstract":"In the theory of complete Pick spaces, the column-row property has appeared in a variety of contexts. We show that it is satisfied by every complete Pick space in the following strong form: each sequence of multipliers that induces a contractive column multiplication operator also induces a contractive row multiplication operator. In combination with known results, this yields a number of consequences. Firstly, we obtain multiple applications to the theory of weak product spaces, including factorization, multipliers and invariant subspaces. Secondly, there is a short proof of the characterization of interpolating sequences in terms of separation and Carleson measure conditions, independent of the solution of the Kadison–Singer problem. Thirdly, we find that in the theory of de Branges–Rovnyak spaces on the ball, the column-extreme multipliers of Jury and Martin are precisely the extreme points of the unit ball of the multiplier algebra.","PeriodicalId":50895,"journal":{"name":"Acta Mathematica","volume":"8 1","pages":""},"PeriodicalIF":3.7,"publicationDate":"2023-12-19","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"138745254","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":1,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
引用次数: 0
Khintchine’s theorem and Diophantine approximation on manifolds 流形上的Khintchine定理和丢番图近似
IF 3.7 1区 数学
Acta Mathematica Pub Date : 2023-09-29 DOI: 10.4310/acta.2023.v231.n1.a1
Victor Beresnevich, Lei Yang
{"title":"Khintchine’s theorem and Diophantine approximation on manifolds","authors":"Victor Beresnevich, Lei Yang","doi":"10.4310/acta.2023.v231.n1.a1","DOIUrl":"https://doi.org/10.4310/acta.2023.v231.n1.a1","url":null,"abstract":"In this paper we initiate a new approach to studying approximations by rational points to points on smooth submanifolds of $mathbb{R}^n$. Our main result is a convergence Khintchine type theorem for arbitrary non-degenerate submanifolds of $mathbb{R}^n$, which resolves a longstanding problem in the theory of Diophantine approximation. Furthermore, we refine this result using Hausdorff $s$-measures and consequently obtain the exact value of the Hausdorff dimension of $tau$-well approximable points lying on any non-degenerate submanifold for a range of Diophantine exponents $tau$ close to $1/n$. Our approach uses geometric and dynamical ideas together with a new technique of ‘generic and special parts’. In particular, we establish sharp upper bounds for the number of rational points of bounded height lying near the generic part of a non-degenerate manifold. In turn, we give an explicit exponentially small bound for the measure of the special part of the manifold. The latter uses a result of Bernik, Kleinbock and Margulis.","PeriodicalId":50895,"journal":{"name":"Acta Mathematica","volume":"3 4","pages":""},"PeriodicalIF":3.7,"publicationDate":"2023-09-29","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"138494787","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":1,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
引用次数: 0
Sharp well-posedness results of the Benjamin–Ono equation in $H^s (mathbb{T}, mathbb{R})$ and qualitative properties of its solutions $H^s (mathbb{T}， mathbb{R})$中Benjamin-Ono方程的清晰适定性结果及其解的定性性质
1区 数学
Acta Mathematica Pub Date : 2023-01-01 DOI: 10.4310/acta.2023.v231.n1.a2
Patrick Gérard, Thomas Kappeler, Peter Topalov
{"title":"Sharp well-posedness results of the Benjamin–Ono equation in $H^s (mathbb{T}, mathbb{R})$ and qualitative properties of its solutions","authors":"Patrick Gérard, Thomas Kappeler, Peter Topalov","doi":"10.4310/acta.2023.v231.n1.a2","DOIUrl":"https://doi.org/10.4310/acta.2023.v231.n1.a2","url":null,"abstract":"We prove that the Benjamin--Ono equation on the torus is globally in time well-posed in the Sobolev space $H^{s}(mathbb{T},mathbb{R})$ for any $s > - 1/2$ and ill-posed for $s le - 1/2$. Hence the critical Sobolev exponent $s_c=-1/2$ of the Benjamin--Ono equation is the threshold for well-posedness on the torus. The obtained solutions are almost periodic in time. Furthermore, we prove that the traveling wave solutions of the Benjamin-Ono equation on the torus are orbitally stable in $H^{s}(mathbb{T},mathbb{R})$ for any $ s > - 1/2$. Novel conservation laws and a nonlinear Fourier transform on $H^{s}(mathbb{T},mathbb{R})$ with $s > - 1/2$ are key ingredients into the proofs of these results.","PeriodicalId":50895,"journal":{"name":"Acta Mathematica","volume":"285 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2023-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"135844837","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":1,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
引用次数: 3
The extremals of the Alexandrov–Fenchel inequality for convex polytopes 凸多面体的Alexandrov-Fenchel不等式的极值
1区 数学
Acta Mathematica Pub Date : 2023-01-01 DOI: 10.4310/acta.2023.v231.n1.a3
Yair Shenfeld, Ramon van Handel
{"title":"The extremals of the Alexandrov–Fenchel inequality for convex polytopes","authors":"Yair Shenfeld, Ramon van Handel","doi":"10.4310/acta.2023.v231.n1.a3","DOIUrl":"https://doi.org/10.4310/acta.2023.v231.n1.a3","url":null,"abstract":"The Alexandrov-Fenchel inequality, a far-reaching generalization of the classical isoperimetric inequality to arbitrary mixed volumes, lies at the heart of convex geometry. The characterization of its extremal bodies is a long-standing open problem that dates back to Alexandrov's original 1937 paper. The known extremals already form a very rich family, and even the fundamental conjectures on their general structure, due to Schneider, are incomplete. In this paper, we completely settle the extremals of the Alexandrov-Fenchel inequality for convex polytopes. In particular, we show that the extremals arise from the combination of three distinct mechanisms: translation, support, and dimensionality. The characterization of these mechanisms requires the development of a diverse range of techniques that shed new light on the geometry of mixed volumes of nonsmooth convex bodies. Our main result extends further beyond polytopes in a number of ways, including to the setting of quermassintegrals of arbitrary convex bodies. As an application of our main result, we settle a question of Stanley on the extremal behavior of certain log-concave sequences that arise in the combinatorics of partially ordered sets.","PeriodicalId":50895,"journal":{"name":"Acta Mathematica","volume":"68 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2023-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"136008443","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":1,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
引用次数: 17
Daily Exercise Patterns and Their Differences between Parkinson's Disease Patients with and without Postural Instability. 帕金森病患者的日常锻炼模式及其与姿势不稳患者的差异。
IF 6.7 1区 数学
Acta Mathematica Pub Date : 2022-05-19 eCollection Date: 2022-01-01 DOI: 10.1155/2022/3191598
Joomee Song, Jun Kyu Mun, Jong Hyeon Ahn, Jinyoung Youn, Inyoung Choi, Jin Whan Cho
{"title":"Daily Exercise Patterns and Their Differences between Parkinson's Disease Patients with and without Postural Instability.","authors":"Joomee Song, Jun Kyu Mun, Jong Hyeon Ahn, Jinyoung Youn, Inyoung Choi, Jin Whan Cho","doi":"10.1155/2022/3191598","DOIUrl":"10.1155/2022/3191598","url":null,"abstract":"<p><strong>Background: </strong>Due to the clinical impact of exercise in patients with Parkinson's disease (PD), management should include personalized and effective exercises according to patient's PD stage. We investigated the detailed exercise behaviors of patients with mild to advanced PD and compared their patterns between PD with and without postural instability (PI).</p><p><strong>Methods: </strong>We enrolled PD patients from September to December 2019. Clinical data on parkinsonism, exercise behaviors, and Physical Activity Scale of the Elderly (PASE) scores were collected and compared between mild PD without PI (Hoehn-Yahr (HY) stages 1 and 2) and advanced PD with PI (HY stages 3 and 4).</p><p><strong>Results: </strong>In total, 263 PD patients were recruited. The mean exercise frequency was 4.7 ± 2.1 times/week, and the average duration was 7.8 ± 6.7 hours/week. The most common exercise was an aerobic exercise (71.9%) of mild-to-moderate intensity, with active walking being the most common (49.0%). The mild PD patients demonstrated a higher duration and intensity of exercise and more physical activity than the advanced PD patients. However, the frequency of exercise was not significantly different between the two groups. The PASE score was significantly higher in mild PD patients than in advanced PD patients (<i>p</i> < 0.001).</p><p><strong>Conclusion: </strong>PD patients focused mostly on aerobic exercises, especially active walking. With the disease progression, the amount and intensity of exercise decreased while frequency remained. Higher intensity of exercise is needed in the mild PD group, while the advanced PD group requires the increment of duration for each exercise session.</p>","PeriodicalId":50895,"journal":{"name":"Acta Mathematica","volume":"57 1","pages":"3191598"},"PeriodicalIF":6.7,"publicationDate":"2022-05-19","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://www.ncbi.nlm.nih.gov/pmc/articles/PMC9135569/pdf/","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"73899935","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":1,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"OA","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
引用次数: 0
Gravitational instantons with faster than quadratic curvature decay. I 比二次曲率衰减更快的引力瞬子。我
IF 3.7 1区 数学
Acta Mathematica Pub Date : 2022-01-10 DOI: 10.4310/acta.2021.v227.n2.a2
Gao Chen, Xiuxiong Chen
{"title":"Gravitational instantons with faster than quadratic curvature decay. I","authors":"Gao Chen, Xiuxiong Chen","doi":"10.4310/acta.2021.v227.n2.a2","DOIUrl":"https://doi.org/10.4310/acta.2021.v227.n2.a2","url":null,"abstract":"In this paper, we study gravitational instantons (i.e., complete hyperkähler $4$‑manifolds with faster than quadratic curvature decay). We prove three main theorems: (1) Any gravitational instanton must have one of the following known ends: ALE, ALF, ALG, and ALH. (2) In the ALG and ALH non-splitting cases, it must be biholomorphic to a compact complex elliptic surface minus a divisor. Thus, we confirm a long-standing question of Yau in the ALG and ALH cases. (3) In the ALF‑$D_k$ case, it must have an $O(4)$‑multiplet.","PeriodicalId":50895,"journal":{"name":"Acta Mathematica","volume":"4 8","pages":""},"PeriodicalIF":3.7,"publicationDate":"2022-01-10","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"138494783","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":1,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
引用次数: 7
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