Acta Mathematica最新文献

英文中文

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 Q1 MATHEMATICS

Acta Mathematica

Pub Date : 2024-05-10 DOI: 10.4310/acta.2024.v232.n1.a2

Jeremy Kahn, François Labourie, Mozes Shahar

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.

我们证明，一些半简单群（尤其是复群）中的均匀网格允许阿诺索夫曲面子群。这一结果还有一个定量版本：我们引入了一个概念，称为 $K$-Sullivan 映射，它概括了双曲几何中的 $K$-quasi-circles 概念，并特别证明了 Sullivan 映射是 Hölder 映射。利用这一概念，我们展示了曲面子群定理的定量版本，特别是可以得到 $K$-Sullivan 极限映射，与光滑圆非常接近。所有这些结果都使用了某一旗流形中 "三角形路径 "的粗几何学，我们还证明了在此背景下准大地线的莫尔斯定理。

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 Q1 MATHEMATICS

Acta Mathematica

Pub Date : 2024-05-10 DOI: 10.4310/acta.2024.v232.n1.a1

James Gabe, Gábor Szabó

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.

假设 $G$ 是一个二次可数局部紧密群。在本文中，我们将研究基希贝格数组上的可与外$G-$作用，这些作用可以近似地中心嵌入康兹代数$mathcal{O}_{^infty}$上的经典准无作用。如果 $G$ 是离散的，那么它就与基希贝格代数上的可amenable 和外$G-$作用类重合。我们证明，由此产生的 $G-C^ast$ 动力系统是由等变卡斯帕罗夫理论分类的，直到环共轭。这是第一个适用于任意局部紧凑群作用的分类理论。在各种应用中，我们的主要结果解决了 Izumi 关于离散可逆无扭群作用的猜想，并恢复了 Izumi-Matui 最近关于多-$mathbb{Z}$ 群作用的主要结果。

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 Q1 MATHEMATICS

Acta Mathematica

Pub Date : 2023-12-19 DOI: 10.4310/acta.2023.v231.n2.a3

Karl-Theodor Sturm

This is a correction to $href{https://dx.doi.org/10.1007/s11511-006-0002-8}{[11]}$ (Acta Math.), 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 J. Funct. Anal.).

这是对 $href{https://dx.doi.org/10.1007/s11511-006-0002-8}{[11]}$ (Acta Math.) 以及后续出版物 $href{https://doi.org/10.1016/j.jfa.2010.03.024}{[3]}$ 和 $href{https://doi.org/10.1016/j.jfa.2011.02.026}{[5]}$ (both in J. Funct. Anal.) 的更正。

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 Q1 MATHEMATICS

Acta Mathematica

Pub Date : 2023-12-19 DOI: 10.4310/acta.2023.v231.n2.a1

Robert Burklund, Jeremy Hahn, Andrew Senger

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.

在斯托尔兹工作的基础上，我们证明了对于整数$0 leqslant d leqslant 3$和$k gt 232$，$(k-1)$连接的、几乎封闭的$(2k+d)$流形的边界也是可平行流形的边界。在有限多维之外，这解决了华尔（C.T.C. Wall）的长期问题，确定了所有斯坦因可填充同调球，并证明了加拉蒂乌斯和兰道尔-威廉姆斯的猜想。这对高连接流形的分类以及通过克雷克和克兰尼奇的工作计算其映射类群都有意义。我们的技术是用某个托达括号的消失来重构加拉蒂乌斯和兰道尔-威廉斯猜想，然后通过限定所有素数 $p$ 的 $mathrm{H}mathbb{F}_p$-Adams 滤波来分析这个托达括号。此外，我们还证明了球面和摩尔谱的 $mathrm{H}mathbb{F}_p$-Adams 谱序列中的新消失线，这些消失线很可能具有独立的意义。其中几条消失线依赖于罗伯特-伯克伦（Robert Burklund）的附录，该附录回答了马修关于基于 $mathrm{BP}{langle n rangle}$ 的亚当斯谱序列中的消失曲线的问题。

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 Q1 MATHEMATICS

Acta Mathematica

Pub Date : 2023-12-19 DOI: 10.4310/acta.2023.v231.n2.a2

Michael Hartz

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.

在完全皮克空间理论中，列-行性质出现在不同的语境中。我们证明，每一个完整 Pick 空间都满足以下强形式：每一个诱导收缩列乘法算子的乘法序列也诱导收缩行乘法算子。结合已知结果，这将产生一系列结果。首先，我们获得了弱乘空间理论的多种应用，包括因式分解、乘法器和不变子空间。其次，我们还简短地证明了插值序列在分离和卡列松度量条件下的特征，这与卡迪森-辛格问题的解无关。第三，我们发现在球上的 de Branges-Rovnyak 空间理论中，Jury 和 Martin 的列极值乘数正是乘数代数单位球的极值点。

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 Q1 MATHEMATICS

Acta Mathematica

Pub Date : 2023-09-29 DOI: 10.4310/acta.2023.v231.n1.a1

Victor Beresnevich, Lei Yang

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.

本文提出了一种研究光滑子流形$mathbb{R}^n$上有理点对点逼近的新方法。我们的主要成果是$mathbb{R}^n$的任意非退化子流形的收敛Khintchine型定理，它解决了丢芬图近似理论中一个长期存在的问题。此外，我们使用Hausdorff $s$-测度来改进这一结果，从而得到在任何非简并子流形上的$tau$-在Diophantine指数$tau$接近$1/n$的范围内的$tau$-可近似点的Hausdorff维数的精确值。我们的方法将几何和动力学的思想与“通用和特殊零件”的新技术结合在一起。特别地，我们建立了在非退化流形的一般部分附近有界高度的有理点数目的明显上界。然后，我们给出了流形特殊部分测度的显式指数小界。后者使用了Bernik, Kleinbock和Margulis的结果。

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 Q1 MATHEMATICS

Acta Mathematica

Pub Date : 2023-01-01 DOI: 10.4310/acta.2023.v231.n1.a2

Patrick Gérard, Thomas Kappeler, Peter Topalov

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.

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 Q1 MATHEMATICS

Acta Mathematica

Pub Date : 2023-01-01 DOI: 10.4310/acta.2023.v231.n1.a3

Yair Shenfeld, Ramon van Handel

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.

亚历山德罗夫-芬切尔不等式是经典等周不等式对任意混合体积的深远推广，是凸几何的核心。它的极端体的特征是一个长期悬而未决的问题，可以追溯到亚历山德罗夫1937年的原始论文。已知的极值已经形成了一个非常丰富的家族，甚至由于施耐德对其一般结构的基本推测也是不完整的。本文完全解决了凸多面体的Alexandrov-Fenchel不等式的极值问题。特别地，我们表明极值产生于三种不同机制的组合:翻译、支持和维度。这些机制的表征需要发展各种各样的技术，以揭示非光滑凸体混合体积的几何形状。我们的主要结果在许多方面进一步超越了多面体，包括任意凸体的quermass积分的设置。作为我们的主要结果的一个应用，我们解决了Stanley关于部分有序集合组合中出现的某些对数凹序列的极值性问题。

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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 Q1 MATHEMATICS

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

Background: 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).

Methods: 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).

Results: 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 (p < 0.001).

Conclusion: 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.

背景：鉴于运动对帕金森病（PD）患者的临床影响，管理应包括根据患者的帕金森病阶段进行个性化和有效的运动。我们调查了轻度至晚期帕金森病患者的详细运动行为，并比较了有体位不稳（PI）和无体位不稳（PI）帕金森病患者的运动模式：我们在 2019 年 9 月至 12 月期间招募了帕金森病患者。收集了帕金森病、运动行为和老年人体力活动量表（PASE）评分的临床数据，并对无 PI 的轻度 PD（Hoehn-Yahr（HY）1 期和 2 期）和有 PI 的晚期 PD（HY 3 期和 4 期）进行了比较：共招募了 263 名帕金森病患者。平均运动频率为 4.7 ± 2.1 次/周，平均持续时间为 7.8 ± 6.7 小时/周。最常见的运动是轻度至中度强度的有氧运动（71.9%），其中最常见的是主动步行（49.0%）。与晚期帕金森病患者相比，轻度帕金森病患者的运动时间和强度更长，运动量更大。然而，两组患者的运动频率并无明显差异。轻度帕金森病患者的 PASE 评分明显高于晚期帕金森病患者（P < 0.001）：结论：帕金森病患者主要以有氧运动为主，尤其是主动步行。结论：帕金森病患者主要以有氧运动为主，尤其是主动步行。随着病情的发展，运动量和强度都有所下降，但运动频率保持不变。轻度帕金森病组需要更高强度的运动，而晚期帕金森病组则需要增加每次运动的持续时间。

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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 Q1 MATHEMATICS

Acta Mathematica

Pub Date : 2022-01-10 DOI: 10.4310/acta.2021.v227.n2.a2

Gao Chen, Xiuxiong Chen

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.

在本文中，我们研究了引力瞬子(即，比二次曲率衰减更快的完全hyperkähler $4$ -流形)。我们证明了三个主要定理:(1)任何引力瞬子必须有以下已知端点之一:ALE, ALF, ALG和ALH。(2)在ALG和ALH非分裂情况下，它必须是生物全纯的紧复椭圆曲面减去一个除数。因此，我们在ALG和ALH病例中确认了一个长期存在的问题。(3)在ALF - $D_k$的情况下，它必须有一个$O(4)$ - multiplet。

引用次数: 7

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