Advances in Mathematics最新文献

英文中文

A counterexample to the singular Weinstein conjecture 奇异温斯坦猜想的反例

IF 1.5 1区数学 Q1 MATHEMATICS

Advances in Mathematics

Pub Date : 2024-11-05 DOI: 10.1016/j.aim.2024.109998

Josep Fontana-McNally , Eva Miranda , Cédric Oms , Daniel Peralta-Salas

In this article, we study the dynamical properties of Reeb vector fields on b-contact manifolds. We show that in dimension 3, the number of so-called singular periodic orbits can be prescribed. These constructions illuminate some key properties of escape orbits and singular periodic orbits, which play a central role in formulating singular counterparts to the Weinstein conjecture and the Hamiltonian Seifert conjecture. In fact, we prove that the above-mentioned constructions lead to counterexamples of these conjectures as stated in [20]. Our construction shows that there are b-contact manifolds with no singular periodic orbits and no regular periodic orbits away from Z. We do not know whether there are constructions with no generalized escape orbits whose α and ω-limits both lie on Z (a generalized singular periodic orbit). This is the content of the generalized Weinstein conjecture.

本文研究了 b-contact 流形上 Reeb 向量场的动力学性质。我们证明，在维度 3 中，所谓奇异周期轨道的数量是可以规定的。这些构造阐明了逸出轨道和奇异周期轨道的一些关键性质，它们在提出韦恩斯坦猜想和汉密尔顿塞弗猜想的奇异对应猜想中起着核心作用。事实上，我们证明了上述构造导致了 [20] 中所述这些猜想的反例。我们的构造表明，存在没有奇异周期轨道和远离 Z 的规则周期轨道的 b-contact 流形。我们不知道是否存在没有广义逸出轨道的构造，其 α 和 ω 极限都位于 Z 上（广义奇异周期轨道）。这就是广义韦恩斯坦猜想的内容。

引用次数: 0

On the Steiner inequality for the Lp surface area 关于 Lp 表面积的斯坦纳不等式

IF 1.5 1区数学 Q1 MATHEMATICS

Advances in Mathematics

Pub Date : 2024-10-31 DOI: 10.1016/j.aim.2024.109997

Zixin Cao , Tuo Wang , Yanhui Wang

The Steiner inequalities for the

L_{p}

surface area are established for

p > 1

. As a consequence, we give a new proof of Lutwak's

L_{p}

isoperimetric inequalities for

p > 1

together with their equality conditions.

因此，我们给出了 p>1 的 Lutwak Lp 等周不等式的新证明及其相等条件。

引用次数: 0

Conformal measure rigidity for representations via self-joinings 通过自连接实现表征的共形测量刚度

IF 1.5 1区数学 Q1 MATHEMATICS

Advances in Mathematics

Pub Date : 2024-10-31 DOI: 10.1016/j.aim.2024.109992

Dongryul M. Kim, Hee Oh

Let Γ be a Zariski dense discrete subgroup of a connected simple real algebraic group

G_{1}

. We discuss a rigidity problem for discrete faithful representations

ρ : Γ \to G_{2}

and a surprising role played by higher rank conformal measures of the associated self-joining group. Our approach recovers rigidity theorems of Sullivan, Tukia and Yue, as well as applies to Anosov representations, including Hitchin representations.

More precisely, for a given representation ρ with a boundary map f defined on the limit set Λ, we ask whether the extendability of ρ to

G_{1}

can be detected by the property that f pushes forward some Γ-conformal measure class

[ν_{Γ}]

to a

ρ (Γ)

-conformal measure class

[ν_{ρ (Γ)}]

. When Γ is of divergence type in a rank one group or when ρ arises from an Anosov representation, we give an affirmative answer by showing that if the self-joining

Γ_{ρ} = (id \times ρ) (Γ)

is Zariski dense in

G_{1} \times G_{2}

, then the push-forward measures

{(id \times f)}_{⁎} ν_{Γ}

and

{(f^{- 1} \times id)}_{⁎} ν_{ρ (Γ)}

, which are higher rank

Γ_{ρ}

-conformal measures, cannot be in the same measure class.

假设Γ 是连通简单实代数群 G1 的一个扎里斯基密集离散子群。我们讨论了离散忠实表示 ρ:Γ→G2 的刚性问题，以及相关自接群的高阶共形度量所起的惊人作用。我们的方法恢复了沙利文、图基亚和岳的刚性定理，并适用于阿诺索夫表示，包括希钦表示。更确切地说，对于一个给定表示ρ，其边界映射 f 定义在极限集Λ上，我们问ρ到 G1 的可扩展性是否可以通过 f 将某个Γ-共形度量类 [νΓ] 推向ρ(Γ)-共形度量类 [νρ(Γ)]这一性质来检测。当 Γ 在秩为 1 的群中属于发散类型时，或者当 ρ 来自阿诺索夫表示时，我们给出了肯定的答案，证明了如果自连接 Γρ=(id×ρ)(Γ) 在 G1×G2 中是扎里斯基密集的、那么作为高阶Γρ-共形度量的前推度量（id×f）⁎νΓ 和（f-1×id）⁎νρ(Γ)不可能属于同一个度量类。

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引用次数: 0

Irregular sampling for hyperbolic secant type functions 双曲正割型函数的不规则采样

IF 1.5 1区数学 Q1 MATHEMATICS

Advances in Mathematics

Pub Date : 2024-10-30 DOI: 10.1016/j.aim.2024.109981

Anton Baranov , Yurii Belov

We study Gabor frames in the case when the window function is of hyperbolic secant type, i.e.,

g (x) = {(e^{a x} + e^{- b x})}^{- 1}

Re a, Re b > 0

. A criterion for half-regular sampling is obtained: for a separated

Λ \subset R

the Gabor system

G (g, Λ \times α Z)

is a frame in

L^{2} (R)

if and only if

D^{-} (Λ) > α

where

D^{-} (Λ)

is the usual (Beurling) lower density of Λ. This extends a result by Gröchenig, Romero, and Stöckler which applies to the case of a standard hyperbolic secant. Also, a full description of complete interpolating sequences for the shift-invariant space generated by g is given.

我们研究的是窗口函数为双曲正割类型时的 Gabor 框架，即 g(x)=(eax+e-bx)-1, Rea,Reb>0。我们得到了一个半规则采样的标准：对于分离的Λ⊂R，Gabor 系统 G(g,Λ×αZ) 是 L2(R) 中的一个框架，当且仅当 D-(Λ)>α 其中 D-(Λ) 是Λ的通常（Beurling）低密度。这扩展了格罗切尼格、罗梅罗和斯托克勒的一个结果，该结果适用于标准双曲正割的情况。此外，还给出了由 g 生成的移位不变空间的完整插值序列的完整描述。

引用次数: 0

Curvature operators and rational cobordism 曲率算子和有理共线性

IF 1.5 1区数学 Q1 MATHEMATICS

Advances in Mathematics

Pub Date : 2024-10-30 DOI: 10.1016/j.aim.2024.109995

Renato G. Bettiol , McFeely Jackson Goodman

We determine linear inequalities on the eigenvalues of curvature operators that imply vanishing of the twisted

\hat{A}

genus on a closed Riemannian spin manifold, where the twisting bundle is any prescribed parallel bundle of tensors. These inequalities yield surgery-stable curvature conditions tailored to annihilate further rational cobordism invariants, such as the Witten genus, elliptic genus, signature, and even the rational cobordism class itself.

我们确定了曲率算子特征值上的线性不等式，这些不等式意味着封闭黎曼自旋流形上的扭曲 Aˆ 属消失，其中扭曲束是任何规定的张量平行束。这些不等式产生了手术稳定曲率条件，这些条件专门用于湮灭更多的有理共线性不变式，例如维滕属、椭圆属、签名，甚至有理共线性类本身。

引用次数: 0

Prandtl-Batchelor flows on an annulus 环面上的普氏流

IF 1.5 1区数学 Q1 MATHEMATICS

Advances in Mathematics

Pub Date : 2024-10-30 DOI: 10.1016/j.aim.2024.109994

Mingwen Fei , Chen Gao , Zhiwu Lin , Tao Tao

For steady two-dimensional Navier-Stokes flows with a single eddy (i.e. nested closed streamlines) in a simply connected domain, Prandtl (1905) and Batchelor (1956) found that in the inviscid limit, the vorticity is constant inside the eddy. In this paper, we consider the generalized Prandtl-Batchelor theory for the forced steady Navier-Stokes equations on an annulus. First, we observe that in the limit of infinite Reynolds number, if the streamlines of forced steady Navier-Stokes solutions on an annulus are nested closed, then the inviscid limit is a rotating shear flow uniquely determined by the external force and boundary conditions. We call solutions of steady Navier-Stokes equations with the above property Prandtl-Batchelor flows. Then, by constructing higher order approximate solutions of the forced steady Navier-Stokes equations and establishing the validity of Prandtl boundary layer expansion, we give a rigorous proof of the existence of Prandtl-Batchelor flows on an annulus with the wall velocities slightly different from the rigid-rotations along the same direction.

对于简单连接域中具有单一涡流（即嵌套封闭流线）的稳定二维纳维-斯托克斯流，Prandtl（1905 年）和 Batchelor（1956 年）发现，在不粘性极限中，涡流内部的涡度是恒定的。在本文中，我们考虑了环面上受迫稳定纳维-斯托克斯方程的广义普朗特-巴彻勒理论。首先，我们观察到，在无限雷诺数极限下，如果环面上的强制稳定 Navier-Stokes 解的流线是嵌套封闭的，那么不粘性极限就是由外力和边界条件唯一决定的旋转剪切流。我们将具有上述性质的稳定纳维-斯托克斯方程的解称为普朗特-巴切洛流。然后，通过构建受迫稳定纳维-斯托克斯方程的高阶近似解，并建立普朗特边界层扩展的有效性，我们给出了在环面上存在普朗特-巴歇尔流的严谨证明，其壁面速度与沿同一方向的刚体旋转速度略有不同。

引用次数: 0

The inequalities of Chern classes and Riemann-Roch type inequalities 车恩类不等式和黎曼-罗赫型不等式

IF 1.5 1区数学 Q1 MATHEMATICS

Advances in Mathematics

Pub Date : 2024-10-25 DOI: 10.1016/j.aim.2024.109982

Xing Lu, Jian Xiao

<div><div>Motivated by Kollár-Matsusaka's Riemann-Roch type inequalities, applying effective very ampleness of adjoint bundles on Fujita conjecture and log-concavity given by Khovanskii-Teissier inequalities, we show that for any partition λ of the positive integer d there exists a universal bivariate polynomial <math><msub><mrow><mi>Q</mi></mrow><mrow><mi>λ</mi></mrow></msub><mo>(</mo><mi>x</mi><mo>,</mo><mi>y</mi><mo>)</mo></math> which has <math><mi>deg</mi><mo>⁡</mo><msub><mrow><mi>Q</mi></mrow><mrow><mi>λ</mi></mrow></msub><mo>≤</mo><mi>d</mi></math> and whose coefficients depend only on n and λ, such that for any projective manifold X of dimension n and any ample line bundle L on X,<math><mrow><mo>|</mo><msub><mrow><mi>c</mi></mrow><mrow><mi>λ</mi></mrow></msub><mo>(</mo><mi>X</mi><mo>)</mo><mo>⋅</mo><msup><mrow><mi>L</mi></mrow><mrow><mi>n</mi><mo>−</mo><mi>d</mi></mrow></msup><mo>|</mo></mrow><mo>≤</mo><mfrac><mrow><msub><mrow><mi>Q</mi></mrow><mrow><mi>λ</mi></mrow></msub><mo>(</mo><msup><mrow><mi>L</mi></mrow><mrow><mi>n</mi></mrow></msup><mo>,</mo><msub><mrow><mi>K</mi></mrow><mrow><mi>X</mi></mrow></msub><mo>⋅</mo><msup><mrow><mi>L</mi></mrow><mrow><mi>n</mi><mo>−</mo><mn>1</mn></mrow></msup><mo>)</mo></mrow><mrow><msup><mrow><mo>(</mo><msup><mrow><mi>L</mi></mrow><mrow><mi>n</mi></mrow></msup><mo>)</mo></mrow><mrow><mi>d</mi><mo>−</mo><mn>1</mn></mrow></msup></mrow></mfrac><mo>,</mo></math> where <math><msub><mrow><mi>K</mi></mrow><mrow><mi>X</mi></mrow></msub></math> is the canonical bundle of X and <math><msub><mrow><mi>c</mi></mrow><mrow><mi>λ</mi></mrow></msub><mo>(</mo><mi>X</mi><mo>)</mo></math> is the monomial Chern class given by the partition λ. As a special case, when <math><msub><mrow><mi>K</mi></mrow><mrow><mi>X</mi></mrow></msub></math> or <math><mo>−</mo><msub><mrow><mi>K</mi></mrow><mrow><mi>X</mi></mrow></msub></math> is ample, this implies that there exists a constant <math><msub><mrow><mi>c</mi></mrow><mrow><mi>n</mi></mrow></msub></math> depending only on n such that for any monomial Chern classes of top degree, the Chern number ratios satisfy the following inequality<math><mrow><mo>|</mo><mfrac><mrow><msub><mrow><mi>c</mi></mrow><mrow><mi>λ</mi></mrow></msub><mo>(</mo><mi>X</mi><mo>)</mo></mrow><mrow><msub><mrow><mi>c</mi></mrow><mrow><mn>1</mn></mrow></msub><msup><mrow><mo>(</mo><mi>X</mi><mo>)</mo></mrow><mrow><mi>n</mi></mrow></msup></mrow></mfrac><mo>|</mo></mrow><mo>≤</mo><msub><mrow><mi>c</mi></mrow><mrow><mi>n</mi></mrow></msub><mo>,</mo></math> which recovers a recent result of Du-Sun. The main result also yields an asymptotic version of the sharper Riemann-Roch type inequality. Furthermore, using similar method we also obtain inequalities for

受 Kollár-Matsusaka 的 Riemann-Roch 型不等式的启发，应用藤田猜想上的邻接束的有效放大性和 Khovanskii-Teissier 不等式给出的对数凹性，我们证明了对于正整数 d 的任意分区 λ，存在一个普遍的双变量多项式 Qλ(x. y)，其系数仅依赖于 n 和 λ，从而对于任意投影流形 X 上的 X,|cλ(x. y)，存在一个degQλ≤d 的普遍的双变量多项式 Qλ(x. y)、y)的系数只取决于 n 和 λ，因此对于维数为 n 的任何投影流形 X 和 X 上的任何充裕线束 L，|cλ(X)⋅Ln-d|≤Qλ(Ln,KX⋅Ln-1)(Ln)d-1，其中 KX 是 X 的典型束，cλ(X) 是分割 λ 给出的单项式切尔恩类。作为特例，当 KX 或 -KX 是充裕的时，这意味着存在一个仅取决于 n 的常数 cn，从而对于任何顶阶的单核切尔恩类，其切尔恩数比满足以下不等式|cλ(X)c1(X)n|≤cn，这恢复了杜逊的一个最新结果。主要结果还得到了更尖锐的黎曼-罗赫型不等式的渐进版本。此外，利用类似的方法，我们还得到了对数切线束的切恩类不等式。

引用次数: 0

The Brown-Peterson spectrum is not E2(p2+2) at odd primes 奇数素数下的布朗-彼得森谱不是 E2(p2+2)

IF 1.5 1区数学 Q1 MATHEMATICS

Advances in Mathematics

Pub Date : 2024-10-25 DOI: 10.1016/j.aim.2024.109996

Andrew Senger

We show that the odd-primary Brown-Peterson spectrum BP does not admit the structure of an

E_{2 (p^{2} + 2)}

ring spectrum and that there can be no map

MU \to BP

E_{2 p + 3}

ring spectra. We also prove the same results for truncated Brown-Peterson spectra

BP 〈 n 〉

of height

n \geq 4

. This extends results of Lawson at the prime 2.

我们证明奇初布朗-彼得森谱 BP 不具有 E2(p2+2) 环谱的结构，并且不可能存在 E2p+3 环谱的映射 MU→BP。我们还证明了高度为 n≥4 的截断布朗-彼得森谱 BP〈n〉的相同结果。这扩展了劳森在素数 2 时的结果。

引用次数: 0

Hamilton cycles in pseudorandom graphs 伪随机图中的汉密尔顿循环

IF 1.5 1区数学 Q1 MATHEMATICS

Advances in Mathematics

Pub Date : 2024-10-25 DOI: 10.1016/j.aim.2024.109984

Stefan Glock , David Munhá Correia , Benny Sudakov

Finding general conditions which ensure that a graph is Hamiltonian is a central topic in graph theory. An old and well-known conjecture in the area states that any d-regular n-vertex graph G whose second largest eigenvalue in absolute value

λ (G)

is at most

d / C

, for some universal constant

C > 0

, has a Hamilton cycle. In this paper, we obtain two main results which make substantial progress towards this problem. Firstly, we settle this conjecture in full when the degree d is at least a small power of n. Secondly, in the general case we show that

λ (G) \leq d / C {(\log n)}^{1 / 3}

implies the existence of a Hamilton cycle, improving the 20-year old bound of

d / \log^{1 - o (1)} n

of Krivelevich and Sudakov. We use in a novel way a variety of methods, such as a robust Pósa rotation-extension technique, the Friedman-Pippenger tree embedding with rollbacks and the absorbing method, combined with additional tools and ideas.

Our results have several interesting applications. In particular, they imply the currently best-known bounds on the number of generators which guarantee the Hamiltonicity of random Cayley graphs, which is an important partial case of the well known Hamiltonicity conjecture of Lovász. They can also be used to improve a result of Alon and Bourgain on additive patterns in multiplicative subgroups.

寻找确保图是哈密顿图的一般条件是图论的一个核心课题。该领域有一个古老而著名的猜想，即对于某个通用常数 C>0，任何 d 规则 n 顶点图 G 的第二最大特征值的绝对值 λ(G) 至多为 d/C，则该图具有汉密尔顿循环。在本文中，我们获得了两个主要结果，在解决这一问题上取得了实质性进展。其次，在一般情况下，我们证明了 λ(G)≤d/C(logn)1/3 意味着汉密尔顿循环的存在，从而改进了克里夫列维奇（Krivelevich）和苏达科夫（Sudakov）20 年前提出的 d/log1-o(1)n 约束。我们以一种新颖的方式使用了多种方法，如稳健的波萨旋转扩展技术、带有回滚的弗里德曼-皮彭格树嵌入法和吸收法，并结合了其他工具和思想。我们的结果有几种有趣的应用，特别是它们暗示了目前最著名的关于保证随机 Cayley 图哈密尔顿性的生成器数量的边界，这是众所周知的洛瓦兹哈密尔顿性猜想的一个重要的部分情况。它们还可用于改进阿隆和布尔甘关于乘法子群中加法模式的结果。

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引用次数: 0

Spectral invariants over the integers 整数上的谱不变式

IF 1.5 1区数学 Q1 MATHEMATICS

Advances in Mathematics

Pub Date : 2024-10-24 DOI: 10.1016/j.aim.2024.109976

Yusuke Kawamoto , Egor Shelukhin

Spectral invariants are quantitative measurements in symplectic topology coming from Floer homology theory. We study their dependence on the choice of coefficients in the context of Hamiltonian Floer homology. We discover phenomena in this setting which hold for

Z

-coefficients and fail for all field coefficients. For example, we prove that the spectral norm, an important metric derived from spectral invariants, is unbounded over

Z

for complex projective spaces, while it is uniformly bounded over all fields. This allows us to answer a symplectic version of a question of Hingston, originally asked in the setting of the energy functional on the loop space. We also provide applications to Hamiltonian dynamics and Hofer's geometry.

谱不变式是来自弗洛尔同调理论的交错拓扑定量测量。我们在汉密尔顿弗洛尔同调的背景下研究了它们对系数选择的依赖性。我们发现了在此背景下 Z 系数成立而所有场系数失效的现象。例如，我们证明了对于复杂投影空间来说，由谱不变式衍生出的重要度量--谱规范在 Z 上是无界的，而在所有场上则是均匀有界的。这使我们能够回答兴斯顿问题的交映版本，这个问题最初是在环空间上的能量函数的背景下提出的。我们还提供了哈密顿动力学和霍弗几何的应用。

引用次数: 0

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