辛极值附近Hamilton流的周期轨道

Critical symplectic connections on surfaces

The space of symplectic connections on a symplectic manifold is a symplectic affine space. M. Cahen and S. Gutt showed that the action of the group of Hamiltonian diffeomorphisms on this space is Hamiltonian and calculated the moment map. This is analogous to, but distinct from, the action of Hamiltonian diffeomorphisms on the space of compatible almost complex structures that motivates study of extremal Kähler metrics. In particular, moment constant connections are critical, where a symplectic connection is \textit{critical} if it is critical, with respect to arbitrary variations, for the $L^…

辛分层空间中的正规模

Normal modes in symplectic stratified spaces

我们推广了Weinstein-Moser关于非线性正规模存在性的定理（即，周期轨道）的一个定理的存在性附近的一个相对平衡的哈密顿系统连续对称性。更具体地说，我们显着改善的结果证明较早联合Tokieda（数学。SG/9901102）：我们删除了一个技术假设，相当于一个假设，对称群是一个环面。

盘丛中的弗洛尔同调与辛扭测地流

Floer homology in disc bundles and symplectically twisted geodesic flows

我们证明，如果K：P \to R是辛流形（P，Ω）上的一个自治Hamilton算子，它沿辛子流形M达到0作为Morse-Bott非退化极小值，并且如果c_1（TP）|_M在实上同调中为零，则K的Hamilton流在所有充分小的能级上都有周期有界的可缩周期轨道。作为一种特殊情况，如果M的余切丛上的测地流按辛磁场形式扭转，则所得到的流在所有低能级上都有可收缩的周期轨道.这些结果被Ginzburg和Gürel证明，当Ω|_M是球理性的，我们的证明建立在他们的工作上;这个论点涉及到在链水平上构造和仔细分析M的辛正规圆盘丛中的过滤弗洛尔同调的一个版本。

突变周期颤动的离散动力系统：例子和约化

Discrete dynamical systems from mutation-periodic quivers: examples and reduction

介绍了几种新的周期大于1的突变周期抖动以及相关的离散动力系统。减少这些系统的开发使用前辛或泊松方法。前辛方法导致一个减少系统的迭代映射是辛的关于一个日志辛形式。在Poisson方法中，我们建立了迭代映射下的Poisson结构不变量，从而得到一个约化系统，其变量是这种结构的Casimirs。

辛路的指数迭代理论及其在非线性哈密顿系统中的应用

Index iteration theory for symplectic paths with applications to nonlinear Hamiltonian systems

近年来，我们建立了辛矩阵路指数的迭代理论，并将其应用于非线性哈密顿系统的周期解问题。本文是对这些结果的综述。

辛坐标下复曲面流形的Kahler几何

Kahler geometry of toric manifolds in symplectic coordinates

Delzant的一个定理指出，任何维数为$2n$的辛流形$(M,\om)$，如果具有标准$n$ -环面$\T^n = \R^{n}/2π\Z^n$的有效哈密顿作用，则是一个光滑的投射环面簇，它完全由矩映射$ϕ:M\to\R^n$的像（一个凸多面体$P=ϕ(M)\subset\R^n$）确定（作为一个哈密顿$\T^n$ -空间）。在本文中，我们显示，使用辛（作用角）坐标$P\times \T^n$，如何所有$\om$兼容复曲面复杂结构$M$上可以有效地参数化$P$上的光滑函数。我们还讨论了一些主题适合应用这种辛坐标方法的凯勒环面几何，即：显式建设极值凯勒度量，频谱性质的环面流形和组合的多面体。

Relative Hofer-Zehnder capacity and positive symplectic homology

We study the relationship between a homological capacity $c_{\mathrm{SH}^+}(W)$ for Liouville domains $W$ defined using positive symplectic homology and the existence of periodic orbits for Hamiltonian systems on $W$: If the positive symplectic homology of $W$ is non-zero, then the capacity yields a finite upper bound to the $π_1$-sensitive Hofer-Zehnder capacity of $W$ relative to its skeleton and a certain class of Hamiltonian diffeomorphisms of $W$ has infinitely many non-trivial contractible periodic points. En passant, we give an upper bound for the spectral capacity of $W$ in terms of th…

The Calabi invariant and the least number of periodic solutions of locally Hamiltonian equations

In this paper we prove a lower bound for the least number of one-periodic solutions of nondegenerate locally Hamiltonian equations on compact symplectic manifolds in terms of the Betti numbers of the Novikov homology associated to the Calabi invariant of the locally Hamiltonian equations. Our result improves lower bounds obtained by Lê-Ono and Ono for the least number of nondegenerate locally Hamiltonian symplectic fixed points. Our result also generalizes the homological Arnold conjecture proved by Fukaya-Ono and Liu-Tian.

四维流形上的周期Hamilton流

Periodic Hamiltonian flows on four dimensional manifolds

我们将紧致四维辛流形上的周期Hamilton流分类为Hamilton S^1空间的同构。此外，我们表明，所有这些空间是Kaehler，每一个这样的空间是从一个简单的模型通过一系列的辛爆破，如果不动点是孤立的，那么空间是一个环面品种。

Periodic solutions of the sinh-Gordon equation and integrable systems

We study the space of periodic solutions of the elliptic $\sinh$-Gordon equation by means of spectral data consisting of a Riemann surface $Y$ and a divisor $D$. We show that the space $M_g^{\mathbf{p}}$ of real periodic finite type solutions with fixed period $\mathbf{p}$ can be considered as a completely integrable system $(M_g^{\mathbf{p}},Ω,H_2)$ with a symplectic form $Ω$ and a series of commuting Hamiltonians $(H_n)_{n \in \mathbb{N}}$. In particular we relate the gradients of these Hamiltonians to the Jacobi fields $(ω_n)_{n\in \mathbb{N}_0}$ from the Pinkall-Sterling iteration. Moreove…