Symplectic Topology of Integrable Hamiltonian Systems, II: Topological Classification

Some compatible Poisson structures and integrable bi-Hamiltonian systems on four dimensional and nilpotent six dimensional symplectic real Lie groups

We provide an alternative method for obtaining of compatible Poisson structures on Lie groups by means of the adjoint representations of Lie algebras. In this way, we calculate some compatible Poisson structures on four dimensional and nilpotent six dimensional symplectic real Lie groups. Then using Magri-Morosi's theorem we obtain new bi-Hamiltonian systems with four dimensional and nilpotent six dimensional symplectic real Lie groups as phase spaces.

可积系统中$A_n$奇点普适开折的辛分类

Symplectic classification for universal unfoldings of $A_n$ singularities in integrable systems

本文研究可积哈密顿系统中$A_n$奇点普适开折的辛分类问题，得到局部、半局部辛正规形及半全局拉格朗日叶状结构分类，明确了不同情形下的辛不变量与结构稳定性。

Topology of spaces of equivariant symplectic embeddings

We compute the homotopy type of the space of T^n-equivariant symplectic embeddings from the standard 2n-dimensional ball of some fixed radius into a 2n-dimensional symplectic-toric manifold M, and use this computation to define a Z-valued step function on the positive real line which is an invariant of the symplectic-toric type of M. We conclude with a discussion of the partially equivariant case of this result.

粗全纯曲线与辛拓扑

Coarsely Holomorphic Curves and Symplectic Topology

研究具有面积界的几乎复流形与驯服辛结构的关系，证明反之有面积界的流形允许驯服辛结构，还刻画了驯服辛结构锥，证明复二维周期可由粗全纯曲线逼近并给出低能量界，证实格罗莫夫推测。

磁场中的哈密顿可积系统与辛-哈安特耶斯几何

Hamiltonian integrable systems in a magnetic field and Symplectic-Haantjes geometry

本文研究三维黎曼空间中磁场内经典哈密顿系统的几何，证明其具有辛-哈安特耶斯结构，并利用该结构算法求解分离变量及构造新的可积模型。

Some pseudo-Kähler Einstein $4$-symmetric spaces with a "twin" special almost complex structure

On $4$-symmetric symplectic spaces, invariant almost complex structures -- up to sign -- arise in pairs. We exhibit some $4$-symmetric symplectic spaces, with a pair of "natural" compatible (usually not positive) invariant almost complex structures, one of them being integrable and the other one being maximally non integrable (i.e. the image of its Nijenhuis tensor at any point is the whole tangent space at that point). The integrable one defines a pseudo-Kähler Einstein metric on the manifold, and the non integrable one is Ricci Hermitian (in the sense that the almost complex structure preser…

Katok 示例的定量辛拓扑与等变辛嵌入

Quantitative symplectic topology of Katok's examples and equivariant symplectic embeddings

本文计算了 Katok 示例中 Randers 度量族对应的单位余余切丛辛嵌入圆余余切丛的参数集合，发现其呈无限阶梯结构，并建立了与等变椭球嵌入的等价关系。

Transverse Hilbert Schemes and Completely Integrable Systems

Via the transverse Hilbert scheme construction, we associate a holomorphic completely integrable system to a surface $S$ endowed with a holomorphic symplectic form $ω$ and a projection onto $\mathbb{C}$. We provide a full characterization of the completely integrable systems that arise in this way.

Note on Floer theory and integrable hierarchies

In this short note we show how Dubrovin's integrable hierarchies, defined using the Gromov-Witten theory of a closed symplectic manifold, generalizes to Hamiltonian Floer theory. In particular, we show how the required generalization of the PSS isomorphism, relating Gromov-Witten theory and Hamiltonian Floer theory, can be constructed in the framework of Eliashberg-Givental-Hofer's symplectic field theory.

Corrigenda to: Introduction to Symplectic Topology

This contains a list of (mostly very minor) corrections to the book Introduction to Symplectic Topology, Clarendon Press, Oxford, (1995), together with rewritten versions of two lemmas and some additional comments.

The topology of Bott integrable fluids

We construct non-vanishing steady solutions to the Euler equations (for some metric) with analytic Bernoulli function in each three-manifold where they can exist: graph manifolds. Using the theory of integrable systems, any admissible Morse-Bott function can be realized as the Bernoulli function of some non-vanishing steady Euler flow. This can be interpreted as an inverse problem to Arnold's structure theorem and yields as a corollary the topological classification of such solutions. Finally, we prove that the topological obstruction holds without the non-vanishing assumption: steady Euler fl…

On the group of strong symplectic homeomorphisms

We generalize the "hamiltonian topology" on hamiltonian isotopies to an intrinsic "symplectic topology" on the space of symplectic isotopies. We use it to define the group $SSympeo(M,ω)$ of strong symplectic homeomorphisms, which generalizes the group $Hameo(M,ω)$ of hamiltonian homeomorphisms introduced by Oh and Muller. The group $SSympeo(M,ω)$ is arcwise connected, is contained in the identity component of $Sympeo(M,ω)$; it contains $Hameo(M,ω)$ as a normal subgroup and coincides with it when $M$ is simply connected. Finally its commutator subgroup $[SSympeo(M,ω),SSympeo(M,ω)]$ is contained…

The topology of the space of symplectic balls in rational 4-manifolds

We study in this paper the rational homotopy type of the space of symplectic embeddings of the standard ball $B^4(c) \subset \R^4$ into 4-dimensional rational symplectic manifolds. We compute the rational homotopy groups of that space when the 4-manifold has the form $M_λ= (S^2 \times S^2, μω_0 \oplus ω_0)$ where $ω_0$ is the area form on the sphere with total area 1 and $μ$ belongs to the interval $[1,2]$. We show that, when $μ$ is 1, this space retracts to the space of symplectic frames, for any value of $c$. However, for any given $1 < μ< 2$, the rational homotopy type of that space changes…

Reidemeister torsion and Integrable Hamiltonian systems

In this paper we compute the Reidemeister torsion of a isoenergetic surface for the integrable Hamiltonian system on the four-dimensional symplectic manifold. We use the spectral sequence defined by the filtration and following Witten-Floer ideas we bring into play the orbits connecting the critical submanifolds.

On some examples in Symplectic Topology

Article is devoted to the Examples 2 and 3 of the symplectic solvable Lie groups $R$ with some special cohomological properties, which have been constructed by Benson and Gordon. But they are not succeeded in constructing corresponding compact forms for symplectic structures on these Lie groups. Recently A.Tralle proved that there is no compact form in the Example 3. But his proof is rather complicated and uses some very special topological methods. We propose much more simpler (and purely algebraic) method to prove the main result of the Tralle's paper. Moreover we prove that for Example 2 th…

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…

辛拓扑中的特征类

Characteristic Classes in Symplectic Topology

从上同调的观点来看，辛流形的辛同胚群$Sympl (M)$比辛同胚群“更驯服”。李代数$\frak {sympl }(M)$中不变多项式的存在性、辛Chern-Weil理论和Chern-Simons型次级类的存在性是这一原理的第一个表现。在更深的层次上，生活在周期上同调和辛霍奇分解的辛作用的特征类。本文旨在介绍上述理论和结构，并提出许多具体的应用。这些措施包括：辛同构群上同调的非零结果（作为拓扑空间，作为拓扑群和作为离散群），Chern类的辛刚性，Lagrange合痕体积的下界，由Givental，Kleiner和Oh开始的课题，Torelli群的新特征和单关系群的自同构群的推广，Witten zeta函数特殊值的算术性质及Brylinski猜想的解附录，由L。Katzarkov，研究了模空间中有限群作用的不动点集.

On the integrability of symplectic Monge-Ampére equations

Let u be a function of n independent variables x^1, ..., x^n, and U=(u_{ij}) the Hessian matrix of u. The symplectic Monge-Ampere equation is defined as a linear relation among all possible minors of U. Particular examples include the equation det U=1 governing improper affine spheres and the so-called heavenly equation, u_{13}u_{24}-u_{23}u_{14}=1, describing self-dual Ricci-flat 4-manifolds. In this paper we classify integrable symplectic Monge-Ampere equations in four dimensions (for n=3 the integrability of such equations is known to be equivalent to their linearisability). This problem ca…

The topology of Liouville foliation for the Kovalevskaya integrable case on the Lie algebra so(4)

In this paper we study topological properties of an integrable case for Euler's equations on the Lie algebra $\textrm{so}(4)$, which can be regarded as an analogue of the classical Kovalevskaya case in rigid body dynamics. In particular, for all values of the parameters of the system under consideration the bifurcation diagrams of the momentum mapping are constructed, the types of critical points of rank $0$ are determined, the bifurcations of Liouville tori are described and the loop molecules for all singular points of the bifurcation diagram are computed. It follows from the obtained result…

中心仿射 $\mathbb{R}^m$ 中扭曲多边形的可积演化

Integrable evolutions of twisted polygons in centro-affine $\mathbb{R}^m$

本文通过中心仿射空间中弧长参数化多边形的几何实现，证明离散 $W_m$ 格是双哈密顿的，并利用预辛形式简洁地证明了两个哈密顿结构的相容性，进而研究其核与相关可积系统。