Symplectic Group Actions on Homotopy Elliptic Surfaces

In this paper, the authors study the homologically trivial symplectic group actions on homotopy elliptic surfaces E（n） and get some rigidity results.     

椭圆曲面上的辛循环作用

Let X = E（n） be the relatively minimal elliptic surface with rational base, where n 〉 2. In this paper, several pseudofree, homologically trivial, symplectic cyclic actions by groups whose orders are 2, 3, 5 and 7 on X are studied.     

Frankel's theorem in the symplectic category

We prove that if an -dimensional torus acts symplectically on a -dimensional symplectic manifold, then the action has a fixed point if and only if the action is Hamiltonian. One may regard it as a symplectic version of Frankel's theorem which says that a Kähler circle action has a fixed point if and only if it is Hamiltonian. The case of is the well-known theorem by McDuff.

     

A wall-crossing formula for the signature of symplectic quotients

We use symplectic cobordism, and the localization result of Ginzburg, Guillemin, and Karshon to find a wall-crossing formula for the signature of regular symplectic quotients of Hamiltonian torus actions. The formula is recursive, depending ultimately on fixed point data. In the case of a circle action, we obtain a formula for the signature of singular quotients as well. We also show how formulas for the Poincaré polynomial and the Euler characteristic (equivalent to those of Kirwan can be expressed in the same recursive manner.

     

Symplectic group actions and covering spaces

For symplectic group actions which are not Hamiltonian there are two ways to define reduction. Firstly using the cylinder-valued momentum map and secondly lifting the action to any Hamiltonian cover (such as the universal cover), and then performing symplectic reduction in the usual way. We show that provided the action is free and proper, and the Hamiltonian holonomy associated to the action is closed, the natural projection from the latter to the former is a symplectic cover. At the same time we give a classification of all Hamiltonian covers of a given symplectic group action. The main properties of the lifting of a group action to a cover are studied.     

Maslov P-Index Theory for a Symplectic Path with Applications

Abstract The Maslov P-index theory for a symplectic path is defined. Various properties of this index theory such as homotopy invariant, symplectic additivity and the relations with other Morse indices are studied. As an application, the non-periodic problem for some asymptotically linear Hamiltonian systems is considered. *Project supported by the National Natural Science Foundation of China (No.10531050) and FANEDD.     

Symplectic geometry and dissipative differential operators

Everitt and Markus characterized the domains of self-adjoint differential operators in terms of Lagrangian subspaces of complex symplectic spaces. In this paper we define Dissipative and strictly Dissipative subspaces for complex symplectic spaces and characterize the domains of dissipative and strictly dissipative differential operators in terms of these subspaces.     

Maslov P-Index Theory for a Symplectic Path with Applications

The Maslov P-index theory for a symplectic path is defined. Various properties of this index theory such as homotopy invariant, symplectic additivity and the relations with other Morse indices are studied. As an application, the non-periodic problem for some asymptotically linear Hamiltonian systems is considered.     

Moving Symplectic Curves in Kähler-Einstein Surfaces

We derive a parabolic equation for the K?hler angle of a real surface evolving under the mean curvature flow in a K?hler-Einstein surface and show that a symplectic curve remains symplectic with the flow. Received March 29, 2000, Revised April 29, 2000, Accepted May 16, 2000     

辛对称Hamilton算子值域的闭性

利用扰动理论和算子矩阵的因式分解,研究了辛对称Hamilton算子值域的闭性.针对对角占优、上行占优等情形,在一定条件下给出了值域闭性的若干描述,并得到了一般情形的结果.     

Elliptic Fibrations and Symplectic Automorphisms on K3 Surfaces

Nikulin has classified all finite abelian groups acting symplectically on a K3 surface and he has shown that the induced action on the K3 lattice U ³ ⊕ E ₈(?1)² depends only on the group but not on the K3 surface. For all the groups in the list of Nikulin we compute the invariant sublattice and its orthogonal complement by using some special elliptic K3 surfaces.     

Generalized Symplectic Mean Curvature Flows in Almost Einstein Surfaces

The authors mainly study the generalized symplectic mean curvature flow in an almost Einstein surface, and prove that this flow has no type-I singularity. In the graph case, the global existence and convergence of the flow at infinity to a minimal surface with metric of the ambient space conformal to the original one are also proved.     

Reissner板弯曲的辛求解体系

基于Reissner板弯曲问题的Hellinger-Reissner变分原理,通过引入对偶变量,导出Reissner板弯曲的Hamilton对偶方程组．从而将该问题导入到哈密顿体系,实现从欧几里德空间向辛几何空间,拉格朗日体系向哈密顿体系的过渡．于是在由原变量及其对偶变量组成的辛几何空间内,许多有效的数学物理方法如分离变量法和本征函数向量展开法等均可直接应用于Reissner板弯曲问题的求解．这里详细求解出Hamilton算子矩阵零本征值的所有本征解及其约当型本征解,给出其具体的物理意义．形成了零本征值本征向量之间的共轭辛正交关系．可以看到,这些零本征值的本征解是Saint-Venant问题所有的基本解,这些解可以张成一个完备的零本征值辛子空间．而非零本征值的本征解是圣维南原理所覆盖的部分．新方法突破了传统半逆解法的限制,有广阔的应用前景．     

Circle Patterns on Singular Surfaces

We consider “hyperideal” circle patterns, i.e., patterns of disks appearing in the definition of the weighted Delaunay decomposition associated with a set of disjoint disks, possibly with cone singularities at the centers of those disks. Hyperideal circle patterns are associated with hyperideal hyperbolic polyhedra. We describe the possible intersection angles and singular curvatures of those circle patterns on Euclidean or hyperbolic surfaces with cone singularities. This is related to results on the dihedral angles of ideal or hyperideal hyperbolic polyhedra. The results presented here extend those in Schlenker (Math. Res. Lett. 12(1), 82–112, [2005]), however, the proof is completely different (and more intricate) since Schlenker (Math. Res. Lett. 12(1), 82–112, [2005]) used a shortcut which is not available here. The author would like to thank the RIP program at Oberwolfach, where part of the research presented here was conducted. Partially supported by the “ACI Jeunes Chercheurs” Métriques privilégiés sur les variétés à bord, 2003-06, and the ANR program Representations of surface groups, 2007-09.     

Elliptic K3 Surfaces with Abelian and Dihedral Groups of Symplectic Automorphisms

We analyze K3 surfaces admitting an elliptic fibration ? and a finite group G of symplectic automorphisms preserving this elliptic fibration. We construct the quotient elliptic fibration ?/G comparing its properties to the ones of ?.

We show that if ? admits an n-torsion section, its quotient by the group of automorphisms induced by this section admits again an n-torsion section, and we describe the coarse moduli space of K3 surfaces with a given finite group contained in the Mordell–Weil group.

Considering automorphisms coming from the base of the fibration, we find the Mordell–Weil lattice of a fibration described by Kloosterman, and we find K3 surfaces with dihedral groups as group of symplectic automorphisms. We prove the isometries between lattices described by the author and Sarti and lattices described by Shioda and by Greiss and Lam.     

一族离散的可积Hamilton方程与可积的辛映射

该文讨论一个新的离散特征值问题，导出了相应的离散的Hamilton系统的保谱族，并且证明了它们是Liouville可积系。通过谱问题的双非线性化，导出一个新的可积的辛映射 。      

A Lagrangian camel

We prove the Lagrangian analogue of the symplectic camel theorem: there are compact Lagrangian submanifolds of that cannot be moved through a small hole by a global Hamiltonian isotopy with compact support. Received: November 9, 1998.     

求解大规模Hamilton矩阵特征问题的辛Lanczos算法的误差分析

对求解大规模稀疏Hamilton矩阵特征问题的辛Lanczos算法给出了舍入误差分析.分析表明辛Lanczos算法在无中断时,保Hamilton结构的限制没有破坏非对称Lanczos算法的本质特性.本文还讨论了辛Lanczos算法计算出的辛Lanczos向量的J一正交性的损失与Ritz值收敛的关系.结论正如所料,当某些Ritz值开始收敛时.计算出的辛Lanczos向量的J-正交性损失是必然的.以上结果对辛Lanczos算法的改进具有理论指导意义.     

SYMPLECTIC SCHEMES FOR NONAUTONOMOUS HAMILTONIAN SYSTEM

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