The Classification of Exponential Paths in Sp(2)

众所周知，四阶以上辛群中指数型道路的Ｍａｓｌｏｖ指标可以取任意整数．本文给出了二阶辛群中指数型道路的分类及其Ｍａｓｌｏｖ指标的取值范围，指出了与高阶情形的不同之处．     

次二次Hamilton系统的对偶Morse指标及其闭特征的稳定性

本文通过Ｇｅｌｅｒｋｉｎ逼近方法,在没有任何凸的条件下,研究了次二次Ｈａｍｉｌ－ｔｏｎ系统的ｋ对偶Ｍｏｒｓｅ指标理论．作为应用,在本文研究了Ｒ２ｎ中的凸超曲面上的闭特征的稳定性,证明了在一个较宽松的夹条件下,这类超曲面上至少有一条椭圆闭特征．     

The Morse Index of Wente Tori

We find various lower and upper bounds for the index of Wente tori that contain a continuous family of planar principal curves. We then prove a result that gives an algorithm for computing the index sharply.     

Maslov Index and a Central Extension of the Symplectic Group

In this paper we show that over any field K of characteristic different from 2, the Maslov index gives rise to a 2-cocycle on the stable symplectic group with values in the Witt group. We also show that this cocycle admits a natural reduction to I ²(K) and that the induced natural homomorphism from K ₂ Sp(K)I ²(K) is indeed the homomorphism given by the symplectic symbol {x, y} mapping to the Pfister form 1, -x 1, –y.     

Morse Index of Multiple Blow-up Solutions to the Two-Dimensional Sinh-Poisson Equation

In this paper we consider the Dirichlet problemwhere $\rho$ is a small parameter and $\Omega$ is a $C^2$ bounded domain in $\mathbb{R}^2$. In [1], the author proves the existence of a $m$-point blow-up solution $u_\rho$ jointly with its asymptotic behaviour. We compute the Morse index of $u_\rho$ in terms of the Morse index of the associated Hamilton function of this problem. In addition, we give an asymptotic estimate for the first $4m$ eigenvalues and eigenfunctions.     

The symplectomorphism group of a blow up

We study the relation between the symplectomorphism group Symp M of a closed connected symplectic manifold M and the symplectomorphism and diffeomorphism groups Symp and Diff of its one point blow up . There are three main arguments. The first shows that for any oriented M the natural map from to is often injective. The second argument applies when M is simply connected and detects nontrivial elements in the homotopy group that persist into the space of self-homotopy equivalences of . Since it uses purely homological arguments, it applies to c-symplectic manifolds (M, a), that is, to manifolds of dimension 2n that support a class such that . The third argument uses the symplectic structure on M and detects nontrivial elements in the (higher) homology of BSymp, M using characteristic classes defined by parametric Gromov–Witten invariants. Some results about many point blow ups are also obtained. For example we show that if M is the four-torus with k-fold blow up (where k > 0) then is not generated by the groups as ranges over the set of all symplectic forms on . Partially supported by NSF grants DMS 0305939 and 0604769.     

The Minimal Length of a Closed Geodesic Net on a Riemannian Manifold with a Nontrivial Second Homology Group

Let Mⁿ be a closed Riemannian manifold with a nontrivial second homology group. In this paper we prove that there exists a geodesic net on Mⁿ of length at most 3 diameter(Mⁿ). Moreover, this geodesic net is either a closed geodesic, consists of two geodesic loops emanating from the same point, or consists of three geodesic segments between the same endpoints. Geodesic nets can be viewed as the critical points of the length functional on the space of graphs immersed into a Riemannian manifold. One can also consider other natural functionals on the same space, in particular, the maximal length of an edge. We prove that either there exists a closed geodesic of length ≤ 2 diameter(Mⁿ), or there exists a critical point of this functional on the space of immersed θ-graphs such that the value of the functional does not exceed the diameter of Mⁿ. If n=2, then this critical θ-graph is not only immersed but embedded.Mathematics Subject Classifications (2000). 53C23, 49Q10     

Geometrical Decomposition of the Free Loop Space on a Manifold with Finitely Many Closed Geodesics

In Morse theory an isolated degenerate critical point can be resolved into a finite number of nondegenerate critical points by perturbing the totally degenerate part of the Morse function inside the domain of a generalized Morse chart. Up to homotopy we can admit pertubations within the whole characteristic manifold. Up to homotopy type a relative CW-complex is attached, which is the product of a big relative CW-complex, representing the degenerate part, and a small cell having the dimension of the Morse index.     

On a product formula for the Conley–Zehnder index of symplectic paths and its applications

Using invariance by fixed-endpoints homotopies and a generalized notion of symplectic Cayley transform, we prove a product formula for the Conley–Zehnder index of continuous paths with arbitrary endpoints in the symplectic group. We discuss two applications of the formula, to the metaplectic group and to periodic solutions of Hamiltonian systems.      

*-Polynomial Identities of Matrices with the Symplectic Involution: The Low Degrees

Abstract

The *-polynomial identities of minimal degree of M _n (F) are determined for n = 2, 4, * the symplectic involution.     

Hodge equations with change of type

A geometric interpretation is given for certain elliptic-hyperbolic systems in the plane. Among several examples, one which reduces to the equations for harmonic 1-forms on the projective disc is studied in detail. A boundary-value problem for this example is formulated and shown to possess weak solutions. Received: May 16, 2001; in final form: September 24, 2001?Published online: October 30, 2002