轨形Riemann面的Gromov-Witten不变量沿光滑点的加权涨开公式

本文考虑当一个紧辛轨形(orbifold)Riemann面(X,ω)沿着光滑点作加权涨开时,它的轨形Gromov-Witten不变量的变化情形和辛一致规则(uniruledness)性质的变化情形;证明了如下的结果:<α_1,...,α_m>~X _g,A=<p*α_1,...,p*α_m,■,<α_1,...,α_m,[pt]>~X _g,A=I_A·■.第一个公式表明,当(X,ω)是辛一致规则的(uniruled)时,它的沿光滑点的加权涨开■也是辛一致规则的.

Riemann面——平面上的空间结构

Riemann面是复变函数课程中的一个很有特色的内容,也是学生学习中的一个难点。本文从代数和几何这两个视度,简单分析了Riemann面的作用。文中的描述,试图简单、直观、易懂,为初学者了解Riemann面,提供点滴直接的帮助。

黎曼面上的黎曼边值问题及其在物理中的应用

国家自然科学基金资助项目《黎曼面上的黎曼边值问题及其在物理中的应用》是研究Riemann面上的Riemann边值问题：讨论一般（紧和非紧）黎曼面上的向量值、矩阵值和Hopf代数值解析边值问题，利用这些结果处理Alain Connes重整化理论，把它们化为奇异积分方程，并利用已有的奇异积分数值方法的技巧实现数值计算。然后把这些工作用于引力场论，其重点是对应曲面为非紧情形的真空静态轴对称Einstein方程的求解。

高亏格Riemann面上的世界面孤子与自由能

本文通过采用世界面孤子作用量计算了高亏格Riemann面上的自由能。这类孤子作用量对应于弦坐标耦合两维标量场。

Submanifolds with Parallel Mean Curvacture

In this paper, we consider a class of submanifolds with parallel mean curvacture vector fields. We obitain the suffitient conditions that the above submanifolds is of tatall umbilical and that its codimension is decrease.

Periodic Points on the Parabolic Domain

The papers proves that there is one and only one rational neutral periodic points on the bound ary of any parabolic domain.

C~∞ Compactness for a Class of Riemannian Manifolds with Parallel Ricci Curvature

In this paper we prove that tile set of Riemannian manifolds with parallel Ricci curvature, lower bounds for sectional curvature and injectivity radius and a upper bound for volume is coo compact in Gromov-Hausdroff topology. As an application we also prove a pinching result which states that a Ricci flat manifold is flat under certain conditions.

Chaplygin气体带有三片常状态初值的二维Riemann问题

本文研究了具有三片常状态初值的Chaplygin气体的Euler方程组的二维Riemann问题.三片常状态初值由y轴的正半轴和x轴来划分.假设原点外的初始数据的每一次间断都恰好只产生一个平面激波、中心稀疏波或滑移面,我们利用广义特征分析的方法详细给出了解的结构.事实上,我们将其分为十种情形进行讨论,并说明其中只有四个子情形是合理的.

On the Tangent Bundle of a Hypersurface in a Riemannian Manifold

Let （Mn, g） and （N^n＋1, G） be Riemannian manifolds. Let TMn and TN^n＋1 be the associated tangent bundles. Let f ： （M^n, g） → （N^＋1, G） be an isometrical immersion with g = f^＊G, F = （f, df） ： （TM^n,g） → （TN^n＋1, Gs） be the isometrical immersion with g= F＊Gs where （df）x ： TxM → Tf（x）N for any x∈ M is the differential map, and Gs be the Sasaki metric on TN induced from G. This paper deals with the geometry of TM^n as a submanifold of TN^n＋1 by the moving frame method. The authors firstly study the extrinsic geometry of TMn in TN^n＋1. Then the integrability of the induced almost complex structure of TM is discussed.

Hyperbolic Mapping Classes and Their Lifts on the Bers Fiber Space

Let S be a Riemann surface with genus p and n punctures. Assume that 3p- 3 ＋ n 〉 0 and n 〉 1. Let a be a puncture of S and let S = S∪{α}. Then all mapping classes in the mapping class group Mods that fixes the puncture a can be projected to mapping classes of Mod$ under the forgetful map. In this paper the author studies the mapping classes in Mods that can be projected to a given hyperbolic mapping class in Mode.

Embedded Surfaces for Symplectic Circle Actions

The purpose of this article is to characterize symplectic and Hamiltonian circle actions on symplectic manifolds in terms of symplectic embeddings of Riemann surfaces.More precisely, it is shown that(1) if(M, ω) admits a Hamiltonian S^1-action, then there exists a two-sphere S in M with positive symplectic area satisfying c1(M, ω), [S] > 0,and(2) if the action is non-Hamiltonian, then there exists an S^1-invariant symplectic2-torus T in(M, ω) such that c1(M, ω), [T] = 0. As applications, the authors give a very simple proof of the following well-known theorem which was proved by Atiyah-Bott,Lupton-Oprea, and Ono: Suppose that(M, ω) is a smooth closed symplectic manifold satisfying c1(M, ω) = λ· [ω] for some λ∈ R and G is a compact connected Lie group acting effectively on M preserving ω. Then(1) if λ < 0, then G must be trivial,(2) if λ = 0, then the G-action is non-Hamiltonian, and(3) if λ > 0, then the G-action is Hamiltonian.

A magnetically impermeable and electrically permeable interface crack with a contact zone in a magnetoelectroelastic bimaterial under concentrated magnetoelectromechanical loads on the crack faces

An interface crack with a frictionless contact zone at the right crack-tip between two dissimilar magnetoelectroelastic materials under the action of concentrated magnetoelectromechanical loads on the crack faces is considered. The open part of the crack is assumed to be magnetically impermeable and electrically permeable. The Dirichlet-Riemann boundary value problem is formulated and solved analytically. Stress, magnetic induction and electrical displacement intensity factors as well as energy release rate are thus found in analytical forms. Analytical expressions for the contact zone length have been derived. Some numerical results are presented and compared with those based on the other crack surface conditions. It is shown clearly that the location and magnitude of the applied loads could significantly affect the contact zone length, the stress intensity factor and the energy release rate.

Local Gromov-Witten invariants and tautological sheaves on Hilbert schemes

We study the local Gromov-Witten invariants of O（k）⊕O（-k-2） → P1 by localization techniques and the Marino-Vafa formula, using suitable circle actions. They are identified with the equivariant Riemann-Roch indices of some power of the determinant of the tautological sheaves on the Hilbert schemes of points on the affine plane. We also compute the corresponding Gopakumar-Vafa invariants and make some conjectures about them.

On Reduced Lantern Relations in Mapping Class Groups

Let S be a hyperbolic Riemann surface with a finite area. Let G be the covering group of S acting on the hyperbolic plane H. In this paper, the author studies some algebraic relations in the mapping class group of S for S = S／{a point}. The author shows that the only possible relations between products of two Dehn twists and products of mapping classes determined by two parabolic elements of G are the reduced lantern relations. As a consequence, a partial solution to a problem posed by J. D. McCarthy is obtained.

VARIATIONAL PROPERTIES OF THE INTEGRATED MEAN CURVATURES OF TUBES IN SYMMETRIC SPACES

Let Pt denote the tubular hypersurface of radius t around a given compatible submanifold in a symmetric space of arbitrary rank. The authors will obtain some relations between the integrated mean curvatures of P, and their derivatives with respect to f. Moreover, the authors will emphasize the differences between the results obtained for rank one and arbitrary rank symmetric spaces.

MINIMAL SURFACES IN 3-DIMENSIONAL SOLVABLE LIE GROUPS

The author studies minimal surfaces in 3-dimensional solvable Lie groups with left invariantRiemannian metrics. A Weierstraβ type integral representation formula for minimal surfaces isobtained.