The Yamabe Problem and Applications on Noncompact Complete Riemannian Manifolds

We let (M,g) be a noncompact complete Riemannian manifold of dimension n 3 whose scalar curvature S(x) is positive for all x in M. With an assumption on the Ricci curvature and scalar curvature at infinity, we study the behavior of solutions of the Yamabe equation on –u+[(n–2)/(4(n–1))]Su=qu ^{(n+2)/(n–2)} on (M,g). This study finds restrictions on the existence of an injective conformal immersion of (M,g) into any compact Riemannian n -manifold. We also show the existence of a complete conformal metric with constant positive scalar curvature on (M,g) with some conditions at infinity.     

Eigenvalue Comparison on Bakry-Emery Manifolds

We prove a comparison theorem on the modulus of continuity of the solution of a heat equation with a drifting term on Bakry-Emery manifolds. A direct consequence of the result is an alternate proof of an eigenvalue comparison result of Bakry-Qian. Examples are given to show that the estimate is sharp. Discussions on an explicit lower estimate for the corresponding ODE and an application to the diameter lower bound for gradient shrinking solitons are also included.     

Inverse Eigenvalue Problem in Structural Dynamics Design

1 Introduction Structural dynamics design is to design a structure subject to the dynamic characteristics re- quirement, i.e., determine physical and geometrical parameters such that the structure has the given frequencies and (or) mode shapes. This problem often arises in engineering connected with vibration. Recently, Joseph [1], Li et al. [2,3] converted the structural dynamics design to the following inverse eigenvalue problem. GIEP Let x = (x1, , xm)T , and let A(x) and B(x) be real n…     

小Excess与具负下界Ricci曲率开流形的拓扑

In this paper,we study the relation between the excess of open manifolds and their topology by using the methods of comparison geometry.We prove that a complete open Riemmannian manifold with Ricci curvature negatively lower bounded is of finite topological type provided that the conjugate radius is bounded from below by a positive constant and its Excess is bounded by some function of its conjugate radius,which improves some results in [4].     

On Strictly Convex Subsets in Negatively Curved Manifolds

In a complete simply connected Riemannian manifold X of pinched negative curvature, we give a sharp criterion for a subset C to be the ??-neighbourhood of some convex subset of X, in terms of the extrinsic curvatures of the boundary of C.     

An Obstruction to Fundamental Groups of Positively Ricci Curved Manifolds

In this note we propose a conjecture concerning fundamental groups of Riemannian n-manifolds with positive Ricci curvature. We prove a partial result under an extra condition on a lower bound of sectional curvature. Our main tool is the theory of Hausdorff convergence. We also extend Fukaya and Yamaguchi's resolution of a conjecture of Gromov to limit spaces which may have singular points.     

Finsler Manifolds with Positive Constant Flag Curvature

It is shown that a Finsler metric with positive constant flag curvature and vanishing mean tangent curvature must be Riemannian. As applications, we also discuss the case of Cheng's maximal diameter theorem and Green's maximal conjugate radius theorem in Finsler manifolds.     

边界条件含特征参数的Sturm-Liouville问题的特征值比率

给出修正Prfer变换的几何意义以及相关性质,并利用上述性质讨论Sturm-Liouville问题-(py′)′+gy=λωy,x∈[0,1],参数边界条件为y(0)=0与((py′)(1))/(y(1))=aλ+b,得到了当a>0,q≥0的特征值比率上界,这把Ashbaugh等人的结果推广到边界条件含参数的情形.     

紧致流形的第一特征值下界

对紧致Riemannian流形(无边或带有凸边界)的第一(Neumann)特征值,用流形的直径和Ricci曲率的下界,给出一些新的下界估计.     

Examples of Non-Homeomorphic Harmonic Maps Between Negatively Curved Manifolds

Let M and N be closed non-positively curved manifolds, and letf:MN be a smooth map. Results of Eells and Sampson [1] showthat f is homotopic to a harmonic map , and Hartman [6] showedthat this is unique when N is negatively curved and f_*(₁ M)is not cyclic. Lawson and Yau conjectured that if f:MN is ahomotopy equivalence, where M and N are negatively curved, thenthe unique harmonic map homotopic to f would be a diffeomorphism.Counterexamples to this conjecture appeared in [2], and laterin [7] and [5]. There remains the question of whether a ‘topological’Lawson–Yau conjecture holds. 1991 Mathematics SubjectClassification 53C20, 55P10, 57C25, 58E20.     

具有相对迷向平均Landsberg曲率的流形的一个整体刚性定理

讨论了具有相对迷向平均Landsberg曲率的度量的一些几何性质.证明了任一闭的具有负旗曲率与相对迷向平均Landsberg曲率的流形一定是Riemann流形.     

关于3维负曲率流形的拓扑型

本文的主要定理是:对每一个正数V,存在一个D>0,如果一个3维黎曼流形M的体积小于V,截面曲率在-1和0之间,而且直径大于D,那么M允许一个双曲结构。     

关于Kaehler-Einstein流形上Rastogi联络的一点注记

研究Kaehler-Einstein流形M上Rastogi;联络的拟共形曲率张量场W-,证明了若-W是平行的,则M是拟共形对称的.也得到关于M共圆对称的对应条件和结果,推广了Rastogi,贾兴琴等的工作.     

On the General Algebraic Inverse Eigenvalue Problems

A number of new results on sufficient conditions for the solvability and numerical algorithms of the following general algebraic inverse eigenvalue problem are obtained: Given $n+1$ real $n\times n$ matrices $A=(a_{ij}),A_k=(a_{ij}^{(k)})(k=1,2,\cdots,n)$ and $n$ distinct real numbers $\lambda_1,\lambda_2,\cdots,\lambda_n,$ find $n$ real number $c_1,c_2,\cdots,c_n$ such that the matrix $A(c)=A+\sum\limits_{k=1}^{n}c_k A_k$ has eigenvalues $\lambda_1,\lambda_2,\cdots,\lambda_n.$     

Positively Curved Manifolds with Almost Maximal Symmetry Rank

The symmetry rank of a Riemannian manifold is the rank of the isometry group. We determine precisely which closed simply connected 5-manifolds admit positively curved metrics with (almost maximal) symmetry rank two. We also determine the precise Euler characteristic and the fundamental groups of all closed positively curved n-manifolds with almost maximal symmetry rank [(n–1)/2] (n 6, 7).     

斜对称双对角矩阵特征值反问题

讨论了关于斜对称双对角矩阵的特征值反问题．即：已知一个n阶斜对称双对角矩阵的特征值和两个n-1阶子矩阵的部分特征值,则可求得该矩阵．最后给出了数值例子．     

Eigenvalue solution to the electron-collisional effect on ion-acoustic and entropy waves

The linearized electron Fokker- Planck and cold-ion fluid equations are solved as an eigenvalue problem in the quasineutral limit for ionization state, Z = 1, 8, and 64 for ion-acoustic and entropy waves. The perturbed electron distribution function is written as a moment expansion of eigenvectors, and is used to compute collisionality-dependence macroscopic quantities in the plasma such as the generalized specific heat ratio, and the electron thermal conductivity.     

一类特殊矩阵的逆特征值问题

本文主要讨论如下形式矩阵的逆特征值问题:即对给定n个实数λ_1>λ_2>…>λ_2与n-1个实数μ_1>μ_2>…>μ_(n-1),满足λ_1>μ_1>λ_2>…>λ_(n-1)>μ_(n-1)>λ_n,在α_2>α_3>…>α_(n-1)的条件下,存在唯一的一个矩阵A_n是以λ_i为其特征值;且其截边矩阵的特征值为μ_1,μ_2,…,μ_(n-1).     

Stieltjes Sturm-Liouville equations: Eigenvalue dependence on problem parameters

We examine properties of eigenvalues and solutions to a 2n-dimensional Stieltjes Sturm-Liouville eigenvalue problem. Existence and uniqueness of a solution has been established previously. An earlier paper considered the corresponding initial value problem and established conditions which guarantee that solutions depend continuously on the coefficients [L.E. Battle, Solution dependence on problem parameters for initial value problems associated with the Stieltjes Sturm-Liouville equations, Electron. J. Differential Equations 2005 (2) (2005) 1-18]. Here, we find conditions which guarantee that the eigenvalues and solutions depend continuously on the coefficients, endpoints, and boundary data. For a simplified two-dimensional problem, we find conditions which guarantee the eigenvalues to be differentiable functions of the problem data.