黎曼流形上的谱理论与微分几何，偏微分方程，随机过程，数学物理等众多学科领域有着密切的联系，因此相关问题的研究目前是国内外的热点课题。本课题的研究范围是针对完备的黎曼流形，重点围绕三种常曲率空间形式展开；主要的研究对象是黎曼流形上的几类特殊的椭圆算子（包括Laplace算子，二重Laplace算子，Shrodinger算子等）; 研究的背景是从微分方程，几何，物理等角度提出来的相关的Dirichlet问题、 Newmann 问题以及闭特征值问题; 其研究目的是在前人的工作基础之上，考虑上述问题的椭圆算子的特征值的上下界估计，建立起对应的Yang型不等式，讨论相邻两个特征值的间隙以及相邻两个特征值的比率，研究Clamped plate问题所对应的Rayleigh-Faber-Krahn型不等式。同时，拟将这些相应的结果拓展到更一般的椭圆算子上去。

The spectrum theory on Riemannain manifolds is closely related to many of fields of science, such as differential geometry, partial differential equations, stochastic process, mathematical physics and so on. Therefore, research of the corresponding problem is a very popular topic among the domestic and overseas mathematician recently. In this project, the scope of research will be the complete Riemannian manifolds, mainly considering three kinds of constant curvature space; As the main research objects, we would like to involve in some special elliptic operators, including Laplacian, bi-Laplacian, Shrodinger operator and so on; The background of studies will be the Dirichlet problem, Newmann problem and the closed eigenvalue problem which are represented from differential equation，geometry, physics and so on; Its purpose is, based on former work, to consider the estimates for the lower and upper bounds, to establish the universal inequality of Yang type, to discuss the gap and ratio between the consecutive eigenvalues, and to investigate the inequality of the Rayleigh-Faber-Krahn type of the clamped plate problem. In addition, we would like to extend those related results to the general elliptic operators.