Riemann 流形和复流形上特征值问题是几何和分析中的重要研究方向之一，与流形的拓扑、偏微分方程、数学物理有着密切的联系。流形上椭圆算子特征值问题的研究有助于理解流形的几何和拓扑性质。本项目中，我们计划运用几何分析、微分方程、泛函分析等理论知识，以及特征值研究中的极大极小原理的技巧，研究与Polya 猜想相关的Laplace算子特征值问题，关于特征值的比值的PPW猜想以及相邻特征值的间隙渐进阶的估计。这些问题的研究将丰富流形上的特征值理论。我们还会运用多复变和复几何的理论知识，研究具有负全纯双截曲率Kahler-Einstein曲面的分类问题，得到曲面的几何和拓扑性质。

The eigenvalue problems on Riemannian manifolds and complex manifolds is one of the important research fields in Geometry and Analysis, which is related to the topology on manifold, partial differential equation and mathematical physics. The research on the eigenvalue problem of elliptic operators on manifolds is helpful to understand the geometry and topology of manifolds. In this proposal, we propose to study the eigenvalue problem related to Polya Conjecture, PPW conjecture related to the ratio of the eigenvalues and the asymptotic order of the gaps of consecutive eigenvalue, in terms of the theory about geometric analysis, differential equations and functional analysis, and the techniques of the minimum-maximum principle. The research on the problems will contribute to the eigenvalue theory on manifolds. From the theory on the several complex variables and complex geometry, we will also study the classification of Kahler-Einstein surfaces with negative holomorphic bisectional curvature and then obtain the geometric and topological properties of the complex surfaces.