Riemann面上的共形度量是微分几何中的重要研究对象。该项目主要研究和Riemann面上奇异与非奇异共形度量相关的一些问题。这些问题包括：奇异extremal Kaehler度量，带奇点的常Gauss曲率度量和共形极小曲面问题。对于奇异extremal Kaehler度量，我们将用几何与复分析相结合的办法研究非常Gauss曲率的HCMU度量的存在性，用PDE的办法研究extremal Hermitian度量的性质。对于带奇点的常Gauss曲率度量，我们将研究球面上带锥奇点的常Gauss曲率度量的性质和存在性，还将研究既带锥奇点又带cusp奇点的常Gauss曲率度量的性质。对于共形极小曲面，我们将研究复Grassmann流形以及HP^n中极小曲面的几何。

Conformal metrics on Riemann surfaces are important objects in the differential geometry.This project focuses on some problems about singular and nonsingular conformal metrics on Riemann surfaces. These problems consist of singular extremal Kaehler metrics, constant Gauss curvature metrics with singularities and some related problems about conformal minimal surfaces.For singular extremal Kaehler metrics, we will study the existence of HCMU metrics whose Gauss curvatures are not constant by the methods of geometry and complex analysis and we will study the properties of extremal Hermitian metrics by the PDE method. For constant Gauss curvature metrics with singularities, we will study the properties and the existences of constant Gauss curvature metrics on 2-spheres with conical singularities and we will study the properties of constant Gauss curvature metrics with conical singularities and cusp singularities. For conformal minimal surfaces, we will study the geometries of minimal surfaces in complex Grassmann manifolds and HP^n.