在固定Kahler类中，Calabi能量泛函的驻点称为极值度量。我们研究Riemann曲面上极值度量若干问题，包括给定边值条件极值度量的存在性、正则性，极值度量在空间形式的实现，带锥奇点的Kahler-Ricci 孤立子存在性等问题。极值度量涉及一个4阶椭圆方程，边值条件为单位圆盘给定边值、或者在闭Riemann曲面的若干点上给定锥角度。我们需要研究的是单位圆盘上极值度量弱解的正则性，闭曲面上带锥奇点极值度量的存在性，特殊极值度量在三维欧氏空间和二维复射影空间的等距嵌入，Kahler-Ricci孤立子解的存在和几何结构，等具体问题。

A extremal metric is a critical point of Calabi functional in a fixed Kahler class in Kahler geometry. What we will study are the existence and regularity of extremal metric with given boundary conditions, and the existence of Kahler-Ricci soliton with conical singularities, in certain Riemannian surfaces. The Euler-Lagrange equation of extremal metric is a 4th-order elliptic equation, and the boundary conditions are boundary data on unit disc, and pre-described singular angle on closed Riemannian surfaces. We will study the regularity of weak solution of extremal metric equation on unit disc, the existence of extremal metric on closed surface with conical singularity, the isometric embedding of ertremal metric into space forms, and Kahler-Ricci soliton.