本课题的主要研究目标为：.1.完成S^n中的Willmore球面分类，并利用pluri-调和映射以及Uhlenbeck的工作，给出一个比较好的几何解释，利用这些方法讨论具体的Willmore泛函值分布。.2．讨论具有对称性的Willmore曲面，特别是Equivariant Willmore环面的模空间刻画和新的对称Willmore曲面例子。.3. 通过对共形Gauss映射和Helein-Ma调和映射的深入分析，讨论这两个映射之间关系，并给出Willmore曲面的奇点性质的更多刻画，尝试给出伴随变换的可积系统解释。

The main aims of this project are as follows:.1. Solve the classification problem of Willmore 2-spheres in S^n and provide an interesting goemetric interpretation, by use of the methods of pluri-harmonic maps and the factorization theorems of Uhlenbeck. Discuss the value distribution of Willmore functional and the corresponding module space..2. Derive new Willmore surfaces with some kind of symmetry. In particular, construct equivariant Willmore tori and discuss the module space of them..3. Consider the relations between the conformal Gauss map and Helein-Ma's harmonic map, by using detailed calculation. Provide more informations on the sigularities of Willmore surfaces. And try to derive an explaination of Ma's adjoint transform from the view of intergrable system.