本项目希望通过几何、分析及可积系统方法，研究如下3个方向的重要问题: (1) S^n中环面和高亏格曲面的Willmore猜想问题，目标为在特殊情形下通过一些稳定Willmore曲面的分类来证明这些猜想; (2) 给出S^n中Willmore二维球面的完全几何分类，并分类稳定Willmore二维球面; (3) 研究S^n中极小子流形的特征值问题与子流形的广义Willmore猜想问题，给出一些新的第一特征浸入极小子流形的例子及讨论其Willmore能量稳定性。

We study the important problems in the following three directions, in terms of geometric, analytic and integrable system methods: (1) The Willmore conjecture for 2-tori or higher genus surfaces in S^n, aiming at proving them in special cases via classifying special stable Willmore surfaces; (2)give classifications of Willmore two-spheres in S^n and stable Willmore two-spheres; (3) Study the spectral geometry of minimal submanifolds in S^n and relations with the generalized Willmore conjecture problem for submanifolds, by giving examples of minimal submanifolds immersed by 1st eigenfunctions and discussion/characterization of their Willmore energy.