具有闭Moebius形式的子流形的若干研究

The study about Moebius geometry of submanifolds is an important subject in differential geometry. This program will concentrate on Moebius form, which is one of four basic Moebius variants of submanifolds. We will study the classification problem and global analysis problem about topology structure and geometry structure of hypersurfaces with closed Moebius form in a space form. Find new examples of Willmore hypersurfaces in this class of hypersurfaces. We will also study the relation between higher order Willmore hypersurfaces and hypersurfaces with closed Moebius form. All problems mentioned above will enrich our knowledge about submanifolds with non-vanishing Moebius form.

子流形的Moebius几何是微分几何的重要研究方向之一，在子流形的Moebius几何中存在四个基本的Moebius不变量，本项目将致力于研究其中的Moebius形式。研究球中超曲面在具有闭Moebius形式情形下的分类问题、拓扑结构与几何结构的整体分析问题，在这类超曲面中寻找Willmore超曲面的新例子，研究高阶Willmore超曲面与具有闭Moebius形式的超曲面之间的关系。这些研究将极大地丰富我们对Moebius形式非消失的子流形的认识。

本项目研究具有闭Moebius形式的超曲面，得到具有三个不同常Moebius主曲率情形的完整分类结果，构造出此情形下的Willmore超曲面，从而得到Willmore超曲面的新例子。