Lie球几何及其子几何中子流形的局部分类与整体刚性问题

子流形的局部分类和整体刚性问题是子流形几何的核心问题之一，历来受到几何学家高度关注。传统的几何主要是射影几何和Lie球几何，其他几何均是它们的子几何。同射影几何相比，Lie球几何中的子流形理论的研究还处于前期发展阶段，它的基本问题，即子流形的Lie球几何完全不变量系统还没有建立。同射影几何的子几何仿射几何相比，Lie球几何的子几何：Moebius几何和Laguerre几何发展比较滞后，有许多重要问题有待解决。本项目主要研究Lie球几何及其子几何中子流形的一些基本和重要问题，它们包括：构造子流形的Lie球几何的不变量系统；Dupin超曲面局部分类与整体刚性问题；广义Willmore猜想和Willmore子流形的整体刚性问题；Laguerre极小超曲面的局部行为问题；Lie球几何中齐性超曲面的分类问题。我们的研究目标是试图解决这些基本重要问题，推动Lie球几何及其子几何的发展。

Local classification and global rigidity of Submanifolds are one of the core problems of submanifold geometry, which have always attracted great attention of geometrists. Traditional geometry is mainly divided into projective geometry and Lie sphere geometry, and other geometries are their subgeometries. Compared with projective geometry, the research on the Submanifold theory in Lie sphere geometry is still in the early stage of development. The basic problem of Lie sphere geometric, i.e. the Lie sphere complete invariant system of submanifolds, has not yet been established. Compared with the subgeometry of projective geometry affine geometry, the subgeometry of Lie sphere geometry: Moebius geometry and Laguerre geometry also lag behind, and there are many important problems to be solved. This project will mainly study some basic and important problems in submanifolds of Lie sphere geometry and its sub-geometry, they include: constructing a complete Lie sphere invariant system of submanifolds; local classification and global rigidity of Dupin hypersurfaces; generalized Willmore conjecture and global rigidity of Willmore submanifolds; local behavior of Laguerre minimal hypersurfaces; classification problems of Moebius homogeneous hypersurfaces and Laguerre homogeneous hypersurfaces. Our research goal is to try to solve these basic problems and promote the development of submanifold geometry of Lie sphere geometry and its subgeometry.