单位球面中子流形的刚性和分类问题研究

Geometry of submanifolds is an important research branch of global differential geometry, which plays important roles in analysis, topology and differential equations. In particular, the rigidity problem of submanifolds is an important research subject in geometry of submanifolds. Since S. S. Chern proposed Chern conjecture in 1968, only partial results exist in particular for hypersurfaces with low dimensions or with additional curvature conditions during the past half of the century. In this project, we will study the global geometric properties and rigidity problems of closed minimal hypersurfaces in a unit sphere from the view of distinct principal curvatures. Especially, we will study Chern conjecture for the Willmore hypersurfaces with four distinct principal curvatures, study Chern conjecture for the hypersurfaces with more than six distinct principal curvatures uner some geometric conditions, and study the corresponding rigidity problems for hypersurfaces with constant mean curvature. By establishing a frame field, we will study the geometric structure of minimal surfaces or CMC surfaces in a unit sphere with higher codimension, find the relationships of geometric quantities, and study the pinching problems of the Gauss curvature and the normal curvature. This project would supply some further methods and partial answers to the Chern conjecture and Simon conjecture.

子流形几何是整体微分几何的一个重要组成部分,在分析、拓扑和方程中扮演了重要的角色，其中子流形的刚性问题是子流形几何中一个重要的研究课题。自从1968年陈省身提出著名的Chern猜想以来，经过近半个世纪的研究，Chern猜想只有在低维超曲面和特殊的曲率函数条件下被解决了。项目从互异主曲率个数角度研究单位球中任意维闭极小超曲面的整体几何性质和刚性问题，特别是研究Willmore假设下具有四个互异主曲率超曲面的Chern猜想，在一定几何条件下研究具有四个以上互异主曲率超曲面的Chern猜想，研究该情形下具有常数量曲率和常平均曲率超曲面的刚性问题。从建立标架场着手，研究单位球中高余维数的极小曲面和常平均曲率曲面的几何结构，寻找几何量之间的关系，研究高斯曲率和法曲率的拼挤问题。项目拟对Chern猜想和Simon猜想的解决给出进一步的思路和部分答案。

子流形几何是整体微分几何的一个重要组成部分,在分析、拓扑和方程中扮演了重要的角色，其中子流形的刚性和分类问题是子流形几何中一个重要的研究课题。本项目研究了极小曲面和几类相关子流形的分类和几何性质。包括：解决了球空间中的极小曲面关于高斯曲率和法曲率的第三个拼挤问题；给出了几类极小曲面的分类结果，对Simon猜想给出部分进展；研究了球空间中线性Weingarten子流形关于第二基本形式的刚性问题；证明了欧氏和伪欧氏空间中lambda双调和超曲面在一定条件下具有常平均曲率的性质；证明了一类曲线流的存在性，并给出了各种几何不变量在流下的发展方程；对三维Minkowski空间中广义Biconservative曲面进行了完整分类。由于国内外研究领域的发展和变化，研究内容略有调整，但完成了项目预期研究目标。