复Grassmann流形中的子流形几何

It is an important problem in geometry of submanifolds to study the homegeneity and rigidity of specific surfaces and submanifolds in symmetric spaces. In this project, we will focus on studying the geometry and rigidity properties of minimal 2 and 3 dimensional spheres immersed in complex Grassmann manifolds. By using harmonic sequences and moving frames, we will study the geometry and rigidity of minimal 2 spheres in complex Grassmann manifolds, the classification of minimal 2 spheres with constant curvature and the value distribution of Gaussian curvature of minimal 2 spheres with constant curvature in complex Grassmann manifolds. We will also develop new approach to construct minimal 3 spheres with special geometric properties in complex projective spaces and complex Grassmann manifolds by using representations of Lie Groups. Furthermore, we will use Lie group theory and moving frames to study geometry and rigidity of minimal 3 spheres in complex projective spaces and complex Grassmann manifolds. The results of this project will enrich our knowledge about geometry of submanifolds in complex Grassmann manifolds and provide new ideas and methods to study geometry of submanifolds in symmetric spaces.

研究对称空间中特殊曲面和子流形的齐性和刚性是子流形几何中的重要问题。本项目主要研究复Grassmann流形中2维和3维的浸入极小球面的几何和刚性性质。利用调和序列和活动标架法研究复Grassmann流形中极小2维球面的几何和刚性性质，常曲率极小2维球面的分类和高斯曲率值分布；利用李群表示构造复射影空间和复Grassmann流形中具有特殊几何性质的极小3维球面；利用活动标架法和李群理论研究复射影空间和复Grassmann流形中极小3维球面的几何和刚性性质。该项目的研究成果将丰富我们对复Grassmann流形中子流形几何的了解，为进一步研究更一般的对称空间中的子流形几何提供新的思路和方法。

子流形的齐性和刚性是子流形几何研究的经典课题。复Grassmann流形是复射影空间的推广，其几何结构比复射影空间要复杂很多。本项目主要研究复Grassmann流形中极小曲面的几何和刚性。首先我们证明了复Grassmann流形中调和序列保齐性的性质，利用这一结果，我们完全分类了四元数射影空间HP^n中的齐性极小2维球面，当n是奇数时解决了Ohnita的猜想，这里HP^n视为复Grassmann流形G(2,2n+2)中的全测地子流形。其次我们证明了G(2,6)中全纯曲线的局部刚性定理，这一结果推广了P. Griffiths关于G(2,4)中全纯曲线的局部刚性定理。利用李群理论和SU(2)的复不可约表示，我们给出了复Grassmann 流形G(2,n)和G(3,n)中齐性全纯2维球面的完全分类，这些结果给我们提供了大量复Grassmann 流形中常曲率全纯2维球面的例子，有助于我们进一步研究复Grassmann 流形中常曲率全纯2维球面的分类。