复 Grassmann 流形中极小曲面的几何与分析

The geometry of minimal surfaces immersed in Riemann symmetric spaces is an important part of differential geometry, the research on its classification is closely related to Topology, Algebraic geometry and so on, and it attracts extensive attention from geometers and physicists at home and abroad all the time. The complex Grassmann manifold G(2,n) and its totally geodesic submanifold hyperquadric Q_{n-2}, as specific symmetric spaces, they have special geometric structures, the research on their minimal surfaces is closely related to the construction of Grassmann sigma model in theoretical physics, and it is a significant research subject in the field of submanifold geometry. Based on current research results, the following questions will be further studied in this project: Firstly, improve the current algorithm of constructing harmonic two-spheres in G(2,n;R), try to give a complete classification of conformal minimal two-spheres immersed in hyperquadric under the condition of constant Gauss curvature; Secondly, characterize and classify conformal minimal two-spheres in G(2,n) with parallel second fundamental form; And then,study the geometry of totally flat minimal two-toris in hyperquadric , mainly consider the related problem of construction; At last,discuss the 1-1 correspondence between minimal surfaces in G(2,n) and the open and affine Toda equations.

黎曼对称空间中极小曲面的几何是微分几何的重要内容，关于它的分类问题的研究与拓扑学、代数几何等有着密切联系，一直受到国内外几何学家和物理学家的广泛关注。复Grassmann流形G(2,n)以及它的全测地子流形超二次曲面Q_{n-2}，作为特定的对称空间，具有特殊的几何结构，其上极小曲面的研究与理论物理中Grassmann sigma-模的构造息息相关，是子流形几何研究领域的重大课题。本项目将在已有成果的基础上，进一步深入研究以下问题：首先，改进现有的G(2,n;R)中调和2维球面的构造算法，致力对Q_{n-2}中的共形极小2维球面在高斯曲率为常值条件下给出完整分类；其次，刻画并分类G(2,n)中第二基本形式平行的共形极小2维球面；再次，研究Q_{n-2}中全实平坦极小2维环面的几何，着重考虑其构造问题；最后，讨论G(2,n)中极小曲面与开的以及仿射的Toda方程组的一一对应关系。

黎曼对称空间中极小曲面的几何是微分几何的重要内容，关于它的分类问题的研究与拓扑学、代数几何等有着密切联系，一直受到国内外几何学家和物理学家的广泛关注。复 Grassmann 流形G(2,n)以及它的全测地子流形超二次曲面Q_{n-2}，作为特定的对称空间，具有特殊的几何结构，其上极小曲面的研究与理论物理中 Grassmann sigma-模的构造息息相关，是子流形几何研究领域的重大课题。本项目在已有成果的基础上，进一步深入研究以下问题：首先，改进现有的G(2,n;R)中调和2维球面的构造算法，对Q_4中的共形极小2维球面在高斯曲率为常值条件下给出完整分类；其次，刻画并分类G(2,n)中第二基本形式平行的共形极小2维球面；再次，研究Q_{n-2}中全实平坦极小2维环面的几何，着重考虑其构造问题；最后，讨论G(2,n)中极小曲面与开的以及仿射的Toda方程组的一一对应关系。