超二次曲面中的极小曲面的几何

Minimal surfaces is an important topic in differentiable geometry, many geometricians studied minimal surfaces in a complex projective space CP(n). As is well known, conformal minimal two-sphere of constant curvature in CP(n) belongs to the Veronese sequence. A complex hyperquadric Q(n) is a complex submanifold of CP(n+1), because the inclusion is not totally geodesic, so that geometries of minimal surfaces in Q(n) are different from the one in CP(n+1). In this subject, by the harmonic sequences, we will consider geometries of minimal surfaces in complex hyperquadrics, including relative problems about Gauss curvature, Kaehler angle, the seconde fundamental form and so on. When the surfaces are two-spheres, we will consider distribution of their constant Gauss curvature and rigidity. We will characterize minimal surfaces with constant curvature or constant Kaehker angle in complex hyperquadrics of low dimensional. As the image about Gauss mapping of immersed surfaces in Euclidean spaces is a complex hyperquadrics, we will study some calssical topic about minimal surfaces in Eucildean spaces.

极小曲面是微分几何的重要研究课题之一，许多几何学家们研究过复射影空间CP(n)中的极小曲面。众所周知，复射影空间中常曲率极小二维球面是Veronese 序列。超二次曲面Q(n)是复射影空间CP(n+1)中的复子流形，由于其包含映射不是全测地的，所以极小曲面在Q(n)中的几何与它在CP(n+1)中的几何有很大的不同。在本项目中，我们将利用调和序列研究超二次曲面中极小曲面的几何，包括Gauss曲率，Kaehler角和第二基本形式等几何量之间的关系的相关问题。特别，我们将考虑常高斯曲率极小二维球面的曲率值的分布和刚性。在低维情形下，我们将研究常曲率或者常Kaehler 角的极小曲面的分类。由于欧氏空间中浸入曲面的高斯映射的像是超二次曲面，我们也将研究欧氏空间中曲面的一些经典问题。

在该基金的支持下，我完成了3篇论文。“Totally real minimal surfaces in the complex hyperquadrics”发表在Differential Geometry and its Applications，我们构造了一类Qn中全实的极小常曲率的二维球面，在低维情形下，研究了Q2，Q3，Q4，Q5中全实常曲率极小二维球面的刚性；“Totally real conformal minimal tori in the hyperquadric Q2”发表在SCIENCE CHINA Mathematics，我们给出了Qn中极小曲面在CP（n+1）中仍极小的充分必要条件，并给出了Q2中全测地闭曲面的分类；“Geometry of three-dimensional SU(2)-orbits in the comples Grassmannians”发表在中国科学园大学学报，我们利用Cartan 嵌入和活动标架研究了G（k,n）中的三维SU(2)轨道的几何。