曲面到对称空间中调和映射的几何

Studying the geometry of harmonic map is an important object in differential geometry. In this project we will study the geometry of harmonic maps from Riemann surface into symmetric space. The research topics including: geometry problems of minimal two-spheres in symmetric spacce; properties of non-isotropic harmonic sequences in the general complex Grassmannians and other related problems; construction of conformal minimal tori with special geometry properties in some symmetric space. We are going to obtain: classification of minimal two-spheres with constant curvature in the general Grassmann manifolds, the quaternionic projective spaces and the complex hyperquadric; the cyclic properties of the energy and degree of the non-isotropic harmonic sequences in general complex Grassmann manifolds; construction of the conformal minimal tori in quaternionic projective space HP^n and complex hyperquadric Q_n.

研究调和映射的几何是微分几何的重要内容之一。本项目研究曲面到对称空间中调和映射的几何，研究内容包括：几类常见对称空间中极小球面的几何问题；复Grassmann流形中非迷向调和序列的几何性质；构造对称空间中具有特殊几何性质的极小环面等相关问题。我们希望得到：一般Grassmann流形、四元素投影空间、复二次超曲面中常曲率极小S^2的分类；一般复Grassmann流形中非迷向调和序列的能量、degree的循环性等结论；低维四元素复投影空间、复二次超曲面空间中共性极小环面的几何构造。

本项目主要研究曲面到对称空间中调和映射的几何。项目组成员严格按照项目计划书执行，得到了大部分预期研究成果，主要有：给出了超二次曲面与四元素投影空间中齐性球面以及极小球面的完全分类，相关结果给出了四元素投影空间中常曲率球面的Ohnita猜想若干反例；证明了闭流形上向量丛在联络度量下它的球丛都是其等参超曲面，以及在一定条件下它的Ricci曲率是正的，作为应用证明了底流形Ricci曲率为正那么它的double-流形的Ricci曲率也为正；证明了所有degree为2的Lawson-Osserman型锥都是面积最小的，作为特殊情形给出Lawson-Osserman的文章[Acta Math.'77]中遗留问题的肯定回答。