流形上等距浸入曲面的几何与分析

The research on classification of isometric immersions of surfaces into Riemann manifolds has arouse wide attention by geometric and physical researchers . The research on classification of isometric immersions of surfaces into Euclid space, with the background of the research on mean curvature flow, is closely related to Topology, Algebraic geometry and so on. The research on classification of isometric immersions of surfaces into complex Grassmann manifold, which is a special Riemann manifold, is closely related to the construction of Grassmann sigma model of string theory in theoretical physics. The quaternionic projective space, as a submanifold of complex Grassmann manifold, has some special geometric structures. The research on classification of isometric immersions of surfaces into quaternionic projective space is a significant research subject in the research of submanifold geometry. Based on the current research results, this project will further study several problems as follows: Firstly, complete the theory on classification of isometric immersions of minimal 2-spheres into quaternionic projective space , improve the current algorithm of constructing harmonic (under the condition of conformal immersion, that a map is harmonic is equivalent to that it is minimal ) 2-spheres in quaternionic projective space, and classify the corresponding immersions compltetely under the condition of constant Gauss curvature; Secondly, study the problem on classification of isometric immersions of 2-toris into quaternionic projective space; At last, consider the problem on classification of isometric immersions of self-shrinker and soliton surfaces into Euclid space .

黎曼流形上等距浸入曲面的分类研究一直都受到几何和物理研究者广泛的关注。以平均曲率流为研究背景的关于欧式空间中等距浸入曲面的分类研究与拓扑、代数几何等有着密切的联系。关于特殊黎曼流形复Grassmann流形中等距浸入曲面的分类研究与理论物理弦理论中Grassmann sigma-模的构造息息相关。四元射影空间作为复Grassmann流形的子流形，具有较特殊的几何结构，其上等距浸入曲面的研究是子流形几何研究领域的重大课题。本项目将在已有成果基础上，进一步深入研究以下几个方面：首先，完善四元射影空间中等距浸入极小2维球面的分类理论，改进现有的构造四元射影空间中调和（在共形浸入的条件下，极小等价于调和）2维球面的算法，在高斯曲率为常值的条件下对其完全分类；其次，研究四元射影空间中等距浸入2维环面的分类问题；最后，考虑欧式空间中等距浸入曲面self-shrinkers和solitons的分类问题。

黎曼流形中等距浸入曲面的构造和分类研究一直都备受几何学家和物理学家关注。我们综合运用分析，代数和多项式方程构造调和2维球面的方法研究了复Grassmann流形G(2,5)中常高斯曲率全纯2维球面的分类问题，该方法能够有效地构造一般复Grassmann流形中的常高斯曲率调和2维球面，这与理论物理弦理论中Grassmann Sigma-模的构造息息相关。我们根据调和序列保持齐性的特点，对四元射影空间中线性满的齐性极小2维球面进行了完全分类，该结果以及后续的研究表明四元射影空间中的常高斯曲率极小2维球面的分类问题要从模空间的角度考虑。此外，我们定义了四元凯勒角，分类了四元射影空间中的全实平坦极小曲面，这将丰富四元凯勒流形中曲面几何的研究。我们获得了四次K3曲面上满足某一特殊性质球面的非存在性，这有利于研究K3曲面上triholomorphic映射的奇点集问题，该结果有助于更深入地研究超凯勒流形间triholomorphic映射的奇点集问题，而超凯勒流形间triholomorphic映射与物理学规范场论中八元素瞬子相互对应。我们获得平行单位平均曲率向量辛临界曲面的分类，该方法有助于研究四维欧式空间中常平均曲率曲面以及self-shrinkers和solitons曲面的分类问题。