黎曼流形间的双调和映射的若干问题研究

Biharmonic map is an important branch of the field of harmonic map and harmonic morphism. It not only has important theory significance, but also is widely used in some fields,such as elasticity, hydrodynamics, the classification theory of Riemannian manifolds, computational geometry, geometric design. In this project, by absorbing the existing theories and methods of biharmonic maps between Riemannian manifolds and using the basic theories and techniques of some disciplines, such as differential equation, Geometry and analysis, and topology, we investigate some theoretical nature, classifications and constrctions for biharmonic maps between Riemannian manifolds. The main contents are as follows:. (1) we study biharmonic maps from surfaces. In particular, we focus on some constructions and classifications of biharmonic maps from a 2-sphere and a 2-torus; (2) We study biharmonicity of isometric immersions. In particular, we focus on some constructions and classifications of biharmonic isometric immersions; (3) We study biharmonicity of Riemannian submersions, In particular, we focus on some constructions and classifications of biharmonic isometric immersions .

双调和映射是调和映射与调和态射领域的一个重要研究分支,不仅有重要的理论意义,而且在弹性力学、流体力学、黎曼流形分类理论、计算几何及曲面几何设计等领域有广泛应用。本项目以黎曼流形间的双调和映射为研究对象, 通过消化吸收黎曼流形间的双调和映射的已有理论和方法, 综合应用微分方程、几何分析和拓扑学等领域的知识展开研究。研究内容: (1)曲面出发的双调和映射, 特别是球面、环面出发的的双调和映射的构造与分类;(2)等距嵌入的双调和性的研究, 特别是双调和等距嵌入的构造与分类;(3)黎曼淹没的双调和性的研究, 特别是双调和黎曼淹没的构造与分类。

双调和映射是调和映射与调和态射领域的一个重要研究分支，不仅有重要的理论意义， 而且在某些领域有广泛应用。本项目主要研究了双调和等距嵌入(即双调和子流形)、双调和黎曼淹没的相关问题，内容包括：（1）曲面出发的双调和等距嵌入的一些分类与构造。例如对3维Berger 球、某些积空间的具有常平均曲率的双调和曲面进行了刻画，即曲面出发到这些空间的等距嵌入的双调和性刻画。（2）3维模型空间出发的双调和黎曼淹没的一些分类、构造。主要获得了从3维BCV空间、Sol空间、Berger 球面、某些积空间出发的双调和黎曼淹没(包括调和黎曼淹没）的完全分类与构造。本项目成功利用积分数据手段对3维某些“比较好的”模型空间做了双调和黎曼淹没的分类与构造，并且在充实双调和映射理论成果、探索新的思路与方法上均取得了实质性进展。