A study of higher dimensional geometric variational problems

本研究的目的是为定义在拓扑非平凡函数空间上的高维几何变分问题提供泛函分析基础。我们得到的主要结果是：1)我们对定义在流形之间的Sobolev类映射的奇异性进行了分类。特别地，我们给出了Sobolev映射的整体奇性的一般理论。2)研究了Sobolev丛的拓扑性和解析性之间的关系。我们将这些结果应用于高维规范理论。3)证明了三维球面上Faddeev-Skyrme模型的Hopf孤子的强稳定性。4)证明了临界维上一类退化的Yang-Mills联络弱解的正则性。5)建立了带边界的四维流形上的Yang-Mills-Dirichlet问题的Morse理论。6)证明了定义在四维流形上的Yang-Mills-Dirac方程的正则性和能量量子化。

The purpose of this research is to give a functional analytic foundation for higher dimensional geometric variational problems defined on topologically non-trivial function spaces. The main results we have obtained are 1) We have classified singularities of Sobolev class mappings defined between manifolds. In particular, we have given a general theory of global singularities of Sobolev mappings. 2) We studied interrelations between topological and analytical properties of Sobolev bundles. We applied these results to higher dimensional gauge theory. 3) We proved the strong stability of Hopf soliton for the Faddeev-Skyrme model on the 3-sphere. 4) We proved a regularity of weak solutions for a certain kind of degenerate Yang-Mills connections in critical dimensions. 5) We established a Morse theory for the Yang-Mills-Dirichlet problem on 4-manifolds with boundary. 6) We proved a regularity and energy quantization for the Yang-Mills-Dirac equations defined on 4-manifolds.