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一个泛函分析的基础。我们得到的主要结果是：1)我们分类了流形间定义的Sobolev类映射的奇异性。特别地，我们给出了Sobolev映射全局奇异性的一般理论。2)研究了Sobolev束的拓扑性质与解析性质之间的相互关系。我们将这些结果应用于高维规范理论。3)证明了faddev - skyrme模型在3球上Hopf孤子的强稳定性。4)证明了一类临界维简并Yang-Mills连接的弱解的正则性。5)建立了具有边界的4流形上Yang-Mills-Dirichlet问题的Morse理论。6)证明了定义在4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.