Variational Problems in Differential Geometry

本文研究了紧致4流形上Yang-Mills热流解的一个性质。我们得到了一种“小数据全局存在”的结果。众所周知，紧致4流形上光滑连接的能量存在下界。该界由主束的拓扑不变量决定。在本研究中，我们证明了如果初始数据的能量接近边界，则存在Yang-Mills热流的时间全局光滑解。

In this research, we study a behavior of solutions of Yang-Mills heat flow on compact 4-manifolds. We obtain result which is a type of "small data global existence".It is well-known that there exists the lower bound of the energy of smooth connections over compact 4-manifolds. The bound is determined by the topological invariant of the pricipal bundle. In this research, we prove that if the energy of the initial data is close to the bound, then there exists a time-global smooth solution of the Yang-Mills heat flow.