On The Gauge Fixing Yang-Mills Heat Flow And Related Problems

In this paper we will mainly study the gauge fixing Yang-Mills heat flow of a principal bundleFor a principle bundle with a compact seme-simple Lie group as its structure group over a compact Riemannian manifold without boundary, the evolutions of the curvature and its higher derivatives under the flow above will be derived, and the energy inequality and the Bochner type estimates will be obtained. Then, the monotonicity formula and the small action regularity theorem can be proved. We will give the locally uniform estimates for the higher derivatives of the curvature. After deriving the evolution of the derivatives of a, we will have the maximum estimate for a and the locally uniform estimates for its higher derivatives, then we will give a kind of description of the long time existence.As the complement to the work above, we will discuss the gauge fixing Yang-Mills equation. Using the continuity method, we will obtain a theorem of existence of solutions to the equation on 4-dimensional compact Riemannian Manifolds.Finally, we will consider a kind of Kazdan-Warner type equation on compact complex manifolds. By mean of the method of upper and lower solutions, we will obtain a existence theorem like Kazdan-Warner's. Using this theorem, we will study the vortex equation in holomorphic line bundle over complex manifold and prove a theorem of existence of Hermitian metric with a prescribed holomorphic section.