The Affine Hermitian-Yang-Mills Flow On Affine Manifolds

In this thesis,we study the affine Hermitian-Yang-Mills flow on affine Gauduchon manifolds and the curvature estimate of the Yang-Mills-Higgs flow on Kahler manifold-s.In the first part,we first introduce the affine Hermitian-Yang-Mills flow on affine manifolds.Then we study the existence of the long-time solution of the flow.By using this heat flow,we solve the Dirichlet problem of the affine Hermitian-Yang-Mills flow on compact affine Gauduchon manifolds with nonempty boundary.In the second part,we study the affine Hermitian-Yang-Mills flow on noncompact affine Gauduchon manifolds.We prove a crucial equality instead of the Donaldson’s functional in the affine Gauduchon case.By using the exhaustion method,we obtain the existence of the long-time solution of the heat flow and some related estimates.As an application,we prove that the stability implies the existence of the affine Hermitian-Einstein metric on noncompact affine Gauduchon manifolds.In the third part,we study the curvature estimate of the Yang-Mills-Higgs flow on Kahler manifolds.Firstly,under some assumptions,we obtain the local uniform C0-estimate of the rescaled metrics HS(t)=e2(λS-λE)tHS(t)andHQ(t)=e2(λQ-λE)tHQ(t).Then by using the above local C0-estimates,we prove that the norms of 丨γ(t)丨H(t),丨β(t)丨H(t),丨TS(t)丨HS(t)and 丨TQ(t)丨HQ(t)are local uniformly bounded.Finally,by choos-ing suitable test functions and using the maximum principle,we obtain a uniform local estimate of the curvature of the flow.