Some Studies On Yang-Mills-Higgs Equations Over Hermitian Manifolds

The main results of this paper consists three parts.In the first part,by using Uhlenbeck-Yau's continuity method,we prove that semistable holomorphic pairs over compact Gauduchon manifolds must admit approximate Hermitian-Einstein structures.In the second part,by using the method of heat flow,we prove an existence theorem for the Hermitian Yang-Mills-Higgs metrics on holomorphic line bundles over a class of non-compact Gauduchon manifolds.In the third part,firstly,using the method of heat flow,we prove that,the Dirichlet problem for the perturbed Hermitian Yang-Mills-Higgs equation on holomorphic pair over compact Hermitian manifold with boundary has a unique solution.Then,by exhaustion method,we prove that the perturbed Hermitian Yang-Mills-Higgs equation on holomoiphic pair over non-compact Hermitian manifold has a solution.At last,by using Uhlenbeck-Yau's continuity method,we prove that,over a class of non-compact Gauduchon manifolds there exists Hermitian Yang-Mills-Higgs metrics on stable holomorphic pairs and approximate Hermitian-Einstein structures on semi-stable holomorphic pairs.