Some Study Of Hermitian-Yang-Mills Equations Over Non-k?hler Manifolds

This paper mainly contains four parts.In the second chapter,we list some related preparations.In the third chapter,we consider semistable Higgs bundles.By using the continuity method,we obtain that a Higgs bundle over a compact Gauduchon manifold is semi-stable if and only if it admits an approximate Hermitian-Yang-Mills metric.In the forth chapter,we study the filtration of semi-stable Higgs bundles with van-ishing first and second Chern numbers over compact Kahler manifolds.Over projective manifolds,by using algebraic methods,Simpson showed it must admit a filtration whose quotients are Hermitian flat Higgs bundles.By using the Yang-Mills-Higgs flow,we show that the above result is valid for general compact Kahler manifold.In the fifth chapter,we consider the convergence of Hermitian-Yang-Mills flow over compact non-Kahler manifold(X,?).When ? satisfies(?)?n-1=(?)?n-2=0,we obtain the energy inequality,monotonicity formula,small energy regularity and the limiting behaviour of the following heat flow which is gauge equivalent to Hermitian-Yang-Mills flow.