Some Studies On Higgs Bundles Over Non-compact Hermitian Manifolds

The main results of this paper consist two parts.In the first part,we study the Higgs bundles over the asymptotically cylindrical Kahler manifold.Let D be a compact Kahler manifold,V an asymptotically cylindrical Kahler manifold with asymptotic cross-section D,(ED,?D)a stable Higgs bundle over D,and(E,?)a Higgs bundle over V which is asymptotic to(ED,?D).Using the continuity method of Uhlenbeck and Yau,we prove that there exists an asymptotically translation-invariant projectively Hermitian-Einstein metric on(E,?).In the second part,we study the Higgs bundles over a class of non-compact Gaudu-chon manifolds.Firstly,using the method of heat flow,we prove that,the Dirichlet problem for the perturbed Hermitian-Einstein equation on Higgs bundles over compact Hermitian manifold with boundary has a unique solution.Then,by exhaustion method,we prove that the perturbed Hermitian-Einstein equation on Higgs bundles over non-compact Hermitian manifold has a solution.Moreover,we prove that,over a class of noncompact Gauduchon manifolds there exists Hermitian-Einstein metrics on stable Higgs bundles and approximate Hermitian-Einstein structures on semi-stable Higgs bundles.