Hermitian-Einstein metrics on holomorphic vector bundles over Hermitian manifolds

In this paper, we prove the long-time existence of the Hermitian-Einstein flow on a holomorphic vector bundle over a compact Hermitian (non-Kähler) manifold, and solve the Dirichlet problem for the Hermitian-Einstein equations. We also prove the existence of Hermitian-Einstein metrics for holomorphic vector bundles on a class of complete non-compact Hermitian manifolds.