Hermitian–Einstein metrics on holomorphic vector bundles over Hermitian manifolds

Abstract

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.