Hermitian-Einstein equations on noncompact manifolds

This paper first investigates solvability of Hermitian-Einstein equation on a Hermitian holomorphic vector bundle on the complement of an arbitrary closed subset in a compact Hermitian manifold. The uniqueness of Hermitian-Einstein metrics on a Zariski open subset in a compact K\"{a}hler manifold was only figured out by Takuro Mochizuki recently, for this model the second part of this paper gives an affirmative answer to a question proposed by Takuro Mochizuki and it leads to an alternative approach to the unique issue. We also prove stability from solvability of Hermitian-Einstein equation, which together with the classical existence result of Carlos Simpson in particular establish a Kobayashi-Hitchin bijective correspondence. The argument is also effective in more general settings, including basic models of Takuro Mochizuki, as well as non-K\"{a}hler and semi-stable contexts.