Hermitian–Einstein equations on noncompact manifolds

Abstract This paper first investigates solvability of Hermitian–Einstein (HE) equation on a Hermitian holomorphic vector bundle on the complement of an arbitrary closed subset in a compact Hermitian manifold. The uniqueness of HE metrics on a Zariski open subset in a compact Kähler manifold was only figured out by Mochizuki recently. For this model, the second part of this paper gives an affirmative answer to a question proposed by Mochizuki and it leads to an alternative approach to the uniqueness issue without involving the deep regularity result of weakly holomorphic subbundles due to Uhlenbeck–Yau. We also prove stability from solvability of HE equation, which together with the classical existence result of Simpson in particular establishes a Kobayashi–Hitchin bijective correspondence. The argument is also effective in more general settings, including basic models of Mochizuki, as well as non‐Kähler and semi‐stable contexts.