On the equivalence between local and global existence of complete K\"ahler metrics with plurisubharmonic potentials

Like the classical potential theory, it was conjectured that there exists equivalence between locally and globally pluripolar and complete pluripolar sets, namely, Problem I of Lelong, and was solved by Josefson, Bedford - Taylor and Col\c{t}oiu. In this article, we consider complements of complete K\"ahler domains as the generalization of closed complete pluripolar sets and prove that there exists an equivalence between local and global existence of these sets.