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

TL;DR

This paper establishes an equivalence between local and global existence of complete K"ahler metrics with plurisubharmonic potentials, generalizing classical results on pluripolar sets and extending potential theory.

Contribution

It proves the equivalence between local and global existence of complements of complete K"ahler domains, generalizing known results on pluripolar sets.

Findings

Proved the equivalence between local and global existence of certain K"ahler sets.

Extended classical potential theory results to K"ahler geometry.

Generalized the concept of complete pluripolar sets.

Abstract

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.