Negative holomorphic bisectional curvature of some bounded domains

Abstract

We prove that a bounded domain in $\mathbb{C}^n$ admitting a complete K\"ahler metric with negatively pinched holomorphic bisectional curvature near the boundary, admits a complete K\"ahler metric with negatively pinched holomorphic bisectional curvature everywhere. As a consequence we prove that strictly pseudoconvex bounded domains with $C^2$ boundary and bounded domains with squeezing function tending to 1 at every point of the boundary, admit a complete K\"ahler metric with negatively pinched holomorphic bisectional curvature everywhere.