  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.
