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We show the theorem which provides some sufficient condition to thenon-existence of a complete K\"ahler--Einstein metric of negative scalarcurvature whose holomorphic sectional curvature is negatively pinched:  Let $\Omega$ be a bounded weakly pseudoconvex domain in $\mathbb{C}^n$ with aK\"ahler metric $\omega$ whose holomorphic sectional curvature is negative nearthe topological boundary of $\Omega$ (with respect to relative topology of$\mathbb{C}^n$) and $\omega$ admits the quasi-bounded geometry. Then $\omega$is uniformly equivalent to the Kobayashi--Royden metric and the followingdichotomy holds:  1. $\omega$ is complete, and $\omega$ is uniformly equivalent to the completeK\"ahler--Einstein metric with negative scalar curvature.  2. $\omega$ is incomplete, and there is no complete K\"ahler metric withnegatively pinched holomorphic sectional curvature. Moreover, $\Omega$ isCarath\'eodory incomplete.  Our approach is based on the construction of a K\"ahler metric of negativelypinched holomorphic sectional curvature and applying the implication ofequivalence of invariant metrics inspired by Wu-Yau.

Non-existence of complete K\"ahler metric of negatively pinched holomorphic sectional curvature