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

Abstract

We show the theorem which provides some sufficient condition to the non-existence of a complete K\"ahler--Einstein metric of negative scalar curvature whose holomorphic sectional curvature is negatively pinched: Let $\Omega$ be a bounded weakly pseudoconvex domain in $\mathbb{C}^n$ with a K\"ahler metric $\omega$ whose holomorphic sectional curvature is negative near the 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 following dichotomy holds: 1. $\omega$ is complete, and $\omega$ is uniformly equivalent to the complete K\"ahler--Einstein metric with negative scalar curvature. 2. $\omega$ is incomplete, and there is no complete K\"ahler metric with negatively pinched holomorphic sectional curvature. Moreover, $\Omega$ is Carath\'eodory incomplete. Our approach is based on the construction of a K\"ahler metric of negatively pinched holomorphic sectional curvature and applying the implication of equivalence of invariant metrics inspired by Wu-Yau.