Non-existence of complete Kähler metric of negatively pinched holomorphic sectional curvature

Abstract

We show the theorem which provides some sufficient condition to the non-existence of a complete Kähler--Einstein metric of negative scalar curvature whose holomorphic sectional curvature is negatively pinched: Let $Ω$ be a bounded weakly pseudoconvex domain in $\mathbb{C}^n$ with a Kähler metric $ω$ whose holomorphic sectional curvature is negative near the topological boundary of $Ω$ (with respect to relative topology of $\mathbb{C}^n$) and $ω$ admits the quasi-bounded geometry. Then $ω$ is uniformly equivalent to the Kobayashi--Royden metric and the following dichotomy holds: 1. $ω$ is complete, and $ω$ is uniformly equivalent to the complete Kähler--Einstein metric with negative scalar curvature. 2. $ω$ is incomplete, and there is no complete Kähler metric with negatively pinched holomorphic sectional curvature. Moreover, $Ω$ is Carathéodory incomplete. Our approach is based on the construction of a Kähler metric of negatively pinched holomorphic sectional curvature and applying the implication of equivalence of invariant metrics inspired by Wu-Yau.