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

Abstract

We prove a theorem which provides a sufficient condition for the non-existence of a complete Kähler–Einstein metric of negative scalar curvature of which holomorphic sectional curvature is negatively pinched: Let $\Omega$ be a bounded weakly pseudoconvex domain in $\mathbb {C}^n$ with a Kähler metric $\omega$ whose holomorphic sectional curvature is negative near the topological boundary of $\Omega$ (with respect to the relative topology of $\mathbb {C}^n$) and $\omega$ admits 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 Kähler–Einstein metric with negative scalar curvature.

• 2. $\omega$ is incomplete, and there is no complete Kähler metric with negatively pinched holomorphic sectional curvature. Moreover, $\Omega$ 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.