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 $\Omega$ 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 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 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.

To appear in Complex Analysis and its Synergies