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

TL;DR

The paper establishes conditions under which a complete K"ahler metric with negatively pinched holomorphic sectional curvature cannot exist on certain pseudoconvex domains, linking metric completeness to curvature properties.

Contribution

It provides a sufficient condition for the non-existence of complete negatively pinched K"ahler--Einstein metrics on pseudoconvex domains, extending Wu-Yau type implications.

Findings

Complete negatively pinched K"ahler metrics imply equivalence to K"ahler--Einstein metrics.

Incompleteness of such metrics implies no existence of complete negatively pinched K"ahler metrics.

Domains are Carathéodory incomplete under these conditions.

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.