Negative holomorphic bisectional curvature of some bounded domains

TL;DR

This paper demonstrates that certain bounded domains in complex space, under specific curvature conditions near the boundary, can be equipped with a complete Kähler metric exhibiting uniformly negative holomorphic bisectional curvature.

Contribution

It establishes the existence of complete negatively pinched Kähler metrics on bounded domains with particular boundary and squeezing function conditions, extending previous curvature results.

Findings

Bounded domains with negatively pinched curvature near boundary admit globally negatively pinched metrics.

Strictly pseudoconvex domains with $C^2$ boundary admit such metrics.

Domains with squeezing function tending to 1 at boundary points admit such metrics.

Abstract

We prove that a bounded domain in $\mathbb{C}^n$ admitting a complete K\"ahler metric with negatively pinched holomorphic bisectional curvature near the boundary, admits a complete K\"ahler metric with negatively pinched holomorphic bisectional curvature everywhere. As a consequence we prove that strictly pseudoconvex bounded domains with $C^2$ boundary and bounded domains with squeezing function tending to 1 at every point of the boundary, admit a complete K\"ahler metric with negatively pinched holomorphic bisectional curvature everywhere.