Canonical metrics on Hartogs domains

TL;DR

This paper proves that certain canonical metrics on Hartogs domains are either hyperbolic space or Kähler-Ricci solitons, revealing their geometric rigidity and uniqueness properties.

Contribution

It establishes that extremal or Kähler-Ricci soliton metrics on Hartogs domains are necessarily hyperbolic or isometric to hyperbolic space, demonstrating their rigidity.

Findings

Extremal Kähler metrics on Hartogs domains are hyperbolic space.

Kähler-Ricci solitons on Hartogs domains are hyperbolic space.

Results show geometric rigidity of canonical metrics on Hartogs domains.

Abstract

An $n$-dimensional Hartogs domain $D_F$ with strongly pseudoconvex boundary can be equipped with a natural Kaehler metric $g_F$. This paper contains two results. In the first one we prove that if $g_F$ is an extremal Kaehler metric then $(D_F, g_F)$ is holomorphically isometric to an open subset of the $n$-dimensional complex hyperbolic space. In the second one we prove the same assertion under the assumption that there exists a real holomorphic vector field $X$ on $D_F$ such that $(g_F, X)$ is a Kaehler-Ricci soliton.