Extremal metrics on Hartogs domains

TL;DR

This paper proves that any extremal Kähler metric on a strongly pseudoconvex Hartogs domain must be biholomorphically isometric to complex hyperbolic space, revealing a rigidity property of such metrics.

Contribution

It establishes a uniqueness result for extremal Kähler metrics on Hartogs domains, showing they are necessarily hyperbolic spaces.

Findings

Extremal Kähler metrics on Hartogs domains are isometric to complex hyperbolic space.

The result characterizes the geometry of Hartogs domains with extremal metrics.

Biholomorphic isometry to complex hyperbolic space is the only possibility for extremal metrics.

Abstract

An $n$-dimensional Hartogs domain $D_F$ with strongly pseudoconvex boundary can be equipped with a natural \K metric $g_F$. In this paper we prove that if $g_F$ is an extremal \K metric then $(D_F, g_F)$ is biholomorphically isometric to the $n$-dimensional complex hyperbolic space.