The extremal metric on a class of twisted Fock–Bargmann–Hartogs domains

Jing Chen, Daniel Zhuang-Dan Guan, Shaojun Jing and Yanyan Tang

Vol. 333 (2024), No. 1, 59–80

DOI: 10.2140/pjm.2024.333.59

Abstract

We study the Kähler–Einstein metrics and Calabi extremal metrics on some unbounded domains defined by

$$\mathrm{\Omega }:=\{(\zeta ,\tilde{\zeta },w)\in {ℂ}^s\times {ℂ}^n\times {ℂ}^m:e^{\parallel w{\parallel }^2}\parallel \zeta {\parallel }^2+\parallel \tilde{\zeta }{\parallel }^2<1\},$$

which are type-III cohomogeneity-one manifolds. We introduce a Kähler metric $g$ associated with the Kähler form

$$\omega :=\sqrt{-1}(\partial \overline{\partial }F(X)+a\partial \overline{\partial }\parallel w{\parallel }^2+b\partial \overline{\partial }\log <!--FUNCTION\; APPLICATION-->\parallel \zeta {\parallel }^2)$$

on $\mathrm{\Omega }$. Using the method of cohomogeneity, we find many new Kähler–Einstein metrics and prove that any Calabi extremal metric with the scalar curvature as a linear function of the potential function of the circle action on the first vector $\zeta$ must be a metric with a constant scalar curvature. At last, we give one application of our main results. That is, in the case $s=m=1$, we prove that the Kähler–Einstein metric is equivalent, but not equal, to the Bergman metric.

Keywords

Kähler–Einstein metric, extremal metric, twisted Fock–Bargmann–Hartogs domains, scalar curvature

Mathematical Subject Classification

Primary: 32Q02, 32T27

Milestones

Received: 1 March 2024

Revised: 28 October 2024

Accepted: 30 November 2024

Published: 19 December 2024