Bergman--Einstein Rigidity for Hartogs Domains over Bounded Homogeneous Domains

TL;DR

The paper proves a rigidity theorem for the Bergman metric on Hartogs domains over bounded homogeneous domains, characterizing when these domains are biholomorphic to the unit ball based on metric properties.

Contribution

It establishes a new rigidity result linking Bergman metric conditions to biholomorphic equivalence to the unit ball for a class of Hartogs domains.

Findings

Bergman metric being Kähler--Einstein is equivalent to the domain being biholomorphic to the ball.

The domain is homogeneous if and only if it is biholomorphic to the ball.

The result provides a positive answer to Yau's question within this class.

Abstract

We prove a rigidity theorem for the Bergman metric on Hartogs domains over bounded homogeneous domains. Let $\Omega\subset \mathbb C^n$ be a bounded homogeneous domain, let $K_\Omega$ denote its Bergman kernel, and consider $$ \Omega_{m,s}:=\{(z,\zeta)\in \Omega\times \mathbb C^m:\ \|\zeta\|^2<K_\Omega(z,\bar z)^{-s}\}, \qquad m\ge 1,\quad s>-C_\Omega. $$ For $s\neq 0$, we prove that the following conditions are equivalent: the Bergman metric of $\Omega_{m,s}$ is K\"ahler--Einstein; $\Omega_{m,s}$ is homogeneous; $\Omega_{m,s}$ is biholomorphic to $\mathbb B^{n+m}$; and $\Omega\cong\mathbb B^n$ with $s=\frac1{n+1}$. This gives a positive answer to Yau's question within this class and may be viewed as a Cheng-type rigidity phenomenon beyond the smoothly bounded strictly pseudoconvex setting. The proof combines the explicit formula for the Bergman kernel of $\Omega_{m,s}$ with the structural invariants of the bounded homogeneous base.