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

We prove a rigidity theorem for the Bergman metric on Hartogs domains over bounded homogeneous domains. Let $\mathrm{\Omega }\subset {\mathbb{C}}^n$ be a bounded homogeneous domain, let $K_{\mathrm{\Omega }}$ denote its Bergman kernel, and consider $${\mathrm{\Omega }}_{m,s}:=\{(z,\zeta )\in \mathrm{\Omega }\times {\mathbb{C}}^m:\mathrm{\parallel }\zeta {\mathrm{\parallel }}^2<K_{\mathrm{\Omega }}(z,\bar{z}{)}^{-s}\},m\ge 1,s>-C_{\mathrm{\Omega }}\mathrm{.}$$ For $s\mathrm{\ne }0$, we prove that the following conditions are equivalent: the Bergman metric of ${\mathrm{\Omega }}_{m,s}$ is Kähler–Einstein; ${\mathrm{\Omega }}_{m,s}$ is homogeneous; ${\mathrm{\Omega }}_{m,s}$ is biholomorphic to ${\mathbb{B}}^{n+m}$; and $\mathrm{\Omega }\cong {\mathbb{B}}^n$ with $s=\frac{1}{n+1}$. 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 ${\mathrm{\Omega }}_{m,s}$ with the structural invariants of the bounded homogeneous base.