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 $\Omega\subset \mathbb C^n$ 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.