Complex Hyperbolic Immersions of Cheng--Yau Metrics on Thullen Domains
We classify the homotheties of the complete Cheng--Yau Kähler--Einstein metrics on the two-dimensional Thullen domains $M(μ)$, $μ>0$, that admit Kähler immersions into $\mathbb{CH}^{\infty}$. With $\Ric(g_{\mathrm{CY}})=-3g_{\mathrm{CY}}$, an immersion of $c\,g_{\mathrm{CY}}$ exists exactly when $0<μ\leqμ_+$ and $0 1$ is characterized by an explicit equation, and its endpoint is included. Every admissible metric has a proper global Kähler embedding. The same parameter interval classifies immersibility into flat and projective Hilbert space, where every positive homothety admits a proper global embedding. Beyond the wall, all three targets are excluded at every positive scale. The proof combines an algebraic normalization of the Cheng--Yau equation, all-order coefficient estimates, and complex continuation of a scalar flow. For $μ\neq1$, the resulting complete nonhomogeneous examples disprove proposed Kähler--Einstein rigidity conjectures in the complex-hyperbolic and flat Hilbert settings. We also prove a complementary rigidity theorem for Bergman--Hartogs domains and construct complete $η$-Einstein Sasakian manifolds with nonhomogeneous Riemannian metrics and proper global Sasakian embeddings into the infinite-dimensional Heisenberg space form.