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<