Complex Hyperbolic Immersions of Cheng--Yau Metrics on Thullen Domains

For $\mu >0$, let $$M(\mu )=\{(z,w)\in {\mathbb{C}}^2:\mathrm{\mid }z{\mathrm{\mid }}^2+\mathrm{\mid }w{\mathrm{\mid }}^{2\mathrm{/}\mu }<1\}$$ be the Thullen domain, and let $g_{\mathrm{C}\mathrm{Y}}$ denote its complete Cheng–Yau Kähler–Einstein metric, normalized by $$\mathrm{Ric}(g_{\mathrm{C}\mathrm{Y}})=-3g_{\mathrm{C}\mathrm{Y}}\mathrm{.}$$ We prove that, for every $6\mathrm{/}7\le \mu <1$, a suitable rescaling of $g_{\mathrm{C}\mathrm{Y}}$ admits a global holomorphic isometric immersion into the infinite-dimensional complex hyperbolic space ${\mathbb{C}\mathbb{H}}^{\mathrm{\infty }}$. To the best of our knowledge, these are the first examples of complete nonhomogeneous Kähler–Einstein manifolds admitting such an immersion. By \cite[Lemma~6]{DSIL2012}, the same metrics also admit Kähler immersions into the flat Hilbert space ${\mathrm{\ell }}^2(\mathbb{C})$. Since the Thullen domains considered here are nonhomogeneous, these examples are neither totally geodesic complex hyperbolic spaces nor products of rescaled complex hyperbolic spaces. Consequently, they provide counterexamples to \cite[Conjecture~4.1]{LoiZedda2018}, for both the complex-hyperbolic and flat Hilbert-space alternatives, and also disprove the earlier flat-Hilbert rigidity conjecture formulated in \cite[Remark~10]{LoiZedda2011}. As a further consequence, the flat realization yields complete nonhomogeneous $\eta$-Einstein Sasakian manifolds admitting global Sasakian immersions into the infinite-dimensional Heisenberg space form.