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

Abstract

For $\mu>0$, let \[ M(\mu)=\left\{(z,w)\in\mathbb{C}^2 : |z|^2+|w|^{2/\mu}<1\right\} \] be the Thullen domain, and let $g_{\mathrm{CY}}$ denote its complete Cheng--Yau K\"ahler--Einstein metric, normalized by \[ \operatorname{Ric}(g_{\mathrm{CY}})=-3g_{\mathrm{CY}}. \] We prove that, for every $6/7\leq\mu<1$, a suitable rescaling of $g_{\mathrm{CY}}$ admits a global holomorphic isometric immersion into the infinite-dimensional complex hyperbolic space $\mathbb{CH}^{\infty}$. To the best of our knowledge, these are the first examples of complete nonhomogeneous K\"ahler--Einstein manifolds admitting such an immersion. By \cite[Lemma~6]{DSIL2012}, the same metrics also admit K\"ahler immersions into the flat Hilbert space $\ell^2(\mathbb{C})$. 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.