AI 中文总结
本文证明6/7≤μ<1时，Thullen域的Cheng--Yau度量重标后可浸入无限维复双曲空间，为相关猜想提供反例并构造了新的非齐次流形实例。
AI 中文摘要
对于μ>0，令M(μ)={(z,w)∈ℂ²: |z|²+|w|^(2/μ)<1}为Thullen域，g_CY表示其完备Cheng--Yau Kähler-Einstein度量，满足Ric(g_CY)=-3g_CY的归一化条件。我们证明，对每个6/7≤μ<1，g_CY经适当重标后可全局全纯等距浸入无限维复双曲空间ℂH^∞。据我们所知，这些是首个容许此类浸入的完备非齐次Kähler-Einstein流形实例。根据文献[DSIL2012,引理6]，相同度量也可浸入平坦希尔伯特空间ℓ²(ℂ)。由于本文考虑的Thullen域非齐次，这些实例既不是全测地复双曲空间，也不是重标复双曲空间的乘积。因此，它们为文献[LoiZedda2018,猜想4.1]提供了反例（针对复双曲和平坦希尔伯特空间两种情形），也推翻了文献[LoiZedda2011,注记10]中提出的早期平坦希尔伯特刚性猜想。进一步而言，该平坦实现产生了完备非齐次η-爱因斯坦Sasakian流形，可全局Sasakian浸入无限维海森堡空间形式。
英文摘要
For $μ>0$, let \[ M(μ)=\left\{(z,w)\in\mathbb{C}^2 : |z|^2+|w|^{2/μ}<1\right\} \] be the Thullen domain, and let $g_{\mathrm{CY}}$ denote its complete Cheng--Yau Kähler--Einstein metric, normalized by \[ \operatorname{Ric}(g_{\mathrm{CY}})=-3g_{\mathrm{CY}}. \] We prove that, for every $6/7\leqμ<1$, 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 Kähler--Einstein manifolds admitting such an immersion. By \cite[Lemma~6]{DSIL2012}, the same metrics also admit Kähler 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 $η$-Einstein Sasakian manifolds admitting global Sasakian immersions into the infinite-dimensional Heisenberg space form.
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