Morimoto–Nagano realization problem

Morimoto and Nagano’s classification leaves the realization of the exceptional compact homogeneous hypersurfaces $M_t^3$ as the unresolved case; the $n=7$ family is known not to embed. The problem asks whether the remaining $M_t^3$ can be realized throughout the full parameter range $t>1$.

Known results

• Isaev, 2013: real-analytic CR-embeddings exist for $1<t<\sqrt{(2+\sqrt{2})/3}$; the complementary range was left open. [https://arxiv.org/abs/1309.0279]
• Isaev, 2019: every $M_t^3$, $t>1$, has a polynomial immersion into $\mathbb{C}^3$, and embeddings exist for $1<t<\sqrt{5/2}$; the maps are noninjective for $t\ge\sqrt{2}$. [https://arxiv.org/abs/1904.05566]
• Isaev, 2019: no $M_t^7$ is real-analytically CR-embeddable in $\mathbb{C}^7$. [https://arxiv.org/abs/1904.05566]

August 2026 full-range completion claim

A new preprint claims that explicit entire maps extend Isaev’s realization results to the full parameter range, thereby completing the $M_t^3$ case. This is an unrefereed claim, and the scan found no independent verification, objection, or withdrawal.

Current status (as of August 2026): Earlier partial embedding and immersion results are established, while the full $M_t^3$ realization is only claimed by the August 2026 preprint and remains unverified.