On a family of real hypersurfaces in a complex quadric

We discuss a family Mtn, with n ≥ 2, t > 1, of real hypersurfaces in a complex affine n-dimensional quadric arising in connection with the classification of homogeneous compact simply-connected real-analytic hypersurfaces in Cn due to Morimoto and Nagano. To finalize their classification, one needs to resolve the problem of the embeddability of M_tⁿ in Cⁿ for n = 3, 7. We show that M_t⁷ does not embed in C⁷ for every t and observe that M_t³ embeds in C³ for all t sufficiently close to 1. As a consequence of analyzing a map constructed by Ahern and Rudin, we also conjecture that M_t³ embeds in C³ for all 1<t<√(2+√2)/3.