On the classification of homogeneous hypersurfaces in complex space

We discuss a family M_tⁿ, 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 is not embeddable in ℂ⁷ for every t and that M_t³ is embeddable in ℂ³ for all 1 < t < 1 + 10^-6. As a consequence of our analysis of a map constructed by Ahern and Rudin, we also conjecture that the embeddability of M_t³ takes place for all 1 < t < √(2 + √2)/3 .