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 Mtn in Cn for n=3,7. We show that Mt7 does not embed in C7 for every t and observe that Mt3 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 Mt3 embeds in C3 for all 1