On homogeneous hypersurfaces in C³

We consider 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. In our earlier article we showed that Mt7 is not embeddable in C⁷ for every t and that Mt3 is embeddable in C³ for all 1 < t< 1 + 10 ^{- 6}. In the present paper, we improve on the latter result by showing that the embeddability of Mt3 in fact takes place for 1<t<(2+2)/3. This is achieved by analyzing the explicit totally real embedding of the sphere S³ in C³ constructed by Ahern and Rudin. For t⩾(2+2)/3, the problem of the embeddability of Mt3 remains open.