On homogeneous hypersurfaces in ${\mathbb C}^3$

We consider a family $M_t^n$, with $n\ge 2$, $t>1$, 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 ${\mathbb C}^n$ due to Morimoto and Nagano. To finalize their classification, one needs to resolve the problem of the embeddability of $M_t^n$ in ${\mathbb C}^n$ for $n=3,7$. In our earlier article we showed that $M_t^7$ is not embeddable in ${\mathbb C}^7$ for every $t$ and that $M_t^3$ is embeddable in ${\mathbb C}^3$ for all $1<t<1+10^{-6}$. In the present paper, we improve on the latter result by showing that the embeddability of $M_t^3$ in fact takes place for $1<t<\sqrt{(2+\sqrt{2})/3}$. This is achieved by analyzing the explicit totally real embedding of the sphere $S^3$ in ${\mathbb C}^3$ constructed by Ahern and Rudin. For $t\ge\sqrt{(2+\sqrt{2})/3}$ the problem of the embeddability of $M_t^3$ remains open.