Polynomial immersions in ${\mathbf C}^3$ of the boundaries of Grauert tubes with center $S^3$

We consider a one-parameter family of hypersurfaces $M_t$, $t>1$, in the 3-dimensional affine quadric $Q^3$ in ${\mathbb C}^4$ that is of interest in relation to the problem investigated by Morimoto and Nagano in 1967. The problem is to classify all homogeneous compact real-analytic hypersurfaces in ${\mathbb C}^n$. Identifying $Q^3$ with the tangent bundle $T(S^3)$, one observes that each hypersurface $M_t$ is the boundary of a Grauert tube with center $S^3$. In order to finalize Morimoto-Nagano's classification, one needs to resolve the question of for what values of $t$ the hypersurfaces $M_t$ admit real-analytic embeddings in ${\mathbb C}^3$. Even the question of the existence of immersions of these hypersurfaces turns out to be interesting. We show that each $M_t$ can be immersed in ${\mathbb C}^3$ by means of a polynomial map whose restriction to the totally real sphere $S^3\subset Q^3$ is an embedding. We construct a sequence $\{F_n\}$ of polynomial maps from ${\mathbb C}^4$ to ${\mathbb C}^3$ such that every $M_t$ is immersed in ${\mathbb C}^3$ by means of some of them. Moreover, by studying $F_1$ (which is the simplest map in the sequence), we prove that $M_t$ embeds in ${\mathbb C}^3$ for all $1<t<\sqrt{5}/2$.