On CR embeddings of strictly pseudoconvex hypersurfaces into spheres in low dimensions

It follows from the 2004 work of the first author, X.Huang, and D. Zaitsev that any local CR embedding $f$ of a strictly psedoconvex hypersurface $M^{2n+1}\subset {\text{bC}}^{n+1}$ into the sphere ${\text{bS}}^{2N+1}\subset {\text{bC}}^{N+1}$ is rigid, i.e.\ any other such local embedding is obtained from $f$ by composition by an automorphism of the target sphere ${\text{bS}}^{2N+1}$, {\it provided} that the codimension $N-n<n\mathrm{/}2$. In this paper, we consider the limit case $N-n=n\mathrm{/}2$ in the simplest situation where $n=2$, i.e.\ we consider local CR embeddings $f\text{\hspace{0.17em}⁣}:M^5\to {\text{bS}}^7$. We show that there are at most two different local embeddings, up to composition with an automorphism of ${\text{bS}}^7$. We also identify a subclass of 5-dimensional, strictly pseudoconvex hypersurfaces $M^5$ in terms of their CR curvatures such that rigidity holds for local CR embeddings $f\text{\hspace{0.17em}⁣}:M^5\to {\text{bS}}^7$.