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

Abstract:

It follows from the 2004 work of the first author, X.Huang, and D. Zaitsev that any local CR embedding $f$ of a strictly pseudoconvex hypersurface $M^{2n+1}\subset \mathbb {C}^{n+1}$ into the sphere $\mathbb {S}^{2N+1}\subset \mathbb {C}^{N+1}$ is rigid, i.e. any other such local embedding is obtained from $f$ by composition with an automorphism of the target sphere $\mathbb {S}^{2N+1}$, provided that the codimension $N-n<n/2$. In this paper, we consider the limit case $N-n=n/2$ in the simplest situation where $n=2$, i.e. we consider local CR embeddings $f\colon M^5\to \mathbb {S}^7$. We show that there are at most two different local embeddings up to composition with an automorphism of $\mathbb {S}^7$. 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 $f\colon M^5\to \mathbb {S}^7$.