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

TL;DR

This paper investigates the uniqueness of local CR embeddings of 5-dimensional strictly pseudoconvex hypersurfaces into 7-spheres, revealing conditions under which such embeddings are rigid or admit at most two distinct forms.

Contribution

It extends the understanding of CR embedding rigidity to the critical codimension case for 5-dimensional hypersurfaces, identifying when multiple embeddings can occur.

Findings

At most two local embeddings exist up to automorphism in the critical case.

Rigidity holds for a subclass of hypersurfaces characterized by their CR curvatures.

The paper characterizes conditions for uniqueness of CR embeddings in the borderline codimension scenario.

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 psedoconvex hypersurface $M^{2n+1}\subset\bC^{n+1}$ into the sphere $\bS^{2N+1}\subset \bC^{N+1}$ is rigid, i.e.\ any other such local embedding is obtained from $f$ by composition by an automorphism of the target sphere $\bS^{2N+1}$, {\it 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 $f\colon M^5\to \bS^7$. We show that there are at most two different local embeddings, up to composition with an automorphism of $\bS^7$. 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 $f\colon M^5\to \bS^7$.