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 pseudoconvex hypersurface M2n+1 subset of Cn+1 into the sphere S2N+1 subset of CN+1 is rigid, i.e. any other such local embedding is obtained from f by composition with an automorphism of the target sphere S2N+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 : M-5 --> S-7. We show that there are at most two different local embeddings up to composition with an automorphism of 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 : M5 --> S-7.