Deformations and embeddings of three-dimensional strictly pseudoconvex CR manifolds

Abstract deformations of the CR structure a compact strictly pseudoconvex hypersurface M in $${\mathbb {C}}^2$$ are encoded by complex functions on M. In sharp contrast with higher dimensional case, natural integrability condition for 3-dimensional structures is vacuous, and generic $$M\subseteq {\mathbb not embeddable even {C}}^N$$ any N. A fundamental (and difficult) problem to characterize when function $$M \subseteq gives rise an actual deformation inside . this paper we study embeddability families given embedded 3-manifold, space $$S^3$$ We show that standard 3-sphere Frechet submanifold $$C^{\infty }(S^3,{\mathbb {C}})$$ near origin. establish modified version Cheng–Lee slice theorem which able precisely (in terms spherical harmonics). also introduce canonical family corresponding embeddings starting infinitesimally unit sphere