Deformations and embeddings of three-dimensional strictly pseudoconvex CR manifolds

TL;DR

This paper investigates the conditions under which deformations of 3D strictly pseudoconvex CR manifolds, especially the 3-sphere, are embeddable into complex space, revealing the structure of embeddable deformation spaces.

Contribution

It characterizes the space of embeddable CR deformations of the 3-sphere as a Frechet submanifold and extends the Cheng-Lee slice theorem to this setting.

Findings

The space of embeddable deformations of the standard CR 3-sphere forms a Frechet submanifold.

A modified Cheng-Lee slice theorem characterizes embeddable deformations via spherical harmonics.

A canonical family of embeddable deformations is constructed from infinitesimally embeddable ones.

Abstract

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