Deformations and embeddings of three-dimensional strictly pseudoconvex CR manifolds

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 ${\mathbb{C}}^2$.

46 pages. Accepted version. Math. Ann. (2023). Section 3 has been substantially revised to streamline the presentation; a detailed proof of Theorem 3.1 was also added. Section 4.5 has been expanded and clarified. Some inconsistencies of signs and factors of 2 were fixed in Sections 2 and 3

References in corpus (3)

• Variation of total Q-prime curvature on CR manifolds
• Nonnegativity of the CR Paneitz operator for embeddable CR manifolds
• Bounded strictly pseudoconvex domains in ${\mathbb{C}}^2$ with obstruction flat boundary