Embedding of three dimensional Cauchy-Riemann manifolds (Q1337537)

A compact three-dimensional Cauchy-Riemann manifold consists of a compact smooth manifold \(M\) without boundary, a rank two subbundle \(HM \subset TM\), and an endomorphism \(I : HM \to HM\) satisfying \(I^ 2 = - id\). We say that two CR-manifolds \((M, HM, I)\) and \((M,HM,I')\) are close to each other if \(I\) and \(I'\) are close in the \(C^ \infty\)-topology. The main results of the paper are the two following theorems. Theorem 1.1. If a compact strictly pseudoconvex CR manifold admits a CR embedding into \(\mathbb{C}^ 2\) then this embedding is stable. Theorem 1.2. Suppose a compact strictly pseudoconvex CR manifold \((M, HM, I)\) admits a CR embedding into \(\mathbb{C}^ 2\), and let \(f : (M,HM,I) \to \mathbb{C}\) be a smooth CR function. Let \((M, HM, I')\) be another embeddable CR structure close to \((M, HM, I)\). Then there is a CR function \(f' : (M, HM, I') \to \mathbb{C}\) close to \(f\).