Non-realizable CR structures

Abstract

The examples given by Nirenberg in [5] and [6] show that not all three dimensional CR manifolds may be realized as real hypersurfaces in 1172. These examples are perturbations of the CR structure on the sphere. In this paper we extend his construction and show that any strictly pseudo-convex realizable CR structure of dimension 3 may be perturbed so as to obtain a non-realizable one. This is done in Sect. 1. In Sect. 2 we show non-realizability for higher dimensional structures provided the Levi form is non-degenerate and has only one positive eigenvalue, In our last section we summarize Cartan's approach to CR geometry. From this viewpoint it was important to decide if Nirenberg's example was really limited to perturbations of the sphere. We have made this paper self-contained and have provided proofs of several known results. For instance Lemma 1.l, and the first part of Lemma 1.5 were proved by Lewy [2]. And Lemma 1.2 is the well known observation that a strictly pseudo-convex hypersurface is convex in some coordinate system,