Spiraling and Nonhypoellipticity for Cr Structures Degenerate along Transverse Real Curves

Consider a smooth, pseudoconvex CR structure on some small open set of real dimension three. A fundamental sufficient condition for an associated Cauchy-Riemann operator ∂̄b to be C ∞ hypoelliptic, modulo its nullspace (this notion is defined below in §4) is that the CR structure should be of finite type, but this condition is far from necessary. While some sufficient conditions are known [2] (see also [16] for the higher-dimensional case), no useful complete characterization of hypoellipticity is known, or expected, in its absence. Some sufficient conditions take a quantitative form. For example, one such condition is that the set of weakly pseudoconvex points should be contained in a manifold M of real dimension two that is everywhere transverse to the complex tangent space, and moreover the Levi form Λ(z) should not tend to zero too rapidly as this degenerate manifold is approached; the critical threshold is distance (z,M)| log Λ(z)| → ∞ as z → M . A more qualitative criterion is that points of infinite type should be isolated; then the rate of degeneration does not matter. It would be desirable to find other qualititative criteria. The main result of this paper is an example answering one specific question along these lines posed by J. J. Kohn.