Uniformly Levi degenerate CR manifolds; the 5 dimensional case

Abstract

In this paper, we consider real hypersurfaces $M$ in $\Bbb C^3$ (or more generally, 5-dimensional CR manifolds of hypersurface type) at uniformly Levi degenerate points, i.e. Levi degenerate points such that the rank of the Levi form is constant in a neighborhood. We also require the hypersurface to satisfy a certain second order nondegeneracy condition (called 2-nondegeneracy) at the point. Our first result is the construction of a principal bundle $P\to M$ with an absolute parallelism, uniquely determined by the CR structure on $M$, which reduces the question of whether two such CR manifolds $M$ and $M'$ are CR equivalent to the corresponding equivalence problem for the parallelized bundles $P$ and $P'$. A basic example of a hypersurface of the type under consideration is the tube $Γ_\bC$ over the light cone. Our second result is the characterization of $Γ_\bC$ by vanishing curvature conditions in the spirit of the characterization of the unit sphere as the flat model for strongly pseudoconvex hypersurfaces in $\bC^{n+1}$ in terms of the Cartan-Chern-Moser connection.