2016/08/09 by Isaev, Alexander
#32V05 #32V20 #34A05 #34A26 #35J96 #Complex Variables (math.CV) #FOS: Mathematics
paper · doi:10.48550/arxiv.1608.02919
In our recent article (to appear in the Journal of Differential Geometry in 2016) we studied tube hypersurfaces in \mathbb C3 that are 2-nondegenerate and uniformly Levi degenerate of rank 1. In particular, we discovered that for the CR-curvature of such a hypersurface to vanish it suffices to require that only two coefficients (called Θ221 and Θ210) in the expansion of a single component of the CR-curvature form be identically zero. In this paper, we show that, surprisingly, the vanishing of the entire CR-curvature is in fact implied by the vanishing of a single quantity derived from Θ210. This fact strengthens the main theorem of the earlier article and also leads to a remarkable system of partial differential equations. Furthermore, we explicitly characterize the class of not necessarily CR-flat tube hypersurfaces given by the vanishing of Θ221.