On the CR-curvature of Levi degenerate tube hypersurfaces

TL;DR

This paper demonstrates that for certain Levi degenerate tube hypersurfaces in complex three-space, the entire CR-curvature vanishes if a single derived quantity from a specific curvature coefficient is zero, simplifying previous conditions.

Contribution

It shows that the vanishing of the entire CR-curvature is implied by a single quantity derived from one curvature coefficient, strengthening earlier results and deriving new PDE systems.

Findings

CR-curvature vanishing is implied by a single derived quantity.

Explicit characterization of non-CR-flat tube hypersurfaces with vanishing ^2_{21}.

Introduction of a remarkable PDE system related to CR-curvature conditions.

Abstract

In our recent article (to appear in the Journal of Differential Geometry in 2016) we studied tube hypersurfaces in ${\mathbb C}^3$ 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 $\Theta^2_{21}$ and $\Theta^2_{10}$) 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 $\Theta^2_{10}$. 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 $\Theta^2_{21}$.