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arXiv · 1608.02919

On the CR-curvature of Levi degenerate tube hypersurfaces

Alexander Isaev
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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 $Θ^2_{21}$ and $Θ^2_{10}$) 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 $Θ^2_{10}$. 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 $Θ^2_{21}$.

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Alexander Isaev. 2016-09-25. On the CR-curvature of Levi degenerate tube hypersurfaces. https://arxiv.org/abs/1608.02919

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