On the CR-curvature of Levi degenerate tube hypersurfaces

In the article “Affine rigidity of Levi degenerate tube hypersurfaces” [J. Differential Geom., 104 (2016), pp. 111–141] 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 certain 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 result strengthens the main theorem of “Affine rigidity of Levi degenerate tube hypersurfaces” 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 $\Theta^{2}_{21}$.

Keywords

CR-curvature, the Monge–Ampère equation, the Monge equation

2010 Mathematics Subject Classification

32V05, 32V20, 34A05, 34A26, 35J96

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Published 3 April 2017