Affine rigidity of Levi degenerate tube hypersurfaces

Abstract

Let $\mathfrak{C}_{2,1}$ be the class of connected 5-dimensional CR-hypersurfaces that are 2-nondegenerate and uniformly Levi degenerate of rank 1. In our recent article, we proved that the CR-structures in $\mathfrak{C}_{2,1}$ are reducible to $\mathfrak{so}(3, 2)$-valued absolute parallelisms. In the present paper, we apply this result to study tube hypersurfaces in $\mathbb{C}^3$ that belong to $\mathfrak{C}_{2,1}$ and whose CR-curvature identically vanishes. By explicitly solving the zero CR-curvature equations up to affine equivalence, we show that every such hypersurface is affinely equivalent to an open subset of the tube $M_0$ over the future light cone $\lbrace (x_1, x_2, x_3) \in \mathbb{R}^3 \: \vert \: x^2_1 + x^2_2 - x^2_3= 0 , x_3 \gt 0 \rbrace$. Thus, if a tube hypersurface in the class $\mathfrak{C}_{2,1}$ locally looks like a piece of $M_0$ from the point of view of CR-geometry, then from the point of view of affine geometry it (globally) looks like a piece of $M_0$ as well. This rigidity result is in stark contrast to the Levi nondegenerate case, where the CR-geometric and affine-geometric classifications significantly differ.