Uniformly Levi degenerate CR manifolds: The 5-dimensional case

Abstract

In this paper, we consider real hypersurfaces $M$ in $\mathbb {C}^3$ (or, more generally, $5$-dimensional CR (Cauchy-Riemann) manifolds of hypersurface type) at uniformly Levi degenerate points, that is, Levi degenerate points such that the rank of the Levi form is constant in a neighborhood. We also require that the hypersurface satisfy a certain second-order nondegeneracy condition (called $2$-nondegeneracy) at the point. One of our main results is the construction, near any point $p_0\in M$ satisfying the above conditions, of a principal bundle $P\to M$ and a $\mathbb {R}^{\dim P}$-valued 1-form $\underline {\omega}$, uniquely determined by the CR structure on $M$, which defines an absolute parallelism on $P$. If $M$ is real-analytic, then covariant derivatives of $\underline {\omega}$ yield a complete set of local biholomorphic invariants for $M$. This solves the biholomorphic equivalence problem for uniformly Levi degenerate hypersurfaces in $\mathbb {C}^3$ at $2$-nondegenerate points.