Models of 2-nondegenerate CR hypersurfaces in Cᴺ

Abstract

We show that every point in a uniformly 2-nondegenerate CR hypersurface is canonically associated with a model 2-nondegenerate structure. The 2-nondegenerate models which we introduce are essential CR invariants playing the same fundamental role as quadrics do in the classical Levi nondegenerate case. In particular, we show that each real analytic uniformly 2-nondegenerate hypersurface in Cᴺ is a perturbation of a 2-nondegenerate model. We give a complete characterization of all 2-nondegenerate models and show that the moduli space of such hypersurfaces in Cᴺ is infinite dimensional for each N>3. We derive a normal form for these models’ defining equations that is unique up to an action of a finite dimensional Lie group. We generalize recently introduced CR invariants (modified symbols), and show how to compute these intrinsically defined invariants from a model’s defining equation. Moreover, we show that these models automatically possess infinitesimal symmetries spanning a complement to their Levi kernel and derive explicit formulas for such symmetries.