Classification of simply-transitive Levi non-degenerate hypersurfaces in $\mathbb{C}^3$

TL;DR

This paper completes the classification of simply-transitive Levi non-degenerate hypersurfaces in ^3, introducing a novel Lie algebraic method and a new coordinate-free quartic tensor formula, revealing a unique model with geometric relations.

Contribution

It provides the first classification of these hypersurfaces in ^3 using a new Lie algebraic approach and a coordinate-free quartic tensor formula.

Findings

Identified a unique Levi-indefinite non-tubular model

Developed a new coordinate-free quartic tensor formula

Established geometric relations to planar equi-affine geometry

Abstract

Holomorphically homogeneous CR real hypersurfaces $M^3 \subset \mathbb{C}^2$ were classified by \'Elie Cartan in 1932. In the next dimension, we complete the classification of simply-transitive Levi non-degenerate hypersurfaces $M^5 \subset \mathbb{C}^3$ using a novel Lie algebraic approach independent of any earlier classifications of abstract Lie algebras. Central to our approach is a new coordinate-free formula for the fundamental (complexified) quartic tensor. Our final result has a unique (Levi-indefinite) non-tubular model, for which we demonstrate geometric relations to planar equi-affine geometry.