Homogeneous Hypersurfaces in ℂ³, Associated with a Model CR-Cubic

The model 4-dimensional CR-cubic in ℂ³ has the following “model” property: it is (essentially) the unique locally homogeneous 4-dimensional CR-manifold in ℂ³ with finite-dimensional infinitesimal automorphism algebra gg and non-trivial isotropy subalgebra. We study and classify, up to local biholomorphic equivalence, all gg-homogeneous hypersurfaces in ℂ³ and also classify the corresponding local transitive actions of the model algebra gg on hypersurfaces in ℂ³