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

Abstract

The model 4-dimensional CR-cubic in C 3 has the following "model" property: it is (essentially) the unique locally homogeneous 4-dimensional CR-manifold in C 3 with finite-dimensional infinitesimal automorphism algebra g and non-trivial isotropy subalgebra. We study and classify, up to local biholomorphic equivalence, all g-homogeneous hypersurfaces in C 3 and also classify the corresponding local transitive actions of the model algebra g on hypersurfaces in C 3 .

Classification of homogeneous CR-manifolds in dimension 4

“…Hence it is important now to find out what surfaces in the extended list Ia-VIi are spherical. We firstly note that the sphericity of VIa with α = 2, β = 3 follows from [9]. The sphericity of Ic for α = 2 can be verified from the previous fact by applying the binomial formula for (x + iy) 3 .…”

“…As it follows from the above discussion, the homogeneity of M is provided by some 4-dimensional real Lie algebra of holomorphic vector fields, which coincides with aut M p if M is locally non-equivalent to the cubic C , or is a subalgebra of the 5-dimensional Lie algebra aut M p in case when M is locally CR-equivalent to C (see [9] for precise description of aut C ). We denote this algebra by g(M).…”

“…For our purposes it will be more convenient to single out five types of solvable algebras, which do not contain a 3-dimensional abelian ideal (according to [19], these are types A 4.8 , A 4.7 , A 4.9 , A 2.2 ⊕ A 2.2 and A 4.10 correspondingly). We also denote by Type VI all solvable algebras, which contain a 3-dimensional abelian ideal (according to [19], these are algebras of types 9 and A 4. j , j = 1, . .…”

“…is the unique abelian 3-dimensional subalgebra in the 5-dimensional infinitesimal automorphism algebra of the cubic C (it can be easily verified from the commuting relations in the algebra, see [9] for the details and the notations). As it was shown in [9], the cubic C is polynomially equivalent to the tube surface VIa for α = 2, β = 3 (we denote this surface byC ) and we conclude that…”

“…The cubic is the main example of a homogeneous totally non-degenerate CR-manifold. The cubic is a particular case of a model manifold (see, for example, [4]) and has many remarkable properties (see [5][6][7]9] for details), the main one is given by the following trichotomy for a 4-dimensional locally homogeneous CR-submanifold M in C 3 :…”