Holomorphic classification of four-dimensional surfaces in $ \mathbb C^3$

Classification of homogeneous CR-manifolds in dimension 4

“…In the case of (3.2) the isometry property follows from (6). The symmetry property check in all cases is straightforward.…”

“…. , 6) and denote by Types VII and VIII correspondingly the two non-solvable algebras so 2,1 (R) ⊕ R 1 and so 3 (R) ⊕ R 1 . Now the classification looks as follows:…”

“…where the variables are assigned the weights [z] = 1, [w 2 ] = 2, [w 3 ] = 3 and O (6), O (7) are terms of weights grater than 6 and 7 correspondingly (this is some analogue for the sphericity criterion for a hypersurface in C N , see [10,11]). To apply this criterion to the above list of homogeneous surfaces, we consider all possible holomorphic transformations, preserving the origin, and present them as…”

“…They are also known as Engel-type manifolds (see [7]). As it was shown in [6], the total non-degeneracy condition is also equivalent to the existence of local holomorphic coordinates (z, w 2 , w 3 ) ∈ C 3 , in which p is the origin and M is given as…”

“…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 :…”