Normal forms and symmetries of real hypersurfaces of finite type in $\\mathbb C^2$

Rigid Embeddings of Sasakian Hyperquadrics in $$\mathbb {C}^{n+1}$$

“…Chern-Moser type normal forms of embedded real hypersurfaces in C n+1 have proved to be highly efficient tools in the study of automorphisms that fix the reference point of the normal form (see e.g. [3], [9], [8] [4]). In fact, any such automorphism becomes a projective transformation in suitable Chern-Moser normal coordinates.…”

Normal forms in Cauchy-Riemann geometry

“…As it was shown in [EKS13] in the case of a 2-dimensional Abelian symmetry algebra g of M there exist holomorphic coordinates (z, w) where the two generators of g take the form Notice that the integer m, together with the type k, is a biholomorphic invariant of (M, 0). The hypersurfaces v = y k + εy m can be considered as the models for this type of symmetry.…”

“…Since the action of rotations and dilations on the defining equation is completely straightforward, the remaining real parameter(s) can be easily fixed (see [EKS13]).…”

Infinitesimal CR automorphisms and stability groups of infinite-type models in C2

“…For a real hypersurface in C n , the stability group and the Lie algebra of infinitesimal CR automorphisms are not easy to describe explicitly; besides, they are unknown in most cases. But, the study of Aut(M, p) and aut(M, p) of special types of hypersurfaces is given in [CM], [EKS1], [EKS2], [K1], [K2], [K3], [KM], [KMZ], [S2], and [S1]. For instance, explicit forms of the stability groups of models (see detailed definition in [K1], [KMZ]) have been obtained in [EKS2], [K1], [K2], and [KMZ].…”

“…But, the study of Aut(M, p) and aut(M, p) of special types of hypersurfaces is given in [CM], [EKS1], [EKS2], [K1], [K2], [K3], [KM], [KMZ], [S2], and [S1]. For instance, explicit forms of the stability groups of models (see detailed definition in [K1], [KMZ]) have been obtained in [EKS2], [K1], [K2], and [KMZ]. However, these results are known for Levi nondegenerate hypersurfaces or, more generally, for Levi degenerate hypersurfaces of finite type in the sense of D'Angelo [D].…”