Infinitesimal CR automorphisms of rigid hypersurfaces in \({\mathbb{C}}^ 2\) (Q810205)

Let M be a real-analytic, real hypersurface through the origin in \({\mathbb{C}}^ 2\) given by an equation of the form Im w\(=F(z,\bar z)\); such M is called rigid. A smooth vector field X on M is called an infinitesimal CR automorphism if the local l-parameter group it generates is a local group of CR automorphisms of M. This paper contains several quite explicit results describing the set of such X, based upon a convenient normal form (choice of coordinates) for the function F defining the hypersurface. The approach to describe such X is different according to whether M is homogeneous or not. A rigid hypersurface is called homogeneous if it admits an equation with F a homogeneous polynomial with respect to a non-isotropic group of dilations. Among other things, the author characterises the homogeneous rigid hypersurfaces as those which admit a certain kind of infinitesimal CR automorphism. The proofs are quite direct and involve mainly thorough application of differential calculus and power series calculations.