Rigid hypersurfaces and infinitesimal CR automorphisms

We study a survey on the relations between rigid hypersurfaces and infinitesimal CR automorphisms. After reviewing the case of hypersurfaces of finite type, we study the case of hypersurfaces of infinite type. Some open problems are posed in the last section.

Regularity of infinitesimal CR automorphisms

We study the regularity of infinitesimal CR automorphisms of abstract CR structures which possess a certain microlocal extension and show that there are smooth multipliers, completely determined by the CR structure, such that if [Formula: see text] is such an infinitesimal CR automorphism, then [Formula: see text] is smooth for all multipliers [Formula: see text]. As an application, we study the regularity of infinitesimal automorphisms of certain infinite type hypersurfaces in [Formula: see text].

Lie algebras of infinitesimal CR automorphisms of weighted homogeneous and homogeneous CR-generic submanifolds of CN

We consider the significant class of homogeneous CR manifolds represented by some weighted homogeneous polynomials and we derive some plain and useful features which enable us to set up a fast effective algorithm to compute homogeneous components of their Lie algebras of infinitesimal CR automorphisms. This algorithm mainly relies upon a natural gradation of the sought Lie algebras, and it also consists in treating separately the related graded components. While classical methods are based on constructing and solving some associated PDE systems which become time consuming as soon as the number of variables increases, the new method presented here is based on plain techniques of linear algebra. Furthermore, it benefits from a divide-and-conquer strategy to break down the computations into some simpler subcomputations. Also, we consider the new and effective concept of comprehensive Gr?bner systems which provides us some powerful tools to treat the computations in the much complicated parametric case. The designed algorithm is also implemented in the Maple software, what required also implementing a recently designed algorithm of Kapur et al.