Normal forms for hypersurfaces of finite type in $\mathbb{C}^2$

Abstract

Abstract. We construct normal forms for Levi degenerate hypersurfaces of finite type in C 2 . As one consequence, an explicit solution to the problem of local biholomorphic equivalence is obtained. Another consequence determines the dimension of the stability group of the hypersurface.

Higher Order invariants of Levi Degenerate Hypersurfaces

“…It was proved in [15] that if e < k 2 , the local automorphism group of M H consists of transformations…”

“…The results of [15] give three different complete normal forms, depending on the form of the model. There are two exceptional models, S k and T k , while the generic case covers all remaining models.…”

“…By definition, any local automorphism of M in normal coordinates preserves normal form. In [15], such transformations are completely characterized, and correspond to the action of the local symmetry group of the model.…”

“…In dimension two, a complete set of local CR invariants can be constructed, on the level of formal power series, using analysis of generalized Chern-Moser operators ( [15]). Combined with the result of M. S. Baouendi, P. Ebenfelt and L. P. Rothschild on convergence of formal equivalences, it gives a solution to the local equivalence problem.…”

Local equivalence problem for Levi flat hypersurfaces

“…Levi non-degenerate real hypersurfaces in C n were classified by Cartan (see [6], [7]) for n = 2, and independently by Tanaka [12] and by Chern and Moser (who gave a classification in terms of normal forms in [8]) for n ≥ 2. Later on, the question has been considered under hypotheses of increasingly higher degeneracy-for example, Kolář [10] has provided a normal form for hypersurfaces of finite type in C 2 -but an exhaustive review of the subject would be far beyond the scope of this introduction.…”