A hypersurface in $\mathbb {C}^2$ whose stability group is not determined by $2$-jets

Abstract

Abstract. We give an example of a hypersurface in C 2 through 0 whose stability group at 0 is determined by 3-jets, but not by jets of any lesser order. We also examine some of the properties which the stability group of this infinite type hypersurface shares with the 3-sphere in C 2 .

Analytic differential equations and spherical real hypersurfaces

“…A recent result of the authors [34] showing that formal equivalences between nonminimal hypersurfaces can be actually divergent, proves, in particular, that a formal normal form can no longer be a solution for the equivalence problem for nonminimal hypersurfaces, which further illustrates the difficulties for this class of hypersurfaces. In fact, even the class of nonminimal hypersurfaces spherical at a generic point appears to be highly nontrivial (we refer here to the work [35,17,6,32,33,34] of V. Beloshapka, P. Ebenfelt, M. Kolar, Kowalski, B. Lamel, D. Zaitsev and the authors), as it is not even known whether the moduli space for this class of hypersurfaces is finite dimensional.…”

Local symmetries of finite type hypersurfaces in ℂ2

“…First we will show that f and g may be replaced by f and g when considering terms of weight less or equal to p + k in (9). More precisely, (20)…”

“…The problems of finite jet determination and estimation of the dimension of the stability group on Levi degenerate hypersurfaces have been intensively studied in the last decade (see [13,11,12,10,26], and the survey article [2] for further references). In dimension two, one of the most important results states that uniform finite determination, which holds on finite type hypersurfaces, actually fails for points of infinite type (see [20,26]). More precisely, for any integer k there is an infinite type, non Levi flat hypersurface whose local automorphisms are not determined by their k-jets.…”

On the analyticity of CR-diffeomorphisms

“…We shall also note that the paper contains an important intermediate result which is a complete characterization of all real-analytic hypersurfaces in C 2 , which are nonminimal at the origin and spherical outside the complex locus X 0 (see Theorem 20 and Corollary 22 below). The latter class of hypersurfaces was previously studied in a long sequence of publications [36,21,8,32,33,34,35] and appears to be highly nontrivial. The results of Section 3 below completes the study of hypersurfaces of this class.…”