Convergence and finite determination of formal CR mappings

Abstract

It is shown that a formal mapping between two real-analytic hypersurfaces in complex space is convergent provided that neither hypersurface contains a nontrivial holomorphic variety. For higher codimensional generic submanifolds, convergence is proved e.g. under the assumption that the source is of finite type, the target does not contain a nontrivial holomorphic variety, and the mapping is finite. Finite determination (by jets of a predetermined order) of formal mappings between smooth generic submanifolds is also established.

Finite Jet Determination of Local CR Automorphisms through Resolution of Degeneracies

“…Consequently, Theorem 1.2 may be seen as a generalization of the mentioned result of [16]. On the other hand, this class also contains the important class of real-analytic hypersurfaces containing no analytic discs for which the finite jet determination property is already known to hold in view of the results of [3] and for which the methods of this paper offer a completely different new proof.…”

“…This approach strongly contrasts with that used to deal with the set Σ 2 of minimal points in e.g. [3,6] and, therefore, up to now, two different approaches have been used to study the finite jet determination problem according to the minimality or nonminimality of the base point.…”

The stationary disc method in the unique jet determination of CR automorphisms

“…For a survey on this matter, we point out for instance to the articles of Zaitsev [61] or Baouendi et al [4]. Naturally, after Theorem 1.1, many other situations were investigated; finitely nondegenerate hypersurfaces [1,60]; Ebenfelt et al [26] proved that 2-jet determination holds for hypersurfaces of finite type in ℂ 2 (see also [39]); finite (multi)type in ℂ N [5,40,42]. We note the paper of Juhlin [36] for holomorphically nondegenerate hypersurfaces which settles a conjecture due to Baouendi et al [2].…”

The equivalence theory for infinite type hypersurfaces in $\mathbb C^2$

“…He employed models of the type Im w = P (z, z), where P (z, z) is a nonzero homogeneous polynomial without harmonic terms of degree k ≥ 3. Notably, for the class of finite type hypersurfaces, it is known by the work of Baouendi-Ebenfelt-Rothschild [1] that every formal holomorphic map actually converges. Therefore, remarkably, Kolář's formal normal form for finite type hypersurfaces provides a solution to the biholomorphic equivalence problem for these hypersurfaces.…”