We show that germs of local real-analytic CR automorphisms of a real-analytic hypersurface M in C 2 at a point p ∈ M are uniquely determined by their jets of some finite order at p if and only if M is not Levi-flat near p. This seems to be the first necessary and sufficient result on finite jet determination and the first result of this kind in the infinite type case.If M is of finite type at p, we prove a stronger assertion: the local real-analytic CR automorphisms of M fixing p are analytically parametrized (and hence uniquely determined) by their 2-jets at p. This result is optimal since the automorphisms of the unit sphere are not determined by their 1-jets at a point of the sphere. The finite type condition is necessary since otherwise the needed jet order can be arbitrarily high [Kow1,2], [Z2]. Moreover, we show, by an example, that determination by 2-jets fails for finite type hypersurfaces already in C 3 .We also give an application to the dynamics of germs of local biholomorphisms of C 2 .

We define a new local invariant (called degeneracy) associated to a triple (M, M , H), where M ⊂ C N and M ⊂ C N are real submanifolds of C N and C N , respectively, and H : M → M is either a holomorphic map, a formal holomorphic map, or a smooth CR-map. We use this invariant to find sufficient conditions under which finite jet dependence, convergence and algebraicity results hold.

No abstract

In any positive CR-dimension and CR-codimension we provide a construction of realanalytic holomorphically nondegenerate CR-submanifolds, which are C ∞ CR-equivalent, but are inequivalent holomorphically. As a corollary, we provide the negative answer to the conjecture of Ebenfelt and Huang [20] on the analyticity of CR-equivalences between real-analytic Levi nonflat hypersurfaces in dimension 2.

For any real-analytic hypersurface M ⊂ C N M\subset \mathbb {C}^N , which does not contain any complex-analytic subvariety of positive dimension, we show that for every point p ∈ M p\in M the local real-analytic CR automorphisms of M M fixing p p can be parametrized real-analytically by their ℓ p \ell _p jets at p p . As a direct application, we derive a Lie group structure for the topological group Aut ⁡ ( M , p ) \operatorname {Aut}(M,p) . Furthermore, we also show that the order ℓ p \ell _p of the jet space in which the group Aut ⁡ ( M , p ) \operatorname {Aut}(M,p) embeds can be chosen to depend upper-semicontinuously on p p . As a first consequence, it follows that given any compact real-analytic hypersurface M M in C N \mathbb {C}^N , there exists an integer k k depending only on M M such that for every point p ∈ M p\in M germs at p p of CR diffeomorphisms mapping M M into another real-analytic hypersurface in C N \mathbb {C}^N are uniquely determined by their k k -jet at that point. Another consequence is the following boundary version of H. Cartan’s uniqueness theorem: given any bounded domain Ω \Omega with smooth real-analytic boundary, there exists an integer k k depending only on ∂ Ω \partial \Omega such that if H : Ω → Ω H\colon \Omega \to \Omega is a proper holomorphic mapping extending smoothly up to ∂ Ω \partial \Omega near some point p ∈ ∂ Ω p\in \partial \Omega with the same k k -jet at p p with that of the identity mapping, then necessarily H = Id H=\textrm {Id} . Our parametrization theorem also holds for the stability group of any essentially finite minimal real-analytic CR manifold of arbitrary codimension. One of the new main tools developed in the paper, which may be of independent interest, is a parametrization theorem for invertible solutions of a certain kind of singular analytic equations, which roughly speaking consists of inverting certain families of parametrized maps with singularities.