Parametrization of local CR automorphisms by finite jets and applications

Abstract

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.