Parametrization of local CR automorphisms by finite jets and applications

Abstract:

For any real-analytic hypersurface $M\subset \mathbb {C}^N$, which does not contain any complex-analytic subvariety of positive dimension, we show that for every point $p\in M$ the local real-analytic CR automorphisms of $M$ fixing $p$ can be parametrized real-analytically by their $\ell _p$ jets at $p$. As a direct application, we derive a Lie group structure for the topological group $\operatorname {Aut}(M,p)$. Furthermore, we also show that the order $\ell _p$ of the jet space in which the group $\operatorname {Aut}(M,p)$ embeds can be chosen to depend upper-semicontinuously on $p$. As a first consequence, it follows that given any compact real-analytic hypersurface $M$ in $\mathbb {C}^N$, there exists an integer $k$ depending only on $M$ such that for every point $p\in M$ germs at $p$ of CR diffeomorphisms mapping $M$ into another real-analytic hypersurface in $\mathbb {C}^N$ are uniquely determined by their $k$-jet at that point. Another consequence is the following boundary version of H. Cartan’s uniqueness theorem: given any bounded domain $\Omega$ with smooth real-analytic boundary, there exists an integer $k$ depending only on $\partial \Omega$ such that if $H\colon \Omega \to \Omega$ is a proper holomorphic mapping extending smoothly up to $\partial \Omega$ near some point $p\in \partial \Omega$ with the same $k$-jet at $p$ with that of the identity mapping, then necessarily $H=\textrm {Id}$. 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.